Unit Operations in Environmental Engineering 9781119283638, 1119283639

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Unit Operations in Environmental Engineering

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Unit Operations in Environmental Engineering

By Louis Theodore, R. Ryan Dupont and Kumar Ganesan

This edition first published 2017 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2017 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-28363-8 Cover image: Pixabay.Com Cover design by Russell Richardson Set in size of 10pt and Minion Pro by Exeter Premedia Services Private Ltd., Chennai, India Printed in the USA 10 9 8 7 6 5 4 3 2 1

Dedicated to all the past, present, and future environmental engineering students, with whom may rest both the hopes and future of mankind.

Contents Preface Introduction

xi xvii

Part I: Introduction to the Principles of Unit Operations

1

1 History of Chemical Engineering and Unit Operations

3

2 Transport Phenomena versus the Unit Operations Approach

7

3 The Conservation Laws and Stoichiometry

11

4 The Ideal Gas Law

19

5 Thermodynamics

27

6 Chemical Kinetics

39

7 Equilibrium versus Rate Considerations

51

8 Process and Plant Design

57

Part II: Fluid Flow

69

9 Fluid Behavior

71

10 Basic Energy Conservation Laws

81

11 Law of Hydrostatics

89

12 Flow Measurement

95

13 Flow Classification

107

14 Prime Movers

121

15 Valves and Fittings

135

16 Air Pollution Control Equipment

145

17 Sedimentation, Centrifugation, and Flotation

157 vii

viii

Contents

18 Porous Media and Packed Beds

171

19 Filtration

181

20 Fluidization

193

21 Membrane Technology

205

22 Compressible and Sonic Flow

219

23 Two-Phase Flow

225

24 Ventilation

237

25 Mixing

247

26 Biomedical Engineering

253

Part III: Heat Transfer

265

27 Steady-State Conduction

267

28 Unsteady-State Conduction

275

29 Forced Convection

281

30 Free Convection

289

31 Radiation

299

32 The Heat Transfer Equation

311

33 Double Pipe Heat Exchangers

325

34 Shell and Tube Heat Exchangers

337

35 Finned Heat Exchangers

347

36 Other Heat Transfer Equipment

357

37 Insulation and Refractory

369

38 Refrigeration and Cryogenics

375

39 Condensation and Boiling

391

40 Operation, Maintenance, and Inspection (OM&I)

403

41 Design Principles

411

Part IV: Mass Transfer

419

42 Equilibrium Principles

421

Contents

ix

43 Phase Equilibrium Relationships

427

44 Rate Principles

441

45 Mass Transfer Coefficients

451

46 Classification of Mass Transfer Operations

463

47 Characteristics of Mass Transfer Operations

471

48 Absorption and Stripping

483

49 Distillation

493

50 Adsorption

503

51 Liquid-Liquid and Solid-Liquid Extraction

515

52 Humidification

527

53 Drying

541

54 Absorber Design and Performance Equations

553

55 Distillation Design and Performance Equations

569

56 Adsorber Design and Performance Equations

587

57 Crystallization

595

58 Other and Novel Separation Processes

607

Part V: Case Studies

613

59 Drag Force Coefficient Correlation

615

60 Predicting Pressure Drop with Pipe Failure for Flow through Parallel Pipes

619

61 Developing an Improved Model to Describe the Cunningham Correction Factor Effect

621

62 Including Entropy Analysis in Heat Exchange Design

623

63 Predicting Inside Heat Transfer Coefficients in Double-Pipe Exchangers

627

64 Converting View Factor Graphical Data to Equation Form

629

65 Correcting a Faulty Absorber Design

631

66 A Unique Liquid-Liquid Extraction Unit

633

x

Contents

67 Effect of Plate Failure on Distillation Column Performance

637

Appendix A: Units

639

Appendix B: Miscellaneous Tables

647

Appendix C: Steam Tables

651

Appendix D: Basic Calculations

661

Index

669

Preface Unit operations are several of the basic tenets of not only chemical engineering but also several other engineering disciplines, and contains many practical concepts that are utilized in countless industrial applications. One engineering curriculum that has embraced the unit operations approach is environmental engineering, and interestingly, a comprehensive “overview text” in the subject area is not presently available in the literature. Therefore, the authors considered writing a practical introductory text involving unit operations for environmental engineers. The text will hopefully serve as a training tool for those individuals pursuing degrees that include courses on unit operations. Although the literature is inundated with texts in this area emphasizing theory and theoretical derivations, the goal of this text is to present the subject from a strictly pragmatic introductory point-of-view, particularly for those individuals involved with environmental engineering. As noted in the opening paragraph and in the title of this book - Unit Operations in Environmental Engineering - this work has been written primarily for environmental engineering students. But, who are environmental engineers and what is the environmental engineering profession? The answer, to some degree, depends on who one talks to since this profession has undergone dramatic changes over the last half century. The term environmental engineering came into existence in the mid-1960s when it displaced the perhaps politically incorrect term, sanitary engineering. The reader should keep in mind that during the late 1900s, it was no secret that industry preferred to hire chemical engineers to address real-world environmental engineering problems. This was no doubt brought about because of the chemical engineer’s understanding, and the environmental engineer’s lack of knowledge, of unit operations, particularly those related to mass transfer operations. Interestingly, the early sanitary engineering curriculum was almost exclusively based on primarily “water” topics, e.g., sewage, water supply and usage, sanitation, etc. The expansion of environmental engineering to include air, solid waste, noise, health risk, hazard risk, etc., evolved over time, all of which can today be viewed as a legitimate part of an environmental engineering curriculum. This book’s subject matter of unit operations is therefore only one, but perhaps the most important subject in any interdisciplinary discipline that includes environmental engineering.

xi

xii

Preface

This is a book on unit operations...well, sort of. The principles of unit operations were originally set forth soon after the birth of the chemical engineering profession at the turn of the 20th century and it remains the keystone course in the chemical engineering curriculum. A new kid on the block entered the engineering field around 1950, perhaps spearheaded by the adoption of a sanitary engineering program at Manhattan College, Bronx, NY. The program was later renamed “Environmental Engineering” around 1970. The College also later served as the host of several NSF-funded environmental engineering course development seminars that were directed by the one of the authors, Lou Theodore; and the college also served as home for Lou Theodore (as Professor of Chemical Engineering) for 50 years. Converting the aforementioned chemical engineering principles/approaches/ applications embodied in unit operations to environmental engineering in an optimum manner was not as difficult as the authors originally anticipated. This was, no doubt, due to the clear overlap between the two disciplines. As noted above, this book is concerned with unit operations, fluid flow, heat transfer, and mass transfer. Unit operations, by definition, are physical processes although there are some that include chemical and biological reactions. The unit operations approach allows both the student and practicing engineer to compartmentalize the various operations that constitute a process. As such, it has enabled the engineer of yesterday and today (and tomorrow) to perform more efficiently. This approach has also allowed the environmental engineer to achieve considerable success in the environmental management field. The authors’ approach in presenting unit operations material to environmental engineering students is to primarily key on introductory engineering principles so that the reader could then later satisfactorily predict the performance of the various unit operation equipment. In effect, the reader or instructor is provided the opportunity to expand chapter presentations. Although a chapter on Process and Plant Design is included in the introductory part (Part I) of the book, details on equipment design are treated superficially and the subject of plant design is rarely broached. A comment on chemical reactions is also warranted. Chemical reactions have been defined by some as chemical unit processes. They serve as the backbone of the chemical process industries employing the batch, continuously stirred tank reactors (CSTRs), and tubular flow reactors. However, these chemical unit processes also find application in the wastewater treatment industry. Some of these chemical processes include oxidation, precipitation, neutralization, pH control, disinfection operations, certain coagulation operations, etc. Many of the chemical reactions involve chemicals such as calcium and sodium hydroxide, ferric and aluminum chloride, alum, ferric sulfide, etc. And, as one might suppose, these chemical unit processes are often operated in conjunction with physical unit processes (or unit operations). Biological unit processes represent another class of chemical reactions that are important to the practicing environmental engineer. The major applications of

Preface

xiii

these biochemical reaction pathways are in wastewater treatment and hazardous waste remediation. The principal biological processes used for wastewater treatment can be divided into two main categories: suspended growth and attached growth (or biofilm) processes. Their successful design and operation requires an understanding of the types of microorganisms involved, the specific reactions that occur, and the environmental factors that affect their performance, their nutritional needs, and their biochemical reaction kinetics. The decision as to what units and notations to use was difficult. After much deliberation, the authors chose to use engineering - as opposed to metric/SI units, and chemical engineering notation - as opposed to those of other disciplines. This decision was based, to some extent, on the reality that a good part of the book’s content was drawn from the chemical engineering literature, some of which was written by the primary author, Lou Theodore. The book is divided into five parts. Part I - Introduction to the Principles of Unit Operations Part II - Fluid Flow Part III - Heat Transfer Part IV - Mass Transfer Part V - Case Studies In addition to providing materials on the history of unit operations and a discussion of the relationship among the transport phenomenon/unit operations/unit processes approaches, Part I contains material on traditional introductory engineering principles. These include: thermodynamics, chemical reaction principles, equilibrium versus rate consideration, rate principles, and process and plant design. Part II - Fluid Flow - addresses such subject areas as: fluid classifications, flow mechanisms, flow in conduits, prime movers plus various valve and fittings, sedimentation and centrifugation, porous media and packed beds, filtration, fluidization, ventilation and mixing. Part III - Heat Transfer - contains material concerned with: heat exchangers, waste heat boilers and evaporators, quenchers, psychrometry, humidification, drying, and cooling towers. (Note that the subject of heat transfer was rarely (if ever) included in the environmental engineering curriculum in the early days. For example, in Rich’s classic “Unit Operations in Sanitary Engineering” text, only one of the 15 chapters in the text dealt with heat transfer. That has changed today because of the environmental engineer’s interest in energy, energy conservation, combustion, hazardous waste incineration, global climate change, radiation effects of the sun, etc. In effect, heat transfer has become the new kid on the block in the unit operations arena, and is a topic that every environmental engineer should be proficient in). Part IV of the book - Mass Transfer - covers such topics as: absorption and stripping, adsorption, distillation, liquid-liquid and liquid-solid extraction, and other mass transfer operations. The last part of the book, Part V - Case Studies provides three applications in each of the three unit operations. An Appendix is also included. An outline of the topics can be found in the Table of Contents.

xiv

Preface

The reader will note that there is no separate section, part or chapter devoted to biological processes. Rather, they have been integrated into relevant material presented in Parts II and IV. Biological treatment processes (in alphabetical order) that receive treatment include: Activated Sludge Aerated Lagoons Anaerobic Digestion Composting Enzyme Treatment Trickling Filters Waste Stabilization Ponds Details on the above seven biological methods were provided earlier by Theodore and McGuinn in “Pollution Prevention,” Van Nostrand Reinhold, New York City, NY, 1992. An extensive analysis of these processes (plus many more) is also available in the work of Metcalf and Eddy, “Wastewater Engineering: Treatment and Reuse,” McGraw-Hill, 4th Edition, New York City, NY, 2004 and L. Rich, “Unit Operations of Sanitary Engineering,” John Wiley & Sons, Hoboken, NJ, 1961. The authors cannot claim sole authorship to all of the essay material and examples in this text. The present book has evolved from a host of sources, including: notes, homework problems and exam problems prepared by several faculty for a required one-semester, three-credit, “Principles III: Mass Transfer” undergraduate course offered at Manhattan College; L. Theodore and J. Barden, “Mass Transfer”, A Theodore Tutorial, East Williston, NY, 1995; I. Farag, “Fluid Flow,” A Theodore Tutorials, East Williston, NY, 1994; I Farag and J. Reynolds, “Heat Transfer,” A Theodore Tutorials, East Williston, NY, 1995; J. Reynolds, J. Jeris, and L. Theodore, “Handbook for Chemical and Environmental Engineering Calculations,” John Wiley & Sons, Hoboken, NJ, 2004; and J. Santoleri, J. Reynolds, and L. Theodore, “Introduction to Hazardous Waste Management,” 2nd edition, John Wiley & Sons, Hoboken, NJ, 2000. Although the bulk of the material is original and/or taken from sources that the authors have been directly involved with, every effort has been made to acknowledge material drawn from other sources. It is hoped that this book covers the principles and applications of unit operations in a thorough and clear manner. Upon completion of the text, the reader should have acquired not only a working knowledge of the principles of unit operations, but also experience in their application; and, the reader should find himself/herself approaching advanced texts, engineering literature and industrial applications (even unique ones) with more confidence. The authors strongly believe that, while understanding the basic concepts is of paramount importance, this knowledge may be rendered virtually useless to an environmental engineer if he/she cannot apply these concepts in real-world situations. This is the essence of engineering.

Preface

xv

Last, but not least, the authors believe that this modest work will help the majority of individuals working and/or studying in the field of environmental engineering to obtain a more complete understanding of unit operations. If you have come this far and read through most of the Preface, you have more than just a passing interest in this subject. The authors are indebted to the pioneers in the sanitary/environmental engineering field, including such notables as Don O’Connor, Linvil Rich, Wes Eckenfelder, Ross McKinney, Perry McCarty, etc. Pioneers in the environmental management field include James Fenimore Cooper, John Muir, Howard Hesketh, Charlie Pratt, Art Stern, Werner Strauss, etc. Sincere and special thanks are extended to Haley Seiler of the Civil and Environmental Engineering Department at Utah State University for her invaluable help in the preparation of the draft of the text of this manuscript, and to Ivonne Harris of the Utah Water Research Laboratory for her assistance in preparing all of the figures for the text. Louis Theodore East Williston, New York R. Ryan Dupont Smithfield, Utah Kumar Ganesan Butte, Montana March, 2017 NOTE: The authors are in the process of preparing an additional resource for this text. An accompanying website containing 15 hours of exams and solutions for the exams will soon be available for those who adopt the book for training and/or academic purposes.

Introduction How are unit operations related to a unit process? Consider the flow diagram in the figure below. There are three unit operations, 1, 2, and 3. The combination of the three operations that reside in the dashed box is the process or what has come to be referred to as a unit process. Fluid flow, heat transfer, and mass transfer operations fit into the description/definition of unit operations. Chemical and biological operations are, in line with the accepted definitions of unit operations, not considered unit operations. As such, they are reviewed only superficially in this book since they are both treated extensively in the literature. The reader should note that many engineering activities can be classified as: 1. Physical unit processes, 2. Chemical unit processes, and/or 3. Biological unit processes. Physical unit processes involve the application of physical forces, while chemical and biological unit processes are brought about by the addition of chemicals or chemical reactions, and biochemical reactions, respectively. Physical unit processes have come to be defined as unit operations, the subject title of this book. The similarity of the physical changes occurring in widely differing industries led to the study of the many steps common to both industry and environmental applications/systems, as the aforementioned unit operations. The unit operations Feed B

MakeUp D Recycle

Feed A

2

1

3

Product F

Bypass Product C Figure I.1 Unit operations versus unit processes.

xvii

xviii

Introduction

came to be regarded as special cases or combinations of fluid flow, heat transfer and mass transfer. The chemical reactor is usually at the heart of many processes and it is here that the engineer may simultaneously utilize the principles of fluid flow, heat transfer, and mass transfer, as well as chemical kinetics and thermodynamics, to carry out desired transformations, whether it be for the production of materials or the removal of undesirable pollutants. However, reactions of a chemical or biochemical nature, and the associated equipment, have traditionally not been considered to reside in the unit operations domain. Underlying nearly every step of a unit process are the principles of fluid flow and heat transfer; the fluid must be transported, and its temperature must be controlled. In a chemical process, where composition is a variable, the principles of mass transfer enter the design of separation and reaction equipment. This book deals with physical processes, referred to as the aforementioned unit operations, which are common to many chemical and environmental systems. By examining these operations apart from a particular application, students are encouraged to concentrate on fundamental principles. The duplication of topic material normally encountered in “compartmentalized” curricula is avoided, and time should be available to consider a larger variety of operations. The unit operations approach has been employed in chemical engineering education for nearly a century with considerable success, and has recently become an integral part of the environmental engineering curricula. The book has been written for students with a typical undergraduate background in engineering and the sciences. Comprehension requires only an understanding of freshman chemistry, engineering physics, calculus, and to a lesser extent, differential equations. Details of equipment design are discussed only briefly since several books already published treat this aspect of unit operations with thoroughness and clarity. References to these sources are commonplace in the text. Furthermore, several operations of a less complex nature have been omitted to make room for those ordinarily not considered in courses in environmental engineering but which are of growing importance in the field. As noted earlier, chemical and biochemical operations received minimal treatment. The material used in this book was taken from both the environmental and chemical engineering field. The use of mixed notation can be confusing, and the choice then was between two alternatives, environmental notation or those of the chemical engineer. As noted in the Preface, the latter was chosen. The use of standard notation in chemical engineering is thought to better serve students in making them familiar with the standard notation used in the literature of the process engineering field. The decision on a unit convention can also be a problem, and the authors have chosen to use English (or engineering) as opposed to SI units as is the standard in much of the process and environmental engineering fields. Comprehensive conversion tables for units are included in the Appendix. In conclusion, it must be emphasized that this book is not a treatise. Rather, it should be viewed as an introductory textbook dealing primarily with unit operations.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

Part I INTRODUCTION TO THE PRINCIPLES OF UNIT OPERATIONS

The purpose of this Part can be found in its title. The book itself offers the reader the principles of unit operations with appropriate practical applications, and serves as an introduction to the specialized and more sophisticated texts in this area. The reader should realize that the contents are geared not only toward practitioners in this field, but also students of science and engineering. Topics of interest to all practicing engineers have been included. It should also be noted that the microscopic approach of unit operations is not covered here. The approach taken in the text is to place more emphasis on real-world and design applications. However, microscopic approach material is available in the literature, as noted in the ensuing chapters. The chapters in this Part provide an introduction and overview of unit operations. Part I chapter content includes: 1. 2. 3. 4. 5. 6.

History of Chemical Engineering and Unit Operations Transport Phenomena versus Unit Operations Approach The Conservation Laws and Stoichiometry The Ideal Gas Law Thermodynamics Chemical Kinetics 1

2

Unit Operations in Environmental Engineering 7. Equilibrium versus Rate Considerations 8. Process and Plant Design

Topics covered in the first two introductory chapters include a history of chemical engineering and unit operations, and a discussion of transport phenomena versus unit operations. The remaining chapters are concerned with introductory engineering principles.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

1 History of Chemical Engineering and Unit Operations

A discussion of the field of chemical engineering is warranted before proceeding to some specific details regarding unit operations and the contents of this first chapter. A reasonable question to ask is: What is chemical engineering? An outdated, but once official definition provided by the American Institute of Chemical Engineers is: Chemical Engineering is that branch of engineering concerned with the development and application of manufacturing processes in which chemical or certain physical changes are involved. These processes may usually be resolved into a coordinated series of unit physical “operations” (hence part of the name of the chapter and book) and chemical processes. The work of the chemical engineer is concerned primarily with the design, construction, and operation of equipment and plants in which these unit operations and processes are applied. Chemistry, physics, and mathematics are the underlying sciences of chemical engineering, and economics is its guide in practice.

The above definition was appropriate up until a few decades ago when the profession branched out from the chemical industry. Today, that definition has changed. Although it is still based on chemical fundamentals and physical principles, these principles have been de-emphasized in order to allow for the expansion 3

4

Unit Operations in Environmental Engineering

of the profession to other areas (biotechnology, semiconductors, fuel cells, environment, etc.). These areas include environmental management, health and safety, computer applications, and economics and finance. This has led to many new definitions of chemical engineering, several of which are either too specific or too vague. A definition-proposed here is simply that “chemical engineers solve problems”. Unit operations is the one subject area that historically has been the domain of the chemical engineer. It is often present in the curriculum and includes fluid flow [1], heat transfer [2] and mass transfer [3] principles. Although the chemical engineering profession is usually thought to have originated shortly before 1900, many of the processes associated with this discipline were developed in antiquity. For example, filtration operations were carried out 5,000 years ago by the Egyptians. MTOs such as crystallization, precipitation, and distillation soon followed. During this period, other MTOs evolved from a mixture of craft, mysticism, incorrect theories, and empirical guesses. In a very real sense, the chemical industry dates back to prehistoric times when people first attempted to control and modify their environment. The chemical industry developed as did any other trades or crafts. With little knowledge of chemical science and no means of chemical analysis, the earliest chemical “engineers” had to rely on previous art and superstition. As one would imagine, progress was slow. This changed with time. The chemical industry in the world today is a sprawling complex of raw-material sources, manufacturing plants, and distribution facilities which supply society with thousands of chemical products, most of which were unknown only a century ago. In the latter half of the 19th century, an increased demand arose for engineers trained in the fundamentals of chemical processes. This demand was ultimately met by chemical engineers. The first attempt to organize the principles of chemical processing and to clarify the professional area of chemical engineering was made in England by George E. Davis. In 1880, he organized a Society of Chemical Engineers and gave a series of lectures in 1887 which were later expanded and published in 1901 as A Handbook of Chemical Engineering. In 1888, the first course in chemical engineering in the United States was organized at the Massachusetts Institute of Technology by Lewis M. Norton, a professor of industrial chemistry. The course applied aspects of chemistry and mechanical engineering to chemical processes [4]. Chemical engineering began to gain professional acceptance in the early years of the 20th century. The American Chemical Society had been founded in 1876 and, in 1908, it organized a Division of Industrial Chemists and Chemical Engineers while authorizing the publication of the Journal of Industrial and Engineering Chemistry. Also in 1908, a group of prominent chemical engineers met in Philadelphia and founded the American Institute of Chemical Engineers [4]. The mold for what is now called chemical engineering was fashioned at the 1922 meeting of the American Institute of Chemical Engineers when A. D. Little’s committee presented its report on chemical engineering education. The 1922 meeting marked the official endorsement of the unit operations concept and

1892

Pennsylvania University begins its Chemical Engineering curriculum

1888

The Massachusetts Institute of Technology begins “Course X,” the first 4 yr Chemical Engineering program in the U.S. Tulane begins its Chemical Engineering curriculum

1894

Figure 1.1 Chemical Engineering time line [4].

George Davis provides the blueprint for a new profession with 12 lectures on Chemical Engineering in Manchester, England

George Davis proposes a “Society of Chemical Engineers” in England

1880

The American Institute of Chemical Engineers is formed

1908

William H. Walker and Warren K. Lewis, two prominent professors establish a School of Chemical Engineering Practice

1915

The Massachusetts Institute of Technology starts an independent Department of Chemical Engineering

1920

Manhattan College begins its Chemical Engineering curriculum; adoption of R. Bird et al. “Transport Phenomena”a pproach [5]

1960

ABET stresses once again the emphasis on practical design approach

1990

Unit Operations versus Transport Phenomena; the profession at a crossroad

2010

History of Chemical Engineering and Unit Operations 5

6

Unit Operations in Environmental Engineering

saw the approval of a “declaration of independence” for the profession [4]. A key component of this report included the following: Any chemical process, on whatever scale conducted, may be resolved into a coordinated series of what may be termed “unit operations,” as pulverizing, mixing, heating, roasting, absorbing, precipitation, crystallizing, filtering, dissolving, and so on. The number of these basic unit operations is not very large and relatively few of them are involved in any particular process… An ability to cope broadly and adequately with the demands of this (the chemical engineer’s) profession can be attained only through the analysis of processes into the unit actions as they are carried out on the commercial scale under the conditions imposed by practice.

It also went on to state that: Chemical Engineering, as distinguished from the aggregate number of subjects comprised in courses of that name, is not a composite of chemistry and mechanical and civil engineering, but is itself a branch of engineering…

A classical approach to chemical engineering education, which is still used today, has been to develop problem solving skills through the study of several topics. One of the topics that has withstood the test of time is mass transfer operations (MTOs). In many MTOs, one component of a fluid phase is transferred to another phase because the component is more soluble in the latter phase. The resulting distribution of components between phases depends upon the equilibrium of the system. MTOs may also be used to separate products (and reactants) and may be used to remove byproducts or impurities to obtain highly pure products. Finally, they can be used to purify raw materials. A time line of the history of chemical engineering between the profession’s founding to 2010 is shown in Figure 1.1 [4]. It can be seen from the time line that the profession has reached a crossroads regarding the future education/curriculum for chemical engineers. This is highlighted by the differences of Transport Phenomena and Unit Operations, a topic that is discussed in the next chapter.

References 1. Abulencia, P. and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 2. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 3. Theodore, L. and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 4. Serino, N., 2005 Chemical Engineering 125th Year Anniversary Calendar, term project, submitted to L. Theodore, Manhattan College, Bronx, NY, 2004. 5. Bird, R., Stewart, W., and Lightfoot, E., Transport Phenomena, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2002.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

2 Transport Phenomena versus the Unit Operations Approach

The history of unit operations is interesting. As indicated in the previous chapter, chemical engineering courses were originally based on the study of unit processes and/or industrial technologies. However, it soon became apparent that the changes produced in equipment from different industries were similar in nature, i.e., there was commonality in the mass transfer operations in the petroleum industry and the chemical. These similar operations became known as unit operations. This approach to chemical engineering was promulgated in the 1922 Little report discussed earlier, and has, with varying degrees and emphasis, dominated the profession to this day. The unit operations approach was adopted by the profession soon after its inception. During the more than 135 years (since 1880) that the profession has been in existence as a branch of engineering, society’s needs have changed tremendously and so has chemical engineering. The teaching of unit operations at the undergraduate level has remained relatively unchanged since the publication of several early – to mid-1900 texts. However, by the middle of the 20th century, there was a slow movement from the unit operation concept to a more theoretical treatment called transport phenomena or, more simply, engineering science. The focal point of this science is the rigorous mathematical description of all physical rate processes in terms of 7

8

Unit Operations in Environmental Engineering

mass, heat, or momentum crossing phase boundaries. This approach took hold of the education/curriculum of the profession with the publication of the first edition of the Bird et al. book [1] in 1960. Some, including the authors of this text, feel that this concept set the profession back several decades since graduating chemical engineers were being trained more as applied physicists than traditional chemical engineers. There has fortunately been a return to the traditional approach to chemical engineering, primarily as a result of the efforts of ABET (Accreditation Board for Engineering and Technology). The more traditional approach replaced some theoretical material normally covered in transport phenomena courses in part with material emphasizing the solution of design and open-ended problems. This design-oriented approach is emphasized in this text. The following paragraphs attempt to qualitatively describe the differences between the above two approaches. Both deal with the transfer of certain quantities (momentum, energy, and mass) from one point in a system to another. There are three basic transport mechanisms which can potentially be involved in a process. They are: 1. Radiation 2. Convection 3. Molecular Diffusion The first mechanism, radiative transfer, arises as a result of wave motion and is not considered, since it may be justifiably neglected in most engineering applications. The second mechanism, convective transfer, occurs simply because of bulk motion. The final mechanism, molecular diffusion, can be defined as the transport mechanism arising as a result of gradients. For example, momentum is transferred in the presence of a velocity gradient; energy in the form of heat is transferred because of a temperature gradient; and, mass is transferred in the presence of a concentration gradient. These molecular diffusion effects are described by phenomenological laws [1]. Momentum, energy, and mass are all conserved. As such, each quantity obeys the conservation law within a system. The conservation law may be applied at the macroscopic, microscopic, or molecular level. One can best illustrate the differences in these methods with an example. Consider a system in which a fluid is flowing through a cylindrical tube (see Figure 2.1) and define the system as the fluid contained within the tube between Points 1 and 2 at any time. If one is interested in determining changes occurring at the inlet and outlet of a system, the conservation law is applied on a “macroscopic” level to the entire system. The resultant equation (usually algebraic) describes the overall changes occurring to the system (or equipment). This approach is usually applied in Unit Operation (or its equivalent) courses, an approach which is highlighted in this and three companion texts [2–4]. In the microscopic/transport phenomena approach, detailed information concerning the behavior within a system is required; this is occasionally requested of

Transport Phenomena versus the Unit Operations Approach 9 1

2

Fluid out

Fluid in

1

2

Figure 2.1 Fluid flow through a cylinder tube.

and by the engineer. The conservation law is then applied to a differential element within the system that is large compared to an individual molecule, but small compared to the entire system. The resulting differential equation is then expanded via an integration in order to describe the behavior of the entire system. The molecular approach involves the application of the conservation laws to individual molecules. This leads to a study of statistical and quantum mechanics – both of which are beyond the scope of this text. In any case, the description at the molecular level is of little value to the practicing engineer. However, the statistical averaging of molecular quantities in either a differential or finite element within a system can lead to a more meaningful description of the behavior of a system. Both the microscopic and molecular approaches shed light on the physical reasons for the observed macroscopic phenomena. Ultimately, however, for the practicing engineer, these approaches may be valid but are akin to attempting to kill a fly with a machine gun. Developing and solving these differential equations (in spite of the advent of computer software packages) is typically not worth the trouble. Traditionally, the applied mathematician has developed differential equations describing the detailed behavior of systems by applying the appropriate conservation law to a differential element or shell within the system. Equations were derived with each new application. The engineer later removed the need for these tedious and error-prone derivations by developing a general set of equations that could be used to describe systems. These have come to be referred to by some as the transport equations. In recent years, the trend toward expressing these equations in vector form has gained momentum (no pun intended). However, the shell-balance approach has been retained in most texts where the equations are presented in componential form, i.e., in three particular coordinate systems – rectangular, cylindrical, and spherical. The componential terms can be “lumped” together to produce a more concise equation in vector form. The vector equation can be, in turn, re-expanded into other coordinate systems. This information is available in the literature [1,5]. It should be noted that the macroscopic approach has been primarily adapted by undergraduate environmental engineering educators. Any attempt to include the microscopic approach has been essentially reserved solely for graduate studies in this area. Only the macroscopic approach is employed in this text.

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Unit Operations in Environmental Engineering

References 1. Bird, R., Stewart, W., and Lightfoot, E., Transport Phenomena, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 1960. 2. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 3. Abulencia, P. and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 4. Theodore, L. and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 5. Theodore, L., Introduction to Transport Phenomena, International Textbook Co., Scranton, PA, 1970.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

3 The Conservation Laws and Stoichiometry

This chapter is primarily concerned with the conservation laws, and to a lesser degree, chemical stoichiometry. As with all remaining chapters in Part I, there are several sections: an overview, specific technical topics, illustrative examples where appropriate, and references.

3.1 Overview In order to better understand the design, as well as the operation and performance of equipment in the environmental industry, it is necessary for engineers (as well as applied scientists) to understand the fundamentals and principles underlying the conservation laws and stoichiometry. How can one predict what products will be emitted from effluent streams? At what temperature must a unit be operated to ensure the desired performance? How much energy in the form of heat is given off? Is it economically feasible to recover this heat? Is the design appropriate? The answers to these questions are rooted not only in the subject matter in this chapter but also in the various theories of chemistry, physics, and applied economics.

11

12

Unit Operations in Environmental Engineering The remaining topics covered in this section include: 1. The Conservation Law 2. The Conservation Laws for Mass, Energy and Momentum 3. Stoichiometry

Note: the bulk of the material in this chapter has been drawn from the original work of Reynolds [1].

3.2 The Conservation Law Mass, energy and momentum are all conserved. As such, each quantity obeys the general conservation law below, as applied within a system.

quantity into system

quantity out of system

quantity generated in system

quantity accumulated in system

(3.1)

Equation 3.1 may also be written on a time rate basis:

rate into system

rate out of system

rate generated in system

rate accumulated in system

(3.2)

The conservation law may be applied by the practitioner at the macroscopic, microscopic, or molecular level. The differences in these methods was illustrated with an example in the previous chapter in Figure 2.1 by considering fluid flow through a cylindrical tube. The microscopic approach is employed when detailed information concerning the behavior within the system is of interest. If one is interested in determining changes occurring at the inlet and outlet of the system, the conservation law is applied on a macroscopic level to the entire system. The resultant equation describes the overall changes occurring to the system without regard for internal variations within the system, and it is this approach that is usually applied by the practicing engineer. The macroscopic approach is primarily adopted and applied in this text, and little to no further reference to microscopic or molecular analyses will be made. This chapter’s aim, then, is to express the laws of conservation for mass, energy, and momentum in algebraic or finite difference form.

3.3 Conservation of Mass, Energy, and Momentum The conservation law for mass can be applied to any process, equipment, or system. The general form of this law is given by Equations 3.3 and 3.4.

The Conservation Laws and Stoichiometry 13

mass in

mass out

I

O

mass generated

mass consumed

G

C

+

mass accumulated =

(3.3)

A

or on a time rate basis by

rate of mass in I

rate of mass out O

+

rate of mass generated

rate of mass consumed

G

C

rate of mass (3.4) accumulated =

A

The law of conservation of mass states that mass can neither be created nor destroyed. Nuclear reactions, in which interchanges between mass and energy are known to occur provide a notable exception to this law. Even in chemical reactions, a certain amount of mass-energy interchange takes place. However, in normal environmental engineering applications, nuclear reactions do not occur and the mass-energy exchange in chemical reactions is so minuscule that it is not worth taking into account. The law of conservation of energy, which like the law of conservation of mass, applies for all processes that do not involve nuclear reactions, states that energy can neither be created nor destroyed. As a result, the energy level of the system can change only when energy crosses the system boundary, i.e.,

Δ (Energy level of system) = Energy crossing boundary

(3.5)

(Note: The symbol “Δ” means “change in”). Energy crossing the boundary can be classified in one of two different ways: heat, Q, or work, W. Heat is energy moving between the system and the surroundings by virtue of a temperature driving force, and heat flows from high temperature to low temperature. The entire system is not necessarily at the same temperature; neither are the surroundings. If a portion of the system is at a higher temperature than a portion of the surroundings and as a result, energy is transferred from the system to the surroundings, that energy is classified as heat. If part of the system is at a higher temperature than another part of the system and energy is transferred between the two parts, that energy is not classified as heat because it is not crossing the boundary. Work is also energy moving between the system and surroundings, but the driving force here is something other than temperature difference, e.g., a mechanical force, a pressure difference, gravity, a voltage difference, a magnetic field, etc. Note that the definition of work is a force acting through a distance. All of the examples of driving forces just cited can be shown to provide a force capable of acting through a distance [2].

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Unit Operations in Environmental Engineering

The energy level of a system has three contributions: kinetic energy, potential energy, and internal energy. Any body in motion possesses kinetic energy. If the system is moving as a whole, its kinetic energy, Ek, is proportional to the mass of the system and the square of the velocity of its center of gravity. The phrase “as a whole” indicates that motion inside the system relative to the system’s center of gravity does not contribute to the Ek term, but rather to the internal energy term. The terms external kinetic energy and internal kinetic energy are sometimes used here. An example would be a moving railroad tank car carrying liquid waste. (The liquid waste is the system). The center of gravity of the waste is moving at the velocity of the train, and this constitutes the system’s external kinetic energy. The liquid molecules are also moving in random directions relative to the center of gravity, and this constitutes the system’s internal energy due to motion inside the system, i.e., internal kinetic energy. The potential energy, Ep, involves any energy the system as a whole possesses by virtue of its position (more precisely, the position of its center of gravity) in some force field, e.g., gravity, centrifugal, electrical, etc., that provides the system with the potential for accomplishing work. Again, the phase “as a whole” is used to differentiate between external potential energy, Ep, and internal potential energy. Internal potential energy refers to potential energy due to force fields inside the system. For example, the electrostatic force fields (bonding) between atoms and molecules provide these particles with the potential for work. The internal energy, U, is the sum of all internal kinetic and internal potential energy contributions [2]. The law of conservation of energy, which is also called the first law of thermodynamics, may now be written as:

Δ(U + Ek + Ep) = Q + W

(3.6)

ΔU + ΔEk + ΔEp = Q + W

(3.7)

or equivalently as

It is important to note the sign convention for Q and W adapted for the above equation. Since any term is always defined as the final minus the initial state, both the heat and work terms must be positive when they cause the system to gain energy, i.e., when they represent energy flowing from the surroundings to the system. Conversely, when the heat and work terms cause the system to lose energy, i.e., when energy flows from the system to the surroundings, they are negative in sign. This sign convention is not universal and the reader must take care to check what sign convention is being used by a particular author when referring to the literature. For example, work is often defined in some texts as positive when the system does work on the surroundings [2, 3]. The application of the conservation laws to both environmental equipment and process design and analysis is presented in Chapter 8.

The Conservation Laws and Stoichiometry 15

3.4 Stoichiometry When chemicals react, they do so according to a strict proportion. When oxygen and hydrogen combine to form water, the ratio of the amount of oxygen to the amount of hydrogen consumed is always 7.94 by mass and 0.500 by moles. The term stoichiometry refers to this phenomenon, which is sometimes called the chemical law of combining weights. The reaction equation for the combining of hydrogen and oxygen is:

2H2 + O2 = 2H2O

(3.8)

In chemical reactions, atoms are neither generated nor consumed, merely rearranged with different bonding partners. The manipulation of the coefficients of a reaction equation so that the number of atoms of each element on the left of the equation is equal to that on the right is referred to as balancing the equation. Once an equation is balanced, the whole number molar ratio that must exist between any two components of the reaction can be determined simply by observation; these are known as stoichiometric ratios. There are three such ratios (not counting the reciprocals) in the above reaction. These are: 2 mol H2 consumed/mol O2 consumed 1 mol H2O generated/mol H2 consumed 2 mol H2O generated/mol O2 consumed The unit mole represents either the gmol or the lbmol. Using molecular weights, these stoichiometric ratios (which are molar ratios) may easily be converted to mass ratios. For example, the first ratio above may be converted to a mass ratio by using the molecular weights of H2 (2.016) and O2 (31.999) as follows: (2 gmol H2 consumed) (2.016 g/gmol) = 4.032 g H2 consumed (1 gmol O2 consumed) (31.999 g/gmol) = 31.999 g O2 consumed The mass ratio between the hydrogen and oxygen consumed is therefore: 4.032/31.999 = 0.126 g H2 consumed/g O2 consumed These molar and mass ratios are used in material balances to determine the amounts or flow rates of components involved in chemical reactions. Multiplying a balanced reaction equation through by a constant does nothing to alter its meaning. The reaction used as an example above is often written:

H2 + ½ O2 = H2O

(3.9)

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Unit Operations in Environmental Engineering

In effect, the stoichiometric coefficients of Equation (3.8) have been multiplied by 0.5. There are times, however, when care must be exercised because the solution to a problem may depend on the manner or form the reaction is written. This is the case with chemical equilibrium problems and problems involving thermochemical reaction equations. These are addressed in Chapters 5 and 6. There are two different types of material balances that may be written when a chemical reaction is involved: the molecular balance and the atomic balance. It is a matter of convenience which of the two types is used. Each is briefly discussed below. The molecular balance is the same as that described earlier. Assuming a steadystate continuous reaction, the accumulation term, A, is zero for all components involved in the reaction, Equation 3.3 then becomes:

I+G=O+C

(3.10)

If a total material balance is performed, the above form of the balance equation must be used if the amounts or flow rates are expressed in terms of moles, e.g., lbmol or gmol/h, since the total number of moles can change during a chemical reaction. If, however, the amounts or flow rates are given in terms of mass, e.g., kg or lb/h, the G and C terms may be dropped since mass cannot be gained or lost in a chemical reaction. Thus,

I=O

(3.11)

In general, however, when a chemical reaction is involved, it is usually more convenient to express amounts and flow rates using moles rather than mass. A material balance that is not based on the chemicals (or molecules), but rather on the atoms that make up the molecules, is referred to as an atomic balance. Since atoms are neither created nor destroyed in a chemical reaction, the G and C terms equal zero and the balance once again becomes:

I=O

(3.11)

As an example, consider once again the combination of hydrogen and oxygen to form water:

2H2 + O2 = 2H2O

(3.8)

As the reaction progresses, O, and H, molecules (or moles) are consumed while H2O molecules (or moles) are generated. On the other hand, the number of oxygen atoms (or moles of oxygen atoms) and the number of hydrogen atoms (or moles of hydrogen atoms) do not change. Care must also be taken to distinguish between molecular oxygen and atomic oxygen. If, in the above reaction, one starts

The Conservation Laws and Stoichiometry 17 out with 1000 lbmol of O2 (oxygen molecules), one may replace this with 2000 lbmol of O (oxygen atoms). Thus, a chemical equation provides a variety of qualitative and quantitative information essential for the calculation of the quantity of reactants reacted and products formed in a chemical process. As noted, a balanced chemical equation must have the same number of atoms of each type in the reactants on the left hand side of the equation and in the products on the right hand side of the equation. Thus a balanced equation for butane combustion (reaction with oxygen to form oxidized end products CO2 and H2O) is:

C4H10 + (13/2) O2 = 4CO2 + 5H2O

(3.12)

Note that: Number of carbons in reactants = number of carbons in products = 4 Number of oxygens in reactants = number of oxygens in products = 13 Number of hydrogens in reactants = number of hydrogens in products = 10 Number of moles of reactants is 1 mol C4H10 + 6.5 mol O2 = 7.5 mol total Number of moles of products is 4 mol CO2 + 5 mol H2O = 9 mol total The reader should note that although the number of moles on both sides of the equation do not balance, the masses of reactants and products (in line with the conservation law for mass) must balance.

3.5 Limiting and Excess Reactants Limiting and excess reactants involve an extension of the stoichiometric calculations provided above. Consider the following example. When methane is combusted completely, the stoichiometric equation for the reaction is:

CH4 + 2O2 = CO2 + 2H2O

(3.13)

The stoichiometric ratio of the oxygen to the methane is:

0.5 mol methane consumed/mol oxygen consumed If one starts out with 1 mol of methane and 3 mol of oxygen in a reaction vessel, only 2 mol of oxygen would be used up, leaving an excess of 1 mol of oxygen in the vessel. In this case, the oxygen is called the excess reactant and methane is the limiting reactant. The limiting reactant is defined as the reactant that would

18

Unit Operations in Environmental Engineering

be completely consumed if the reaction went to completion. All other reactants are excess reactants. The amount by which a reactant is present in excess of stoichiometric requirements (i.e., the exact number of moles needed to react completely with the limiting reactant) is defined as the percent excess, and is given by Equation 3.14:

% excess

n ns ns

100

(3.14)

where n = number of moles of the excess reactant at the start of the reaction; and ns = the stoichiometric number of moles of the excess reactant. In the example above, the stoichiometric amount of oxygen is 2 mol, since that is the amount that would react with the 1 mol of methane. The excess amount of oxygen is 1 mol, which is a percentage excess of 50% or a fractional excess of 0.50. These definitions are employed in Chapter 6, which is concerned with chemical kinetics. A detailed and expanded treatment of stoichiometry is available in references [3] and [4].

References 1. Reynolds, J., Material and Energy Balances, A Theodore Tutorial, East Williston, NY, originally published by the USEPA/APTI, RTP, NC, 1992. 2. Theodore, L., and Reynolds, J., Thermodynamics, A Theodore Tutorial, East Williston, NY, originally published by the USEPA/APTI, RTP, NC, 1991. 3. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York City, NY, 2008. 4. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2014.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

4 The Ideal Gas Law

There are numerous environmental science and engineering applications that involve air pollution. These applications require an understanding of the role the Ideal Gas Law plays in fundamental calculations and relationships related to gas volumes, densities, and concentrations.

4.1 Overview Observations based on physical experimentation often can be synthesized into simple mathematical equations called laws. These laws are never perfect and hence are only an approximate representation of reality. The Ideal Gas Law (IGL) is one such law derived from experiments in which the effects of pressure and temperature on gaseous volumes were measured over moderate temperature and pressure ranges. This law works well in the pressure and temperature ranges that were used in collecting the data; extrapolations outside of the ranges have been found to work well in some cases and poorly in others. As a general rule, this law works best when the molecules of the gas are far apart, i.e., when the pressure is low and the temperature is high. Under these conditions, the gas is said to behave ideally, i.e., its behavior is a close approximation to the so-called perfect or ideal gas: a hypothetical entity that obeys the IGL exactly. For environmental calculations, the 19

20

Unit Operations in Environmental Engineering

ideal gas law is often assumed to be valid since it generally works well (usually within a few percent of the correct result) up to the highest pressures and down to the lowest temperatures used in many industrial environmental applications [1]. The two precursors of the ideal gas law were Boyle’s Law and Charles’ Law. Boyle found that the volume of a given mass of gas is inversely proportional to its absolute pressure if the temperature is kept constant:

P1V1 = P2V2

(4.1)

where V1 = volume of gas at absolute pressure P1 and temperature T; and V2 = volume of gas at absolute pressure P2 and absolute temperature T. Charles found that the volume of a given mass of gas varies directly with the absolute temperature at constant pressure:

V1/T1 = V2/T2

(4.2)

where V1 = volume of gas at pressure P and absolute temperature T1; and V2 = volume of gas at pressure P and temperature T2. Boyle’s and Charles’ laws may be combined into a single equation in which neither temperature nor pressure need be held constant:

P1V1/T1 = P2V2/T2

(4.3)

For Equation 4.3 to hold, the mass of gas must be constant as the conditions change from (P1, T1) to (P2, T2). This equation indicates that for a given mass of a specific gas, PV/T has a constant value. Since, at the same temperature and pressure, volume and mass must be directly proportional, this statement may be extended to:

PV/mT = C

(4.4)

where m = mass of a specific gas and C = a constant that depends on the gas. Moreover, experiments with different gases showed that Equation 4.4 can be expressed in a far more generalized form. If the number of moles (n) is used in place of the mass (m), the constant is the same for all gases:

PV/nT = R

(4.5)

where R = the universal gas constant. Equation 4.5 is referred to as the Ideal Gas Law. Numerically, the value of R depends on the units used for P, V, T and n (see Table 4.1).

4.2

Other Forms of the Ideal Gas Law

Other useful forms of the IGL are shown in Equations 4.6 and 4.7. Equation 4.6 applies to gas flow rather than to gas confined in a container.

The Ideal Gas Law

21

Table 4.1 Values of R in various units. R

Temperature scale

Units of V

Units of n

Units of P

Units of PV (energy)

10.73

°R

ft3

lbmol

psia

–

0.7302

°R

3

ft

lbmol

atm

–

21.85

°R

ft3

lbmol

In Hg

–

555.0

°R

ft3

lbmol

mm Hg

–

297.0

°R

ft3

lbmol

in H20

–

0.7398

°R

ft3

lbmol

bar

–

1545.0

°R

3

ft

lbmol

psfa

–

24.75

°R

ft3

lbmol

ft H20

–

1.9872

°R

–

lbmol

–

Btu

0.0007805

°R

–

lbmol

–

hp–h

0.0005819

°R

–

lbmol

–

kW–h

500.7

°R

–

lbmol

–

cal

3

1.314

K

ft

lbmol

atm

–

998.9

K

ft3

lbmol

mm Hg

–

19.32

K

3

ft

lbmol

psia

–

62.361

K

L

gmol

mm Hg

–

0.08205

K

L

gmol

atm

–

0.08314

K

L

gmol

bar

–

8314

K

L

gmol

Pa

–

8.314

K

m3

gmol

Pa

–

3

82.057

K

cm

gmol

atm

–

1.9872

K

–

gmol

–

cal

8.314

K

–

gmol

–

J

Pq nRT

(4.6)

where P = absolute pressure (psia); q = gas volumetric flow rate (ft3/h); n = molar flow rate (lbmol/h); R = 10.73 psia-ft3/lbmol-°R; and T = absolute temperature (°R). Equation 4.7 combines n and V from Equation 4.6 to express the law in terms of density:

P(MW) = RT where MW= molecular weight of the gas (lb/lbmol) and (lb/ft3).

(4.7) = density of the gas

22

Unit Operations in Environmental Engineering

Volumetric flow rates are often not given at the actual conditions of pressure and temperature but at arbitrarily chosen standard conditions (STP, standard temperature and pressure). To distinguish between flow rates based on the two conditions, the letters “a” and “s” are often used as part of the unit. The units acfm and scfm represent actual cubic feet per minute and standard cubic feet per minute, respectively. The ideal gas law can be used to convert from standard to actual conditions, but, since there are many standard conditions in use, the STP employed must be known. Standard conditions most often used in environmental applications are shown in Table 4.2. The reader is cautioned on the incorrect use of acfm and/or scfm. Employing standard conditions is a convenience; when predicting the performance of or designing equipment, the actual conditions must be employed. Designs based on standard conditions can lead to disastrous results, with the unit usually under designed. For example, for a gas stream at 2140 °F, the ratio of acfm to scfm (standard temperature = 60 °F) is 5.0. Equation 4.8, which is a form of Charles’ law, can be used to correct flow rates from standard to actual conditions:

qa = qs (Ts/Ta)

(4.8)

where qa = volumetric flow rate at actual conditions (ft3/h); qs = volumetric flow rate at standard conditions (ft3/h); Ts = standard absolute temperature (°R); and Ta = actual absolute temperature (°R). The reader is again reminded that absolute temperatures and absolute pressures must be employed in all IGL calculations. In engineering practice, mixtures of gases are more often encountered than single or pure gases. The IGL is based on the number of molecules present in the gas volume; the type of molecules is not a significant factor, only the number. The IGL applies equally well to mixtures and pure gases alike. Dalton and Amagat both applied the IGL to mixtures of gases. Since pressure is caused by gas molecules colliding with the walls of a container, it seems reasonable that the total pressure of a gas mixture is made up of pressure contributions due to each of the component gases. These pressure contributions are called partial pressures. Dalton defined the partial pressure of a component as the pressure that would be exerted if the same Table 4.2 Common standard conditions. System

Temperature

Pressure

Molar volume

SI

273 K

101.3 kPa

22.4 m3/kmol

Universal scientific

0 °C

760 mmHg

22.4 L/gmol

Natural gas industry

60 °F

14.7 psia

379 ft3/lbmol

American engineering

32 °F

1 atm

359 ft3/lbmol

Environmental industry

60 °F

1 atm

379 ft3/lbmol

70 °F

1 atm

387 ft3/lbmol

The Ideal Gas Law

23

mass of the component gas occupied the same total volume alone at the same temperature as the mixture. The sum of these partial pressures equals the total pressure: n

P

pa

pb

pc

pn

pi

(4.9)

i 1

where P = total pressure; pi = partial pressure of component i; and n = number of components in the mixture. Equation 4.9 is known as Dalton’s law of partial pressures. Applying the ideal gas law to one component (A) only, yields:

pAV = nART

(4.10)

where nA = number of moles of component A. Eliminating R, T, and V between Equations 4.5 and 4.10 yields:

pA/P = nA/n = yA

(4.11)

where yA = mole fraction of component A. Amagat’s law is similar to Dalton’s law. Instead of considering the total pressure to consist of partial pressures where each component occupies the total container volume, Amagat considered the total volume to consist of the partial volumes in which each component is at (or is exerting) the total pressure. The definition of the partial volume is therefore the volume occupied by a component gas alone at the same temperature and pressure as the mixture. For this case: n

V

VA VB VC

Vn

Vi

(4.12)

i 1

Applying Equation 4.10 as before, one obtains:

VA/V = nA/n = yA

(4.13)

where VA = partial volume of component A. It is common in environmental engineering practice to describe low concentrations of components in gaseous mixtures in parts per million by volume, ppmv. Since partial volumes are proportional to mole fractions, it is necessary only to multiply the mole fraction of the component by 1 million to obtain the concentration in parts per million. For liquids and solids, parts per million (ppm) is also used to express concentration, although it is usually on a mass basis rather than a volume basis. The terms ppmv and ppmw are sometimes used to distinguish between the concentration of a component on a volume or mass basis, respectively.

24

4.3

Unit Operations in Environmental Engineering

Non-Ideal Gas Behavior

Some environmental applications require that deviations from ideality be included in the analysis. Many of the non-ideal correlations involve the critical temperature Tc, the critical pressure Pc, and a term defined as the acentric factor, w. An abbreviated list of these properties is available in the literature [2, 3]. These reduced quantities find wide application in thermodynamic analyses of non-ideal systems (see also Chapter 5). The critical temperature and pressure are employed in the calculation of the reduced temperature, Tr, and the reduced pressure, Pr, as provided in Equations 4.14 and 4.15:

Tr = T/Tc

(4.14)

Pr = P/Pc

(4.15)

Both reduced properties are dimensionless and play important roles in nonideal gas behavior. Many physical and chemical properties of elements and compounds can be estimated from models (equations) that are based on the reduced temperature and pressure of the substance in question. These reduced properties have also served as the basis for many equations that are employed in practice to describe non-ideal gas (and liquid) behavior. Although a rigorous treatment of this material is beyond the scope of this book, information is available in the literature [2,3]. Highlights of this topic are presented below. No real gas conforms exactly to the IGL, but it can be used as an excellent approximation for most gases at pressures about or less than 5 atm and near ambient temperatures. One approach to account for the previously mentioned deviations from ideality is to include a correction factor, Z, which is defined as the compressibility coefficient or compressibility factor. The ideal gas law is then modified to the following form:

PV = ZnRT

(4.16)

Note that Z approaches 1.0 as P approaches 0.0. For an ideal gas, Z is exactly unity. This equation may also be written as:

Pv = ZRT

(4.17)

where v is now the specific molar volume (not the total volume) with units of volume/mole. Regarding gas mixtures, the ideal gas law can be applied directly for ideal gas mixtures. However, the molecular weight of the mixture is based on a mole fraction average, MW , of the n components: n

MW

( yi )( MWi ) i 1

(4.18)

The Ideal Gas Law

25

One approach to account for deviations from ideality is to assume the aforementioned compressibility coefficient for the mixture, Zm , is a linear mole fraction combination of the individual component Z values: n

Zm

y i Zi

(4.19)

i 1

Furthermore, Kay [4] has shown that the deviations arising in using this approach can be reduced by employing pseudocritical values for T and P where: n

Tc

yiTci

(4.20)

yi Pci

(4.21)

i 1 n

Pc i 1

In lieu of other information, the authors suggest employing Equation 4.22 for the pseudocritical value of v: n

v

y i vi

(4.22)

i 1

These pseudocritical values – Tc,Pc,and v – are then employed in the appropriate pure component equation of state. This approach has been defined by some as Kay’s rule, an approach that has unfortunately been abandoned in recent years. Another equation of state available to account for observations from ideality is that proposed by van der Waal. This equation attempts to correct for intermolecular forces of attraction (a/V2) and the volume occupied by the molecules themselves (b) in the following manner [1]:

(P + a/V2) (V – b) = RT

(4.23)

where a = (27R2Tc2)/(64 Pc), and b = (RTc)/(8Pc).

References 1. Theodore, L., Ricci, F., and VanVliet, T., Thermodynamics for the Practicing Engineer, John Wiley and Sons, Hoboken, N.J., 2009. 2. Smith, J., Van Ness, H., and Abbott, M., Introduction to Chemical Engineering Thermodynamics, 6th Edition, McGraw-Hill, New York City, NY, 2005. 3. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York City, NY, 2008. 4. Kay, W., Density of hydrocarbon gases and vapors. Ind. Eng. Chem., 28, 1014, 1936.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

5 Thermodynamics

Prior to undertaking the writing of this text, one of the authors co-authored a text entitled Thermodynamics for the Practicing Engineer [1]. It soon became apparent that some overlap existed between thermodynamics (the subject of this chapter) and heat transfer (the subject of –Part III of this text). Even though the former topic is broadly viewed as engineering science, heat transfer is one of the unit operations and can justifiably be classified as an engineering subject. But what are the similarities and what are the differences? The similarities that exist between thermodynamics and heat transfer are grounded in the three conservation laws: mass, energy, and momentum. Both are primarily concerned with energy-related subject matter and both, in a very real sense, supplement each other. However, thermodynamics deals with the transfer of energy and the conversion of one form of energy into another (e.g., heat into work), with consideration generally limited to systems in equilibrium. The topic of heat transfer deals with the transfer of energy in the form of heat; the applications almost exclusively occur within heat exchangers that are employed in chemical, petrochemical, petroleum (refinery) and environmental engineering processes and applications.

27

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Unit Operations in Environmental Engineering

5.1 Overview Thermodynamics was once defined as “the science that deals with the intertransformation of heat and work.” The fundamental principles of thermodynamics are contained in the first, second, and third laws of thermodynamics. These principles have been defined as “pure” or “theoretical” thermodynamics. These laws were developed and extensively tested in the latter half of the 19th Century and are essentially based on experience. (The third law was developed later in the 20th Century). Practically all thermodynamics, in the ordinary meaning of the term, is “applied thermodynamics” in that it is essentially the application of these three laws, coupled with certain facts and principles of mathematics, physics, and chemistry, to problems in engineering and science. The fundamental laws are of such generality that it is not surprising that these laws find application in other disciplines, including physics, chemistry, plus environmental, chemical and mechanical engineering. The first law of thermodynamics is a conservation law for energy transformations. Regardless of the types of energy involved in processes – thermal, mechanical, electrical, elastic, magnetic, etc. – the change in energy of a system is equal to the difference between energy input and energy output. The first law also allows free “convertibility” from one form of energy to another, as long as the overall energy quantity is conserved. Thus, this law places no restriction on the conversion of work into heat, or on its counterpart – the conversion of heat into work. Because work is 100% convertible to heat, whereas the reverse situation is not true, work is a more valuable form of energy than heat. This leads to an important second-law consideration – i.e., that energy has quality as well as quantity. Although it is not as obvious, it can also be shown through the second-law principles and arguments that heat has quality in terms of its temperature. The higher the temperature at which heat transfer occurs, the greater the potential for energy transformation into work. Thus, thermal energy stored at higher temperatures is generally more useful to society than that available at lower temperatures. While there is an immense quantity of energy stored in the oceans and the earth’s core, for example, its present availability to society for performing useful tasks is essentially nonexistent. Theodore et al. [1] provide additional qualitative reviews of the second law. The choice of topics to be reviewed in this chapter was initially an area of debate, and after some deliberation, it was decided to provide an introduction to five areas that many have included in this broad engineering subject. These are detailed below: 1. 2. 3. 4. 5.

The First Law of Thermodynamics Enthalpy Effects Second Law Calculations Phase Equilibrium Chemical Reaction Equilibrium

Thermodynamics 29 The reader should note that the bulk of the material in this chapter has been drawn from L. Theodore, Thermodynamics, A Theodore Tutorial, originally published by the USEPA/APTI, RTP, NC in 1991 [2].

5.2 The First Law of Thermodynamics For many environmental processes, the energy requirement represents a major item of the cost of operation and one cannot arrive at a proper systems analysis and/or economic evaluation without performing an energy balance as well as a material balance. Just as practicing engineers rely on the law of conservation of mass for a material balance, they depend on the law of the conservation of energy for energy balance calculations. These energy balance considerations are based on thermodynamics, the branch of science founded on laws of experience which deal with both energy and its conversion, as well as the transfer of energy in terms of heat and work as a system passes from one equilibrium state to another. Joule’s experiments cleared the way for the enunciation of the first law of thermodynamics; namely, when a closed system is taken through a cyclic process, the work done on the surroundings equals the heat absorbed from the surroundings. This law for batch processes, can be represented as:

ΔE = Q + W

(5.1)

where potential, kinetic, and other energy effects have been neglected and E (often denoted as U), is the internal energy of the system, ΔE is the change in the internal energy of the system, Q is energy in the form of heat transferred across the system boundaries, and W is energy in the form of work transferred across system boundaries. In accordance with a recent change in convention, both Q and W are treated as positive terms if added to the system. For practical purposes, the total work term, W, in the first law may be regarded as the sum of shaft work, Ws, and flow work, Wf

W = Ws + Wf

(5.2)

where Ws is work done on the fluid by some moving solid part within the system such as the vanes of a centrifugal pump. Note that in Equation 5.2, all other forms of work such as electrical, surface tension, and so on are neglected. The first law of thermodynamics for steady-state flow processes is then:

H = Q + Ws

(5.3)

where H is the enthalpy of the system and ΔH is the change in the system’s enthalpy.

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Unit Operations in Environmental Engineering

The internal energy and enthalpy in Equations 5.1 and 5.2, as well as other equations in this section may be on a mass basis, on a mole basis, or represent the total internal energy and enthalpy of the entire system. They may also be written on a time-rate basis as long as these equations are dimensionally consistent. For the sake of clarity, upper case letters (e.g., H, E) represent properties on a mole basis, while lower-case letters (e.g., h, e) represent properties on a mass basis. Properties for the entire system will rarely be used and therefore require no special symbols. Perhaps the most important thermodynamic function the engineer works with is the above mentioned enthalpy. This is a term that requires additional discussion. The enthalpy is defined by the equation

H = E + PV

(5.4)

where P is once again the pressure of the system and V is the volume of the system. The terms E and H are state or point functions. By fixing a certain number of variables upon which the function depends, the numerical value of the function is automatically fixed; that is, it is single-valued. For example, fixing the temperature and pressure of a one-component single-phase system immediately specifies its enthalpy and internal energy.

5.3

Enthalpy Effects

There are many different types of enthalpy effects. These include: 1. 2. 3. 4. 5.

Sensible (temperature) Latent (phase) Dilution (with water), e.g., HCl with H2O Solution (nonaqueous), e.g., HCl with a solvent other than H2O Reaction (chemical)

This section is only concerned with Effects 1, 2 and 5. Details on Effects 3 through 4 are available in the literature [1–3].

5.3.1 Sensible enthalpy effects Sensible enthalpy effects are associated with temperature. There are methods that can be employed to calculate these changes. These methods include the use of: 1. enthalpy values 2. average heat capacity values 3. heat capacity as a function of temperature

Thermodynamics 31 If enthalpy values are available, the enthalpy change is given by

h = h2 – h1; mass basis

(5.5)

H = H2 – H1; mole basis

(5.6)

If average molar heat capacity data are available,

H

Cp T

(5.7)

where C p = average molar value of Cp in the temperature range ΔT. Average molar heat capacity data are provided in the literature [1–5]. A more rigorous approach to enthalpy calculations can be provided if the heat capacity variation with temperature is available. If the heat capacity is a function of the temperature, the enthalpy change is written in differential form:

dH = CpdT

(5.8)

If the temperature variation of the heat capacity is given by

Cp =

+ T + T2

(5.9)

Equation 5.8 may be integrated directly between some reference or standard temperature (To) and the final temperature (T1) employing Equation 5.9:

H = H1 – Ho

(5.10)

H = (T1 – To) + ( /2)(T12 – To2) + ( /3)(T13 – To3)

(5.11)

Equation 5.8 may also be integrated if the heat capacity is a function of temperature of the form:

Cp = a + bT + cT

2

(5.12)

The enthalpy change is then given by

H = a(T1 – To) + (b/2)(T12 – To2) + c(T1–1 – To–1)

(5.13)

Tabulated values of , , , and a, b, c for a host of compounds (including some chlorinated organics) are available in the literature [2].

5.3.2 Latent Enthalpy Changes It has been observed that there is absorption of heat at constant temperature and pressure that accompanies the transition or equilibrium phase change from solid

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Unit Operations in Environmental Engineering

to liquid and liquid to gas. In terms of molecular theory, this latent enthalpy represents the energy required to overcome inter-molecular forces of attraction and to permit molecules to pass from a more highly restrained condensed phase to a more mobile phase. This relationship indicates that the latent heat is the sum of a change in internal energy and energy of expansion (or contraction). Common types of transition and the terminology applied to them are: Type of transition: solid liquid liquid gas solid vapor solid solid

Designation of energy change: enthalpy of fusion enthalpy of vaporization enthalpy of sublimation enthalpy of transition

The last case represents a change from one crystalline modification to another. The enthalpy change involved is small compared with those accompanying the other three types, and rarely finds application in environmental engineering. Since there is a relatively small difference in volume between a solid and the liquid to which it melts, the heat of fusion represents mainly an increase in internal energy. If the molar heat of fusion is not known, it may be approximated from the absolute temperature of fusion, Tf , using the following equation:

Hf/Tf = 2–3 for solid elements; 5–7 for inorganic solids; 9–11 for organic solids

(5.14)

The ratio may be expressed in any consistent molar units, such as cal/(gmol-K) or Btu/(lbmol-°R). Since there is a very substantial increase in volume in passing from the liquid to the vapor state, usually from a few hundred to over a thousand-fold, the heat of vaporization is large and important in many environmental engineering applications involving water. Information on properties of water are available in steam tables in the literature [1, 2] and in Appendix C.

5.3.1 Chemical Reaction Enthalpy Effects The standard enthalpy (heat) of reduction can be calculated from standard enthalpy of formation data. To simplify the presentation that follows, examine the authors’ favorite equation:

aA + bB = cC + dD

(5.15)

If the above reaction is assumed to occur at a standard (or reference) state, the standard enthalpy of reaction, Ho, is given by:

Ho = c( H°f )C + d( H°f )D – a(ΔH°r)A – b( H°f )B where ( Hf°)i.= standard enthalpy of formation of species i.

(5.16)

Thermodynamics 33 Thus, the (standard) enthalpy of a reaction is obtained by taking the difference between the (standard) enthalpy of formation of products and reactants multiplied by their respective stoichiometric coefficients. If the (standard) enthalpy of reaction or formation is negative (exothermic), as is the case with most combustion reactions, then energy is liberated due to the chemical reaction. Energy is absorbed and Ho is positive (endothermic). Tables of enthalpies of formation and reaction are available in the literature (particuiariy thermodynamics text/reference books) for a wide variety of compounds [1]. It is important to note that these are valueless unless the stoichiometric equation and the state of the reactants and products are included. Theodore, et al. [1, 2] provide equations to describe the effect of temperature on the enthalpy of reaction. For heat capacity data in , , form:

HTo

o H 298

(T 298)

1 2

1 3

(T 2 2982 )

(T 3 2983 )

(5.17)

For the reaction presented in Equation 5.15:

= c c+ d

D

–a

A

–b

= c c+ d

D

–a

A

–b

= c c+ d

D

–a

A

–b

(5.18)

B

(5.19)

B

(5.20)

B

For heat capacity in a, b, c form,

HTo

5.4

o H 298

a(T 298)

1 b(T 2 2982 ) 2

c(T

1

298 1 )

(5.21)

Second Law Calculations [2]

The law of conservation of energy has already been defined as the first law of thermodynamics. Its application allows calculations of energy relationships associated with a wide variety of processes. The “limiting” law is called the second law of thermodynamics. Applications involve calculation of the maximum power output from a power plant and equilibrium yields of chemical reactions. In principle, this law starts that water cannot flow uphill and heat cannot flow from a cold to a hot body of its own accord. Other defining statements for this law that have appeared in the literature are provided below: 1. Any process, the sole net results of which is the transfer of heat from a lower temperature level to a higher one, is impossible.

34

Unit Operations in Environmental Engineering 2. No apparatus, equipment, or process can operate in such a way that its only effect (on system and surroundings) is to convert heat absorbed completely into work. 3. It is impossible to convert the heat taken into a system completely into work in a cyclical process.

The second law also serves to define another important thermodynamic function called entropy. It is normally designated as S. The change in S for a reversible adiabatic process is always zero:

S=0

(5.22)

For liquids and solids, the entropy change for a system undergoing an absolute temperature change from T1 to T2 is given by:

S = Cp ln(T2/T1); Cp = constant

(5.23)

The entropy change of an ideal gas undergoing a physical change of state from Pl to P2 at a constant temperature T is given by:

ST = R ln(P1/P2); Btu/lbmol-°R

(5.24)

The entropy change of one mole of an ideal gas undergoing a physical change of state from absolute temperatures T1 to T2 at a constant pressure is given by:

Sp = Cp ln(T2/T1); Cp(gas)= constant

(5.25)

Correspondingly, the entropy change for an ideal gas undergoing a physical change from (P1, T1) to (P2, T2) is

S = R ln(P1/P2) + Cp ln(T2/Tl)

(5.26)

Some fundamental facts relative to the entropy concept are discussed below. The entropy change of a system may be positive (+), negative ( ), or zero (0); the entropy change of the surroundings during this process may likewise be positive, negative, or zero. However, note that the total entropy change, S must be equal to or greater than zero:

S≥0

(5.27)

The equality sign applies if the change occurs reversibly and adiabatically. The third law of thermodynamics is concerned with the absolute values of entropy. By definition; the entropy of all pure crystalline materials at absolute zero temperature is exactly zero. Note however, that chemical and environmental

Thermodynamics 35 engineers are usually concerned with changes in thermodynamic properties, including entropy, rather than their absolute values.

5.5 Phase Equilibrium Relationships governing the equilibrium distribution of a substance between two phases, particularly gas and liquid phases, are the principal subject matter of phase-equilibrium thermodynamics. These relationships form the basis of calculational procedures that are employed in the design and prediction of the performance of several unit operation equipment and processes involving mass transfer. The most important equilibrium phase relationship is that between a liquid and a vapor. Raoult’s and Henrys Laws theoretically describe liquid-vapor behavior and, under certain conditions, are applicable in practice. Raoult’s Law is sometimes useful for mixtures of components of similar structure. It states that the partial pressure of any component in the vapor phase is equal to the product of the vapor pressure of the pure component and the mole fraction of that component in the liquid:

pi = pi xi

(5.27)

where pi = partial pressure of component i in the vapor, pi = vapor pressure of pure component i at the same temperature, and xi = mole fraction of component i in the liquid. This expression may be applied to all components. If the gas phase is ideal, this equation becomes:

yi = (pi /P)xi

(5.28)

where yi = mole fraction of component i in the vapor, and P = total system pressure. Unfortunately, relatively few mixtures follow Raoult’s law, but it can be applied to estimating the vapor pressure of individual compounds above a hydrocarbon mixture, i.e., benzene above gasoline in underground storage tank and soil and groundwater remediation applications. Henrys law is a more empirical relation used for representing data for many environmental engineering systems where the liquid phase is water. Henry’s law is expressed as:

pi = Hixi

(5.29)

where Hi = Henrys law constant for component i (in units of pressure). If the gas behaves ideally, the above equation may be written as:

yi = KH xi where KH = dimensionless Henry’s law constant.

(5.30)

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Unit Operations in Environmental Engineering

In some environmental engineering applications, mixtures of condensable vapors and non-condensable gases must be handled. A common example is water vapor and air; a mixture of organic vapors and air is another such example that often appears in air pollution applications. Condensers can be used to control organic emissions to the atmosphere by lowering the temperature of the gaseous stream, although an increase in pressure will produce the same result. The calculation for this is often accomplished using the phase equilibrium constant, K. This constant has also been referred to in the chemical industry as a phase componential split factor since it provides the ratio of the mole fractions of a component in the two equilibrium phases. The defining equation is

Ki = yi/xi

(5.31)

where Ki = phase equilibrium constant for component i (dimensionless). As a first approximation, Ki is generally treated as a function only of the temperature and pressure. For ideal gas conditions, Ki may be approximated by

Ki = pi /P

(5.31)

where pi is the vapor pressure of component i, and P is the total system pressure. Many of the phase equilibrium calculations involve hydrocarbons. Fortunately, most hydrocarbons approach ideal gas behavior over a fairly wide range of temperatures and pressures. Values for Ki for a large number of hydrocarbons are provided in two DePriester nomographs, which are available in the literature. These two nomographs were originally developed by DePriester in 1953 [6]. These DePriester charts are a valuable source of vapor-liquid equilibrium data for many hydrocarbons that approach ideal behavior. However, it should be noted that the DePriester charts are based on experimental data. The fact that these compounds approach ideal behavior allows the data to be presented in a simple form, i.e., as a function solely of temperature and pressure.

5.6 Chemical Reaction Equilibrium With regard to chemical reactions, two important questions are of concern to the engineer: (1) how far will the reaction go; and (2) how fast will the reaction go? Chemical thermodynamics provides the answer to the first question; however, it tells nothing about the second. Reaction rates fall within the domain of chemical reaction kinetics. To illustrate the difference and importance of both questions in an engineering analysis of chemical reactions, consider the following process: Substance A, which costs 1 cent/ton, can be converted to Substance B, which is worth 1 million dollars/mg, by the reaction A B. Chemical thermodynamics will provide information on the maximum amount of Substance B that can be formed. If 99.99% of Substance A can be converted to Substance B, the reaction

Thermodynamics 37 would then appear to be highly economically feasible, from a thermodynamic point of view. However, a kinetic analysis might indicate that the reaction is so slow that, for all practical purposes, its rate is vanishingly small. For example, it might take 106 years to obtain 10–6 % conversion of Substance A. The reaction is then economically unfeasible. Thus it can be seen that both equilibrium and kinetic effects must be considered in an overall engineering analysis of a chemical reaction [3]. A rigorous, detailed presentation of this equilibrium topic is beyond the scope of this chapter and this text. However, some discussion is presented to provide at least a qualitative introduction to chemical reaction equilibrium. As will be shown in Chapter 6, if a chemical reaction is carried out in which reactants go to products, the products will be formed at a rate governed (in part) by the concentration of the reactants and conditions such as temperature and pressure. Eventually, as the reactants form products and the products react to form reactants, the net rate of reaction must equal zero. At this point, equilibrium will have been achieved. Chemical reaction equilibrium calculations are structured around a thermodynamic term referred to as free energy, G. This so called Gibbs free energy is a thermodynamic property that represents the maximum useful work (excluding PV work associated with volume changes of the system) that a system can do on the surroundings when the process occurs reversibly at constant temperature and pressure. Note that free energy has the same units as enthalpy, and may be used on a mole or total mass basis. Consider the equilibrium reaction presented earlier as Equation 5.15:

aA + bB = cC + dD

(5.15)

For this reaction

Go = c( G°f )C + d( G°f )D – a( G°r)A – b( G°f )B

(5.32)

The standard free energy of reaction Go may thus be calculated from standard free energy of formation data in a manner similar to that for the standard enthalpy of reaction. The following equation is used to calculate the chemical reaction equilibrium constant K at a temperature T:

ΔGT° = RT ln(K)

(5.33)

The effect of temperature on the standard free energy of reaction, ΔGT°, and the chemical reaction equilibrium constant, K, is available in the literature [1, 3, 5, 7] and has been developed in a manner similar to that presented earlier for the effect of temperature on the enthalpy of reaction. Once the chemical reaction equilibrium constant (for a particular reaction) has been determined, one can proceed to estimate the quantities of the participating species at equilibrium [1, 3, 5, 7]. A detailed and expanded treatment of chemical reaction equilibrium is available in the literature in references [1, 3, 5, 7].

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Unit Operations in Environmental Engineering

References 1. Theodore, L., Ricci, F., and VanVliet, T., Thermodynamics for the Practicing Engineer, John Wiley and Sons, Hoboken, N.J., 2009. 2. Theodore, L., Thermodynamics, A Theodore Tutorial, East Williston, NY, originally published by the USEPA/APTI, RTP, NC, 1991. 3. Smith, J., Van Ness, H., and Abbott, M., Chemical Engineering Thermodynamics, 6th Edition, McGraw-Hill, New York City, NY, 2001. 4. Pitzer, K., Thermodynamics, 3rd Edition, McGraw-Hill, New York City, NY, 1995. 5. Perry, R., and Green, D. (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York City, NY, 2008. 6. DePriester, C., Light-hydrocarbon vapor-liquid distribution coefficients. Chem Eng. Prog. Symp. Ser., 49(7), 1–43, 1953. 7. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2014.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

6 Chemical Kinetics

6.1 Overview Almost every chemical process is designed to economically produce a desired product from a variety of starting materials through a succession of treatment steps. The raw materials may first undergo a number of physical treatment steps to put them in the form in which they can chemically react. They then pass through a reactor. The discharge, or product of the reaction then usually undergoes additional physical treatment steps, i.e., separations, purifications, etc., before the desired final products are obtained. This chapter is concerned with the kinetics of chemical reactions and reactors, particularly as they relate to environmental engineering. An objective of this chapter is to prepare the reader to solve real-world engineering and design problems that involve chemical reactors. There are several classes of reactors. The three that are most often encountered in practice are: batch (B), continuous stirred tank (CST), and plug flow (PF) reactors. As such they receive the bulk of the treatment that follows. A final topic reviewed is catalytic reactors, an extension of which includes biochemical reactions of significant importance to environmental engineering. The contents of the chapter therefore include: 1. Chemical Kinetic Principles 2. Batch Reactors (BRs) 39

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Unit Operations in Environmental Engineering 3. Continuous Stirred Tank Reactors (CSTRs) 4. Plug Flow Reactors (PFRs) 5. Catalytic Reactors

The reader should note that the bulk of the material in this chapter has been drawn from L. Theodore, Chemical Reaction Kinetics, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by the USEPA/APTI, RTP, NC, 1995 [1]. Also note that in an attempt to be consistent with the chemical reactor literature, the volumetric flow rate is represented by Q (not q, as in most other chapters in this text).

6.2 Chemical Kinetics Principles Chemical kinetics involves the study of reaction rates and the variables that affect these rates. It is a topic that is critical for the analysis of reacting systems and chemical reactors. The rate of a chemical reaction can be described in any of several different ways. The most commonly used definition involves the time rate of change in the amount of one of the components participating in the reaction; the rate is also based on some arbitrary factor related to the reacting system size or geometry, such as volume, mass, and interfacial area. Based on experimental evidence, the rate of reaction, rA, is often a function of: 1. the concentration of components existing in the reaction mixture (this includes reacting and inert species) 2. temperature 3. pressure 4. catalyst variables (if applicable) This may be put in equation form as:

rA = rA (C,P,T, catalyst variables)

(6.1)

rA = ±kAf(C)

(6.2)

or simply

where kA incorporates all the variables other than concentration. The ± notation is included to account for the consumption or formation of A. One may think of kA as a constant of proportionality. It is defined as the specific reaction rate or more commonly the reaction velocity constant. It is a “constant” which is independent of concentration but dependent on the other variables. This approach has, in a sense, isolated one of the variables. The reaction velocity constant, k, like the rate of reaction, must refer to one of the species in the reacting system. However, k

Chemical Kinetics 41 almost always is based on be same species as the rate of reaction. Consider once again the reaction

aA + bB

cC + dD

(6.3)

The notation represents an irreversible reaction; i.e., if stoichiometric amounts of A and B are initially present, the reaction will proceed to the right until all of A and B have reacted (disappeared) and C and D have been formed. If the reaction is elementary, the rate of the above reaction is given:

k AC AaCBb

rA

(6.4)

where the negative sign is introduced to account for the consumption of A, and the product concentrations do not affect the rate. For elementary reactions, the reaction mechanism for rA is simply obtained by multiplying the molar concentrations of the reactants raised to powers of their respective stoichiometric coefficients (power law kinetics). For non-elementary reactions, the mechanism can take any form. The order of the above reaction with respect to a particular species is given by the exponent of that concentration term appearing in the rate expression. The above reaction is, therefore a order with respect to A, and b order with respect to B. The overall order n, usually referred to as “the order,” is the sum of the individual exponents; i.e.,

n=a+b

(6.5)

All real and naturally occurring reactions are reversible. A reversible reaction is one in which both reactants form products, and products react to form reactants. Unlike irreversible reactions, which proceed to the right until completion, reversible reactions achieve an equilibrium state, in which rates of forward and reverse-reactions are equal for an infinite period of time. Reactants and products are both present in the system. At this (equilibrium) state, the reaction rate is zero. For example, consider the following reversible reaction:

aA + bB

cC + dD

(6.6)

where the notation is a reminder that the reaction is reversible. The overall rate of the reaction for this elementary equilibrium reaction is then given by:

rA

k AC AaCBb kACCc CDd forward reverse reaction reaction

(6.7)

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Unit Operations in Environmental Engineering

The two most common conversion variables employed by practitioners are denoted by and X. The term is employed to represent the change in the number of moles of a particular species due to chemical reaction. The conversion variable X is used to represent the change in the number of moles of a particular species (say A) relative to the number of moles of A initially present or initially introduced to a flow reactor. Thus,

XA = moles of A reacted/initial moles of A = NA/NAo

(6.8)

Another conversion (related) variable includes CA, the concentration of A at some later time (or position). Also note that all of the conversion variables can be based on mass, but this is rarely employed in practice. The conversion variable of choice is almost always X. The yield of a reaction is defined as a measure of how much of the desired product is formed relative to how much would have been formed if only the desired reactions occurred, and if that reaction went to completion. Alternatively, selectivity is a measure of how a desired reaction is completed relative to one of the side reactions. To drive a chemical reaction to completion it is common practice to add an excess of one of the reactants, especially an inexpensive one. In most reaction mixtures, there is a reactant that is present in the lowest number of chemical equivalents. This is the limiting reactant since it sets the absolute limit upon the extent of chemical change and the quantities of products that can form. The degree of completion of a reaction is the percentage of the limiting reactant that undergoes the reaction in question. The degree of completion is also called the extent of reaction. Once having identified the limiting reactant, it is possible to express in a precise way the amount to which any other reactant is present in excess. The percentage excess for any reactant is the total amount added less the theoretical amount, divided by the amount theoretically required for complete reaction (stoichiometric requirement) with the limiting reactant, multiplied by 100 (Equation 6.9). % Excess

( Amount of Reactant A Stoichiometric Requirement for Reactant A) 100 Stoichiometric Requirement for Reactant A

(6.9)

If half again as much as that theoretically required is added, the percentage excess is 50 percent; if a double quantity is added, the excess is 100 percent, etc. The addition of a triple quantity often introduces a semantic “booby trap” that the reader should be aware of, i.e., a 200 percent excess means that a three-fold quantity of the reactant is being used, not a double quantity. As noted earlier the reaction rate is affected not only by the concentration of species in a reacting system but also by the temperature. An increase in

Chemical Kinetics 43 temperature results in an increase in the rate of reaction. For biological systems, a 10 °C increase in reaction temperature will generally double the reaction velocity constant. Physical reactions, i.e., aeration, are less temperature sensitive and change less drastically with an increase or decrease in temperature than biological reactions. The Arrhenius equation relates the reaction velocity constant, k, to temperature, T. It is given by [1,2]:

k

Ae

Ea / RT

(6.10)

where A = collision frequency factor and is a constant for a given reacting system over a wide temperature range; Ea=activation energy of the specific reaction under consideration kJ/mol; R = universal gas constant, 8.314 kJ/K-mol; and T = absolute reaction temperature, K. With these variables as defined, it is suggested that the reaction velocity constant is directly proportional to A. Moreover, because of the minus sign in the exponent, the reaction velocity constant decreases with increasing activation energy, and increases with increasing temperature.

6.3 Batch Reactors (BRs) Before studying the quantitative aspects of reactors, it is necessary to have a clear understanding of various reaction terms. Batch processes have one thing in common. They either come to an end or are heading to an end during the operating interval. This results through failure to maintain or replenish the supply of all the materials essential to the process. Cutting off the supply of only one essential material is sufficient to lead the process to ultimate cessation and mark it as a batch process. Batch reactors (BRs) are commonly used in experimental studies. Their industrial applications are somewhat limited. They are rarely used for gas phase, e.g., combustion reactions, since small quantities (mass) of product are produced with even a very large-sized reactor. BRs are used for liquid phase reactions when small quantities of reactants are to be processed. They find their major application in the pharmaceutical industry, but new wastewater system designs have been developed using sequencing batch reactor (SBR) technology. As a rule, BRs are less expensive to purchase but more expensive to operate than either continuous stirred tank or plug flow reactors [1–3]. The extent of a chemical reaction and/or the amount of product produced can be affected by the relative quantities of reactants introduced into the reactor. For two reactants, each is normally introduced through separate feed lines normally located at or near the top of the reactor. Both are usually fed simultaneously over a short period of time. Mixing is accomplished with the aid of an impeller. The reaction is assumed to begin after both reactants come into contact. No spatial variations in concentration, temperature, etc., are generally assumed, and in most liquid-phase batch systems, the reactor volume is considered to be constant.

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The describing equation for chemical reaction mass transfer is obtained by applying the conservation law for either mass or moles on a time-rate basis to the contents of a batch reactor. It is best to work with moles rather than mass since the rate of reaction is most conveniently described in terms of molar concentrations. The describing equation for species A in a batch reactor takes the form:

dNA/dt = – rAV

(6.11)

where NA = moles A at time t; rA = rate of reaction of A in moles A/time-volume; and V = reactor volume. The above equation may also be written in terms of the conversion variable X since

NA = NAo – NAoX

(6.12)

where NAo = initial moles of A. Thus (setting X = XA),

NAo (dX/dt) = – rAV

(6.13)

The integral form of Equation 6.11 is: x

t

N Ao 0

1 dX rAV

(6.14)

If V is constant (as with most liquid phase reactions) then

t

N Ao V

x

0

1 rA

x

dx C Ao 0

1 dX rA

(6.15)

6.4 Continuously Stirred Tank Reactors (CSTRs) Continuous processes are marked by a flow of all essential materials from the beginning to the end of the operating interval. By definition, continuous processes represent open systems. They may be divided into variable and constant flow processes. The constant flow process is subdivided into unsteady state constant flow and steady state flow. CSTRs, like the batch reactor, also consists of a tank equipped with an agitator. It may be operated under steady or transient conditions. Reactant(s) are fed continuously, and the product(s) are withdrawn continuously. The reactant(s) and product(s) may be liquid, gas, or solid, or a combination of these. If the contents

Chemical Kinetics 45 are perfectly mixed, the reactor design problem is greatly simplified for steady conditions because the mixing results in uniform concentrations, temperature, etc., throughout the reactor. This means that the rate of reaction is constant and the describing equations are not differential in form, and therefore, do not require integration. In addition, since the reactor contents are perfectly mixed, the concentration and/or conversion within a CSTR is exactly equal to the concentration and/or conversion leaving the reactor. The describing equation for a CSTR can then be shown to be:

V = FAXA/( rA)

(6.16)

where V= volume of reacting mixture required to yield a specified conversion rate of A; FA = inlet molar feed rate of A; XA = conversion of A; and – rA = rate of reaction of A. If the volumetric flow rate through the CSTR, Q, is constant, the above equation becomes:

V/Q = (CAi – CAo)/rA = (CAo – CAi)/( rA)

(6.17)

where CAo = the inlet molar concentration of A; and CAi = the exit molar concentration of A. The left-hand side of Equation 6.17 has units of time, and represents the average residence time of liquid within the reactor. However, there is a distribution around this average, and in certain types of systems it is often important to include this distribution effect in any system analysis [4]. In many chemical and environmental engineering applications, residence time is denoted by the symbol . In chemical engineering applications, the reciprocal of is defined as the space velocity (SV) and finds wide application in chemical engineering applications of plug flow reactors. In general, CSTRs are used for liquid phase reactions. High reactor concentrations can be maintained with low flow rates so that conversion approaching 100% can often be achieved. However, the overall economics of the system is reduced because of the low throughput. CSTRs (as well as the plug flow reactors described next) are often connected in series in such a manner that the exit stream of one reactor is the feed stream for the next reactor. Under these conditions, it is convenient to define the conversion at any point downstream in the series of CSTR reactors in terms of inlet conditions, rather than with respect to any one of the reactors in the series. The conversion X is then the moles of A that have reacted up to that point per mole of A fed to the first reactor. However, this definition should only be employed if there are no side stream withdrawals and the only feed stream enters the first reactor in the series. The conversion from Reactors 1, 2, 3, … in the series are usually defined as X1, X2, X3, … respectively, and effectively represent the overall conversion for that reactor relative to the feed stream to the first reactor.

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6.5 Plug Flow Reactors (PFRs) The last “traditional” reactor to be examined is the plug flow reactor. The most common type in chemical engineering applications is the single-pass cylindrical tube. Another is one that consists of a number of tubes in parallel. The reactor(s) may be vertical or horizontal. The feed is charged continuously at the inlet of the reactor, and the products are continuously removed at the outlet. If heat exchange with surroundings is required, the reactor setup includes a jacketed tube. If the  reactor is empty, a homogenous reaction, i.e., one phase is present, usually occurs. If the reactor contains catalyst particles, the reaction is said to be heterogeneous. This catalytic type reactor is considered in the next section. Plug flow reactors are usually operated under steady conditions so that physical and chemical properties do not vary with time. Unlike the BR and CST reactors, there is no axial (longitudinal) mixing. Thus, the state of the reacting fluid will vary spatially from point to point along the flow path (axial or longitudinal direction) in the system. The describing equations are then differential, with position as the independent variable. The reacting system for the describing equations presented below is assumed to move through the reactor via plug flow (no velocity variation through the crosssection of the reactor). It is further assumed that there is no mixing in the axial direction so that the concentration, temperature, etc., do not vary through the cross-section of the reactor. Thus, the reacting fluid flows through the reactor as an undisturbed plug of mass. For these conditions, the describing equation for a plug flow reactor is: x

V

1

FAo

dX A

rA

0

(6.18)

Since FAo = CAo Qo x

V /Q C Ao 0

1 rA

dX A

(6.19)

The left hand side (LHS) of the above equation represents the residence time in the reactor, , based on inlet conditions. If Q does not vary through the reactor then:

= V/Qo

(6.20)

In actual practice, plug flow reactors deviate from plug flow conditions because of shortcircuiting, and velocity variations in the radial direction. This is particularly true for environmental applications of wastewater treatment that utilize large

Chemical Kinetics 47 rectangular, baffled tanks that are baffled to encourage plug flow conditions. For this non-ideal condition, the residence time for annular elements of fluid within the reactor will vary from a value substantially greater than the mean, to some minimal value at a point where the velocity approaches zero. Thus, the concentration and temperature profiles, as well as the velocity profile, are not constant across the reactor and the describing equation based on plug flow assumptions are then not applicable. This situation can be further complicated if the reaction occurs in the gas phase. In a gas phase reaction, volume effect changes that impact on the concentration term(s) in the rate equation also need to be taken into account [1–3]. From a qualitative point of view, as the length of the reactor approaches infinity, the concentration of a (single) reactant approaches zero for irreversible reactions (except zero order) and the equilibrium concentration for reversible reactions. Since infinite time is required to achieve equilibrium conversion, this value is approached as the reactor length approaches infinity. For reactors of finite length, where the reaction is reversible, some fraction of the equilibrium conversion is achieved. Note that the time for a hypothetical plug of material to flow through a PF reactor is the same as the contact or reaction time in a batch reactor. Under these conditions, the same form of the describing equation for batch reactors will also apply to PF reactors. Another design variable for PF reactors is pressure drop. This effect is usually small for most liquid and gas phase reactions conducted in short or small reactors. However, this effect can be estimated using Fanning’s Equation [5]:

P = 4fLV2/(2gcD)

(6.21)

where P = pressure drop; f = Fanning friction factor; L = reactor length; V = average flow velocity; gc = gravitational constant; and D = reactor diameter.

6.6 Catalytic Reactors Metals in the platinum family are recognized for their ability to promote reactions at low temperatures. Other catalysts include various oxides of copper, chromium, vanadium, nickel, and cobalt. These catalysts are subject to poisoning, particularly from halogens, sulfur compounds, zinc, arsenic, lead, mercury, and particulates. High temperatures can also reduce catalyst activity. It is therefore important that catalyst surfaces are clean and active to insure optimum performance. For example, catalysts can be regenerated with superheated steam. Catalyst may be porous pellets, usually cylindrical or spherical in shape, ranging from 1/16 to ½ inch in diameter. Small sizes are recommended, but the pressure drop through the reactor increases with decreasing catalyst size. Other shapes include honeycombs, ribbons, wire mesh, etc. Since catalysis is a surface phenomenon, an important physical property of these particles is that the total internal

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pore surface be many magnitudes greater than the outside surface. The reader is referred to the literature for more information on catalyst preparation, properties, comparisons, costs, and impurities [1]. Some of the advantages of catalytic reactors are: 1. 2. 3. 4. 5.

low fuel requirements lower operating temperatures little or no insulation requirements reduced fire hazards reduced flashback problems

The disadvantages include: 1. 2. 3. 4. 5.

high initial cost catalyst poisoning large particles must first be removed some liquid droplets must first be removed catalyst regeneration problems

In many catalytic reactions, the rate equation is extremely complex and cannot be obtained either analytically or experimentally. A number of rate equations may result and some simplification is warranted. It is safe in many cases to assume that the rate expression may be satisfactorily expressed by the rate of reaction of a single step [3]. It is common practice to write the describing equations for mass and energy transfer for homogeneous and heterogeneous flow reactors in the same way. However, the (units of the) rate of reaction may be expressed as either:

r = (moles reacted/time)/(volume of reactor)

(6.22)

r = (moles reacted/time)/(mass of catalyst)

(6.23)

or

The latter is normally the preferred method employed in industry since it is the mass of catalyst present in the reactor that significantly impacts the reactor design. As noted, the rate expression is often more complex for a catalytic reaction than for a non-catalytic (homogeneous) one, and this can make the design equation of the reactor difficult to solve analytically. Numerical solution of the reactor design equation is usually required when designing plug flow reactors for catalytic reactions. The principal difference between reactor design calculations involving homogeneous reactions and those involving catalytic (fluid-solid) heterogeneous

Chemical Kinetics 49 reactions is that the reaction rate for the latter is based on the mass of solid catalyst, W, rather than on the reactor volume, V. The reaction of a substance A for a fluid-solid heterogeneous system is then defined as:

r A = (moles A reacted/mass catalyst) (time)

(6.24)

A brief discussion of the two major classes of catalytic reactors follows.

6.6.1 Fluidizied Bed Reactors This type of catalytic reactor is in common use in chemical and environmental engineering. The fluidized-bed reactor is analogous to the CSTR in that its contents, though heterogeneous, are well mixed, resulting in a uniform concentration and temperature distribution throughout the bed. The fluidized-bed reactor can therefore be modeled, as a first approximation, as a CSTR. For the ideal CSTR, the reactor design equation based on volume is:

V = FAoXA/( rA)

(6.25)

The companion equation for catalytic or fluid-solid reactors, with the rate based on the mass of solid, W, is:

W = FAoXA/( r A)

(6.26)

The reactor volume is simply the catalyst weight, W, divided by the fluidized bed density, rfb, of the catalyst in the reactor.

V = W/

(6.27)

fb

The fluid bed catalyst density is normally expressed as some fraction of the catalyst bulk density rB.

6.6.2

Fixed bed reactors

A fixed-bed (packed-bed) reactor is essentially a PF reactor that is packed with solid catalyst particles. This type of heterogeneous reaction system is most frequently used to catalyze gaseous reactions. The design equation for a PF reactor was previously shown to be: x

V

FAo 0

1 rA

dX A

(6.18)

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The companion equation based on the mass of catalyst, W, for a fixed-bed reactor is: x

W

FAo 0

1

dX A

rA

(6.28)

The volume of the reactor, V, is then:

V = W/

B

(6.29)

where B = bulk density of the catalyst. The Ergun equation [6] is normally employed to estimate the pressure drop for fixed bed units.

References 1. Theodore, L., Chemical Reaction Kinetics, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published in the USEPA/APTI, NC. 1995. 2. Theodore, L., Chemical Reactor Analysis and Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2012. 3. Fogler, S., Elements of Chemical Reaction Engineering, 4th Edition, Prentice Hall, Upper Saddle River, NJ, 2006. 4. Shaefer, S., and Theodore, L., Probability and Statistics Applications in Environmental Science, CRC Press/Taylor & Francis Group, Boca Raton, FL, 2007. 5. Abulencia, P. and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 6. Ergun, S., Fluid Flow Though Packed Columns, CEP, 48:49, New York City, NY, 1952.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

7 Equilibrium versus Rate Considerations

There are two important factors that need to be taken into consideration when analyzing unit operations: equilibrium and process rate. Although these two subjects have been segmented and treated separately below, both need to be considered together when analyzing most unit operations and their associated equipment. Because of the importance of equilibrium and process rate, both receive treatment in this chapter. However, the question often asked is: Which is the more important of the two when discussing unit operations involving mass transfer? This is best answered by noting that most, but not all, mass transfer calculations assume equilibrium conditions apply. A correction factor or an efficiency term is then included to adjust/upgrade the result/prediction to actual (as opposed to equilibrium) conditions in order to accurately describe the phenomena in question. As one would expect, the correction or efficiency factor is usually based on experimental data, past experience, similar designs, or just simply good engineering judgement. Although this method of analysis has been maligned by many theoreticians and academicians, the approach has merit and has been routinely used by practicing engineers. This pragmatic approach is primarily employed in this textbook.

51

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7.1 Overview It will be helpful to review certain definitions and concepts associated with equilibrium and the phase rule of J. Willard Gibbs. A homogeneous system is one for which all parts (on a macro scale) have the same physical and chemical properties. Such a system has a uniform composition and concentration throughout. Air in a cylinder is an example of such a homogeneous system. A solution completely occupying a closed container is another. A heterogeneous system has non-uniform composition. Its physical and chemical properties vary from one location to another. A phase is a finite part of a system that is homogeneous throughout and is physically separated from other phases of the system by distinct boundaries called interfaces. When a liquid stands in contact with its own vapor, the surface of the liquid is the interface between the liquid and vapor phases. In stating and applying the phase rule it is essential to determine the number of so-called components. By definition, the number of components is the minimum number of independently variable chemical species needed to express the total composition of the system or any phase present in the system. The phase rule applies to all systems in which a condition of equilibrium exists. According to Gibbs:

F=C+2–P

(7.1)

where P = the number of phases present; C = the number of components as previously defined: and F = the so-called degrees of freedom or variability of the system. The degrees of freedom, F, may be defined as the number of independent variables, such as temperature, pressure, or concentration, that must be specified in order to completely define the system. A second definition of F is the number of variables, such as temperature and pressure, that may be changed independently without causing the appearance or disappearance of a phase. The number 2 is valid only when there are two variables in addition to concentration. These two are commonly temperature and pressure. If, for example, conditions are such that pressure may be regarded as fixed throughout, then there is only one effective variable in addition to concentration and, in such a condensed system

F=C+1

P

(7.2)

It is obvious from the phase rule equation that in any equilibrium system involving a fixed number of components, the greater the number of phases present, the fewer are the degrees of freedom.

Equilibrium versus Rate Considerations

53

7.2 Equilibrium A state of equilibrium exists when the forward and reverse rates of a process are equal. From a macroscopic point of view, there is no change in the system with time. Equilibrium represents a limiting value for the practicing engineer as demonstrated in the following example. Whenever a substance distributes itself between two other materials, the system will attempt to approach equilibrium. Consider the following mass transfer unit operation. If NH3 gas in air is brought into contact with water at 60 °F, the NH3 will begin to dissolve in the water until its concentration in the water has reached a maximum value (its aqueous solubility) at that temperature. This condition represents equilibrium, and no further dissolution of the NH3 will occur unless this equilibrium is disturbed. Alternatively, if water containing NH3 is brought into contact with air containing no NH3, the NH3 will escape from the water and pass into the gas phase until the value of the concentrations in the two phases has reached the system’s equilibrium state. This particular scenario will be revisited in Chapter 48, a chapter that is concerned with absorption and stripping. If one wishes to remove NH3 from an air mixture by using water, it is obvious that equilibrium can set a limit on the maximum amount of NH3 that can be removed from the air to the water. It can also set the lower limit on the amount of water necessary for a particular degree of NH3 removal. Therefore, equilibrium is a vital factor in the design and operation of these type of mass transfer systems. In the situation described above, equilibrium is fortunately simple to represent. Occasionally, however, the relationships are more complicated to describe and model. Specific cases of importance will be discussed later in the book.

7.3 Transfer Process Rates The companion to equilibrium in unit operations is process rate. In mass transfer, the transfer of a substance from one phase to another obviously requires time to complete. The rate of transfer is proportional to the surface of contact between phases, the resistance to the transfer, and the driving force present for mass transfer to occur. This phenomenon is described in Equations (7.3) and (7.4). Employing the macroscopic approach, a transfer process, whether it be mass, energy, or momentum, can be simply described as shown below:

Rate of Transfer

(Driving Force)(Area Available for Transfeer) (Transfer Resistance) (7.3)

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Unit Operations in Environmental Engineering For mass transfer, Equation 7.3 becomes:

Rate of Mass Transfer (Concentration Driving Force)(Area Avvailable for Mass Transfer) (Mass Transfer Resistance) (7.4) Other things being equal, Equations 7.3 and 7.4 indicate that the rate of transfer can be increased by: 1. increasing the transfer area between phases 2. increasing the driving force, and/or 3. decreasing the transfer resistance These three factors are almost always considered in the design of mass transfer equipment. As one might intuitively expect, any increase in the transfer rate leads to a more compact mass transfer device that is generally either more economical or operates more efficiently. Consider again the comments provided earlier in the section on equilibrium concerning the NH3-air-H20 system. No mention was made of the rate of transfer of NH3 at that time. But in line with Equation 7.4, during the initial stages of the transfer process, the NH3 concentration in the air is high while its concentration in water is low (or even zero). This produces a high concentration gradient or driving force that leads to a high initial NH3 transfer rate. However, later in the transfer process, the NH3 concentrations in both phases tend to “equilibrate,” leading to a lower concentration gradient and thus a lower driving force, and a corresponding lower transfer rate. When the system ultimately reaches/achieves equilibrium, the driving force becomes zero and the rate of mass transfer also becomes zero. With all mass transfer processes, the true driving force for mass transfer is the chemical potential of the substance to be transferred. Just as temperature acts as a thermal potential for heat transfer, every substance has a chemical (or “mass”) potential which “drives” it from one phase into another. A state of equilibrium is only achieved in a non-reacting system when the temperature, pressure, and chemical potential of every species is equal in all phases. While the study of chemical potential is best left to a thermodynamicist, the subject of mass transfer tends to use concentration as a substitute for representing a substance’s chemical potential. Obviously, the process rate as well as equilibrium play a role in the analysis and design of unit operations and processes. Both effects need to be considered in designing, predicting performance, and operating process equipment. Which is more important? It depends. Normally, equilibrium information is required. Moreover, if the rate of transfer is extremely high, rate considerations can in some instances be neglected.

Equilibrium versus Rate Considerations

55

7.4 Chemical Reaction Process Rates As was seen earlier in Chapter 5, two important questions are of concern to the engineer when chemical reactions are considered: (1) how far will the reaction go; and (2) how fast will the reaction go? Chemical thermodynamics provides the answer to the first question as was discussed in Chapter 5; however, it tells nothing about the second. Reaction rates fall within the domain of chemical reaction kinetics that were discussed in Chapter 6. The difference and importance of both questions was illustrated by the example of low cost Substance A that could be converted to an extremely high value Substance B, worth 100 million times more per unit weight than Substance A, but at a rate that was infinitesimally small, making the reaction economically unfeasible. Thus, it can be seen that both equilibrium and kinetic effects must both be considered in an overall analysis of a chemical reaction [1, 2] to clearly identify those processes that might be theoretically desirable from those that are practical. The same principle applies, for example, to gaseous mass transfer separation, e.g., absorption, processes [3]. Equilibrium and rate are both important factors to be considered in the design and prediction of performance of equipment employed for chemical reactions just as they are for phase transfer processes as described above. If one is conducting a chemical reaction in which reactants go to products, the products will be formed at a rate governed in part by the concentration of the reactants and reaction conditions such as the temperature and pressure. As with transfer processes, in chemical reaction processes, as the reactants form products and the products react to form reactants, the net rate of reaction eventually must equal zero. At this point, equilibrium will have been achieved.

References 1. Theodore, L., Chemical Reaction Kinetics, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published in the USEPA/APTI, NC. 1995. 2. Fogler, S., Elements of Chemical Reaction Engineering, 4th Edition, Prentice Hall, Upper Saddle River, NJ, 2006. 3. Theodore, L., Air Pollution Control Equipment Calculations, John Wiley & Sons, Hoboken, NJ, 2009.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

8 Process and Plant Design

Current process and environmental plant design practices can be categorized as state-of-the-art and pure empiricism. Past experience with similar applications is commonly used as the sole basis for the design procedure. The vendor (seller) maintains proprietary files on past installations, and these files are periodically revised and expanded as new installations are evaluated. During the design of a new unit, the files are consulted for similar applications and old designs are heavily relied upon for effectively modifying existing technology to new operating systems, regulations, and societal constraints. By contrast, the engineering profession in general, and the chemical engineering profession in particular, have developed fairly well-defined procedures for the design, construction, and operation of chemical plants. These techniques are routinely used by today’s chemical engineers. These same procedures may also be used in the design of other facilities, especially those related to environmental engineering practice.

8.1 Overview The purpose of this chapter is to introduce the reader to some process design fundamentals. Such an introduction to design principles can provide the reader with 57

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a better understanding of the major engineering aspects of new or modified facilities, including some of the operational and economic factors, controls, instrumentation for safety and regulatory requirements, and environmental factors associated with the process. The authors have simplified the process by keying in on five critical topics that are summarized in the acronym SCORE: Safety, Costs, Operability, Reliability, and Environmental aspects. It should also be noted that process plant location and layout considerations are only briefly reviewed. However, in a general sense, physical plant considerations should include process flow, construction, maintenance, operator access, site conditions, space limitations, future expansion, special piping requirements, structural supports, storage space. utility requirements, and (if applicable) energy conservation. The topics covered in this chapter include preliminary studies, process schematics. material and energy balances, equipment design, instrumentation and controls, design approach, and the design report. The reader should note that no attempt is made in these sections that follow to provide extensive coverage of this topic; only general procedures and concepts are presented and discussed. Coverage of these topics, even in an introductory manner, and their consideration in process and plant design is still highly valuable as it can provide early signals as to when a proposed process or process change is technically unfeasible or economically prohibitive.

8.2 Preliminary Studies A process engineer is usually involved in one of two principal activities: building a plant or facility, or deciding whether to do so. The skills required in both cases are quite similar, but the money, time, and detail involved are not as great in the latter situation. It has been estimated that only one out of 15 proposed new processes ever reaches the construction stage [1]. In general design practice, there are usually five levels of sophistication for evaluating projects and estimating costs. Each is discussed in the following list, with particular emphasis on cost: 1. The first level of analysis requires little more than identification of products, raw materials, and utilities. This is what is known as an order-of-magnitude estimate and is often made by extrapolating or interpolating from data on similar existing processes. The evaluation can be done quickly and at minimum cost, but with a probable error exceeding ±50%. 2. The next level of sophistication is called a study estimate and requires a preliminary process flowchart as a first attempt at identification of equipment, utilities, materials of construction, and other processing units that will be required in the project.

Process and Plant Design

59

Estimation accuracy improves to within ±30% probable error, but more time is required and the cost of the evaluation can escalate to, for example, over $30,000 for a $5 million plant. 3. A scope or budget authorization is the next level of economic evaluation. It requires a more defined process definition, detailed process flowcharts, and pre-final equipment design. The information required is usually obtained from pilot plant, marketing, and other studies. The scope authorization estimate could cost upward of $80,000 for a $5 million facility with a probable error potentially exceeding ±20%. 4. If the evaluation is positive at this stage, a project control estimate is then prepared. Final flowcharts, site analyses, equipment specifications, and architectural and engineering sketches are employed to prepare this estimate. The accuracy of this estimate is about ± 10% probable error. Because of the increased intricacy and precision, the cost of preparing such an estimate for the process can approach $150,000. 5. The fifth and final economic analysis is called a firm or contractor’s estimate. It is based on detailed specifications and actual equipment bids. It is employed by the contractor to establish a project cost and has the highest level of accuracy of ±5% probable error. The cost of preparation resulting from engineering, drafting, support, and management-labor expenses can amount to 3 to 5% of total estimated project costs. Because of unforeseen contingencies, inflation, and changing political and economic trends, it is impossible to ensure actual costs for even the most precise estimates.

8.3 Process Schematics To the practicing engineer, the process flowchart is the key document for defining, refining, and documenting a physical/chemical/biological process. The process flow diagram is the authorized process blueprint, the framework for specifications used in equipment designation and design; it is the single, authoritative document employed to define, construct, and operate environmental processes [1]. Beyond equipment symbols and process stream flow lines, there are several essential constituents contributing to a detailed process flowchart. These include equipment identification numbers and names; temperature and pressure designations; utility designations; mass, molar, and volumetric flow rates for each process stream; and, a material balance table pertaining to process flow lines. The process flow diagram may also contain additional information such as energy requirements, major instrumentation, environmental equipment (and concerns),

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and physical properties of the process streams. When properly assembled and employed, a process schematic provides a coherent picture of the overall process; it can pinpoint some deficiencies in the process that may have been overlooked earlier in the study, e.g., instrumentation overkill, by-products (undesirable or otherwise), and recycle needs. Basically, the flowchart symbolically and pictorially represents the interrelation between the various flow streams and equipment, and permits easy calculation of material and energy balances. Controls and instrumentation must also be considered in the overall requirements of the system. A process flow diagram for a chemical or petroleum plant is usually significantly more complex than that for a simple environmental facility. For the latter case, the flow sequence and determinations reduce to an approach that generally employs a “railroad” or sequential type of calculation that does not require iterative calculations. Various symbols are universally employed to represent equipment, equipment parts. valves, piping, etc. Some of these are depicted in the schematic in Figure 8.1. Although a significant number of these symbols are used to describe some of the chemical and petrochemical processes, only a few are needed for most environmental facilities. These symbols obviously reduce, and in some instances replace detailed written descriptions of the process. Note that many of the symbols are pictorial, which helps in better describing process components, units, and equipment. The degree of sophistication and details of a process flow diagram usually vary with both the preparer and level of sophistication desired to describe the process. It may initially consist of a simple freehand block diagram with limited information that includes only the equipment (Figure 8.2). Later versions may include line drawings with pertinent process data such as overall and componential flow rates, utility and energy requirements, environmental equipment, and instrumentation. During the later stages of the design project, the flow diagram will consist of a highly detailed P&I (piping and instrumentation) diagram. The reader is referred to the literature [2] for detailed information on P&I diagrams. In a sense, process flow diagrams are the international language of the process engineer. Process engineers conceptually view a (chemical) plant as consisting of a series of interrelated building blocks that are defined as units or unit operations. Process flow diagrams essentially tie together the various pieces of equipment that make up the process. Flow schematics follow the successive steps of a process by indicating where the pieces of equipment are located, and the material streams entering and leaving them [3–5].

8.4

Material and Energy Balances

Overall and componential material balances have already been described in some detail in Chapter 3. Material balances may be based on mass, moles, or volume, usually on a rate (time-rate of change) basis. Care should be exercised here since moles and volumes are not conserved; i.e., these quantities may change during the

Process and Plant Design

61

Steam Flue gases

Feed Boiler feed water

Fuel and air

Ash Chimney or stack

Incinerator Packed column

Spray column

Venturi ccrubber

Electrotastic precipitator or bag filter

Utilities Raw water Water treatment system

Treated water

Sludge Centrifugal Pump

Process fluid on tube side heat exchanger Counterflow Figure 8.1 Typical process flow diagram symbols.

Water treatment plant

Process fluid on shell side heat exchanger

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Unit Operations in Environmental Engineering

Process fluid on tube side heat exchanger

Process fluid on shell side heat exchanger Parallel flow

Process Fluid on Shell Side

Process Fluid on Tube Side Condensers

Figure 8.1 (Continued).

Influent

Primary clarifier

Air

Aeration tank

Secondary Clarifier Disinfection

Primary sludge

Return activated sludge

Effluent

Waste activated sludge

Figure 8.2 Simple process flow diagram for an activated sludge wastewater treatment plant.

course of a reaction. Thus, it is preferred that the initial material balance calculations be based on mass. Mole balances and molar information are important in not only stoichiometric calculations but also chemical reaction and phase equilibria calculations. Volume rates play an important role in equipment sizing calculations.

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The units (pound, kilogram, etc.) employed may also create a problem. Despite earlier efforts by the Environmental Protection Agency (EPA) and other government agencies, industry still primarily employs the British and/or American engineering units in practice; there has been a reluctance to accept standard international units, commonly referred to as SI, that employ a modified set of metric units. As indicated earlier, this text uses primarily engineering units. However, for those individuals who are more comfortable with the SI system, a short writeup on conversion of units has been prepared by the authors and included in the appendix. Some design calculations in industry today include transient effects that can account for process upsets, start-ups, shutdowns, etc. The describing equations for these time-varying (unsteady-state) systems are differential. The equations usually take the form of a first-order derivative with respect to time, where time is the independent variable. However, design calculations for most facilities assume steady-state conditions, with the ultimate design based on worst-case or maximum flow conditions. This greatly simplifies the calculations, since the describing equations (usually algebraic) provide an accounting or inventory of all mass entering and leaving one or more pieces of equipment, or the entire process. The heart of any material balance analysis is the basis selected for the calculation. The usual basis is a unit of time, such as minutes, hours, days, or years. For more complex calculations, and this may include multicomponent systems and recycle streams, one may choose a convenient amount of a key component or element as a basis. Note also that the calculation may be based on either a feed stream, an intermediate stream. or a product stream. Selecting a feed stream as a basis is preferred since it often allows one to follow calculations through the process in a railroad manner, i.e., in sequential order [2–5]. The number of material balance equations can be significant, depending on the number of components in the system, process chemistry, and pieces of equipment. These are critical calculations since (as noted above) the size of the equipment is often linearly related to the quantity of material being processed. This can then significantly impact – often linearly or even exponentially – capital and operating costs. In addition, componential rates can have an impact on (other) equipment needs, energy considerations. materials of construction, and so on. Once the material balance is completed, one may then proceed directly to energy calculations, some of which play a significant role in the design of a facility. As indicated earlier, energy calculations are also usually based on steady-state conditions. An extensive treatment of this subject has already been presented earlier in this text, and need not be repeated here. However, a thorough understanding of thermodynamic principles – particularly the enthalpy calculations – is required for most of the energy (balance) calculations. Entropy calculations are employed in meaningful energy conservation analysis; this subject is beyond the scope of this presentation but is a topic that may be given more serious attention by the environmental engineering community in the not-so-distant future (see also Case Study in Chapter 62).

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Industry has always recognized that wasting energy reduces profits. But, because the cost of energy was often a negligible part of the overall process cost, immense operational inefficiencies were tolerated before the 1973 Arab oil embargo. The sudden decrease in the availability of natural gas and oil resulting from the embargo significantly raised the cost of energy and encouraged the elimination of unnecessary energy consumption. These energy-saving measures have continued to be employed, even as energy costs have dropped and energy supplies have increased, and will continue in the future, driven by concerns related to continued process improvements, and pressures for life cycle cost reductions and life cycle environmental impact considerations. One of the principal jobs of an environmental engineer involved in the design of a facility is to account for the energy that flows into and out of each process unit and to determine the overall energy requirement and environmental footprint for the process. This is accomplished by performing energy balances on each process unit and on the overall process. These energy balances play as important a role in facility design as material balances. They find particular application in determining fuel requirements, heat exchanger design, heat recovery systems, specifying materials of construction, calculating fan and pump power requirements, and in conducting carbon footprint calculations for a process.

8.5 Equipment Design Environmental engineers describe the application of any piece of equipment that operates on the basis of mass, energy, and/or momentum transfer as a unit operation. A combination of two or more of these operations is defined as a unit process. A whole chemical process can be described as a coordinated set of unit operations and unit processes. This subject matter has received much attention in recent years and, as a result, is adequately covered in the literature. From details on these unit operations and processes, it is therefore possible to design new plants and facilities more efficiently by coordinating a series of unit actions, each of which operates according to certain laws of physics regardless of the other operations being performed along with it. The unit operation of combustion, for example, is used in many different types of industries; many of the critical design parameters for the combustion processes, however, are common to all combustion systems and independent of the particular industry. For example, in a hazardous waste incineration, the major pieces of equipment that must be considered include the following [6]: 1. Storage and handling facilities (feed and residuals) 2. Incinerator (rotary kiln or liquid injection) 3. Waste-heat boiler (primary quench system or energy recovery, if economically practical) 4. Quench system

Process and Plant Design 5. 6. 7. 8. 9. 10. 11. 12.

65

Wet scrubber – venturi scrubber (particulate removal scrubber) Absorber (packed tower for acid gas absorption) Spray dryer (quench and acid gas absorption) Baghouse or electrostatic precipitator (ESP) particulate removal Peripheral equipment (cyclone) Fan(s) and blower(s) Stack Pumps (feed, recycle, and scrubber)

Since design calculations are generally based on the maximum throughput capacity for the proposed process or for each piece of equipment, these calculations are never completely accurate. It is usually necessary to apply reasonable safety factors when setting the final design. Safety factors vary widely and are a strong function of the accuracy of the data involved, calculation procedures, and past experience. Attempting to justify these by a chemical or environmental process engineer is a difficult task. Unlike many of the problems encountered and solved by the environmental engineer, there is absolutely no correct solution to a design problem; however, there is usually a better solution. Many alternative designs, when properly implemented, will function satisfactorily, but one alternative will usually prove to be economically more efficient and/or attractive than the others. This leads to the general subject of optimization, a topic which is often covered in graduate level engineering programs.

8.6 Instrumentation and Controls The control of a system or process requires careful consideration of all operational and regulatory requirements. The system is usually designed to process materials. Safety should be a primary concern of all individuals involved with the handling, treatment, or movement of the materials. The safe operation of any unit requires that the controls keep the system operating within a safe operating envelope. The envelope is based on many of the design, process, regulatory, and societal constraints. Thus, instrumentation and controls are placed on the unit to ensure proper operation. Additional controls may be installed to operate additional equipment needed for energy recovery, neutralization, or other peripheral operations. The control system should also be designed to vary one or more of the process variables to maintain the appropriate conditions with the unit. These variations are programmed into the system on the basis of past experience of the process in question. The operational parameters that may vary include the temperature and system pressure. The proper control system should also be subjected to extensive analysis of operational problems and items that could go wrong. A hazard-and-operability (HAZOP) analysis is often conducted on the control system to examine and

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identify all possible failure mechanisms [7]. It is important that all of the identified failure modes yield appropriate responses by the control system. Several of the failure mechanisms that must be addressed within the appropriate control system response are excess or minimal temperature excursions, excessive or subnormal flow rate excursions, equipment failure, sticking or inoperable components, and broken circuits. All are usually examined, including the response time of the control system to the problem, as well as the appropriate response of the system to the problem. The control system must identify the problem and integrate necessary actions and alarms into the process. A complete review of the maximum and minimum process (variable) time rate of change and the equipment time rate of change must be identified before the control system can be defined. The system must also be reviewed to define the primary control parameters for operation and safety. Finally, the regulatory limitations imposed on the unit must be identified and monitored to ensure that they are not exceeded.

8.7 Plant Location and Layout [adopted from 8] The geographical location of a plant must be taken into consideration when a process design is developed. If the plant is located in a cold climate, costs may be increased by the necessity for constructing protective shelters around the process equipment. Cooling water costs may be very high if the plant is not located near an ample water supply. Electrolytic processes require a cheap source of electricity, and plants of this type ordinarily cannot operate economically unless they are located near large hydroelectric installations. The effects of the following factors on production costs are of importance in considering plant location: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Source of raw materials Markets for finished products Transportation facilities Labor supply Power and fuel Water supply and waste disposal Utilization of by-products Taxation and legal restrictions Water table Flood and fire protection Room for expansion Building and land costs Climatic conditions

The physical layout of a plant should be designed to permit coordination between the operation of the process equipment and the use of storage and

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materials-handling equipment. Scale drawings complete with elevation indications are useful for determining the best locations for the equipment and facilities. The following items should be taken into consideration in setting up the plant layout: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

8.8

Storage facilities readily available for use Need for possible changes in the future Process operations arranged for operators’ convenience Space available Time interval between successive operations Automatic or semiautomatic controls Safety considerations Availability of utilities and services Waste disposal Necessary control tests Auxiliary equipment Personnel working and break locations

Plant Design [adopted from 9]

The plant-design group translates results from the process-design engineer into complete plans and specifications which will be used by constructors to build the plant or facility. The plant design must be complete in every detail. It is used to make a firm estimate of the cost of the plant, and it may serve as a basis for a contract between the client and the construction firm. The plant-design group may include environmental, chemical, mechanical, electrical, and civil engineers. The group is supervised by a Project Engineer with knowledge of the over-all process. The Project Engineer must coordinate the activities of the various specialists in the design group; carefully analyzes the data supplied to them by the process-design engineer; and may make suggestions for modifying the fundamental process if leading to substantial savings. The Process Engineer and the Project Engineer work closely in analyzing such suggestions. The Project Engineer must concern themselves with peripheral problems such as supply of water and other utilities, waste disposal, and safety. They may be the only person who has a thorough understanding of and responsibility for the entire design. Working closely with the Project Engineer are the designer and draftsman. The designer is often a specialist in a particular phase of plant design. For example, after a chemical engineer has determined the number, size, and spacing of plates in a distillation column, a mechanical designer may specify the physical details of the column, the electrical designer may specify the location and type of instrumentation and control, the structural designer considers the support framework and foundations for the column and auxiliary equipment, and the

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environmental engineer considers environmental compliance and health and safety operating and control constraints. The designers make suggestions to the Project Engineer on specific points where money might be saved. The designers supervise the draftsmen who make the detailed drawings of each unit of the process. Because of the complexity of some plants, some design groups historically used small-scale models of the plant to study the physical layout of equipment and piping. Current practice is to use 3-D plant design and visualization for the same purpose. Such models often bring out weaknesses in a layout which would make the plant difficult to operate, maintain, or repair. The Project Engineer is ultimately responsible for all these details. The Project Engineer works closely with the contractor who is building the plant. Many questions arise during construction: materials specified may be unavailable in the time limit set; pilot-plant work may show that a change in one unit is essential; the contractor may suggest changes to improve constructability, etc. It should be realized that the design of a facility is an iterative process that involves consideration of all viable process alternatives that meet the design objectives. These design considerations must be factored into process decisions to ensure that the facility meets most if not all design objectives.

References 1. Ulrich, G., A Guide to Chemical Engineering Process Design and Economics, John Wiley & Sons, Hoboken, NJ, 1984. 2. Shen, T., Choi, Y., and Theodore, L., Hazardous Waste Incineration Manual, USEPA/APTI, Research Triangle Park, N.C. 1985. 3. Felder, R., and Rousseau, R., Elementary Principles of Chemical Processes, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 1986. 4. Perry, R., and Green, D. (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York City, NY, 2008. 5. McCabe, M., Smith, J., and Harriott, P., Unit Operations of Chemical Engineering, 5th Edition McGraw-Hill, New York, NY, 1993. 6. Santoleri, J., Reynolds, J., and Theodore, L., Introduction to Hazardous Waste Incineration, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2000. 7. Theodore, L., and Dupont, R.R., Environmental Health and Hazard Risk Assessment: Principles and Calculations, CRC Press/Taylor & Francis Group, Boca Raton, Fla., 2012. 8. Anderson, L., and Wenzel, Z., Introduction to Chemical Engineering, McGraw-Hill, New York, NY, 1969. 9. Peters, M., Elementary Chemical Engineering, McGraw-Hill, New York, NY, 1954.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

Part II FLUID FLOW

The second part of this text provides material addressing various aspects of fluid flow. It contains three sections and a total of 18 chapters, and each serves a unique purpose in an attempt to treat important aspects of fluid flow. From a practical point-of-view, systems and plants move liquids and gases from one point to another, hence, environmental students and/or practicing environmental engineers must be concerned with key topics in this area. Part II chapter content includes: II-A Fundamentals 9. Fluid Behavior 10. Basic Energy Conservation Laws 11. Law of Hydrostatics 12. Flow Measurement 13. Flow Classifications II-B Equipment 14. Prime Movers 15. Valves and Fittings 16. Air Pollution Control Equipment 69

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Unit Operations in Environmental Engineering 17. Sedimentation, Centrifugation, and Flotation 18. Porous Media and Packed Beds 19. Filtration 20. Fluidization 21. Membrane Technology II-C Other Considerations 22. Compressible and Sonic Flow 23. Two-Phase Flow 24. Ventilation 25. Mixing 26. Biomedical Applications

Material presented in this Part covers a wide range of topics relevant to both environmental and chemical engineering, and specific environmental engineering applications are highlighted throughout.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

9 Fluid Behavior

This chapter is concerned with fluid behavior. For the purpose of this text, a fluid may be defined as a substance that does not permanently resist distortion. An attempt to change the shape of a mass of fluid will result in layers of fluid sliding over one another until a new shape is attained. During the change in shape, shear stresses (forces parallel to a surface) will result, the magnitude of which depends upon the viscosity (to be discussed shortly) of the fluid and the rate of shearing. However, when a final shape is reached, all shear stresses will have disappeared. Thus, a fluid at equilibrium is free from shear stresses. This definition applies for both liquids and gases. Two types of fluids are considered in this chapter: Newtonian, and non-Newtonian fluids. The bulk of the material in this chapter has been drawn from the work of Farag [1].

9.1 Introduction This chapter is introduced by examining the units of some of the pertinent quantities that are encountered below. The momentum of a system is defined as the product of the mass and velocity of the system.

Momentum = (Mass) (Velocity) 71

(9.1)

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One set of units for momentum is, therefore, lb-ft/s. The units of the time rate of change of momentum (hereafter referred to as the rate of momentum) are simply the units of momentum divided by time, i.e.,

Rate of momentum = lb-ft/s2

(9.2)

The above units can be converted to lbf if divided by an appropriate constant. The conversion constant in this case is a term that was introduced earlier, the gravitational constant, gc,

gc = 32.2 lb-ft/(lbf-s2)

(9.3)

This serves to define the conversion constant gc. If the rate of momentum is divided by gc the following units result:

Rate of momentum =(lb-ft/s2)/{gc, lb-ft/(lbf-s2)} = lbf

(9.4)

One may conclude from the above dimensional analysis that a pound force is equivalent to a rate of change of momentum.

9.2 Newtonian Fluids [2] The above development is now extended to Newton’s law of viscosity. Consider a fluid flowing between the region bounded by two infinite parallel horizontal plates separated by a distance h. The flow is steady and only in the y-direction. Part of the system is represented in Figure 9.1. A sufficient force, F, is being applied to the upper plate at z = h to maintain the upper plate in motion with a velocity vy = Vh. If the fluid density is constant and the flow is everywhere isothermal and laminar, the linear velocity gradient in the two-dimensional representation in Figure 9.2 will result. It has been shown by experiment that the applied force per unit area, F/A, required to maintain the upper plate in motion with velocity Vh, is proportional to the velocity gradient, i.e.,

F/A

Vh/h

(9.5)

For a slightly more general form, one may write

F/A

vy/Δz

(9.6)

The difference term Δ can be removed by applying Equation 9.6 to a differential width, dz:

F/A

dvy/dz

(9.7)

Fluid Behavior 73 z

z=h

F

y=0

z=0

y= L y

x=0

x=w

x

Figure 9.1 Fluid/two plate system.

z

Vh

z=h

F

Vy z

z=0

y

Figure 9.2 Velocity profile.

Equation (9.7) may be written in equation form by replacing the proportionality sign with a proportionality constant, μ:

F/A = μ (dvy/dz)

(9.8)

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The term μ is defined as the coefficient of viscosity, or simply the viscosity of the fluid. The term F/A is a shear stress since F is exerted parallel to the direction of motion. This applied force per unit area is now designated by zy, zy

= μ (dvy/dz)

(9.9)

A fluid whose shear stress is described by Equation 9.9 is defined as a Newtonian fluid. A word of interpretation is in order for Equation 9.9. The applied force at z = h has resulted in a velocity Vh at z = h. The fluid at this point possesses momentum due to this velocity. As z decreases, the momentum of the fluid decreases since the velocity decreases in this direction. It was already shown that the force applied to a fluid is equivalent to the fluid receiving a rate of momentum. Part of the momentum imparted to the fluid at z = h is transferred at the specified rate to the slowermoving fluid immediately below it. This momentum maintains the velocity of the fluid at that point, and is, in turn, transported to the slower fluid below it, and so on. This momentum transfer process is occurring in the z-direction throughout the fluid. One may therefore conclude that the applied force in the positive y-direction has resulted in the transfer of momentum in the negative z-direction. The first subscript in zy is retained as a reminder of this fact. The subscript y indicates the direction of motion. The negative sign in Equation 9.8 was introduced since momentum is transferred in the negative z-direction due to a positive velocity gradient. The force per unit area term, , is equivalent to a rate of momentum per unit area Therefore, the shear stress and its components are also defined as the momentum flux. Referring once again to the shear stress component zy, one may divide the RHS of Equation (9.9) by gc to yield Equation 9.10. zy

= ( μ/gc) (dvy/dz)

(9.10)

If zy has the units lbf/ft2, the viscosity μ assumes units of lb/ft-s. A term that will frequently be employed in this part of the text is the kinematic viscosity, . This is defined as the ratio of the viscosity to the density of the fluid.

= μ/r The units of

(9.11)

can be shown to be ft2/s.

9.3 Strain Rate, Shear Rate, and Velocity Profiles When a fluid flows past a stationary solid wall, the fluid adheres to the wall at the interface between the solid and fluid. This condition is referred to as “no slip.”

Fluid Behavior 75 Therefore, the local velocity, v, of the fluid at the interface is zero. At some distance, y, normal to and displaced from the wall, the velocity of the fluid is finite. Therefore, there is a velocity variation from point to point in the flowing fluid. This causes a velocity field, in which the velocity is a function of the normal distance from the wall, that is, v = f(y). If y = 0 at the wall, v = 0, and v increases with y. The rate of change of velocity with respect to distance is defined as the velocity gradient,

dv/dy = v/ y

(9.12)

This velocity derivative (or gradient) is also referred to as the shear or strain rate, time rate of shear, or rate of deformation. The units of the strain rate are time 1. Strain rate is important in the classification of real fluids. The relationships between shear stress and strain rate are presented in diagrams called rheograms. The velocity profile in a pipe for two classes of flow (laminar and turbulent) is provided in Figure 9.3. In laminar flow, the velocity profile approaches a true parabola slightly pointed in the middle and tangent to the walls of the pipe. The average velocity over the whole cross-section (volumetric flow rate divided by the cross-sectional area) is 0.5 times the maximum velocity. This fact will be discussed again in Chapter 13. In turbulent flow, the profile approaches a flattened parabola and the average velocity is usually approximately 0.8 times the maximum value. Because of its viscosity, a real fluid in contact with a nonmoving wall will have a velocity of zero at the wall. Similarly, a fluid in contact with a wall moving at a velocity, v, will move at the same velocity. This earlier described “no-slip” condition of real fluids flowing in a duct results in a fluid velocity at the wall of zero.

C A

Velocity B

Distance A = Laminar flow B = Turbulent flow C = Random flow

Figure 9.3 Velocity profile in a pipe under laminar, turbulent, and random flow conditions.

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To calculate the volumetric flow rate, q, of the fluid passing through a perpendicular surface, S, one must integrate the product of the component of the velocity, v, that is normal to the area by the area over the whole cross-section of the duct, i.e.,

q

vdS

(9.13)

s

In accordance with the definition of average values, the average velocity of the fluid passing through the surface. S, is then given by:

v

vave

vdS

s

s

S

q S

(9.14)

As noted above, the velocity calculated is an average value. Plug flow, characterized by a uniform velocity distribution, is often assumed. In actual operation, the following velocity profiles might develop (see Figure 9.3): 1. Parabolic – laminar flow. 2. “Flattened” parabola – turbulent flow, wherein velocities are low (often near zero at the perimeter/walls) and high (often near 20% above the average velocity) at the center. 3. Random distribution – following a bend, valve, or disturbance. These profiles are discussed further in later chapters.

9.4 Non-Newtonian Fluids The study of the mechanics of the flow of liquids and suspensions comes under the science of Rheology. The name Rheology was chosen by Prof. John R. Crawford of Lafayette College, PA, and is defined as the study of the flow and deformation of matter. (The name is a combination of the Greek words “Rheo” – flow and “Logos” – theory). The shear-stress equations developed in the previous section were written for fluids with a viscosity that is constant at constant temperature and independent of the rate of shear and the time of application of shear. Fluids with this property were defined as Newtonian fluids. All gases and pure low-molecular-weight liquids are Newtonian. Miscible mixtures of low-molecular-weight liquids are also Newtonian. On the other hand, high-viscosity liquids as well as polymers, colloids, gels, concentrated slurries, and solutions of macromolecules generally do not exhibit Newtonian properties, i.e., a strict proportionality between stress and strain rate. Interestingly, non-Newtonian properties are sometimes desirable.

Fluid Behavior 77 For example, non-Newtonian behavior is exhibited in many paints. During brush working, certain paints flow readily to cover the surface, but upon standing, the original highly viscous condition returns and the paint will not run. The study of non-Newtonian fluids has not progressed far enough to develop many useful theoretical approaches. As will be noted in Chapter 13, if a liquid or suspension is found to be Newtonian, the pressure drop can be calculated from the “Poiseuille” equation for laminar flow (see Chapter 13) and the Fanning equation for turbulent flow (see Chapter l3), using the density and viscosity of the liquid or suspension. For non-Newtonian liquids and suspensions, the viscosity is a variable and the procedure for computing the pressure drop is more involved. The remainder of this section will discuss non-Newtonian liquids and suspensions. Useful engineering design procedures and prediction equations receive treatment that are limited to isothermal laminar (viscous) flow. The turbulent flow of non-Newtonian fluids (as with Newtonian ones) is characterized by the presence of random eddies and whirls of fluid that cause the instantaneous values of velocity and pressure at any point in the system to fluctuate wildly. Because of these fluctuations, flow problems often cannot be easily solved. Fluids can also be classified based on their viscosity. An imaginary fluid of zero viscosity is called a Pascal fluid. The flow of a Pascal fluid is termed inviscid (or non-viscous) flow. Viscous fluids are classified based on their rheological (viscous) properties. These are detailed below: 1. Newtonian fluids, as described in the previous section, obey Newton’s law of viscosity (i.e., the fluid shear stress is linearly proportional to the velocity gradient). All gases are considered Newtonian fluids. Newtonian liquid examples are water, benzene, ethyl alcohol, hexane and sugar solutions. All liquids of a simple chemical formula are normally considered Newtonian fluids. 2. Non-Newtonian fluids do not obey Newton’s law of viscosity. Generally, they are complex mixtures (e.g., polymer solutions, slurries, and so on). Non-Newtonian fluids are classified into three types: a. Time-independent fluids are fluids in which the viscous properties do not vary with time. b. Time-dependent fluids are fluids in which the viscous properties vary with time. c. Visco-elastic or memory fluids are fluids with elastic properties that allow them to “spring back” after the release of a shear force. Examples include egg-white and rubber cement. Additional details on the first two classes of non-Newtonian fluids follow. 1. Time-independent, non-Newtonian fluids are further classified into three types.

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Unit Operations in Environmental Engineering a. Pseudoplastic or shear thinning fluids that are characterized by fluid resistance that decreases with increasing stress (e.g., polymers). b. Dilatant or shear thickening fluids whose resistance increases with increasing velocity gradient or applied stress. These are uncommon, but an example is quicksand. c. Bingham plastics that resist a small shearing stress. At low shear stress these fluids do not move. At high shear these fluids move. These fluids just start moving when sufficient stress is applied. This stress is termed the yield stress. When the applied stress exceeds the yield stress, the Bingham plastic flows. Examples are toothpaste, jelly, and bread dough. 2. Time-dependent, non-Newtonian fluids are further classified into two types. a. Rheopectic fluids are characterized by an increasing viscosity with time. Rubber cement is an example. b. Thixotropic fluids have a decreasing viscosity with time. Examples are slurries or solutions of polymers.

The shear stress equation equivalent to Equation 9.8 for one of the more common types of non-Newtonian fluids is given by the so-called “power law” equation: zy

= K/gc (dvy/dz)n

(9.15)

K is defined as the consistency number and may in special cases equal μ. The exponent n is defined as the flow-behavior index and is a real number that usually assumes a value other than unity. Although n is considered a physical properly of a fluid, it is not necessarily a constant; rather, it may vary with the shear rate, dvy/ dz. Equation 9.15 may be written in terms of the apparent viscosity μa for nonNewtonian fluids (most non-Newtonian fluids have apparent viscosities that are relatively high compared with the viscosity of water):

μa/gc =

/(dvy/dz)

zy

(9.16)

μa = K(dvy/dz)n-1

(9.17)

or

In order to remove the problem arising when the velocity gradient is a negative quantity, Equation 9.15 is rewritten as: zy

= K/gc (dvy/dz) |dvy/dz|n-1

(9.18)

Fluid Behavior 79

ss tre ld s

st

ie ic y

am gh Bin

pla

zy

=

0

–

( g ) ( dvd ) : a

c

y

z

0

= yield stress

Shear stress,

zy

n1 Dilatant

Shear rate,

dvy dz

Figure 9.4 Fluid shear diagrams for a variety of fluid types.

A typical shear stress versus shear rate (dvy/dz) curve (often referred to as a rheogram), is shown for a non-Newtonian fluid in Figure 9.4 on arithmetic coordinates. Newtonian behavior is also depicted in the diagram. Due to the exponential nature of the shear rate of this type of non-Newtonian fluid, a straight line would be obtained on a log-log plot as demonstrated in Equation 9.15 and its log transform, Equation 9.19.

log

zy

= log(K/gc) + n log(dvy/dz)

(9.19)

One notes that a Newtonian fluid yields a slope of 1.0 on log-log coordinates. The slope of a non-Newtonian fluid generally differs from unity. The slope, n, can be thought of as an index of the degree of non-Newtonian behavior in that the farther that n is from unity (above or below), the more pronounced is the nonNewtonian characteristics of the fluid.

References 1. Farag, I., Fluid Flow, A Theodore Tutorial, Theodore Tutorials, East Williston, NY. Originally published by the U.S. EPA APTI, RTP, NC, 1996. 2. Theodore, L., Transport Phenomena for Engineers, Theodore Tutorials, East Williston, NY, originally published by International Textbook Company, Scranton, PA, 1971.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

10 Basic Energy Conservation Laws

This chapter is concerned with describing equations based on the conservation of energy. The presentation includes a review of some key pressure terms. A general introduction to the conservation of energy is in turn followed by the development of a general total energy balance for steady state flow. The total energy equation is then extended to include mechanical energy; this has come to be defined as the mechanical energy balance equation. It is this equation that is employed in the solution of most real-world environmental fluid flow problems. The chapter concludes by extracting the Bernoulli equation from the mechanical energy balance equation.

10.1 Introduction One of the most critical parameters in fluid flow is pressure. Three pressure terms should be defined before proceeding to the body of this chapter. These are static pressure (Ps), velocity pressure (Pv), and the sum of the two, the total pressure (Pt). Any fluid confined in a stationary enclosure has static pressure simply because the molecules of that fluid are in constant random motion and are continually colliding with the container walls. The bulk velocity of this stationary fluid is zero, 81

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and the total pressure is then equal to the static pressure. If the same fluid is flowing and the temperature has not changed, it possesses the same static pressure since its molecules still have the same degree of random motion. Its total pressure is now higher, however, because it also possesses the second pressure component, velocity pressure. If the fluid flow were to suddenly change direction because of a solid obstruction (e.g., a plate), an extra pressure on the plate (over and above the static pressure) would be exerted because of the momentum of the bulk flow against the plate. This extra pressure is the velocity pressure and the total fluid pressure is the sum of the static and velocity pressures. Static pressure is therefore the result of motion on the molecular level, while velocity pressure is due to motion at the macroscopic or bulk level [1]. The difference in total pressure between two different points along the flow stream is called the pressure loss or the pressure drop. Pressure losses from fluid flow are due to any effect that can change fluid momentum at either the molecular or macroscopic levels. The two main contributing factors to momentum changes are skin friction and form friction. Skin friction losses are caused by fluid moving along (parallel to) a solid surface such as a pipe or duct wall. The layers of fluid immediately adjacent to the wall are in laminar flow and moving much slower than the bulk of the fluid. This pressure drop caused by the drag effect of the wall on the fluid is due to skin friction. Form friction losses are due to the acceleration or deceleration of the fluid. These include changes in bulk fluid velocity that occur because of changes in either flow direction or flow velocity. An example of a change in flow direction is fluid flowing through a 90° elbow. Alternatively, a change in flow velocity occurs when the cross-section of a conduit changes. Besides changes in bulk fluid velocity, form friction losses also include changes in velocity that occur locally, i.e., internal to the bulk motion of the fluid. This occurs in turbulent flow (see Chapter l3 for more details), which is characterized by rapidly swirling masses of fluid called eddies [1].

10.2 Conservation of Energy A system may possess energy due to its temperature (internal energy), velocity (kinetic energy), position (potential energy), molecular structure (chemical energy), surface (surface energy), etc. As described earlier, energy, like mass and momentum, is conserved. Application of the conservation law for energy gives rise to the first law of thermodynamics. This law for batch processes, is presented below.

E=Q+W

(10.1)

where potential, kinetic, and other energy effects have been neglected, E (often denoted as U) is the internal energy of the system, ΔE is the change in the internal energy of the system, Q is energy in the form of heat transferred across the system boundaries, and W is energy in the form of work transferred across system

Basic Energy Conservation Laws

83

boundaries. In accordance with the recent change in convention, both Q and W are treated as positive terms if energy is added to the system. By definition, a flow process involves material streams entering and exiting a system. Work is done on the system at the stream entrance when the fluid is forced into the system. Work is also performed by the system to force the fluid out at the stream exit. The net work on the system is called flow work, Wf, and is given by:

Wf

PoutVout

PinVin

PV

(10.2)

where Pout is the pressure of the outlet stream, Vout is the volume of fluid exiting the system during a given time interval, Pin is the pressure of the inlet stream, and Vin is the volume of fluid entering the system during the same time interval. If the volume term is represented as the specific volume (i.e., volume/mass), the work term carries the units of energy/mass. For practical purposes, the total work term, W, in the first law may be regarded as the sum of shaft work, Ws, and flow work, Wf, or

W = Ws + Wf

(10.3)

where Ws is work done on the fluid by some moving solid part within the system such as the rotating vanes of a centrifugal pump. Note that in Equation 10.3, all other forms of work, such as electrical, surface tension, and so on are neglected. The first law for steady-state flow processes is then:

H = Q + Ws

(10.4)

where H is the enthalpy of the system and H is the change in the system’s enthalpy. The internal energy and enthalpy in Equations 10.l and 10.4, as well as other equations in this section may be on a mass basis, on a mole basis, or represent the total internal energy and enthalpy of the entire system. They may also be written on a time-rate basis as long as these equations are dimensionally consistent. For the sake of clarity, upper case letters (e.g., H, E) represent properties on a mole basis, while lower-case letters (e.g., h, e) represent properties on a mass basis. Properties for the entire system will rarely be used and therefore require no special symbols. Although the topics of material and energy balances have been covered separately in this and the previous chapter, it should be emphasized that this segregation does not exist in reality. Many processes are accompanied by heat effects, and one must work with both energy and material balances simultaneously [2, 3].

10.3

Total Energy Balance Equation

Equations 10.1 and 10.4 find application in many chemical and environmental process units such as heat exchangers, reactors, and distillation columns, where

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shaft work plus kinetic and potential energy changes are negligible compared with heat flows and either internal energy or enthalpy changes. Energy balances on such units therefore reduce to

Q = E for closed systems

(10.5)

Q = H for open systems

(10.6)

Another important class of operations is one for which the opposite is true, i.e., heat flows and internal energy changes are secondary in importance to kinetic and potential energy changes and shaft work. Most of these operations involve the flow of fluids to, from, and between tanks, reservoirs, wells, and process units. Accounting for energy flows in such processes is most conveniently accomplished with mechanical energy balances. Consider the steady-state flow of a fluid in the process pictured in Figure 10.1. The mass entering at Location 1 brings in with it a certain amount of energy, existing in various forms. Thus, because of its elevation, z1 ft above any arbitrarily chosen horizontal reference plane, it possesses a potential energy (g/gc) z1 (which can be recovered by allowing the fluid to fall from the height at Location 1 to that of the reference point). Because of its velocity, v1, the mass possesses and brings into Location 1 an amount of kinetic energy, v12/2gc. It also brings its so-called internal energy, E1, because of its temperature. Furthermore, the mass of fluid in question entering at Location 1 is forced into the section by the pressure of the fluid behind it and this form of flow energy must also be included. The amount of this energy is given by the force exerted by the flowing fluid times the distance

Z2

Location 1

Q

Z1

Location 2

Compression

Z2 Pump W2

Heat exchanger TQ

Process flow

Figure 10.1 Fluid flow in a process.

Basic Energy Conservation Laws

85

through which it acts, and this force is clearly the pressure per unit area, P1, times the area, S1 of the cross-section. The distance through which the force acts is the volume, V1, of the fluid divided by the cross-sectional area S1. Since the work is the force times the distance, that is, (P1 S1)(V1/S1) = P1 V1, the energy expended is the product of the pressure times the volume of the fluid. This was referred to earlier as flow work (see Equation 10.2). Two additional energy terms need to be included in the analysis. These two involve energy exchange in the form of heat (Q) and work (W) between the fluid and the surroundings. In the development to follow, it will be assumed (consistent with the notation recently adopted by the scientific community) that any energy in the form of heat or work added to the system is treated as a positive term. Applying the conservation law of energy mandates that all forms of energy entering the system equal that of those leaving. Expressing all terms in consistent units (e.g., energy per unit mass of fluid flowing), results in a total energy balance:

v12 2gc

PV 1 1

g z1 E1 Q Ws gc

P2V2

v22 2gc

g z2 gc

E2

(10.7)

As written, each term in Equation 10.7 represents a mechanical energy effect. For this reason, it is defined as a form of the mechanical energy balance equation and is essentially a special application of the conservation law for energy. Also note that, as written, the volume term, V, (for necessity) is the specific volume, or 1/ . In terms of the density, the above equation becomes:

P1

v12 2gc

g z1 E1 Q Ws gc

P2

v22 2gc

g z2 gc

E2

(10.8)

Note once again that Q and Ws can be written on a time rate basis in the above equation by simply dividing by the mass flowrate though the system; the above equation then dimensionally reduces to an energy/mass balance. Three points need to be made before leaving this subject. 1. The term Q should represent the total net heat added to the fluid, but in this analysis it includes only the heat passing into the fluid across the containing walls from an external source. This excludes heat generated by friction, by the fluid or otherwise, within the unit. However, this effect can normally be safely neglected. 2. The work, Ws, similar to Q, must pass though the containing walls. While it could conceivably enter in other ways, it is supplied in most environmental applications by some form of moving mechanism, such as a pump, or a fan, and is often referred to as shaft work. 3. The internal energy term, E, corresponds to the thermodynamic definition provided earlier. For convenience, the sum of E and

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Unit Operations in Environmental Engineering PV may be treated as the single function defined above as the enthalpy, H,

H = E + PV

(10.9)

It too is a property of the fluid, uniquely determined by point conditions. Like E, its absolute value is arbitrary; differences in value are often given above a reference. With this revision, Equation 10.7 becomes:

v12 2gc

g z1 H1 Q Ws gc

v22 2gc

g z2 H 2 gc

(10.10)

or simply

v2 2gc

g z gc

H

Q Ws

(10.11)

As noted in the presentation of Equation 10.7, each term is dimensional with units of energy/mass. If this equation is multiplied by the fluid flow rate, that is, mass/time, the units of each term become energy/time. In the absence of both kinetic and potential energy effects, the above equation reduces to Equation 10.4. Also note that Δ, the difference term, refers to a difference between the value at Location 2 (the usual designation for the outlet) minus that at Location 1 (the inlet).

10.4

The Mechanical Energy Balance Equation

As noted, the solutions to many fluid flow problems are based on the mechanical energy balance equation. This equation is derived, in part, from the general (or total) energy equation developed in the previous section. Equation 10.7 is shown again below.

PV 1 1

v12 2gc

g z1 E1 Q Ws gc

P2V2

v22 2gc

g z2 gc

E2

Certain “changes” to the above equation can now be made: 1. Assume adiabatic flow, that is, Q = 0. 2. For isothermal, or near isothermal, flow (valid in most applications), the internal energy is constant, so that E1 = E2.

(10.7)

Basic Energy Conservation Laws

87

3. A term, F, representing the total friction arising due to fluid flow, is added to the equation. This is treated as a positive term in Equation 10.12 below. 4. An efficiency (fractional) term, , is combined with the shaft work term, Ws. If work is imparted on the system, the term becomes Ws. If work is extracted (with an engine or turbine) the term appears as Ws/ . The efficiency term needs to be included since part of the work added to or extracted from a system is lost due to irreversibilities associated with the mechanical device. The notation hs will be employed for this term in Chapter 13. Equation 10.7 can be now be rearranged and simplified to become:

P

v2 2gc

g z gc

Ws

F

0

(10.12)

This equation is defined as the mechanical energy balance equation; it will receive extensive attention later in the book.

10.5 The Bernoulli Equation If both work and frictional effects are neglected in Equation 10.12, it reduces to Equation 10.13, a common form of the Bernoulli equation:

P

v2 2gc

g z gc

0

(10.13)

This is also often referred to as the Field equation in other disciplines. This equation, which applies to flow in the absence of friction, has some interesting ramifications. If one of the three terms is increased, either of the other two terms must decrease; alternately, both of the other two terms can change but the sum of the two changes must decrease. For example, if the Bernoulli equation is applied along a horizontal streamline (path) of a fluid, an increase in the velocity results in a decrease in pressure. The phenomenon is “exploited” by birds during flight, and in the design of airplane wings at the industrial level. The above effect can also explain why roofs are lifted off some buildings during a hurricane or tornado; the high velocity on top of the roof creates a lower pressure at the outer surface relative to the inner surface. This difference in pressure – force per unit area – across the roof ’s top surface produces a net upward force lifting the roof off its foundation. This can be prevented in many instants by simply opening all windows and doors; the high velocity within the structure produces a lower pressure and consequently a smaller or zero upward force.

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Unit Operations in Environmental Engineering

The reader might like to test the validity of the proposed explanation by taking a sheet of 8½ inch by 11 inch paper and holding it by its sides while allowing the paper to droop. Blowing across needs a space here not/the top of the paper does in fact result in the paper rising to a near horizontal level.

References 1. Badger, W., and Banchero, J., Introduction to Chemical Engineering, McGraw-Hill, New York, 1955. 2. Felder, R., and Rousseau, R., Elementary Principles of Chemical Processes, 3rd Edition, John Wiley & Sons, Hoboken, NJ, 2000. 3. Santoleri, J, Reynolds, J., and Theodore, L., Introduction to Hazardous Waste Incineration, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2000.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

11 Law of Hydrostatics

11.1 Introduction When a fluid is at rest, there is no shear stress and the pressure at any point in the fluid is the same in all directions. The pressure is also the same across any longitudinal section parallel with the Earth’s surface. It varies only in the vertical direction, i.e., from height to height. This phenomenon gives rise to hydrostatics, the subject of this chapter. Following this introduction, this chapter addresses pressure principles, buoyancy effects (including Archimedes’ Law), and manometry principles. Note, the bulk of this chapter is drawn from the work of Reynolds [1] and Faraq [2].

11.2 Pressure Principles Consider a differential element of fluid of height, dz, and uniform crosssectional area, S. The pressure, P, is assumed to increase with height, z. The pressure at the bottom surface of the differential fluid element is P, and at the top surface, it is P + dP. Thus, the net pressure difference, dP, on the differential

89

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Unit Operations in Environmental Engineering

element is acting downward. A force balance on this element in the vertical direction yields:

downward pressure force

upward pressure force + gravity force = 0 (11.1)

so that

(P dP )S PS

S

g dz gc

0

(11.2)

Simplification and rearrangement yields:

(dP )S

g S(dz ) 0 gc

(11.3)

As described earlier, one has a choice as to whether to include gc in the describing equation(s). As noted, the term gc is a conversion constant with a given magnitude and units, e.g., 32.2 (lb/lbf)(ft/s2) or dimensionless with a value of unity. In this development, gc is retained. Further rearrangement and simplification of Equation 11.3 yields:

dP dz

g gc

(11.4)

The term is the specific weight of fluid with units of lbf/ft3 or N/m3. This equation is the hydrostatic or barometric differential equation. The term dP/dz is often referred to as the pressure gradient. Equation 11.4 is a first-order ordinary differential equation. It may be integrated by separation of variables

dP

g dz gc

g gc

dz

(11.5)

For most engineering applications involving liquids, and many applications involving gases, the fluid density may be considered constant, i.e., the fluid is incompressible. Taking outside the integration sign and integrating between any two limits in the fluid (Location 1 where the pressure equals P1 and the elevation is z1, and Location 2 where pressure is P2 and the elevation is z2), the pressure-height relationship is:

P2 P1

g (z 2 z1 ) gc

(11.6)

Law of Hydrostatics 91 Equation 11.6 may also be written as:

P1 z1 ( g /g c )

P2 z2 ( g /g c )

(11.7)

The term P/( (g/gc)) is defined as the pressure head of the fluid, with units of m (or ft) of fluid. Equation 11.7 states that the sum of the pressure head and potential head is constant in hydrostatic “flow”. Equation 11.7 is sometimes termed Bernoulli’s hydrostatic equation. It is useful in calculating the pressure at any liquid depth. Calculations of fluid pressure force on submerged surfaces is important in selecting the proper material and thickness to be used for that surface. If the pressure on the submerged surface is not uniform, then the pressure force is calculated by integration. Also note that a nonmoving fluid exerts only pressure forces; moving fluids exert both pressure and shear forces.

11.3 Buoyancy Effects: Archimedes’ Law Buoyancy force is the force exerted by a fluid on an immersed or floating body. Archimedes’ Law states that for a body floating in a fluid, the volume of fluid displaced equals the volume of the immersed portion of the body, and the weight (force) of fluid displaced equals the weight (force) of the body. The buoyancy force on the body, FB, is:

FB = (displaced volume of fluid) (fluid density) (g/gc) = (displaced volume of fluid) (fluid specific weight) (l/gc)

(11.8)

Thus,

FB

Vdisp

fl

g gc

Vdisp

fl

(11.9)

where fl is the specific weight of the fluid (see Equation 11.4). In the case where the density of the fluid and the body are equal, the body remains at its point or location in the fluid where it is placed. This is termed neutral buoyancy. In the case where the body density, body is greater than the fluid density, , the body will sink in the fluid until the volume of the displaced fluid equals fluid the weight of the body. A hydrometer is used to indicate a liquid’s specific gravity, with the value being indicated by the level at which the free surface of the liquid intersects the stem of the hydrometer when it is floating in a liquid. Three hydrometer scales are commonly used. The API scale is used for oils, and the two Baumé scales are used

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for liquids – one for liquids heavier than water and the other for liquids lighter than water. The relationship between the hydrometer API scales and the specific gravity, SG, is

SG

141.15 131.5 deg API

(11.10)

The relationship between the hydrometer degree Baumé scale and the specific gravity, SG, of liquids lighter than water is:

SG

140 130 deg BaumØ

(11.11)

For liquids heavier than water the relationship is:

SG

145 145 deg BaumØ

(11.12)

When placed in a liquid, the hydrometer floats at a level which is a measure of the specific gravity of the liquid. The 1.0 mark is the level when the liquid is distilled water ( = 1000 kg/m3). The stem is of constant diameter, and contains a volume of fluid within it with a density greater than water. The hydrometer is a simple device to estimate liquid densities. It is used widely in various industries. By varying the stabilizing weight or the stem diameter it is possible to design a hydrometer to be highly sensitive to different ranges of fluid specific gravities.

11.4 Manometer Principles As noted earlier, the fundamental equation of fluid statics indicates that the change of the pressure, P, is directly proportional to the change of the depth, z, or:

g gc

dP dz

(11.4)

For constant density, the above equation may be integrated to give the hydrostatic equation:

P2

P1

gh gc

(11.13)

where h = z1 – z2. Here Location 2 is located at a distance h below Location 1.

Law of Hydrostatics 93 Manometers, typically U-shaped tubes containing various liquids, are often used to measure pressure differences. This is accomplished by a direct application of the above equation. Pressure differences acting on the two arms of the manometers causes the liquid to reach different heights in the two arms. These height differences can be used to compute pressure differences by systematically applying the above equation to each leg of the manometer. Care should be exercised when providing pressure values in gauge and absolute pressure. The key equation is:

P(gauge) = P(absolute)

P(ambient); all with consistent units (11.14)

The subject of manometry is revisited in the next chapter.

References 1. Reynolds, J., Jeris, J., and Theodore, L., Handbook of Chemical and Environmental Engineering Calculations, John Wiley & Sons, Hoboken, NJ, 2004. 2. Farag, I., Fluid Flow, A Theodore Tutorial, Theodore Tutorials, East Williston, NY. Originally published by the U.S. EPA APTI, RTP, NC, 1996.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

12 Flow Measurement

12.1 Introduction Measurement of a flowing fluid can be difficult since it requires that the mass or volume of material be quantified as it moves through a pipe or conduit. Problems may arise due to the complexity of the dynamics of flow. Further, flow measurements draw on a host of physical parameters that are also often difficult to quantify. This chapter serves to review standard industrial methods that are employed to measure fluid flow rates. Information provided can include the velocity or the amount of fluid that passes through a given cross-section of a pipe or conduit per unit time. Local velocity variations across the cross-section or short-time fluctuations (e.g., turbulence) are not considered. These concerns can be important, particularly the former. For example, in air pollution applications, it is often necessary to traverse a stack to obtain local velocity variations with position. Hydrodynamic methods are primarily used by industry in the measuring the flow of fluids. These methods include the use of the following equipment: 1. Pitot tubes 2. Venturi meters 3. Orifice meters 95

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Other approaches include weighing, direct displacement, and dilution. Weighing involves mass or gravitational approaches which, as one might suppose, cannot be used for gases. Direct displacement can be applied to liquids and is based on a displacement of either a moving part of the unit or the moving fluid. Dilution methods involve adding a second fluid of a known rate to the stream of fluid to be measured and determining the concentration of this second fluid at some point downstream of the second fluids injection point. For accurate gas measurements, there is the vane anemometer that is in effect a windmill consisting of a number of light blades mounted on radial arms attached to a common spindle rotating in two jeweled bearings. When placed parallel to a moving gas stream, the forces on the blades cause the spindle to rotate at a rate depending mainly on the gas velocity. An extension of this unit is the hot-wire anemometer that essentially consists of a fine, electrically-heated wire exposed to the gas stream in which the velocity is being measured. The velocity of the gas determines the cooling effect upon the wire, which in turn affects the electrical resistance. Finally, the rotameter is the most widely used form of area meter that is essentially a vertical tapered glass tube inserted into a pipe by means of special end connections and containing a float that moves up and down as the flow increases or decreases past the float. Graduations are etched onto the side of the rotameter tube to indicate the rate of flow. This chapter will primarily key on the three hydrodynamic methods listed above, introducing the subject with the general topic of manometry and pressure measurement. Also note that the bulk of material in this chapter is from the work of Farag [1].

12.2 Manometry and Pressure Measurement Pressure is usually measured by allowing it to act across some area and opposing it with some type of force (e.g., gravity, compressed spring, electrical, and so on). If the force is gravity, the device is usually a manometer. A very common device to measure pressure is the Bourdon-tube pressure gauge. It is a reliable and inexpensive direct displacement device. It is made of a stiff metal tube bent in a circular shape. One end is fixed and the other is free to deflect when pressurized. This deflection is measured by a linkage attached to a calibrated dial (see Figure 12.1). Bourdon gauges are available with an accuracy of ± 0.1% of the full scale. Other pressure gauges can measure pressure by the displacement of the sensing element electrically via changes in capacitance, resistance, or inductance. However, the interest in this section is primarily with the manometer. Consider the open manometer shown in Figure 12.2. P1 is unknown and Pa is the known atmospheric pressure. The heights za, z1, and z2 are also known. Applying Bernoulli’s hydrostatic equation at Points 1 and 2, and again at Points a and 2 yields:

P1 P2

1

g ( z1 z 2 ) gc

(12.1)

Flow Measurement

Bourdon tube

Sc ale

Pointer

Linkage

Pressure Figure 12.1 Bourdon-tube pressure gauge.

a

1

P1

Pa

za (atm)

z1

P2

2

Figure 12.2 Open manometer pressure gauge.

z2

97

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Gauge pressure

Compressed air

Dip tube

Vent

Air

h=?

Liquid

Figure 12.3 Common depth measurement method for liquids in tanks.

and

P2 Pa

2

g (z 2 z a ) gc

(12.2)

If these two equations are added, one obtains:

P1 Pa

1

g ( z1 z 2 ) gc

2

g (z 2 z a ) gc

(12.3)

The reader should refer to Chapter 11 for more details on manometry. Liquid depths in tanks are commonly measured by the scheme shown in Figure 12.3, a modified form of manometry. Compressed air (or nitrogen) bubbles slowly through a dip tube in the liquid. The flow of the air is so slow that it may be considered static. The tank is vented to the atmosphere. The gauge pressure reading at the top of the dip tube is then primarily due to the liquid depth in the tank.

12.3

Pitot Tubes

Bernoulli’s equation provides the basis to analyze some devices for fluid flow measurement. A common device is the Pitot tube shown in Figure 12.4. It essentially consists of one tube with an opening normal to the direction of flow and a second tube in which the opening is parallel to the flow. It measures both the

Flow Measurement Static pressure

99

2

v

Stagnation pressure 1 5

6

h

3

4

M

Figure 12.4 Pitot tube employed for velocity measurements in pipes.

static pressure (through the side holes at Location 2) and the stagnation, or impact, pressure (through the hole in the front at Location 1). Applying Bernoulli’s equation between Locations 1 and 2, one obtains (after neglecting frictional effects):

P1

v12 2gc

g z1 gc

P2

v22 2gc

g z2 gc

(12.4)

Since v1 = 0 (stagnation); z1 = z2 (horizontal); and v2 = v = fluid velocity, one may rewrite Equation 12.4 as:

v

2(P1 P2 ) g c

(12.5)

This is the Pitot tube formula. The pressure difference (P1 – P2) is often measured by connecting the ends of the Pitot tube to a manometer. The manometer

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liquid (density M ) develops a differential height, h, due to the flowing fluid. Applying Bernoulli’s equation at the manometer as presented in Figure 12.4 yields:

P3

P4

(12.6)

and

P5

g h gc

P5 P6

P1 P2

P6

M

g h gc

(12.7)

In addition

g h( gc

M

)

(12.8)

Equation 12.8 is a modified form of Equation 12.3. Substituting Equation 12.8 into the Pitot tube formula, Equation l2.5 yields:

v

2 gh(

m

)

(12.9)

This equation has also been written as:

v C

2 gh(

m

)

(12.10)

The term C is included to account for the assumption of negligible frictional effects. However, for most Pitot tubes, C is approximately unity, and Equation 12.9 applies. Note that a Pitot tube measures the local velocity at only one point. To obtain the average velocity over the cross-section, it is necessary to measure the velocity at a number of specific locations in the cross-section of the pipe. Also note that when the Pitot tube is used for measuring low-pressure gases, the pressure difference reading is usually extremely small, and can lead to large errors in flow velocity estimates.

12.4

Venturi Meters

The Venturi meter is also a device for measuring a fluid flow rate. As shown in Figure 12.5, it consists of three sections: a converging section to accelerate the flow, a short cylindrical section (called the throat), and a diverging section to

Flow Measurement 2

101

(Throat)

1 P1 ,

P2 ,

L

6 5 h

Μ

3

4

Figure 12.5 Typical Venturi flow meter.

increase the cross-sectional area to its original (upstream) value. There is a change in pressure between the upstream measurement location (Point 1) and the throat (Point 2). This pressure difference is measured (often with a manometer) to determine the fluid flow rate. The Venturi meter can determine the volumetric flow rate from either the pressure difference (P1 P2) or the manometer head, h. The development of pertinent equations is presented below. Referring to Figure 12.5 for point locations, from the conservation law of mass m1 m2 at steady state. If the fluid is assumed incompressible, then: 1

=

=

(12.11)

q1 = q2

(12.12)

2

so that

D12 v1 4

D22 v2 4

(12.13)

and

v1

D22 v2 D12

(12.14)

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Applying Bernoulli’s equation between Points 1 and 2, and assuming no frictional losses yields Equation 12.4:

P1

v12 2gc

g z1 gc

v22 2gc

P2

g z2 gc

(12.4)

For a horizontal Venturi meter, z1 = z2. Therefore, the above equation simplifies to:

v12 2gc

P1

v22 2gc

P2

(12.15)

Rearranging Equation 12.15 and substituting for v1 from Equation 12.14 leads to:

2 g c (P1 P2 ) [1 (D24 /D14 )]

v2

(12.16)

Substituting the manometer equation, Equation 12.8, for (P1 Equation 12.16 yields:

2 gh( M ) 4 4 [1 (D2 /D1 )]

v2

where once again M is the manometer fluid density, and density. By definition, the volumetric flow rate, q, is:

q

P2) in

(12.17) is the flowing fluid

D22 v2 4

(12.18)

Equation 12.17 is often referred to as the Venturi formula. It applies to frictionless flow. To account for the small friction loss between Points 1 and 2, a Venturi discharge coefficient, Cv, is introduced in Equations 12.16 and 12.17, that is:

v2

Cv

2 g c (P1 P2 ) [1 (D24 /D14 )]

Cv

) 2 gh( M 4 4 [1 (D2 /D1 )]

(12.19)

For well-designed Venturi meters, Cv, is approximately 0.96.

12.5

Orifice Meters

Another device used for flow measurement is the orifice meter depicted in Figure 12.6. The pressure difference is measured (often with a manometer) between

Flow Measurement

103

Orifice plate

Flow 2

1

L

Circular drilled hole

h 4

3

Front view of orifice plate Figure 12.6 Typical orifice meter.

an upstream location (Point 1) and the orifice (Point 2). Although it operates on the same principles as a Venturi meter, orifice plates can be easily changed to accommodate a wide range of flow rates. The orifice can be employed to determine either the volumetric flow rate from the pressure difference, (P1 – P2), or from the manometer head, h. For a horizontal orifice meter, the velocity equation is the same as the Venturi meter, i.e.,

v2

Co

2 g c (P1 P2 ) [1 (D24 /D14 )]

Co

2 gh( M ) 4 4 [1 (D2 /D1 )]

(12.20)

The volumetric flow rate for an orifice meter is given by Equation 12.18. The discharge coefficient, Co, for drilled-plate orifices is shown in Figure 12.7, where Co is a function of D2/D1 and the Reynolds number at the throat, Re2. At Re2 values greater than 20,000, the discharge coefficient, Co, is approximately constant at 0.61–0.62.

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Unit Operations in Environmental Engineering

The orifice meter is simpler in construction and less expensive than a Venturi meter, and occupies less space. However, it has a lower pressure recovery (around 70%). Another way to measure the fluid flow rate in a pipe analogous to a pitot tube is to insert a long constriction (of smaller diameter than the pipe) inside the pipe and measure the pressure drop (or head loss) across the constriction. The flow goes through a sudden contraction when it enters the constriction and through a sudden expansion as it exits the constriction. “Major” loss due to friction may be neglected. Only “minor” losses due to sudden expansion and sudden contraction are considered in the calculation of the flow rate. This form of flow meter consists of any restricted opening or tube through which the rate of flow has been determined by calibration. For example, a 6-inch pipe may be tapered down to 2 inches and then enlarged back to 6 inches. The pressure drop through this “opening” provides a measure of the rate of flow, but this relation should be determined by calibration for a specific configuration and pipe material.

12.6

Flow Meter Selection

Several factors should be considered in selecting a flow measurement device. The responsible engineer should consider the following when selecting a flow measurement device for a specific application: 1. Is the fluid phase to be measured a gas or liquid? 2. What is the range (or capacity) of the device and the expected flow rate to be measured? 3. What is the accuracy of the device and that required for the specific application? 4. What is the desired readout format? 5. What are the fluid properties? 6. What is the internal environment in the location of the measurement? 7. What is the external environment in the location of the measurement? 8. What are the capital costs for the device? 9. What are the operating costs for the device? 10. What is the reliability of the device? Considering the complexity and diversity of flow meter measuring devices and the wide range of flow conditions encountered in industrial applications, one should carefully compare the different options that are available before purchasing a device. It should be noted that it is possible that a number of devices may be suitable for a given application, and the final decision on a particular device may hinge on one’s experience with the vendor and/or previous experience with the devices themselves in other similar or related applications.

Flow Measurement 1

102

10

103

D2 = 0.80 D1

1.00 0.95

D2 = 0.70 D1 D2 = 0.65 D1 D2 = 0.60 D1

D2 = 0.75 D1

0.90 0.85 0.80 0.75 C0

104

105

0.95 0.90 0.85 0.80 0.75

0.70

0.70

0.65

0.65

0.60

D2 D2 = 0.10 = 0.20 D1 D1 D2 = 0.30 D1

0.55 D2 = 0.50 D1

0.50 0.45

105

0.60

0.40 0.35 0.30 0.25 0.20 0.15

1

10

102

D2

103 2

104

105

/

Figure 12.7 Discharge coefficients for drilled plate orifices. D2 is the orifice diameter, and D1 is the pipe diameter. The abscissa is the Reynolds number based on the orifice conditions.

Reference 1. Farag, I., Fluid Flow, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by the U.S. EPA APTI, RTP, NC, 1996.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

13 Flow Classification

13.1 Introduction When fluids move through a closed conduit of any cross-section, one of two different types of flow may occur. These two flow types are most easily visualized by referring to a classic experiment first performed by Osborne Reynolds in 1883. In Reynolds’ experiment, a glass lube was connected to a reservoir of water in such a way that the velocity of the water flowing through the tube could be varied. A nozzle was inserted in the inlet end of the tube through which a fine stream of colored dye could be introduced. Reynolds found that when the velocity of the water was low, the “thread” of dye color maintained itself throughout the tube. By locating the nozzle at different points in the cross-section it was shown that there was no mixing of the dye with water and that the dye flowed in parallel, straight lines. At high velocities, it was found that the “line” or “thread” of dye disappeared and the entire mass of flowing water was uniformly colored with the dye. In other words, the liquid, instead of flowing in an orderly manner parallel to the long axis of the tube, was now flowing in an erratic manner and so there was complete mixing.

107

108

Unit Operations in Environmental Engineering

These two forms of fluid motion are known as laminar or viscous flow that occurs at low velocity, and turbulent flow that occurs at high velocity. The velocity at which the flow changes from laminar to turbulent is defined as the critical velocity [1].

13.2 The Reynolds Number Reynolds, in a later study of the conditions under which the two types of flow might occur, showed that the critical velocity depended on the diameter of the tube, the velocity of the fluid, its density, and its viscosity. Further, Reynolds showed that the term representing these four quantities could be combined in a manner that later came to be defined as the Reynolds number. The Reynolds number, Re, is a dimensionless quantity, and can be shown to be the ratio of inertia to viscous forces in the fluid:

Re

L v

Lv v

(13.1)

where L is a characteristic length, is the fluid density, v is the average velocity, μ is the dynamic (or absolute) viscosity, and v is the kinematic viscosity. In flow through round pipes and tubes, L is the diameter of the pipe or tube, D. The Reynolds number provides information on flow behavior, and Reynolds number calculations become important and are necessary for determining flow regimes, i.e., laminar versus turbulent. It is particularly useful in scaling up benchscale or pilot data to full-scale applications. Laminar flow is always encountered at a Reynolds number, Re < 2100 in a circular duct, but it can persist up to higher Reynolds numbers in very smooth pipes. However, the flow is unstable at higher Reynolds numbers, and small disturbances may cause a transition to turbulent flow. Very slow flow (in circular ducts) for which Re < 1 is termed creeping or Stokes flow. Under ordinary conditions of flow (in circular ducts), the flow is turbulent at a Reynolds number above 4000. A transition region is observed between 2100 and 4000, where the type of flow may be either laminar or turbulent, and predictions are unreliable. The Reynolds numbers at which the fluid flow changes from laminar to transition or to turbulent flow are termed critical Reynolds numbers. In other than circular geometries, different critical Re criteria exist. For most environmental engineering applications, one can assume turbulent (high Reynolds number) flow for gases. The reader should also note that the Reynolds number appears in many semi-empirical and empirical equations that involve fluid flow, heat transfer, and mass transfer applications. For flow in noncircular conduits, some other appropriate length (termed the hydraulic radius) replaces the diameter in Re. This is discussed later in this chapter. As noted in Chapter 10, applying the conservation law of energy mandates that all forms of energy entering a system equal that of those leaving. Expressing all

Flow Classification 109 terms in consistent units, e.g., energy per unit mass of fluid flowing, resulted in the total energy balance equation as shown below:

P1

v12 2gc

P2

g z1 E1 Q Ws gc

v22 2gc

g z2 gc

E2 .

(10.8)

An important point needs to be made before leaving this subject. By definition, the kinetic energy of a small parcel of fluid with local velocity, v, is v2/2gc (ft-lbf/ lb). If the local velocities at all points in the cross-section were uniform, vav would be equal to v, and the kinetic energy term can be retained as written. Ordinarily there is a velocity gradient across a pipe or conduit, and this introduces an error in the kinetic energy term, the magnitude of which depends on the nature of the velocity profile and the shape of the cross-section. For the usual case where the velocity is approximately uniform (i.e., turbulent flow), the error is not serious, and since the error tends to cancel because of the appearance of kinetic terms on each side of any energy balance equation, it is customary to ignore the effect of velocity gradients. When the error cannot be ignored, the introduction of a correction factor that is used to multiply the v2/2gc term is needed. The term , called the kinetic energy correction factor, is employed, where is defined as:

v 2 dS

(13.2)

s

3 vav S

where S is the cross-sectional area of a pipe or duct. For most environmental engineering applications, flow is turbulent and may be assumed to be 1.0. Where the velocity distribution is parabolic, as in laminar flow, it can be shown that the exact value of is 2.0. For transition state flow, 1 < < 2 [l].

13.3 Laminar Flow in Pipes The following equation was derived in Chapter 10 as Equation 10.12:

P

v2 2gc

g z gc

Ws

F

0

(10.12)

This was defined as the mechanical energy equation. The above equation was later rewritten without the work and friction terms as Equation 10.13,

P

v2 2gc

g z gc

0

and was defined as the basic form of the Bernoulli equation.

(10.13)

110

Unit Operations in Environmental Engineering

In applying the Bernoulli equation to a prime mover (e.g., a centrifugal pump application) where both work and frictional terms are included, Equation 10.12 can be written as:

P

v2 2gc

g z hs h f gc

0

(13.3)

where hs and hf have effectively replaced Ws and F, as the work energy and total frictional loss, respectively. The units of both hs and hf are ft-lbf/lb. If the equation is expanded, rearranged, and multiplied by gc/g, each term in the resulting equation has units of ft, the common form of the equation used in environmental engineering.

P1 g c g

v12 2g

z1

P2 g c g

v22 2g

z 2 hs

gc g

hf

gc g

(13.4)

The reader should note that in the process of converting Equation 13.3 to 13.4, the Δ term (representing a difference) applies to outlet minus inlet conditions (i.e., P2 – P1, etc.). One can now examine Equation 13.3 at the following extreme or limiting conditions. 1. If only the first (ΔP/ ) and last term (hf) are present, an increase in the latter’s frictional losses would result in a corresponding decrease in the original pressure term P2. Thus, a large pressure drop results, which is to be expected. 2. If only the first (ΔP/ ) and fourth term (hs) are present, an increase in the latter’s input mechanical (shaft) work term would result in a corresponding increase in the output pressure term P2. Thus, a smaller pressure drop results, and again, this is in agreement with what one would expect. 3. If only the latter two terms are present, hs = hf, and this too agrees with one’s expectation since both terms are positive and any frictional effect is compensated by the mechanical (shaft) work introduced to the system. Care should be exercised in the interpretation of the term P. Although the notation Δ represents difference, P can be used to describe the difference between the inlet minus the outlet pressure (i.e., P1 – P2), or it can describe the difference between the outlet minus the inlet pressure (i.e., P2 – P1). When a fluid is flowing in the 1 2 direction, the term P1 – P2 is positive and represents a decrease in pressure that is defined as the pressure drop. The term P2 – P1, however, also represents a pressure change whose difference is negative and is also defined as a pressure drop. One’s wording and interpretation of this pressure change is obviously a choice that is left to the user.

Flow Classification 111 13.3.1

Frictional Losses

As indicated above, the hf term was included to represent the loss of energy due to friction in the system. These frictional losses can take several forms. An important engineering problem is the calculation of these losses. It is known that fluid can flow in either of two modes - laminar or turbulent. For laminar flow, an equation is available from basic theory to calculate friction loss in a pipe. In practice, however, fluids (particularly gases) are rarely moving via laminar flow. Since these two methods of flow are so widely different, a different equation describing frictional resistance is to be expected in the case of turbulent flow from that which applies in the case of laminar flow. On the other hand, it will be shown in the next section that both cases may be handled by one relationship in such a way that it is not necessary to make a preliminary calculation to determine whether the flow is taking place above the critical Reynolds number or below it [1]. One can theoretically derive the hf term for laminar flow [2, 3]. The equation takes the form

32 vL g c D2

hf

(13.5)

for a fluid flowing through a straight cylinder of diameter D and length L. A friction factor, f, termed the Fanning friction factor, that is dimensionless, may now be defined for laminar flow as:

f

16 Re

(13.6)

Equation 13.6 and the definition of Re from Equation 13.1can be substituted into Equation 13.5, realizing that for a circular pipe, L in Equation 13.1 becomes the pipe diameter, D, to yield the following expression for fictional loss during laminar flow:

hf

4 fLv 2 2gcD

(13.7)

Although the above equation describes friction loss or the pressure drop across a conduit of length L, it can also be used to provide the pressure drop due to friction per unit length of conduit, for example, P/L, by simply dividing the above equation by L. It should also be noted that another friction factor term exists that differs from that of Equation 13.6. In this other case, fD is the Darcy or Moody friction factor and is defined as:

fD

64 Re

4f

(13.8)

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Unit Operations in Environmental Engineering

Care should be taken as to which of the friction factors are being used and this will become more apparent in the next section. In general, chemical engineers employ the Fanning friction factor; while it is by convention that environmental engineers employ the Darcy (or Moody) friction factor. This book employs the Fanning friction factor. Manning’s equation to be discussed shortly, normally employs fD. With reference to Equation 13.6 or 13.8, one should note that this is an equation of a straight line with a slope of -1 if f is plotted versus Re on log-log scales. Also note (once again) that the equation for f applies only to laminar flow, i.e., when Re is 200,000 acfm). However, any sticky material and particulates, in general, can accumulate in the slight curvature of the blades and cause imbalance. Efficiencies here typically range from 52 to 74%, with 70% being common. Generally, centrifugal fans are easier to control, more robust in construction, and less noisy than axial units. They have a broader operating range at their highest efficiencies. Centrifugal fans are better suited for operations in which there are flow variations and they can handle dust and fumes better than axial fans. Fan laws are equations that enable the results of a fan test (or operation) at one set of conditions to be used to calculate the performance of the fan at another set of conditions, including differently sized but geometrically similar models of the same fan design. The fan laws can be written in many different ways. The three key laws are provided in the following equations [1]:

qa=kl (rpm) D3

(14.1)

Ps = k2 (rpm)2 D2

(14.2)

hp = k3 (rpm)3 D5

(14.3)

where qa = volumetric flow rate,; kl, k2, k3 = proportionality constants; rpm = revolutions per minute; D = wheel diameter; Ps = static pressure; = gas density; and hp = horsepower. Thus, these three laws may be used to determine the effect of fan speed, fan size, and gas density on flow rate, developed static pressure head, and horsepower. For two conditions where the constants k remain unchanged, Equations 14.1, 14.2, and 14.3 become:

qa qa

rpm rpm

D D

3

(14.4)

Prime Movers

Ps Ps

rpm rpm

hp hp

rpm rpm

2

D D

3

D D

125

2

(14.5)

5

(14.6)

where the prime refers to the new condition. It is also important to note that these fan laws are approximations and should not be used over wide ranges or changes of flow rate, size, etc. It is common practice among fan vendors to publish voluminous data in tabular form providing flow rate, static pressure, fan speed, and horsepower at a standard temperature and gas density. These are often referred to as multi-rating tables. Note: These tables should not be used for fan selection except by those who have experience in this area. For those who do not, the proper course of action to follow is to provide the fan manufacturer with a complete description of the system and allow the manufacturer to select and guarantee the optimum fan choice. To help in the actual selection of fan size, a typical fan rating table is given in Table 14.1. The fan size and dimensions are usually listed at the top of the table. Values of static pressure are arranged as columns that contain the fan speed and Table 14.1 Typical fan multi-rating table. Static pressure (in H2O) ½

1

1½

2

qa (acfm)

rpm

bph

rpm

bph

rpm

bph

rpm

bph

5757

216

0.74

278

1.34

330

2.01

378

2.75

6873

236

0.97

291

1.67

339

2.42

384

3.24

8018

250

1.27

305

2.05

352

2.87

393

3.76

9164

271

1.63

320

2.49

366

3.42

405

4.35

10,309

293

2.12

338

3.05

381

4.06

419

5.06

11,455

315

2.72

356

3.65

396

4.76

432

5.88

12,600

337

3.46

377

4.39

413

5.58

448

6.81

13,746

360

4.39

399

5.25

430

6.48

465

7.82

14,891

382

5.43

421

6.25

451

7.52

481

8.93

16,037

405

6.66

442

7.48

473

8.67

501

10.16

17,182

429

8.08

463

8.97

496

10.05

521

11.54

18,328

451

9.64

486

10.61

517

11.61

543

12.99

Source: Bayler Blower Co. Wheel style – backward incline; wheel diameter – 50.5 in; maximum fan speed – 1134 rpm; performance underlined are those at maximum efficiency; bhp = brake horsepower.

126

Unit Operations in Environmental Engineering

Static pressure

brake horsepower required to produce various volumetric flows. The point of maximum efficiency at each static pressure is usually underlined or printed in special type. In order to select a fan for the exact condition desired, it is sometimes necessary to interpolate between values presented in the multi-rating tables. Straight-line interpolation can be used with negligible error for multi-rating tables based on a single fan size. Some multirating tables attempt to show ratings for a whole series of geometrically similar (homologous) fans in one table. In this case, interpolation is not advised [1]. The selection procedure is, in part, an examination of the fan curve and the system curve. A fan curve, relating static pressure to flow rate, is shown in Figure 14.1. Note that each type of fan has its own characteristic curve. Also note that fans are usually tested in the factory or laboratory with open inlets and long, smooth, straight discharge ducts. Since these conditions are seldom duplicated in the field, actual operation often results in lower efficiency and reduced performance than predicted from a fan curve. A system curve is also shown in Figure 14.1. This curve is calculated prior to the purchase of a fan and provides a best estimate of the pressure drop across the system through which the fan must deliver the gas. (This curve should approach a straight line with an approximate slope of 1.8 on a loglog scale). The system pressure (drop) is defined as the resistance through ducts, fittings, equipment, contractions, expansions, etc. A number of methods are available to estimate the total system pressure change. These vary from very crude approximations to detailed, rigorous calculations. The simplest procedure is to obtain estimates of the pressure change associated with the movement of the gas through all of the resistances described previously. The sum of these pressure changes represents the total pressure drop across the system, and represents the total pressure (change) that must be developed by the fan. This calculation becomes more complex if branches (i.e., combining flows) are involved. However, the same stepwise procedure should be employed.

Operating point System curve

Volumetric flow rate Figure 14.1 System and fan characteristic curves.

Fan curve

Prime Movers

127

When the two curves are superimposed, the intersection is defined as the operating point. The fan should be selected so that it operates just to the right of the peak on the fan curve. The fan operates most efficiently and with maximum stability at this condition. If a fan is selected for operation too close to its peak, it will surge and oscillate. Thus, the point of intersection of the two curves determines the actual volumetric flow rate. If the system resistance has been accurately specified, and the fan properly selected, the two performance curves will intersect at the design flow rate. If system pressure losses have not been accurately specified or if undesirable inlet and outlet conditions exist, design conditions will not be obtained. Dampers and fan speed changes can provide some variability on operating conditions. There are a number of process and equipment variables that are classified as part of a fan specification. These include [2]: Flow rate (acfm) Temperature Density Gas stream characteristics Static pressure that needs to be developed Motor type Drive type Materials of construction Fan location Noise controls (briefly discussed earlier) With respect to drive type, belt drives are usually employed if the power is 1000. The effect of media roughness is less significant than the other variables but may become more important in the highly turbulent region. Generally, for flow in the laminar and early turbulent regions, roughness has little effect on pressure drop and should not be included in most correlations for porous media [3]. Orientation is an important variable in special cases. However, variations in orientation do not occur with random packing as encountered in most industrial practices. Oriented beds have been occasionally used in some absorbers and for other specialized applications where the packing is stacked by hand rather than dumped into the vessel; however, real world applications of this case are rare [3].

18.4 Applications of Porous Media and Packed Beds In the application of porous media and packed beds in environmental engineering, particles being removed can vary in size and shape from relatively large cylinders and rings to small and almost spherical granules. The ratio of the cross-section area of the bed to the projected area of the particles will range from the order of 10 to several thousands. The direction of flow may be either up or down. Such beds are used for a variety of purposes in environmental engineering. Some of the more common units employed in practice are discussed below. As indicated above, the most important application of porous media and packed beds in environmental engineering is the rapid sand filtration of potable water, where water is passed through beds of size-stratified sand and gravel to remove suspended matter. As solids accumulate in these rapid sand filters, head loss increases through them. This requires periodic reversal of the flow to the filters and expansion of the beds to dislodge particulates and rejuvenate the beds via backwashing. Once the beds are cleaned, the backwash flow is stopped, the bed re-stratifies, and downward filtration recommences. The application of slow sand filters is now confined to the treatment of potable water for very small, rural communities, or in applications of point of use filters in remote applications or in developing countries [4]. Raw water is passed through beds of unstratified sand and gravel, and the filter is designed to remain saturated at all times and is not backwashed. Suspended material is trapped at the surface where any organic fraction is decomposed in a biological layer that develops on the filter over time. This biological layer or schmutzdecke is also responsible for removal of pathogens in the raw water. The accumulation of non-decomposable solids, and continuous production of biomass in the schmutzdecke over time necessitates periodic cleaning by scraping off the top layers of sand. Potable water can be demineralized when passed through granular beds of ion exchange materials which are regenerated periodically with strong acid or based regenerant solutions depending on the specific demineralization application. As will be discussed in Chapter 50, the removal of odors and organic vapors from air can be accomplished by forcing the air through towers filled with activated

178

Unit Operations in Environmental Engineering

carbon or similar adsorbents. Towers containing silica gel and activated alumina are also effective in removing water vapor from air (dehumidification). Columns filed with various packing materials are widely used as liquid- and gas-contacting devices (see also Chapters 48 and 49).

18.5 The Carmen-Kozeny Equation In the development of an expression for computing the pressure drag across a bed of solids, it is convenient to visualize the bed as consisting of many parallel channels formed by the voids between the particles. These channels are irregular and indeterminate as far as the geometric configuration of the cross-sections are concerned. Measurements must be taken from actual flow experiments for a reliable evaluation of the shape factor for granular materials. For many types of particles, the porosity or void content will vary depending upon the manner and rate with which the material is introduced to the bed. The usual method of determining porosity is based on an immersion procedure in liquid. When the particles themselves are highly porous, porosity determination becomes complex. When, in the case of circular beds, the bed diameter is small relative to the diameter of the particles, there exists next to the wall a region in which the porosity is greater than that in the middle of the bed. With hollow packing, the orientation of pieces near the wall contributes heavily toward channeling effects. The wall effect becomes significant when the ratio of the bed diameter to particle diameter is less than 20. However, no correction is required if the actual porosity of the bed in question is used. In most instances where compressible fluids flow through porous beds, the frictional pressure drop across the bed is small compared to the upstream pressure. In these cases, the fluid density may be considered as being a constant through the bed. When the pressure drop due to friction is substantial, however, the effect of decreasing density should be taken into account. Equations applicable to the solution of such problems are available. The equation for the pressure drop for flow through porous media and packed beds is referred to as the Carman-Kozeny equation. It takes the same general form as Fanning’s equation provided in Chapter 13, and as with the Fanning equation, the Carman-Kozeny equation is applicable only for laminar flow. The CarmanKozeny equation can be written as follows [5], and allows the determination of pressure drop, Δp, through a porous media of total bed height, L, when various characteristics of the packed media, i.e., diameter, particle shape, and the fluid are known.

p L

180

f 2 s

)2

(1

d p ,e 2

3

vs

(18.14)

Porous Media and Packed Beds 179 where

s

is sphericity of the particles, described by Equation 18.15 as: 1 3 s

2

(6Vp ) 3

(18.15)

ap

References 1. Bennett, C., and Myers, J., Momentum, Heat, and Mass Transfer, 3rd Edition, McGraw-Hill, New York, NY, 1982. 2. Ergun, S., Fluid Flow Through Packed Columns, CEP, New York, 48, 89, 1952. 3. Noyes, R., Unit Operations in Environmental Engineering, Noyes Publications, Park Ridge, NJ, 1994. 4. Centre for Affordable Water and Sanitation Technology, Biosand Filter Manual, Design, Construction, Installation, Operation and Maintenance, CAWST Training Manual, May 2010 Edition, Calgary, Canada, 2010. 5. McCabe, W.L., Smith, J.C., and Harriot, P., Unit Operations of Chemical Engineering, 7th Edition, McGraw-Hill, New York, NY, 2005.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

19 Filtration

19.1 Introduction Filtration is one of the most common environmental applications of the flow of fluids through packed beds. Carried out industrially, it is similar to filtration in a chemical laboratory using a filter paper in a funnel. The object is the separation of a solid from the fluid in which it is carried, and the separation is accomplished by forcing the fluid through a porous filter material. The solids are trapped within the pores of the filter and build up as a layer on the surface of this filter. The fluid, which may be either gas or liquid, passes through the bed of solids and through the retaining filter. Filtration may therefore be viewed as an operation in which a heterogeneous mixture of a fluid and solid particles are separated by a filter medium that permits the flow of the fluid but retains the particles of the solid. Therefore, it primarily involves the flow of fluids through porous media (see Chapter 18). Vacuum filtration finds wide application in the partial separation of liquids from concentrated suspensions, sludges, and slurries. When the liquid phase is highly viscous, or when the solids are so fine that vacuum filtration is too slow, pressure filtration provides a convenient solution to the separation problem. Centrifugal filtration is used when the solids are easy to filter and a filter cake of low moisture content is desired. 181

182

Unit Operations in Environmental Engineering

In all filtration processes, the mixture, or slurry, flows as a result of some driving force, e.g., gravity, pressure (or vacuum), or centrifugal force. In each case, the filter medium supports the particles as they form a porous cake. This cake, supported by the filter medium, retains the solid particles in the slurry with successive layers added to the cake as the filtrate passes through the cake and medium. The several methods for creating the driving force on the fluid, the different methods of cake deposition and removal, and the different means for removal of the filtrate from the cake subsequent to its formation, result in a great variety of filter equipment. In general, filters may be classified according to the nature of the driving force supporting filtration. The various equipment options are described in the material that follows.

19.2 Filtration Equipment There are various types of filtration equipment employed by environmental engineers. Included in this list are gravity filters, plate-and-frame filters, leaf filters, and rotary vacuum filters. Gravity filters are the oldest and simplest type. These filters consist of tanks with perforated bottoms filled with porous sand through which a solid suspension passes. Plate-and-frame filters are very commonly used, and consist of porous plates that are held rigidly together in a frame. Leaf filters are similar to the plate-and-frame filters in that a cake is deposited on each side of the leaf and the filtrate flows to the outlet in the channels provided by the coarse drainage screen in the leaf between the cakes. The leaves are immersed in the slurry. Rotary vacuum filters are used where a continuous operation is desirable, particularly for large-scale municipal solids dewatering operations. The filter drum is immersed in a slurry where a vacuum is applied to the filter medium that causes the cake to deposit on the outer surface of the drum as it rotates through the slurry. Plate-and-frame filter presses are perhaps the most widely used type of filtering devices in the chemical industry. Plate-and-frame filter presses are used in a variety of industries. The chemical industry uses the filter press in order to separate the solid portion of a chemical slurry from the liquid. A chemical, for example zinc, builds up on the frames. The filter press is then opened and the wet cake, containing solid zinc, can be collected, removed and dried. The pharmaceutical industry also uses filter presses in similar applications to concentrate manufactured products from process slurries. The solids collected on the inside of the frames is further dried and sold as pharmaceutical product. The sugar industry also employs filter presses to separate solid sugar from a sugar solution. This solid cake is further dried, crystallized, and sold as sugar. In the pottery industry, filter presses are used to make ceramic pieces. A slurry is sent though a filter press and the solid cake is collected. This cake is then used for the production of various pottery products. Finally, the filter press is also used widely in the wastewater treatment industry for solids dewatering prior to disposal. Once a solid reaches 20% solids or greater, it can be handled as a solid material, and can be disposed of via landfilling, land

Filtration 183 application, or incineration. The liquid separated from wastewater sludges are recycled back to the front of wastewater treatment plants for further treatment before it is discharged from the treatment plant. Regarding the filter press, feed slurry is pumped to the unit under pressure and flows in the press and into the bottom-corner duct of the frame (Figure 19.1). This duct has outlets into each of the frames, so the slurry fills the frames in parallel. The plates and frames are assembled alternately with filter cloths over each side of each plate. The assembly is held together as a unit by mechanical force applied hydraulically or by a screw. The liquid filtrate then flows through the filter media while the solids build up in a layer on the inside of the frame side of the media. The filtrate flows between the filter cloth and the face of the plate to an outlet duct. As filtration proceeds, a cake builds up on the filter cloth until the cake being formed on each face of the frame meet in the center. When this happens, the flow of filtrate, which has been decreasing continuously as the cakes build up, drops off abruptly to a trickle. Filtration is usually stopped well before this occurs [1]. The process of slurry flow through the press can be seen in Figure 19.2. The slurry enters the lower right-hand side of Frame 18. When Frame 18 is filled with slurry, the excess slurry is forced through the filler media to the upper righthand side of Plate 17. When the slurry is in Frame 17, cake builds up on the filter media. It then flows into Plate 16. When in Frame 16, the slurry flows down because of gravity. When the frame is filled with slurry, the cake forms and the filtrate is sent to the upper right-hand side of Frame 16 and out through Plate 15. Plate 15 leads to the filtrate collecting drum outside of the press. A filter operation can be carried out using centrifugal force rather than the pressure force used in the equipment described above. Filters using centrifugal force are generally used for coarse granular or crystalline solids and are available primarily for batch operations. Batch centrifugal filters most commonly consist of a basket with perforated sides rotated around a vertical axis. The slurry is fed into the center of the rotating basket and is forced against the basket sides by centrifugal force. There, the liquid passes through the filter medium, which is placed around the inside surface of the

Frame Figure 19.1 Plate-and-frame schematics.

Inlet

Plate

Outlet

184

Unit Operations in Environmental Engineering Plate 17

Frame 18

Inlet Frame 16

Plate 15

Outlet

Figure 19.2 Flow of the slurry through the filter press.

basket, and is caught in a “shielding” vessel, referred to as a curb, within which the basket rotates. The solid phase builds up a filter cake against the filter medium. When this cake is thick enough to retard the filtration to an uneconomical rate or to endanger the balance of the centrifuge, the operation is halted, and the cake is scraped into a bottom discharge or is scooped out of the centrifuge. In an automatically discharging batch centrifugal filter, unloading occurs automatically while the centrifuge is rotating, but the filtration cycle is still operated in batch mode [1]. Another process device that some have classified as a “hybrid” filtration unit is the mist eliminator. This class of separator allows gas (usually air) to pass through the unit while captured liquid droplets are returned to the emitting process equipment. These units are widely used to prevent the emission of undesirable amounts of liquid droplets from scrubbers and absorbers. An unfortunate consequence of intimate and vigorous liquid-gas contacting in the scrubber is that some of the scrubbing liquor is atomized, entrained by the gas that has been cleaned, and discharged from the unit. The droplets carried over by entrainment generally contain both suspended and dissolved solids. In many cases, excessive entrainment imposes a limitation upon the capacity of the scrubber. The describing equations to follow for the traditional filtration equipment bear no resemblance to those for mist eliminators [2].

Filtration 185 Irrespective of the type of equipment employed, cake washing and dewatering operations are usually accomplished with the filter cake in place. Since the cake thickness is now unchanged, the wash rate usually varies directly with the pressure drop across the cake. If the wash water follows the same path as the slurry and is fed at the same pressure, the wash rate will then be approximately equal to the final filtration rate. In the dewatering part of a filtration cycle, fluid is drawn through the filter cake, pulling the filtrate or wash water remaining in the pores of the cake out ahead of it. As discussed earlier, filter media consisting of cloth, paper, or woven or porous metal may be used. The criteria upon which a filter medium is selected must include the ability to remove the solid phase from the medium after filtration, high liquid throughput for a given pressure drop, mechanical strength, and chemical inertness to the slurry to be filtered and to any wash fluids used for final cake washing. Each of these considerations is tempered by the economics involved, so that the design engineer attempts to choose a medium that meets the required filtration requirement while contributing to the lowest possible overall filtration cost. Filter-cake solids usually penetrate the filter medium and fill some of the pores. As filtration continues, these particles are thought to bridge across the pores as the cake begins to form on the face of the medium. As a result, the resistance to flow through the medium increases sharply. In some cases, the solids fill the filter medium to such an extent that the filtration rate is seriously reduced. Of all the various filtration equipment, the plate-and-frame filter press is probably the cheapest per unit of filtering surface and requires the least floor space. However, the cost of labor for opening and dumping such presses is high, particularly in the large sizes. For this reason, they are not chosen when a large quantity of worthless solid is to be removed from the filtrate. If the solids have high value, as in the pharmaceutical industry, the cost of labor per unit value of product is relatively low and the plate-and-frame filter press proves satisfactory. It has a high recovery of solids, and the solid in the form of a cake may be readily handled in a tray or shelf drier, which is frequently used for valuable products. The leaf filter offers the advantages of ease of handling, minimum labor with efficient washing, and the discharge of cake without removing any leaves from the filter. The rotary continuous filter offers the additional advantages of continuous and automatic operation for feeding, filtering, washing, and cake discharge.

19.3 Describing Equations Regarding the filtration process, two important consideration need to be addressed: pressure drop and filtration efficiency. Industry normally relies on certain simple guidelines and calculations to insure satisfactory separation efficiently. Perhaps the most convenient starting point to describe flow through porous media is with the classic Darcy equation:

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Unit Operations in Environmental Engineering

dP dz

vs K

(19.1)

where vsis the superficial velocity in the filter and is the slurry viscosity. This may be integrated (for constant coefficients) to give:

P L

vs K

(19.2)

where L is the filter thickness. The term K is defined as the permeability coefficient for the filtering medium in consistent units. Darcy’s equation may be rearranged and written as:

K dP dz

vs

(19.3)

Multiplying both sides of the equation by the approach (face) area of the filter, S, gives

vs S q

SK dP dz

(19.4)

where q is the volumetric flow rate of the slurry passing S. Integrating yields

z q KS

P

(19.5)

The term in parentheses represents the resistance to flow. It is a constant for a given fabric since z is simply the filter thickness, L. The pressure drop across the fabric medium is then

P

L q KS

(19.6)

The development for the pressure drop across the filter cake is similar to that for the fabric. The bracketed term in Equation 19.5 is not a constant for the deposited particles since z is a variable. If the collection efficiency is close to 100%, which is usually the case in an industrial operation, then

z

Vs c )S p (1

Vs c BS

(19.7)

Filtration 187 where Vs is the volume of slurry filtered for deposit thickness z, c is the inlet solids loading or concentration (mass/volume), is the void volume or void fraction, is the true density of the solid and B is the bulk density of the cake of deposited p solids. Substituting Equation 19.7 into Equation 19.5 yields an equation for the pressure drop across the cake

c Vs q 2 B KS

P

(19.8)

It is assumed that Equations 19.6 and 19.8 are additive (i.e., the solids and fabric do not interact). Thus, the total or overall pressure drop across the filter system is obtained by combining these two equations:

c Vs q 2 B KS

P

L q KS

(19.9)

Setting

B

c 2 B KS

(19.10)

L KS

(19.11)

and

C then

P

(BVs C )q

(19.12)

This represents the general relationship between P, q, and Vs. During a constant (overall) pressure (drop) operation, the flow rate is a function of time

q(t )

dVs dt

(19.13)

Substituting q(t) into Equation 19.12 yields

P

(BVs C )

dVs dt

(19.14)

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Unit Operations in Environmental Engineering

or

( P )dt

(BVs C )dVs

(19.15)

Integrating this from 0 to t and from 0 to Vs and solving for t leads to:

BVs2 2 P

t

CVs P

(19.16)

This equation can be rearranged so that Vs is an explicit function of t

2( P )t B

Vs

C B

2

C B

(19.17)

Also, Vs can be shown to be related to an instantaneous value of q(t) by rearrangement of Equation 19.12

( P )t Bq(t )

Vs

C B

(19.18)

Equating Equations 19.17 to Equation 19.18 and solving for q yields

t P

q B

2( P )t B

(19.19)

C B

2

Numerical values for design coefficients B and C are usually obtained from experimental data. If both sides of Equation 19.16 are divided by Vs, it can be rewritten as

t Vs

B Vs 2 P

C P

(19.20)

A plot of t/Vs versus Vs will yield a straight line of slope (B/2 P) and an intercept (C/ P). Note that only two data points, (Vs, t), are necessary to provide a first approximation of B and C. Some industrial filter operations are conducted in a manner approaching constant flow rate. For this condition,

dq dt

0

(19.21)

Filtration 189 and

dVs dt

q

constant

(19.22)

qt

(19.23)

Bq 2t Cq

(19.24)

so that

Vs Equation 19.12 now becomes

P

Thus, a plot of ΔP versus t yields a straight line of slope Bq2 and intercept Cq. At t = 0, the only resistance to flow is that of the filter medium; the pressure drop, however, is a linear function of time, and as time increases, the resistance due to the cake may predominate. Some filter operations have both a constant pressure and constant rate period. At the beginning of a normal cycle, the pressure drop is held constant until the flow rate increases to a maximum value that is obtained by experiment. The flow rate is then maintained constant until the pressure drop increases above a predetermined limit that may be dictated by economics. The coefficients B and C above have appeared in revised form in the literature in an attempt to relate them to physically measurable quantities. Equation 19.20 has been rewritten as:

t Vs

Kc Vs 2

1 q0

(19.25)

where Kc and q0 are constants and (in English units) equal

Kc

c S gc P

(19.26)

1 q0

Rm Sg c P

(19.27)

2

and

Note that S (once again) is the total surface of filtration cakes in the system, ft2; is the specific cake resistance, ft/lb; and Rm is the filter medium resistance, ft 1.

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Unit Operations in Environmental Engineering

As before, a plot of t/Vs versus Vs will give a straight line with a slope of Kc/2. This allows the calculation of the specific cake resistance, . Also, the y-intercept of the plot will be 1/q0, which means that filter medium resistance, Rm, can also be calculated. Theodore and Buonicore [3] have developed equations to predict coefficients B and C (or and Rm) from basic principles. Despite the progress in developing pure filtration theory, and in view of the complexity of the phenomena, the most common methods of correlation are based on producing a form of a final equation that can be verified by experiment.

19.4 Compressible Cakes In actual practice, the deposited cake is usually assumed to be incompressible. However, all cakes are compressible to some degree. For large pressure drops, these effects can become important. These large changes in pressure tend to force the solids further into the interstices in the filtering medium, thereby increasing the resistance to flow and the value of . If is not constant, but is a function of P, the cake is referred to as compressible. The cake resistance, , is one of the more important variables in filtration applications; it is dependent on a host of factors including the filter area, pressure drop, viscosity, etc. For the above situation, the following experimental relationship between and P is often assumed to apply: 0

Pb

(19.28)

where 0 and b are empirically assumed constants that can be obtained from a best-fit straight line on a log-log plot of versus P. The term 0 is usually referred to as the specific cake resistance at zero pressure and b is the compressibility factor for the cake. Note that the term b is zero for incompressible (i.e., the cake resistance does not vary with pressure) sludges and is positive for compressible ones. Constant pressure experiments are often used to determine the two coefficients of the cake. The first step in this process is to generate a logarithmic graph of versus P. Note that the logarithmic form of Equation 19.28 indicates that if the data are regressed linearly, the slope of the regression equation will equal the value of b.

log( ) log(

0

) blog( P )

(19.29)

If a pressure drop and corresponding value of cake resistance, , are obtained from the graph then 0 can then be solved mathematically since it is the only unknown.

Filtration 191

19.5 Filtration Unit Selection The choice of filter equipment depends largely on economics, but the economic advantages will vary depending on: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Fluid viscosity, density, and chemical reactivity, etc. Solid particle size, size distribution, and/shape. Flocculation tendencies. Deformability. Feed slurry concentration. Slurry temperature. Amount of material to be handled. Absolute and relative values of liquid and solid products. Completeness of separation required. Relative costs of labor, capital, and power.

References 1. McCabe, W.L., Smith, J.C., and Harriot, P., Unit Operations of Chemical Engineering, 7th Edition, McGraw-Hill, New York, NY, 2005. 2. Calvert, S., Guidelines for Selecting Mist Eliminators, Chemical Engineering, February 27, 1978. 3. Theodore, L., and Buonicore, A., Industrial Air Pollution Control Equipment for Particulates, CRC Press, Boca Raton, FL, 1976.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

20 Fluidization

20.1 Introduction Fluidization is the process in which fine solid particles are transformed into a fluidlike state through contact with either a gas or liquid, or both. Fluidization is normally carried out in a vessel filled with solids. The fluid is introduced through the bottom of the vessel and forced up through the bed. At a low flow rate, the fluid (liquid or gas) moves through the void spaces between the stationary, solid particles and the bed is referred to as fixed. As the flow rate increases, the particles begin to vibrate and move about slightly, resulting in the onset of an expanded bed. When the flow of fluid reaches a certain velocity, the solid particles become suspended because the upward frictional force between the particle and the fluid balances the gravitational force associated with the weight of the particle, i.e., the frictional drag and buoyant force is enough to overcome the downward force exerted on the bed by gravity. This point is termed minimum fluidization or incipient fluidization, and the velocity at this point is defined as the minimum or incipient fluidization velocity. Although the bed is supported at the bottom by a screen, the bed is free to expand upward, as it will if the velocity is increased above the minimum fluidization velocity. At this point, the particles are no longer supported by the screen, but rather are suspended in the fluid in equilibrium and act and behave as the fluid. 193

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Unit Operations in Environmental Engineering

The bed is then said to be fluidized. From a momentum or force balance perspective, the sum of the drag, buoyancy, and gravity forces must be equal to zero at this point. Beyond this stage, the bed enters the fluidization state where the solid particles produce a circulatory and/or mixing pattern [1]. The terminal settling velocity, vt, can be evaluated for the case of flow past one bed particle. By superimposition, this case is equivalent to that of the terminal velocity that a particle would attain flowing through a fluid. Once again, a force balance can be applied and empirical data used to evaluate a friction coefficient. At intermediate velocities between the minimum fluidization velocity and the terminal velocity, the bed is expanded above the volume that it would occupy at the minimum value. Note also that above the minimum fluidization velocity, the pressure drop stays essentially constant. One of the novel characteristics of fluidized beds is the uniformity of temperature found throughout the system. Essentially constant conditions are known to exist in both the horizontal and vertical directions in both short and long beds. This homogeneity is due to the turbulent motion and rapid circulation rate of the solid particles within the fluid stream described above. In effect, excellent fluidparticle contact results. Temperature variations can occur in some beds in regions where quantities of relatively hot or cold particles are present but these effects can generally be neglected. Consequently, fluidized beds find wide application in industry, e.g., oil cracking, zinc coating, coal combustion, gas desulfurization, heat exchangers, plastics cooling and fine powder granulation. Bed fluidization is also the principle associated with the cleaning of rapid sand filters in potable water treatment, termed “backwashing” as described earlier in Chapter 19.

20.2

Fixed Beds

The friction factor f for a “fixed” packed bed is defined as:

P 1 2 v 2 s

L 4f dp

(20.1)

in which dp is the particle diameter (defined presently) and vs is the superficial velocity defined in the previous chapter as the average linear velocity that the fluid would have in the column if no packing were present. The term L is the length of the packed column. The friction factor for laminar flow and that for turbulent flow can now be estimated separately. For laminar flow in circular tubes of radius R, it was shown that

v

PR 2 8 L

(20.2)

Fluidization

195

Now imagine that a packed bed is just a tube with a very complicated crosssectional area with hydraulic radius rh. The average flow velocity in the crosssection available for flow is then

v

Prh2 ; rh 2 L

cross section available for flow wetted perimeter

(20.3)

The hydraulic radius may be expressed in terms of the void fraction and the wetted surface, a, per unit volume of bed in the following way:

rh

volume available for flow total wetted surface

cross section available for flow wetted perimeter volume of voids vollume of bed wetted surface volume of bed

(20.4)

a

The quantity a is related to the specific surface, av (total particle surface/volume of the particles), by

a av (1

)

(20.5)

The quantity av is in turn used to define the mean particle diameter dp:

dp

/av

(20.6)

This definition is chosen because, for spheres, Equation 20.6 reduces to just dp as the diameter of the sphere. Finally, note that the average value of the velocity in the interstices, vI, is not of general interest to the engineer, but rather the superficial velocity, vs, is the main operating parameter of interest. These two velocities are related by

vs

vI

(20.7)

If the above definitions are combined with the modified Hagen-Poiseuille equation, the superficial velocity can be expressed as

vs

Prh2 2 L

P 2 2 La 2

P 2 2 Lav2 (1 )2

Pd 2p 2L(36 ) (1

2

)2

(20.8)

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Unit Operations in Environmental Engineering

or finally

vs

2 P dp L 2(36 ) (1

3

(20.9)

)2

In laminar flow, the assumption of mean hydraulic radius frequently gives a throughput velocity too large for a given pressure gradient. Because of this assumption, one would expect that the right side of Equation 20.9 should be somewhat smaller. A second assumption implicitly made in the foregoing development is that the path of the fluid flowing through the bed is of length L, i.e., it is the same as the length of the packed column. Actually, of course, the fluid traverses a very tortuous path, the length of which may be approximately twice as long as the length L. Because of this, one would again expect that the right side of Equation 20.9 should be somewhat smaller than calculated. Experimental measurements indicate that the above theoretical formula can be improved if the 2 in the denominator on the right-hand side is changed to a value somewhere between 4 and 5. Analysis of a great deal of data has led to a value 25/6, which is accepted here. Insertion of that value into Equation 20.9 then gives

vs

2 P dp L 150 (1

3

(20.10)

)2

which some have defined as the Blake-Kozeny equation. This result is generally good for void fractions less than 0.5 and is valid only in the laminar region where the particle Reynolds number is given by dpGs/ (1 – ) < 10; Gs = vs [2]. Note that the Blake-Kozeny equation corresponds to a bed friction factor of

f

)2

(1 3

75 d p Gs /

(20.11)

Exactly the same treatment can be repeated for highly turbulent flow in packed columns. One begins with the expression for the friction factor definition for flow in a circular tube. This time, however, note that for highly turbulent flow in tubes with any appreciable roughness, the friction factor becomes a function of the roughness only. Assuming that all packed beds have similar roughness characteristics, a unique friction factor, f0, may be used for turbulent flow. This leads to the following results if the same procedure as before is applied:

P L

1 v s2 4 f0 D 2

6 f0

1 v s2 1 dp 2

3

(20.12)

Fluidization

197

Experimental data indicate that 6f0 = 3.50. Hence, Equation 20.12 becomes

P L

3.50

1 pv s2 1 dp 2

1.75

3

pv s2 1 dp

3

(20.13)

which some have defined as the Burke-Plummer equation. This equation is valid for (dpGs/ )(1 – ) > 1000. This result corresponds to a friction factor given by

f0

0.875

1

(20.14)

3

Note that this dependence on is different from that given for laminar flow. When the Blake-Kozeny equation for laminar flow and the Burke-Plummer equation for turbulent flow are simply added together, the result is

P L

1.75 v s2 (1 dp

)2

150 v s (1 d 2p

3

) 3

(20.15)

Equation 20.15 may be rewritten in terms of dimensionless groups (numbers):

P G02

dp

3

150

1

L

1 d pG0 /

1.75

(20.16)

This is the Ergun equation [2]. It has been applied with success to gases by using the density of the gas at the arithmetic average of the inlet and discharge pressures. For large pressure drops, however, it seems more reasonable to use Equation 20.15 with the pressure gradient in differential form. Note that Gs is a constant through the bed whereas vs changes through the bed for a compressible fluid. The dp used in this equation is that defined in Equation 20.6. Equation 20.15 may now be written in the following form

P 150

v s (1 g

)2

L d 2p

3

1.75

v s2 (1 g

3

) L dp

(20.17)

Equation 20.16 may also be written in a similar form. It is important to note that other terms have been used to represent P, including

P

Pf

hf

(20.18)

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Unit Operations in Environmental Engineering

where the subscript f is a reminder that the pressure drop term represent friction due to the flowing fluid. Thus,

hf

150

v s (1 g

)2

L d 2p

3

1.75

v s2 (1 g

3

) L dp

(20.19)

The reader should note that the pressure drop term in Equation 20.17 has units of height of flowing fluid, e.g., in H2O. This may be converted into units of force per unit area (e.g., psf), by applying the hydrostatic pressure equation

gh gc

P

(20.20)

This equation can then be employed to rewrite Equation 20.17 in the following form:

v s (1 gc

P

v s2 (1 )2 L 1 . 75 3 gc d 2p

3

) L dp

(20.21)

The units of P are then those of pressure (i.e., force per unit area).

20.3 Permeability In porous media applications involving laminar flow, the Carmen-Kozeny equation is rewritten as

hf

150

v s (1 g

)2 3

1 vs L k g

L d 2p

(20.22)

where k is the permeability of the medium. The permeability may then be written as

k

10

1 vs 150 g (1

3

)2

d 2p

(20.23)

The permeability may be expressed in units of Darcies where 1 Darcy = 0.99 12 m2 = 1.06 10 11 ft2.

Fluidization

199

20.4 Minimum Fluidization Velocity There are various kinds of contact between solids and a fluid, starting from a packed bed and ending with “pneumatic” transport as shown in Figure 20.1. At a low fluid velocity, one observes a fixed bed configuration of height Lm, a term that is referred to as the slumped bed height. As the velocity increases, fluidization starts, and this is termed the onset of fluidization. The superficial velocity (that velocity which would occur if the actual flow rate passed through an empty vessel) of the fluid at the onset of fluidization is noted again as the minimum fluidization velocity, vmf, and the bed height is Lmf. As the fluid velocity increases beyond vmf, the bed expands and the bed void volume increases. At low fluidization velocities (fluid velocity ≈ vmf ), the operation is termed dense phase fluidization. The onset of fluidization (or minimum fluidization condition) in a packed bed occurs when drag forces due to friction by the upward moving gas equal the gravity force of the particles minus the buoyancy force on the particles. This can be represented in equation form as

FD

Wnet

(W Fbouyant ) particle

(20.24)

where W is the weight of the particle. The drag force, FD, exerted by the upward gas is a product of the friction pressure drop in gas flow across the bed and the bed cross-sectional area. From Bernoulli’s equation, the total pressure drop, P, can be expressed as:

P

Incipient or minimum fluidization

Fixed bed

Pfr

f

g Lmf gc

Aggregative or bubbling fluidization

Particulate or smooth fluidization

(20.25)

Lean phase fluidization with pneumatic Slugging transport

Lf

Lm

Gas or liquid (low velocity)

Lf

L mf

Gas or liquid

Liquid

Figure 20.1 Types of particle-fluid contact in a fixed bed.

Gas

Gas

Gas or liquid (high velocity)

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Unit Operations in Environmental Engineering

The pressure drop due to frictional resistance can then be expressed by rearrangement as

Pfr

P

f

g Lmf gc

(20.26)

with the latter term representing the fluid head. Therefore,

FD

Pfr Sb

P

g Lmf Sb gc

f

(20.27)

The net gravity force, Wnet, is due to the gravity of the solid particles minus the fluid buoyancy:

Wnet

Sb Lmf (1

mf

)(

s

f

)

g gc

(20.28)

Combining Equations 20.27 and 20.28, one obtains the condition for minimum fluidization:

Pfr

f

g hf gc

(1

mf

)(

s

f

f

Lmf

)

g Lmf gc

(20.30)

or

hf

(1

mf

s

)

(20.31)

f

The equations for minimum fluidization are similar to those presented for a fixed bed, i.e., Equations 20.17 through 20.21. For laminar flow conditions (Rep < 10), the Blake-Kozeny equation is used to express h f in terms of the superficial gas velocity at minimum fluidization, vmf, and other fluid and bed properties. This equation is obtained from Equations 20.9 and 20.31.

hf

(1

mf

)

s

f

Lmf

150

vmf

f

f f

g

(1

mf

)2 Lmf

3 mf

d 2p

(20.32)

Rearranging, one obtains the minimum fluidization velocity:

vmf

1 150 1

3 mf mf

g(

s

f f

)d 2p

; for Re p

10

(20.33)

Fluidization

201

with (once again)

d p vmf

Re p

f

(1

mf

(20.34)

)

The Burke-Plummer equation is used to express the head loss, h f . For turbulent flow conditions (Rep < 1000), the result is:

vmf In the absence of

mf

vmf

s

3 mf

f

gd p

(20.35)

f

data, the above equations can be approximated as

1 g( 1650 1 24.5

vmf

1 1.75

s

f

)d 2p

; for Re p

10

(20.36)

1000

(20.37)

f

s

f

gd p ; for Re p

f

where Rep is the particle Reynolds number at minimum fluidization and is equal to:

d p vmf

Re p

(20.38)

f

Both Equations 20.17 and 20.21 may be rewritten in a slightly different form and viewed as contributing terms to a more general equation. An equation covering the entire range of flow rates for various shaped particles can be obtained by assuming that the laminar and turbulent effects are additive. This result is also referred to as the Ergun equation [2]. Thus,

P L

150v0 (1 g c s2 d 2p

)2 3

1.75 v02 (1 g cp s d p

) 3

(20.39)

where Φs is the sphericity or shape factor of the fluidized particles. Typical values for particle sphericity are in the 0.75 to 1.0 range. In lieu of any information on Φs, one should employ a value of 1.0, typical for spheres, cubes and cylinders where L = d.

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Another approach that may be employed to calculate the minimum fluidization velocity is to employ Happel’s equation. Happel’s equation is given by [3]:

vmf

3 4.5(1

mf

)1/3 4.5(1

vt

3 2(1

mf mf

)5/3 3(1

mf

)5/3

)2

(20.40)

20.5 Bed Height, Pressure Drop and Porosity The above development is now extended above and beyond the state of minimum fluidization. As described above, when a fluid moves upward in a packed bed of solid particles, it exerts an upward drag force. Minimum fluidization occurs at a point when the drag force equals the net gravity force. By increasing the fluid velocity above minimum fluidization, the bed expands, the porosity increases, and the pressure drop remains the same. The fluidized bed height, Lf , at any porosity, , can be found from the minimum fluidization conditions (Lmf and mf), or from the bed height at zero porosity, L0, that is,

m

S L0

S Lmf (1

s b

s b

mf

)

S L f (1

s b

)

(20.41)

so that

Lf

Lmf

1

L0

mf

1

(20.42)

1

The pressure drop at minimum fluidization remains constant at any fluidization height, Lf, so that

g ( gc

Pfr

s

f

)L f (1

)

(20.43)

Initially, the bed pressure drop increases rapidly with an increase in velocity. Then, the pressure drop begins to level off. This point, as defined earlier, is called incipient fluidization. Beyond this point, the pressure drop remains fairly constant as the superficial velocity increases. This is the fluidized region. The variation of porosity (and hence bed height) with the superficial velocity is calculated from the Blake-Kozeny equation, assuming laminar flow and f 0.13, then the fluidization is bubbling.

References 1. Kunii, D., and Levenspiel, O., Fluidization Engineering, 2nd Edition, Butterworth-Heinemann, Boston, MA, 1991. 2. Ergun, S., Fluid Flow Through Packed Column, Chemical Engineering Progress, 48, 89-94, 1952. 3. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2014.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

21 Membrane Technology

21.1 Overview Phase separation, as its name implies, simply involves the separation of one (or more) phase(s) from another phase. Most industrial equipment used for this class of processes involves the relative motion of two phases under the action of various external forces (gravitational, electrostatic, etc.). There are basically five phase separation processes: 1. 2. 3. 4. 5.

Gas-solid (GS) Gas-liquid (GL) Liquid-solid (LS) Liquid-liquid (LL) Solid-solid (SS)

Phases 1 to 4 are immiscible processes. As one might suppose, the major phase separation processes encountered in environmental engineering are GS and LS. Traditional equipment for GS separation processes in air pollution control include: (1) gravity settlers, (2) centrifugal separators (cyclones), (3) electrostatic precipitators, (4) wet scrubbers, and (5) baghouses, while LS separation in water 205

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and wastewater treatment applications include: (1) gravity settlers, (2) centrifugal separators, (3) filtration, and (4) flotation. The overall collection-removal process for solid particles in a fluid consists of four steps [1, 2]: 1. An external force (or forces) must be applied that enables the particle to develop a velocity that will displace or direct it to a collection or retrieval section or area or surface. 2. The particle should be retained at this area with sufficiently strong forces that it is not re-entrained. 3. As collected/recovered particles accumulate, they are subsequently removed. 4. The ultimate disposition of the particles completes the process. Obviously, the first is the most important step in the overall process. The particle collection mechanisms discussed below are generally applicable when the fluid is air; however, they may also apply if the fluid is another gas, water, or another liquid. Miscible phases are another matter. For solids separation from a miscible phase, one common separation technique involves the use of membranes. In operations with miscible phases separated by a membrane, the membrane is necessary to prevent intermingling of the phases. Two different membrane separation phase combinations are briefly discussed below: 1. Liquid-Liquid (LL). The separation of a crystalline substance from a colloid by contact of the solution with a liquid solvent with a membrane permeable only to the solvent and the dissolved crystalline substance is known as dialysis. For example, aqueous sugar beet solutions containing undesired colloidal material are freed of the latter by contact with a semipermeable membrane; sugar and water diffuse through the membrane, but the larger colloidal particles cannot. Fractional dialysis for separating two crystalline substances in solution makes use of the difference in the membrane permeability of the substances. If an electronegative force is applied across the membrane to assist in the diffusional transport of charged particles, the operation is electrodialysis. If a solution is separated from the pure solvent by a membrane that is permeable only to the solvent, and the solvent is transported into the solution, this process is termed osmosis. If the flow of solvent is reversed by superimposing a pressure to oppose the osmostic pressure, the process is then defined as reverse osmosis, one of the most commonly used membrane processes for producing potable water in the environmental engineering field. 2. Gas-Gas (GG). Operation in the gas-gas category is known as gaseous diffusion, gas permeation, or effusion. If a gas mixture whose components are of different molecular weights is brought into contact with a porous membrane or diaphragm, the various components

Membrane Technology 207 of the gas will diffuse through the pore openings at different rates. This leads to different compositions on opposite sides of the membranes and, consequently, to some degree of separation of the gas mixture. The large-scale separation of the isotopes of uranium, in the form of uranium hexafluoride, can be carried out in this manner. Membrane processes are today state-of-the-art separation technologies that have shown promise for future technical growth and wide-scale industrial commercialization. They are used in many industries for process stream and product concentration, purification, separation, and fractionation, as well as for potable water production for marginal water sources. The use of membranes in wastewater treatment systems to replace conventional sedimentation tanks is also expanding. The need for membrane research and development is important because of the increasing use of membrane technology in both traditional and emerging chemical and environmental engineering fields. Membrane processes are increasingly finding their way into the growing engineering application areas of biotechnology, green engineering, specialty chemical manufacture, and biomedical engineering, as well as in the traditional chemical and environmental process industries. Membrane technology is also being considered as either a replacement for or supplement to traditional separations such as sedimentation, distillation or extraction. Membrane processes can be more efficient and effective since they can simultaneously concentrate and purify, and can also perform separations at ambient conditions in small or existing reactor volumes. Membranes create a boundary between different bulk gas or liquid mixtures. Different solutes and solvents flow through membranes at different rates; this enables the use of membranes in separation processes. Membrane processes can be operated at moderate temperatures for sensitive components (food, pharmaceuticals, etc.). Membrane processes also tend to have low relative capital and energy costs. Their modular format permits simple and reliable scale-up or retrofitting in existing process tanks. Key membrane properties include their size rating, selectivity, permeability, mechanical strength, chemical resistance, low fouling characteristics, high capacity, low cost, and consistency. Vendors characterize their equipment with ratings indicating the approximate size (or corresponding molecular weight) of components retained by their membranes. Commercial membranes consist of polymers or ceramics. Other membrane types include sintered metal glass and liquid films.

21.2 Membrane Separation Principles There are four major membrane processes of interest to the environmental engineer: 1. Reverse osmosis (hyperfiltration) 2. Ultrafiltration

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Each process is introduced in the sections to follow. As noted above, the main difference between reverse osmosis (RO) and ultrafiltration (UF) is that the overall size and diameter of the particles or molecules in solution to be separated is smaller in RO. In microfiltration (MF), the particles to be separated or concentrated are generally solids or colloids rather than molecules in solution. Figure 21.1 illustrates the different between the processes. Gas permeation (GP) is another membrane process that employs a nonporous semipermeable membrane to fractionate a gaseous stream. Membrane unit operations are often characterized by the following parameters: (1) driving force utilized, (2) membrane type and structure, and (3) species being separated. RO, nanofiltration [3], UF, and MF all utilize a pressure difference as the driving force to separate a liquid feed into a liquid permeate and retentate. They are listed in ascending order in their ability to separate a liquid feed on the basis of solute size. Reverse osmosis uses nonporous membranes and can separate down to the ionic level, e.g., seawater in the rejection of dissolved salt. Nanofiltration performs separations at the nanometer range [3]. Ultrafiltration uses porous membranes and separates components of molecular size ranging from the low thousand to several hundred thousand atoms; an example includes Reverse osmosis Size key

Feed

Solvent Solute (low MW) Solute (high MW)

Permeate

Particle Ultrafiltration Feed

Permeate Microfiltration Feed

Permeate

Figure 21.1 Three common membrane separation processes.

Membrane Technology 209 components in biomedical processing. Microfiltration uses much more porous membranes and is typically employed in the micro – or macromolecular range to remove particulate or larger biological matter from a feed stream (e.g., in the range of 0.05 to 2.5 m) [4]. Gas separation processes can be divided into two categories: gas permeation through nonporous membranes and gas diffusion through porous membranes. Both of these processes utilize a concentration difference across the membranes as the driving force for separation. The gas permeation processes are used extensively in industry to separate air into purified nitrogen and enriched oxygen. Another commercial application is hydrogen recovery in petroleum refineries. As noted above, dialysis membrane processes use a concentration difference as a driving force for separation of liquid feed across a semipermeable membrane, with the major application in the medical field of hemodialysis. Electrodialysis separates a liquid employing an electric charge difference as the driving force, and is widely used in water purification and industrial processing. Naturally, the heart of the membrane process is the membrane itself; it is an ultrathin semipermeable barrier separating two fluids that permits the transport of certain species through the membrane barrier from one fluid to the other. As noted above, a membrane is typically produced from various polymers such as cellulose acetate or polysulfone, but ceramic and metallic membranes are also used in some applications. The membrane is referred to as selective since it permits the transport of certain species while rejecting others. The term semipermeable is frequently used to describe this selective action.

21.3 Reverse Osmosis (RO) The most widely commercialized membrane process by far is RO. It belongs to a family of pressure-driven separation operations for liquids that includes not only reverse osmosis but also ultrafiltration and microfiltration. RO is an advanced separation technique that may be used when low molecular weight (MW) solutes such as inorganic salts or small organic molecules (e.g., glucose) are to be separated from a solvent (usually but not always water). In normal (as opposed to reverse) osmosis, water flows from a less concentrated salt solution to a more concentrated salt solution as a result of driving forces. As a result of the migration of water, an osmotic pressure is created on the side of the membrane to which water flows. In reverse (as opposed to normal) osmosis, the membrane is permeable to the solvent or water and relatively impermeable to the solute or salt. To make water pass through an RO membrane in the desired direction (i.e., away from the concentrated salt solution), a pressure must be applied that is higher than that of the osmotic pressure. RO is widely utilized today by a host of chemical and environmental process industries for a surprisingly large number of operations. Aside from the classic example of RO for seawater desalination, it has found a niche in the food industry

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for concentration of various fruit juices, in the galvanization industry for concentration of waste streams, and in the dairy industry for concentration of milk prior to cheese manufacturing [5, 6]. RO processes are classified into the following two basic categories: 1. Purification of the solvent such as in desalination where the permeate or purified water is the product 2. Concentration of the solute such as in concentration of fruit juices where the retentate is the product The membranes used for RO processes are characterized by a high degree of semi-permeability, high water flux, mechanical strength, chemical stability, and relatively low operating and high capital costs. Early RO membranes were composed of cellulose acetate, but restrictions on process stream pressure, temperature, and organic solute rejection spurred the development of non-cellulosic and composite materials. Reverse osmosis membranes may be configured or designed into certain geometries for system operation: plate-and-frame, tubular, spiral wound (composite), and hollow fiber [7]. In the plate-and-frame configuration, flat sheets of membrane are placed between spacers with heights of approximately 0.5 to 1.0 mm. These are, in turn, stacked in parallel groups. Tubular units are also commonly used for RO. This is a simpler design in which the feed flows inside of a tube whose walls contain the membrane. These types of membranes are usually produced with inside diameters ranging from 12.5 to 25 mm and generally produced in lengths of 150 to 610 cm. There is also the hollow fine-fiber (HFF) arrangement used in approximately 70% of worldwide desalination applications. Millions of hollow fibers are oriented in parallel and fixed in epoxy at both ends. A feed stream is sent through a central distributor where it is forced out radially through the fiber bundle. As the pressurized feed contacts the fibers, the permeate is forced into the center of each hollow fiber. The permeate then travels through the hollow bore until it exits the permeator. A spiral-wound cartridge is occasionally employed in this configuration. Here, the solvent is forced inward towards the product tube while the concentrate remains in the space between the membranes. A flat film membrane is made into a “leaf.” Each leaf consists of two sheets of membrane with a sheet of polyester tricot in between which serves as a collection channel for the water product. Plastic netting is placed between each leaf to serve as a feed channel. Each leaf is then wrapped around the product tube in a spiral fashion. It is no secret that water covers around 70% of Earth’s surface, but 97.5% of it is unfit for human consumption. With the world facing a growing freshwater shortage from which the United States may not be spared, one method of producing freshwater is desalination. The major application of RO is water desalination. Some areas of the world that do not have a ready supply of freshwater have chosen to desalinate seawater or brackish water using RO to generate potable

Membrane Technology 211 drinking water. Because no heating or phase change is required, the RO process is both a relatively low energy and economical water purification process. A typical saltwater RO system consists of an intake, a pretreatment component, a high-pressure pump, membrane apparatus, remineralization, and pH adjustment components, as well as a disinfection step. A pressure difference driving force of about 1.0 to 6.5 mPA is generally required to overcome the osmotic pressure of saltwater. Another important membrane application is dialysis. This technique is used in patients who suffer from kidney failure and can no longer filter waste products (urea) from the blood. In general, RO equipment used for dialysis can reduce ionic contaminants by up to 90%. In this process, the patient’s blood flows in a tubular membrane while a dialysate flows countercurrently on the outside of the feed tube. The concentrations of undesirable salts (e.g., potassium, calcium, urea) are high in the blood (while low or absent in the dialysate). This treatment successfully mimics the filtration capabilities of the kidneys. RO membranes are designed for high salt retention, high permeability, mechanical robustness (to allow module fabrication and withstand operating conditions), chemical robustness (for fabrication materials, process fluids, cleaners, and sanitizers), low extractables, low fouling characteristics, high capacity, low cost, and consistency. The predominant RO membranes used in water applications include cellulose polymers, thin composites consisting of aromatic polyamides, and crosslinked polyetherurea. The membrane operation for the purification of seawater that incorporates a selective barrier can be simply described using the line diagram provided in Figure 21.2. This membrane operation typifies the case in which a feed stream (seawater) is separated by a semipermeable membrane that rejects salt but selectively transports water. A purified stream (the permeate) is therefore produced Feed: seawater, saltwater

Feed

A

Retentate Figure 21.2 Desalination of seawater by RO.

Permeate

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while at the same time, a concentrated salt stream (the retentate) is discharged. With reference to Figure 21.2, a simple material balance can be written on the overall process flows and for that of the solute as:

qf Cf qf

qr q p C r qr C p q p

(21.1) (21.2)

where q = volumetric flow rate, C = solute concentrate, and subscripts f, r, and p refer to the feed, retentate, and permeate, respectively. Osmosis occurs when a concentrated solution is partitioned from a pure solute or relatively lower concentration solution by a semipermeable membrane. The semipermeable membrane allows only the solvent to flow through it freely. Equilibrium is achieved when the solvent from the lower-concentration side ceases to flow through the membrane to the higher-concentration side (thus reducing the concentration) because the mass transfer concentration difference driving force has been reduced. This is shown in Figures 21.3 and 21.4. Osmotic pressure is the pressure needed to terminate the flux of solvent through the membrane or the force that pushes up on the concentrated side of the membrane (see Figure 21.5). Applying a pressure on the concentrated side halts the solvent flux. Reverse osmosis (see Figure 21.6) takes place when an applied force (pressure) overcomes the osmotic pressure and forces the solvent from the concentrated side through the membrane and leaves the solute on the concentrated side.

Membrane

Pure/less concentrated side

Figure 21.3 Pre-osmosis equilibrium.

Concentrated side

Membrane Technology 213 Membrane

Pure/less concentrated side

Concentrated side

Figure 21.4 Osmosis of a solvent.

Osmotic pressure

Pure/less concentrated side

Concentrated side

Figure 21.5 Osmotic pressure.

21.4 Ultrafiltration, Microfiltration, and Gas Permeation 21.4.1 Ultrafiltration (UF) UF is a membrane separation process that can be used to concentrate single solutes or mixtures of solutes. Transmembrane pressure (the membrane pressure drop) is the main driving force in UF operations, and separation is achieved via a sieving mechanism. The UF process can be used for the treatment or concentration of oily

214

Unit Operations in Environmental Engineering Applied pressure

Pure/less concentrated side

Concentrated side

Figure 21.6 Reverse osmosis.

wastewater, for pretreatment of water prior to RO, and for the removal of bacterial contamination (pyrogens). In the food industry, UF is used to separate lactose and salt from cheese whey proteins, to clarify apple juice, and to concentrate milk for ice cream and cheese production [8–10]. The most energy-intensive step in ice cream production is concerned with the concentration of skimmed milk, where membrane processes are more economical for this step than vacuum evaporation [9]. UF processes are also used for concentrating or dewatering fermentation products as well as purifying blood fractions and vaccines. Ultrafiltration may be regarded as a membrane separation technique where a solution is introduced on one side of a membrane barrier while water, salts, and/ or other low-molecular-weight materials pass through the unit under an applied pressure. As noted, these membrane separation processes can be used to concentrate single solutes or mixtures of solutes. The variety in the different membrane materials allows for a wide temperature-pH processing range. The main economic advantage of UF is a reduction in both design complexity and energy usage since UF processes can simultaneously concentrate and purify process streams. The fact that no phase change is required leads to highly desirable energy savings. A major disadvantage is the high capital cost that might be required if low flux rates for purification demand a large system. However, UF processes are usually economically sound in comparison to other traditional separation techniques. In addition to the applications described above, UF membrane processes are utilized in various commercial applications. They are found in the treatment of industrial effluents and process water; in the concentration, purification, and separation of macromolecular solutions in the chemical, food, and drug industries; in the sterilization, clarification, and pre-filtering of biological solutions and

Membrane Technology 215 beverages; and, in the production of ultrapure water and pretreating of seawater in RO processes. The most promising area for the expansion of UF process applications is the biochemical industry. Some of its usage in this area includes purifying vaccines and blood fractions; concentrating gelatins, albumin, and egg solids; and, recovering proteins and starches. The rejection of a solute is a function of the size, size distribution, shape, and surface binding characteristics of the hydrated molecule. It is also a function of the pore size distribution of the membrane. Therefore, MW cutoff values can be used only as a rough guide for membrane selection. The retention efficiency of the solutes depends to a large extent on the proper selection and condition of the membrane. Replacement of highly used membranes and regular inspections of the separation units averts many problems that might otherwise occur because of membrane clogging and gel formation. Ultrafiltration processes use driving forces of 0.2 to 1.0 MPa to drive liquid solvent (usually water) and small solutes through membranes while retaining solutes of 10 to 1,000 Å diameter. The membranes consist primarily of polymeric structures, such as polyethersulfone, regenerated cellulose, poiysuifone, polyamide, polyacrylonitrile, or various fluoropolymers. Hydrophobic polymers are surface modified to render them hydrophilic. The general design factors for any membrane system (including UF), as reported by Wankat [11], are: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Thin, active layer of membrane High permeability for Species A and low permeability for Species B Stable membrane with long service life Mechanical strength Large surface area of membrane in a small volume Elimination or control of concentration polarization Ease in cleaning Low construction costs Low operating costs

Concentration polarization occurs in many separations and for large solutes where the osmotic pressure can be safely neglected. Concentration polarization without gelling should have no effect on flux through the membrane. Therefore, if a flux decline is observed, it is usually attributed to the formation of a gel layer. The gel layer, once formed, usually controls mass transfer [12]. 21.4.2 Microfiltration (MF) MF can separate particles from liquid or gas phase solutions. The usual materials retained by a microfiltration membrane range in size from several micrometers down to 0.2 m. Very large soluble molecules are retained by a microfilter at the low end of this spectrum.

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MF is employed in modern industrial biochemical and biological separation processes. For example, MF can be used instead of centrifugation or pre-coat rotary vacuum filtration to remove yeast, bacteria, or mycelia organisms from fermentation broth in cell harvesting. Both MF and UF are used for cell harvesting. Microfiltration is used to retain cells and colloids, while allowing passage of macromolecules into the permeate stream. UF is also used to concentrate macromolecules, cells, and colloidal material, while allowing small organic molecules and inorganic salts to pass into the permeate stream. As stated above, pore sizes in microfiltration are around 0.10 to 10 m in diameter as compared with 0.001 to 0.02 m for ultrafiltration (ranges vary slightly depending on the manufacturer and specific application) [13, 14]. Similar types of equipment are used for MF and UF; however, membranes with larger pore sizes are generally employed in MF applications. MF processes operate at lower pressure than UF but at a higher pressure difference driving force (PDDF) than does conventional particulate filtration [12]. Ideal membranes possess high porosity, a narrow pore size distribution, and a low binding capacity. When separating microorganisms and cell debris from fermentation broth, a biological cake is formed. Principles of cake filtration [1, 12] apply to MF systems, except that the small size of yeast particles produces a cake with a relatively high resistance to flow, producing a relatively low filtration rate. In dead-end filtration, feed flow is perpendicular to the membrane surface, and the thickness of the cake layer on the membrane surface increases with filtration time. Consequently, the permeation rate decreases. Cross-flow filtration, on the other hand, features feed flowing parallel to the membrane surface which is designed to decrease formation of a cake by sweeping previously deposited solids from the membrane surface and returning them to the bulk feed stream. Crossflow filtration is far superior to dead-end filtration for cell harvesting because the biological cake is highly compressible, which causes the accumulated layer of biomass to rapidly blind the filter surface in dead-end operation. Therefore, MF experiments are often conducted using cross-flow filtration because of the advantages that this filtration mode offers [15–17]. Separation principles and governing equations for MF are similar to those for RO and UF. 21.4.3

Gas Permeation (GP)

Gas separation processes are generally considered as relatively new and emerging technologies because they are not included in the curriculum of many traditional chemical and environmental engineering programs [15–19]. GP is the term used to describe a membrane separation process using a nonporous, semipermeable membrane. In this process, a gaseous feed stream is fractionated into permeate and non-permeate streams. The non-permeating stream is typically referred to as the nonpermeate in gas separation terminology, although it is defined as the retentate in liquid separation. Transport separation occurs by a solution diffusion mechanism. Membrane selectivity is based on the relative

Membrane Technology 217 permeation rates of the components through the membrane. Each gaseous component transported through the membrane has a characteristic individual permeation rate that depends on its ability to dissolve and diffuse through the membrane material. The mechanism for transport is based on solubilization and diffusion; the two describing relationships on which the transport are based are Fick’s law (diffusion) and Henry’s law (solubility), as defined earlier in Chapter 4. Some gas filters, which remove liquids or solids from gases, are MF membranes. Gas permeation systems have gained popularity in both traditional and emerging engineering areas. These systems were originally developed for hydrogen recovery. There are presently numerous applications of gas permeation in industry, and other potential uses of this technology are in various stages of development. Applications today include gas recovery from waste gas streams, landfill gases, and ammonia and petrochemical products. Gas permeation membrane systems are also employed in gas generation and purification, including the production of nitrogen and enriched oxygen gases. There must be a driving force for the process of permeation to occur. For gas separations, that force is the partial pressure of the gas on the feed side of the membrane. Since the ratio of the component fluxes determines the separation, the partial pressure of each component at each point is important in gas permeation systems. There are two ways of driving the process: employing a high partial pressure on the feed side of the membrane, or providing a low partial pressure on the permeate side, the latter which may be achieved by either vacuum or inert-gas flushing [11].

21.5 Pervaporation and Electrodialysis 21.5.1 Pervaporation (PER) PER is a separation process in which a liquid mixture contacts a nonporous, semipermeable membrane. One component is transported through the membrane preferentially. It evaporates on the downstream side of the membrane, leaving as a vapor. The process is induced by lowering the partial pressure of the permeating component, usually by a vacuum or occasionally with an inert gas. The permeated component is then condensed or recovered as the product of interest. 21.5.2

Electrodialysis (ED)

ED is a membrane separation process in which ionic species can be separated from water macrosolutes and all charged solutes. Ions are induced to move by an electric potential, and separation is facilitated by ion-exchange membranes. The membranes are highly selective, passing primarily either anions or cations. In the ED process, the feed solution containing ions enters a compartment whose walls contain either a cation exchange or an anion exchange membrane [11].

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References 1. Theodore, L., Air Pollution Control Equipment Calculations, John Wiley & Sons, Hoboken, NJ, 2008. 2. Reynolds, J., Jeris, J., and Theodore, L., Handbook of Chemical and Environmental Engineering Calculations, John Wiley & Sons, Hoboken, NJ, 2004. 3. Theodore, L., Nanotechnology: Basic Calculations for Engineers and Scientists, John Wiley & Sons, Hoboken, NJ, 2006. 4. Schweitzer, P., Handbook of Separation Techniques for Chemical Engineers, McGraw-Hill, New York City, NY, 1979. 5. Parkinson, G., Reverse Osmosis: Trying for Wider Applications, Chem. Eng., 26–31, 1983. 6. Applegate, L., Membrane Separation Processes, Chem. Eng., 64–89, 1984. 7. Brooks, K., Membranes Push into Separations, Chem. Week, 21–24, Jan. 16, 1985. 8. Kosikowski, F.V., Chapter 9. Membrane Separations in Food Processing, in: Membrane Separations in Biotechnology, W.C. McGregor (Ed.), Marcel Dekker, New York, NY, 1986. 9. Garcia, A., Medina, B., Verhoek, N., and Moore, P., Ice Cream and Components Prepared with Ultrafiltration and Reverse Osmosis Membranes, Biotechnol. Prog., 5, 46–50, 1989. 10. Maubois, J., Recent Developments of Membrane Ultrafiltration in the Dairy Industry, in: Ultrafiltration Membranes and Applications, A.R. Cooper (Ed.), pp. 305–318, Plenum Press, Ney York, NY, 1980. 11. Wankat, P.C., Rate-Controlled Separation, Chapman & Hall, Boston, MA, 1990. 12. Porter, M.C., Handbook of Industrial Membrane Technology, Noyes Publications, Park Ridge, NJ, 1990. 13. Mulder, M., Basic Principles of Membrane Technology, 2nd Edition, Kluwer Academic, Boston, MA, 1996. 14. Hollein, H., Slater, C., D’Aquino, R., and Witt, A., Bioseparation Via Cross Flow Membrane Filtration, Chem. Eng. Educ., 29, 86–93, 1995. 15. Slater, C.S., and Hollein, H., Educational Initiatives in Teaching Membrane Technology, Desalination, Chem. Eng. Educ., 90, 625–634, 1993. 16. Slater, C.S., Hollein, H., Antonechia, P.P., Mazzella, L.S., and Paccione, J., Laboratory Experiences in Membrane Separation Processes, Int. J. Eng. Educ., 5, 369–378, 1989. 17. Slater, C.S., Vega, C., and Boegel, M., Experiments in Gas Permeation Membrane Processes, Int. J. Eng. Educ., 9, 1–7, 1992. 18. Prism Separators, Bulletin PERM-6–008, Permea Inc., St. Louis, MO, 1986. 19. Davis, R., and Sandall, O., A Membrane Gas Separation Experiments for the Undergraduate Laboratory, Chem. Eng. Educ., 87, 10–21, 1990.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

22 Compressible and Sonic Flow

22.1 Introduction Compressibility refers to a condition where the volume or density of a fluid varies with the pressure. In fluid flow applications, it is a consideration only when vapors/gases are involved; liquids can safely be considered incompressible in these and almost all environmental engineering calculations. When the pressure drop in a flowing gas system is less than 10-20% of the absolute pressure in the system, satisfactory engineering accuracy is obtained in pressure drop calculations by assuming the fluid incompressible at conditions corresponding to the average pressure in the system. For larger pressure drops, compressibility effects can become important. The compressible flow of a fluid is further complicated by the fact that the fluid density is dependent on temperature as well as pressure. In such systems, temperature may vary in accordance with thermodynamic principles [1] (see also Chapters 4 in Part I for more details). Although this chapter is primarily concerned with sonic flow, it also addresses the general topic of compressible flow. The presentation that follows first examines compressible flow, which in turn is followed by sonic flow, which in turn is followed by key pressure drop equations that may be employed in environmental engineering flow calculations. 219

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It should be noted that compressible flow is only occasionally encountered in environmental applications. Sonic flow is rarely encountered.

22.2 Compressible Flow As noted above, flowing fluids are typically considered compressible when the density varies by more than 10–20% during a particular application. In practice, compressible flows are normally limited to gases, supercritical fluids, and multiphase flows containing gases; flowing liquids are normally considered incompressible. In industrial applications, one-dimensional gas flow through nozzles or orifices and in pipelines is the most important application of compressible flow. Multidimensional external flows are of interest mainly in aerodynamic applications, a topic beyond the scope of this text [2]. In addition to the factors discussed above, compressible flow calculations are further complicated by other system parameter variations. For example, for a given pipe diameter and mass flow rate, the friction factor depends upon the viscosity of the fluid, which, in turn, depends upon its temperature. This problem does not exist for isothermal flow but can be important during adiabatic operation. However, in adiabatic compressible flow, Reynolds numbers are usually high, indicating turbulent flow, and any variation of the friction factor due to temperature variations along the pipe length is small; thus, the friction factor may be assumed constant. The first step in a compressible-incompressible flow analysis is to classify the flow. One has to specify either steady or unsteady flow as well as whether the flow is compressible or incompressible. Steady and unsteady flow refers to variations with time, while incompressible and compressible flow refers to the aforementioned density variations. If the density is constant, or its variation is very small (i.e., for most liquids, and gases with a Mach number less than 0.3), the flow is deemed incompressible. The Mach number is discussed in the next section.

22.3 Sonic Flow The Mach number, Ma, is a dimensionless number defined as the ratio of fluid velocity to the speed of sound in the fluid, i.e.,

Ma

v c

(22.1)

where v is the average velocity of the fluid and c is the speed of sound. If the Mach number is less than or equal to 0.3, compressibility effects may usually be neglected, and one may safely assume incompressible flow.

Compressible and Sonic Flow

221

The speed of sound in common liquids is given in Table 22.1. The speed of sound, c, in an ideal gas may be calculated from:

c

kRT MW

(22.2)

where k (see Table 22.2) is the ratio of Cp/Cv, R the universal gas constant, T the absolute temperature and MW the molecular weight of the fluid. The derivation of this from basic principles is available in the literature. Note that the k values in Table 22.2 are approximate for 1 atm and 25 °C; a decrease in temperature or an increase in pressure will generally result in higher values [3]. For air, Equation 22.2 simplifies to

c 20 T (K ); m/s

(22.3)

c 20 T ( R); ft/s

(22.4)

and

Equation 22.2 indicates that the square of the velocity of sound is proportional to the absolute temperature of the ideal gas. Thus, the velocity of sound may be Table 22.1 Speed of sound in various liquids. Liquid

Sound velocity (m/s)

Acetone

1174

Benzene

1298

Ethanol

1144

Ethylene glycol

1644

Methanol

1103

Water

1498

Table 22.2 Values of k. Gas

k

C2H6

1.2

CO2, SO2, H2O, H2S, NH3, Cl2, CH4, C2H2, C2H4

1.3

Air, H2, O2, N2, CO, NO, HCl

1.4

Monatomic gases

1.67

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Unit Operations in Environmental Engineering

viewed as being proportional to the internal kinetic energy of the gas. Since the kinetic energy of a flowing gas is proportional to v2, the ratio v2/c2 provides a measure of the ratio of the kinetic energy to the internal energy. The velocity of sound in air at room temperature is approximately 1100 ft/s Thus, for a velocity of 220 ft/s, and noting that v/c is defined as the Mach number, Ma is 0.2 and (Ma)2 is only 0.04. This indicates that kinetic energy effects do not become important until somewhat higher Mach numbers are achieved [3]. Most often, the Mach number is calculated using the speed of sound evaluated at the local pressure and temperature. When Ma = 1, the flow is critical or sonic, and the velocity equals the local speed of sound. For subsonic flow, Ma < 1, while supersonic flow has Ma > 1. A potential error is to assume that compressibility effects are always negligible when the Mach number is small. Proper assessment of whether compressibility is important should be based on relative density changes, not on Mach number alone [2]. However, the Mach number is usually employed in environmental engineering calculations. Equations presented earlier in Chapter 13 for incompressible fluids are applicable to compressible fluids—in a general sense. However, these same equations may often be applied to compressible fluids if the fractional change in pressure is not large. For example, compressibility effects may not be important if there is a change in pressure from 14.7 to 15.7 psia, but could be very important if the change is from 0.1 to 1.0 psia [3]. A detailed treatment of sonic flow through a variety of process units is provided in Perry’s Handbook [1]. Included in the treatment are defining equations for: 1. 2. 3. 4.

Flow through a frictionless nozzle Adiabatic flow with friction in a duct of constant cross-section Compressible flow with friction loss Convergent/divergent nozzles

22.4 Pressure Drop Equations Two equations for pressure drop are presented in this section, one for isothermal flow and one for adiabatic flow. Both can be employed for most real-world applications involving compressible flow. 22.4.1 Isothermal Flow For laminar flow of gases in pipes and other conduits, the pressure drop from P1 to P2 may be estimated from Equation 22.5 for laminar, isothermal flow conditions.

P12 P22

8 RTG 8L g c MWD D

P Re ln 1 P2 3

(22.5)

Compressible and Sonic Flow

223

where Re = Reynolds number, = gas viscosity, T = absolute temperature, G = mass velocity flux, MW = gas molecular weight, D = pipe/conduit diameter, and L = pipe/conduit length. Equation 22.5 may be used for engineering purposes provided that the Mach number is below 0.5 (i.e., Ma < 0.5). One should note that if the flow rate is unknown (this is often the desired quantity), a trial-and-error solution is involved since the friction factor depends on the velocity. However, the value of f does not vary significantly over a very wide range of Reynolds numbers and the solution is therefore not sensitive to the value of f. Consequently, an initial assumption of an average value of 0.004 for f is satisfactory. This will yield a value of the velocity for which the Reynolds number can be calculated and the corresponding value of f determined. An iterative calculation using these new and updated values of f will provide an acceptable answer. In ducts of appreciable length, the second term in the parentheses in Equation 22.5 can be assumed negligible unless the pressure drop is very large. When this term is omitted, Equation 22.5 becomes

P12 P22

2 fLG 2 2 g c avg D

(22.6)

where avg is the density at the average pressure of (P1 + P2)/2 for the mass rate of . flow, m of the system. This equation may also be written as:

m

(P12 P22 ) g c D 5 (MW ) 8 fLRT

(22.7)

It should be noted that in most applications, the flow can more appropriately be described as adiabatic rather than truly isothermal. 22.4.2 Adiabatic Flow The equation for adiabatic flow is based on the condition that the flow arises from the adiabatic expansion of the gas through a frictionless nozzle leading from an inlet source where the velocity is negligible. Such a system is frequently encountered in practice. For this system, the describing equations are given by:

T1 T0 P0 P1

P1 P0

( k 1)/k

RT1 G2 k 1 1 2 g c k ( MW )P12

(22.8) k / ( k 1)

(22.9)

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References 1. Green, D., and Perry, R., (Editors), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGrawHill, New York, NY, 2008. 2. Farag, I., Fluid Flow, A Theodore Tutorial, Theodore Tutorials, East Williston, NY. originally published by the U.S. EPA APTI, RTP, NC, 1996. 3. Bennett, C., and Myers, J., Momentum, Heat, and Mass Transfer, McGraw-Hill, New York, NY, 1962.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

23 Two-Phase Flow

23.1 Introduction The simultaneous flow of two phases in pipes (as well as other conduits) is complicated by the fact that the action of gravity tends to cause settling and “slip” of the heavier phase, with the result that the lighter phase almost always flows at a different velocity in the pipe than does the heavier phase. The results of this phenomena are different depending on the classification of the two phases, the flow regime, and the inclination of the pipe conduit. The reader should note that when pumped through horizontal sections of pipe, emulsions and solid-liquid suspensions have a tendency to separate or settle out unless the velocity of flow is maintained above a minimum value. In the case of emulsions, separation of phases is prevented by keeping the velocity high enough to ensure turbulent flow. There are two velocities of importance in the case of the solid-liquid suspensions. The minimum velocity is that velocity below which the solids settle out freely. The velocity at which the solids are kept dispersed evenly across the pipe diameter is referred to as the standard velocity. At intermediate velocities, concentration gradients can exist in a vertical direction across the pipe or conduit.

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As one might suppose, the major industrial application in this area is gas (G)-liquid (L) flow in pipes. Therefore, the subjects addressed in this chapter key on a G-L flow in pipes. The extension of much of the material to follow for flow in various conduits can be accomplished by employing the equivalent diameter of the conduit in question. The general subject of liquid-solid flow in pipes is not considered in this chapter. Suspensions of solids in liquids fall into two general classes, Newtonian and non-Newtonian. Newtonian suspensions are characterized by a constant viscosity, independent of the rate of shear. In the case of non-Newtonian suspensions, the viscosity is a variable that is a function of the rate of shear and (in some cases) a function of the duration or period of shear for viscous flow. If the suspension is found to be Newtonian in character, the pressure drop can be calculated by standard equations available for both laminar and turbulent flow by employing the average density and viscosity of the mixture. The procedures for computing the pressure drop for non-Newtonian suspensions are more involved but are available from Perry and Green [1]. The general subject of flashing and boiling liquids is also not considered in this chapter. However, when a saturated liquid flows in a pipeline from a given point at a given pressure to another point at a lower pressure, several processes can take place. As the pressure decreases, the saturation or boiling temperature decreases, leading to the evaporation of a portion of the liquid. The net result is that a onephase flowing mixture is transformed into a two-phase mixture with a corresponding increase in frictional resistance in the pipe. Boiling liquids arise when liquids are vaporized in pipelines at approximately constant pressure. Alternatively, the flow of condensing vapors in pipes is complicated due to the properties of the mixture constantly changing with changes in pressure, temperature, and fraction condensed. Further, the condensate, which forms on the walls, requires energy in order to be transformed into spray, and this energy must be obtained from the main vapor stream, resulting in an additional pressure drop. An analytical treatment of these topics is also beyond the scope of this book. However, information is available in the literature [1]. The remainder of the chapter examines the following topics: Gas (G)-Liquid (L) Flow Principles: Generalized Approach, Gas (Turbulent) Flow-Liquid (Turbulent) Flow, Gas (Turbulent) Flow-Liquid (Viscous) Flow, Gas (Viscous) Flow-Liquid (Viscous) Flow, and Gas-Solid Flow.

23.2 Gas (G)-Liquid (L) Flow Principles: Generalized Approach The suggested method of calculating the pressure drop of G-L mixtures flowing in pipes is essentially that originally proposed by Lockhart and Martinelli [2] approximately 70 years ago. The basis of their correlation is that the two-phase pressure

Two-Phase Flow 227 drop is equal to the single-phase pressure drop for either phase (G or L) multiplied by a factor that is a function of the single-phase pressure drops of the two phases. The equations for the total pressure drop Z ( P /Z )T per unit length are written as:

( P /Z )T

YG ( P /Z )G

(23.1)

( P /Z )T

YL ( P /Z )L

(23.2)

The terms YL and YG are functions of the variable X:

YG

FG ( X )

(23.3)

YL

FL ( X )

(23.4)

where

X

( P /Z )L ( P /Z )G

0. 5

(23.5)

The relationship between YL and YG is therefore given by

YG

X 2YL

(23.6)

The single-phase pressure-drop gradients ( P/Z)Land ( P/Z)G can be calculated by assuming that each phase is flowing alone in the pipeline, and the phase in question is traveling at its superficial velocity. The superficial velocities are based on the full cross-sectional area, S, of the pipe so that

vL

qL /S

(23.7)

vG

qG /S

(23.8)

and

where vL = liquid-phase superficial velocity, vG = gas-phase superficial velocity, qL = liquid-phase volume flow rate, qG = gas-phase volume flow rate, and S = pipe cross-sectional area. Note that either Equations 23.1 or 23.2 can be employed to calculate the pressure drop. The functional relationships for YL and YG in Equations 23.3 and 23.4 in terms of X were also provided by Lockhart and Martinelli [2] for the phase classification under different flow conditions. (These relationships are provided later in this

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chapter.) For gas-liquid flows, semi-empirical data were provided for the following three flow categories: 1. Gas (turbulent flow) – liquid (turbulent flow), tt 2. Gas (turbulent flow) – liquid (viscous flow), tv 3. Gas (viscous flow) – liquid (viscous flow), vv The next three subsections address each of the above topics. Note that applications involving gas (viscous flow) – liquid (viscous flow) do not receive treatment since this type of flow rarely occurs in environmental engineering practice; the low viscosity of a gas (or vapor) virtually eliminates the possibility of gas moving in a laminar flow. A variety of the above flow phenomena is possible with the two-phase flow of gases and liquids in horizontal pipes ranging from parallel (two-layer) flow at low velocities to dispersed flow at high velocities (gas carried as bubbles in a continuous liquid phase or liquid carried as spray in the gas). The pressure drop is greater in liquid – gas flow than that for the single-phase flow of either gas or liquid for several reasons. These include the irreversible work done on the liquid by the gas and that the effective cross-sectional area of flow for either phase is reduced by the flow of the other phase in the area. The basis for the Martinelli et al. correlations [2,3] assumes that the pressure drop for the liquid phase must equal the pressure drop for the gas phase for all types of flow, provided that no appreciable pressure differences exist across any pipe diameter and that the volume occupied by the liquid and by the gas at any instant of time must equal the total volume of the pipe. Using these assumptions, the pressure drop due to the liquid flow and that due to the gas flow was expressed in each case by standard pressure drop equations using unknown “hydraulic diameters.” The hydraulic diameters were then expressed in terms of the actual cross-sectional area of flow and the ratio of the actual cross-sectional area of flow to the area of a circle of diameter equal to the unknown hydraulic diameter. The unknown hydraulic diameter for the liquid flow was eliminated in the analysis and an expression was obtained for the pressure drop as a function of the single-phase pressure drop for gas alone. The function, expressed as 2 in their study, was introduced in order to reduce the range of the variables when providing 2 versus X information. Isothermal flow in smooth pipes was assumed. It is important to know what type of flow is occurring, although this can obviously be a difficult task. In order to establish which flow mechanisms applied. Martinelli et al. [2,3] used a set of flow conditions (as noted above) that were functions of the Reynolds number

Re

4w D

(23.9)

where w is the mass flow rate. Martinelli et al. [2,  3] computed the Reynolds number for each using the actual pipe diameter; i.e., a superficial velocity was

Two-Phase Flow 229 employed. For Re < 2000, the flow for that phase was assumed to be viscous (laminar); for Re > 2000, the flow is assumed to be turbulent. The volume fraction or holdup of a phase for two-phase flow in a horizontal pipe is also available [1]: L

F3 ( X )

(23.10)

G

F4 ( X )

(23.11)

where G + L = 1 and G and L are, the fraction (dimensionless) of pipe volume occupied by the liquid phase and gas phase, respectively; X is the aforementioned variable defined by Equation 23.5. The relationship between L and X is approximately provided by [4]: L

0.298 0.117 ln( X )

(23.12)

Gas-liquid flow usually occurs in horizontal pipes. However, when gas-liquid mixtures flow in vertical pipes, there is an increase in liquid concentration or build-up of liquid due to the density difference in the case of upward flow, and a decrease in liquid concentration in the case of downward flow. Since information is available on the upward flow of gas-liquid mixtures, a variety of flow phenomena are possible including gas as the dispersed phase in a continuous liquid phase to gas as the continuous phase with liquid carried as spray. One of the intermediate types of flow is where the liquid flows as an annulus and the gas as a central core. The major applications are gas lifts. A gas lift is a vertical pipe (known as an eduction pipe) open at both ends, part of which is submerged below the surface of the liquid to be pumped. Compressed gas is admitted through a foot-piece inside the lower end; a mixture of liquid and gas is thus formed within the pipe. The gas reduces the average density of the mixture in the eduction pipe to a point where the weight of the mixture is less than the pressure at the foot-piece. With the gas and liquid being supplied at a sufficient rate, the mixture rises upward through the pipe and is discharged at the upper end. Industrial and environmental applications occur with the operation of flowing oil wells. Considerable operating and experimental data have been reported but little attempt has been made to correlate them.

23.3 Gas (Turbulent) Flow-Liquid (Turbulent) Flow, tt This section provides additional details of the original work of Lockhart and Martinelli [2, 3]. This is followed by a simpler approach for predicting pressure drop. The simpler approach is recommended for industrial applications. In the original work (with most of the notation retained), the ratio of the actual cross-sectional area of flow to the area of a circle of diameter equal to the unknown equivalent (or hydraulic) diameter for the gas phase was assumed to be unity and

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the ratio for the liquid phase was determined from experimental data. The following correlations were obtained from the ratio for the liquid phase and the properties of the liquid and gas

P Z where

tt

P Z

tt tt

(23.13) G

is a function of a dimensionless group, Xtt; and, 0.111

Xtt

0.555

L

G

G

L

mL mG

(23.14)

The magnitude of tt for values of Xtt is given in Table 23.1. Results were later expressed in terms of YL and YG, both of which are functions of Xtt (see Equations 23.3 and 23.4 for more details). VanVliet [5] subsequently regressed the data to a model of the form

YG

cXtt2

a bXtt

dXtt3

(23.15)

and

YL

aXttb

(23.16)

The final results for YL and YG for tt are available in VanVliet’s work. As noted earlier, the pressure drop for two-phase flow can be calculated using either Equation 23.1 or 23.2. Longhand calculations have shown [5] that the two equations can, in some cases, produce different results. The authors recommend using either the gas phase value or the average of the two for design purposes. However, if the volume fraction of one phase predominates (see Equation 23.12), the authors suggest employing the pressure drop calculation for that phase.

23.4 Gas (Turbulent) Flow-Liquid (Viscous) Flow, tv The original work of Lockhart and Martinelli [2] is once again reviewed for the turbulent-viscous (tv) case. Using the same procedures as that for the turbulent– turbulent (tt) case, the final correlation took the form:

P Z

2 tv tv

P Z

(23.16) G

Two-Phase Flow 231 Table 23.1

tt

vs

Xtt . ftt

X tt 0

1.00

0.10

1.50

0.20

1.68

0.40

2.13

0.70

3.03

1.00

4.08

2.00

8.30

4.00

19.6

7.00

42.3

10.0

71.0

20.0

222

40.0

770

46.2

1000

for

X tt

46

tt

( X tt )1.8

where tv is a function of a dimensionless group, Xtv. The magnitude of tv for values of Xtv is provided in Table 23.2. The results of Table 23.2, which are functions of Xtv were expressed in terms of YG and YL. VanVliet [5] subsequently regressed data to equation form.

23.5

Gas (Viscous) Flow-Liquid (Viscous) Flow, vv

The same procedure was employed by Lockhart and Martinelli [2, 3] for this flow regime as in the preceding two cases except that both liquid and gas ratios of the actual cross-sectional area of flow to the area of a circle of diameter equal to the unknown hydraulic diameter for the gas phase were determined experimentally in capillary tubes. Their correlation was expressed as

P Z

2 vv vv

P Z

(23.17) G

where vv is a function of a dimensionless group, Xvv. The values of Xvv is provided in Table 23.3.

vv

for values of

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Table 23.2

Xtv versus

tv

. ftv

Xtv 0

1.00

0.07

2.00

0.10

2.14

0.20

2.46

0.40

2.96

0.70

3.42

1.0

3.85

2.0

5.30

4.0

7.87

7.0

11.3

10.0

14.8

20.0

25.4

40.0

46.0

70.0

75.8

100

105

200

203

400

400

1000

1000

for

Xtv > 1000

tv

Xtv

The results in Table 23.3 were later expressed in terms of YL and YG, both of which arc functions of Xvv (see Equations 23.3 and 23.4 for more details). VanVliet [5] subsequently regressed the data to the equation form shown below:

YG(vv) = 1.1241 + 3.7085 Xvv + 6.7318 Xvv2 – 11.541 Xvv3; Xvv < 1 (23.18) YG(vv) = 10 – 10.405 Xvv + 8.6786 Xvv2 – 0.9167 Xvv3; 1< Xvv 10 (23.20) YL(vv) = 3.9794 Xvv YL(vv) = 6.4699 Xvv

1.6583

; Xvv < 1

(23.21)

; 1< Xvv 10

(23.23)

Two-Phase Flow 233 Table 23.3

X vv versus

vv

. f vv

X vv 0.20

1.40

0.40

1.69

0.60

1.93

0.80

2.16

1.0

2.44

2.0

3.81

3.0

5.15

4.0

6.40

6.0

8.70 (limit of experimental data)

vv

X vv

23.6 Gas-Solid Flow There are many gas-solid flow systems in pipes but this section solely addresses pneumatic conveyance. Conveying material pneumatically has been used for many years. The system can be either a pressure system or a suction system. The materials that have been handled via pneumatic conveyance include grain, wood shavings, pulverized coal, cement, staple, plastic chips, small metal parts, and money containers in department stores. Pneumatic conveyors are simple, quiet, convenient, and clean; however, pneumatic conveyors operate at a much lower efficiency than belt or bucket type conveyors. In a pressurized system, material can be fed by a screw conveyor or similar feeder and then forced through the system by compressed air, or the material can be fed into a tank and then forced through the system by compressed air. In a suction system, a fan or blower is installed after the separating system, thereby putting the entire system under vacuum. The material, with sufficient air to keep the material in suspension, is then drawn or “sucked” through the system. Occasionally, it is more convenient to use a combination of pressure and suction systems. In a combination system, the material is drawn in, passes through the fan and then under pressure is forced through the remainder of the system. For cases where the material may damage the fan, an ejector may be used in place of the fan. Also, using an ejector, the material can be fed in the mixing throat of the ejector.

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It should be noted that in the design of pneumatic conveying systems is based on practical operating experience and empirical correlations of test data [6-9]. 23.6.1 Solids Motion The path of solids in a horizontal pipe is somewhat sinusoidal, with the solids striking the bottom of the pipe at intervals and then rising again. The height and length of the rise appears to decrease as the air velocity decreases. The vertical distribution of solids across the pipe diameter is fairly uniform at low concentrations but becomes more dense at the bottom of the pipe as the loading (ratio of weight rate of solids to weight rate of air) increases. Finally, at high loadings, a considerable portion of the solids lie along the bottom of the pipe. The difference between the final average velocity of the solids and that of the air stream is almost constant for both horizontal and vertical conveyors. This difference is the “slip” between the solids and the air and increases with increasing velocity of the air stream. This “slip” velocity is of the order of magnitude of the “choking” velocity and is essentially the minimum transport or conveying velocity. For estimation purposes, the “slip” velocity may be taken as equal to the “choking” velocity. It has been reported that the “choking” velocity is independent of loading for relatively large particles. The minimum transport velocity of a material can be estimated by determining the minimum air velocity to convey the solids in a horizontal tube and the minimum air velocity to just suspend the solids in a vertical tube. The minimum transport velocity of the solids may be several times the free fall velocity. 23.6.2 Pressure Drop The total pressure drop in a system can be considered to consist of the sum of the following pressure drops needed to: 1. accelerate the air to the carrying velocity 2. overcome the friction of the air on the pipe walls 3. supply the loss of momentum of the air in: a. accelerating the solids b. keeping the solids in suspension 4. support the air (vertical pipes) 5. support the solids (vertical pipes) The total pressure drop for horizontal pipes, PTH, is given by

PTH

PAG

PAS

PF

(23.24)

where PAG = pressure drop to accelerate the air, PAS = pressure drop to accelerate the solids, and vPF = pressure drop due to the friction of moving air.

Two-Phase Flow 235 For vertical pipes, the pressure drop PTV is

PTV

PAG

PAS

PF

PV

PTH

PV

(23.25)

where PV = pressure drop to support the air and solid. Details of these calculations are available in the literature [10]. 23.6.3 Design Procedure In an actual design, the quantity of material to be conveyed and the distance are generally known. One can then assume a loading and conveying velocity, and then diameter of the pipe can be computed. Finally, the pressure drop through the system is computed. If the pressure drop is excessive, a smaller loading can be used and the above procedure is repeated until a reasonable pressure drop is obtained. There is no reliable method to accurately calculate the conveying velocity; however, a conveying velocity of 70 ft/s can be assumed in lieu of any other information. Some order of magnitude values of loading, conveying velocities, and pressure drops for various systems are outlined below [11]: 1. Fan systems pressure drop = 10 to 30 in H2O (50 in H2O is a practical maximum) loading = 0.1 to 2.0 (possibly 5.0) lb solids/lb air conveying velocity = 30 to 100 ft/s; usually 50 to 70 ft/s generally used for distances less than 200 ft 2. Vacuum systems pressure drop = 5 to 10 in Hg loading = 5 to 20 lb solids/lb air conveying velocity = (same as for fan systems) 3. Pressure systems pressure drop = 10 to 50 psia (possibly as high as 100 psia) loading = 5.0 to 40 lb solids/lb air conveying velocity = (same as for fan systems) Large radius bends are recommended for all systems as the pressure drop will be less than with tight bends and it will also be less likely for the solids to collect and choke the bend. 23.6.4 Pressure Drop Reduction in Gas Flow Scattered statements in the literature seem to suggest that the pressure to convey a gas can be reduced by the addition of fine particles to the moving stream. This is an area that requires more research since the pressure drop reduction effect is a function of both the particle size (and/or particle size distribution) and concentration.

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References 1. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York City, NY, 2008. 2. Lockhart, R., and Martinelli, R., Generalized Correlation of Two-Phase, Two-Component Flow Data, CEP, 45, 39-48, 1949. 3. Martinelli, R., Putnam, J.A., and Lockhart, R.W., Two-Phase, Two-Component Flow in the Viscous Region, Trans. AIChE, 42, 681-705, 1946. 4. Ricci, F., Princeton University, Princeton, NY, personal correspondence to L. Theodore, 2008. 5. VanVliet, T., personal correspondence to L. Theodore, 2008. 6. Gasterstadt, J., Die Experimentelle Untersuchung des Pneumatischen Fordevorganges, Z.V. D.I., 68, 617-624, 1924. 7. Gasterstadt, J., Die Experimentelle Untersuchung des Pneumatischen Fordervorganges, Forschungsarb Gebiete Ingenieurw, 265, 1924. 8. Wood, S., and Bailey, A., The Horizontal Carriage of Granular Material by an Injector-Driven Air Stream, Proc. Inst. Mech. Engrs. (London), 142, 149-164, 1939. 9. Mills, D., Pneumatic Conveying Design Guide, 2nd Edition, Elsevier, Butterworth-Heinemann, San Francisco, CA, 2004. 10. Dalla Valle, J., Determining Minimum Air Velocities in Exhaust Systems, Heating, Piping & Air Conditioning, 4, 639-641, 1932. 11. Lapple, C., Fluid and Particle Mechanics, John Wiley & Sons, Hoboken, NJ, 1948.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

24 Ventilation

24.1 Introduction Indoor air pollution is rapidly becoming a major health issue in the United States. Indoor pollutant levels are quite often higher than outdoors, particularly where buildings are tightly constructed to save energy. Since most people spend nearly 90% of their time indoors, exposure to unhealthy concentrations of indoor air pollutants is often inevitable. The degree of risk associated with exposure to indoor pollutants depends on how well buildings are ventilated and the type, mixture, and amounts of pollutants in the building [1]. Industrial ventilation is the field of applied science concerned with controlling airborne contaminants to produce healthy conditions for workers and a clean environment for the manufacture of products. However, to claim that industrial ventilation will prevent contaminants from entering the workplace is naïve and unachievable. More to the point, and within the realm of achievement, is the goal of controlling contaminant exposure within prescribed limits. To accomplish this goal, one must be able to describe the movement of contaminants in quantitative terms that take into account: 1. The spatial and temporal rate at which contaminants are generated 2. The velocity field of the air in the workplace 237

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Unit Operations in Environmental Engineering 3. The spatial relationship between source, workers, and openings through which air is withdrawn or added 4. Exposure limits (time-concentration relationships) that define unhealthy conditions [1]

In general, most control/recovery equipment is more economically efficient when handling higher concentrations of contaminants, all else being equal. Therefore, the gas handling system should be designed to concentrate contaminants in the smallest possible volume of air. This is important since, exclusive of the fan, the cost of the control equipment is based principally upon the volume of gas to be handled and not on the quantity of contaminant to be removed. The reduction of emissions by process and system control is an important adjunct to any cleaning technology. Although ventilation does not remove contaminants from the workplace, it does provide an opportunity to either recover/control or dilute (into the atmosphere) any problem emissions. Regarding manufacture and production, some unions and European nations either require or recommend that closed operations be employed. The reader is referred to the classic work of Heinsohn [2] for an excellent treatment of ventilation. In addition, it should be noted that some of the materials in this chapter were excerpted and/or adapted from publications [3,4] resulting from NSF sponsored faculty workshops.

24.2

Indoor Air Quality

Indoor air quality (IAQ) is a major concern because indoor air pollution may present a greater health risk to humans than exposure to outdoor pollutants since people spend nearly 90% of their time indoors. This situation is compounded as sensitive populations, i.e., the very young, the very old, and sick people, who are potentially more vulnerable to disease, spend many more hours indoors than the average population. Indoor air quality problems have become more serious and of greater concern now than in the past because of a number of developments that are believed to have resulted in increased levels of harmful chemicals in indoor air. Some of those developments are: the construction of more tightly sealed buildings to save on energy costs, the reduction of the ventilation rate standards to save still more energy, the increased use of synthetic building materials and synthetics in furniture and carpeting that can release harmful chemicals, and the widespread use of new pesticides, paints, and cleaners. Some of the immediate health effects of IAQ problems are irritation of the eyes, nose and throat, headaches, dizziness and fatigue, asthma, pneumonitis, and “humidifier fever.” Some of the long-term health effects of IAQ problems are respiratory diseases and cancer. These affects are most often associated with radon,

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asbestos, and second-hand tobacco smoke. The U.S. EPA, in a report to Congress, estimated that the costs of IAQ problems were in the tens of billions of dollars per year. The major costs from IAQ problems are direct medical costs, lost productivity due to absence from the job because of illness, decreased efficiency on the job, and damage to materials and equipment.

24.3 Indoor Air/Ambient Air Comparison Outdoor air pollution and indoor air pollution share many of the same pollutants, concerns, and problems. Both can have serious negative impacts on the health of the population. Not too many years ago, it was a common practice to advise people with respiratory problems to stay indoors on days when pollution outdoors was particularly bad. The assumption was that the indoor environment provided protection against outdoor pollutants. Recent studies conducted by the U.S. EPA have found, however, that the indoor levels of many pollutants are often two to five times, and occasionally more than 100 times, higher than corresponding outdoor levels. Indoors is where most people work, attend school, eat, sleep, and even where much of their recreational activity takes place. Among the consumer and commercial products that release pollutants into the indoor air are pesticides, adhesives, cosmetics, cleaners, waxes, paints, automotive products, paper products, printed materials, air fresheners, dry cleaned fabrics, and furniture. In addition to the “active” ingredients in all the products mentioned, many products contain so-called inert ingredients that can also be contaminants when released into indoor air. Examples include solvents, propellants, dyes, curing agents, flame retardants, mineral spirits, plasticizers, perfumes, hardeners, resins, binders, stabilizers, and preservatives. Aerosol products produce droplets which remain in the air long enough to be inhaled. This allows the inhalation of some chemicals that would not be volatile enough to be inhaled otherwise. One consumer product that produces indoor air contaminants and merits special mention is tobacco smoke. The single most important contaminant from building or structural sources is formaldehyde contained in building materials such as plywood, adhesives, insulating materials such as urea formaldehyde foam, floor, and wall coverings. Depending on the individual type of construction and maintenance practices, there can be many other building sources of IAQ problems. Damp or wet wood, insulation, walls, and ceilings can be breeding places for allergens and pathogens that can become airborne. Allergens and pathogens can also originate from poorly maintained humidifiers, dehumidifiers, and air conditioners. If the building has openings to the soil, radon can enter the building in those areas where radon occurs. The building’s heating plant can be the source of contaminants such as carbon monoxide (CO) and nitrogen oxides (NOx). Automobile exhaust from attached garages is another source of CO and NOx. Particulates such as asbestos from crumbling insulation and lead from the sanding of lead-based paints are additional contaminants that can become part of the indoor air.

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Personal activities can be sources of indoor air contaminants such as pathogenic viruses and bacteria, and a number of harmful chemicals such as products of human and animal (pet) respiration. House plants can release allergenic spores. Pet products, such as flea powder, can be sources of pesticides and pets produce allergenic dander when pets groom themselves. Outdoor sources of indoor air contaminants are widely varied. Polluted outdoor air can enter the indoor space through open windows, doors, and ventilation intakes. Most outdoor air is less contaminated than indoor air. In some cases, however, such as with a nearby smokestack, a parking lot, heavy street traffic, or an underground garage, outdoor air can be a significant source of indoor contaminants. Outdoor pesticide applications, barbecue grills, and garbage storage areas can also bring outdoor contaminants into the building if placed close to a window or door, or the intake of a ventilation system. Improper placement of the intake of a ventilation system near a loading dock, parking lots, the exhaust from restrooms, laboratories, manufacturing spaces, and other exhausts of contaminated air is a major source of indoor air pollution. Other outdoor sources of indoor air pollutants include hazardous chemicals entering the structure from the soil. Examples are the aforementioned radon gas, methane and other gases from sanitary landfills, and vapors from leaking underground storage tanks of gasoline, oil, and other chemicals penetrating into basements. Finally, polluted water can give off substantial quantities of harmful chemicals during showering, dishwashing, and similar activities.

24.4 Industrial Ventilation Systems The major components of an industrial ventilation system include the following: Exhaust hood Ductwork Contaminant control device Exhaust fan Exhaust vent or stack Several types of hoods are available. One must select the appropriate hood for a specific operation to effectively remove contaminants from a work area and transport them into the ductwork. The ductwork must be sized such that the contaminant is transported without being deposited within the duct; adequate velocity must be maintained in the duct to accomplish this. Selecting a control device that is appropriate for contaminant removal is important to meet certain regulatorally defined pollution control removal efficiency requirements. The exhaust fan is the workhorse of the ventilation system. The fan must provide the volumetric flow at the required static pressure, and must be capable of handling contaminated air

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characteristics such as dustiness, corrosivity, and high moisture content. Properly venting the exhaust out of the building is equally necessary to avoid contaminant recirculation into the air intake or into the building through other openings. Such problems can be minimized by properly locating the vent pipe in relation to the aerodynamic characteristics of the building. In addition, all or a portion of the cleaned air may be recirculated to the workplace. Primary (outside) air may be added to the workplace and is referred to as makeup air; the temperature and humidity of the makeup air may have to be controlled. It also may be necessary to exhaust a portion of the room air. A line diagram of a typical industrial ventilation system is provided in Figure 24.l. Note that either the control device or the fan (or both) can be located in the room/workplace. Exposure to contaminants in a workplace can be reduced by proper ventilation. Ventilation can be provided either by dilution ventilation or by a local exhaust system. In dilution ventilation, air is brought into the work area to dilute the contaminant sufficiently to minimize its concentration and subsequently reduce worker exposure. In a local exhaust system, the contaminant itself is removed from the source through hoods. A local exhaust is generally preferred over a dilution ventilation system for health hazard control because a local exhaust system removes the contaminants directly from the source, whereas dilution ventilation merely mixes the contaminant with uncontaminated air to reduce the contaminant’s concentration. Dilution ventilation may be acceptable when the contaminant has low toxicity, and the rate of contaminant emission is constant and low enough that the quantity of required dilution air is not prohibitively large. However, dilution ventilation is generally not viable when the acceptable contaminant concentration is less than 100 ppm. To stack

Duckwork

Control equipment Return/ recirculation air

Makeup air Source

Figure 24.1 Industrial ventilation system.

Room/workplace

Discharge/ exhaust air

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In determining the quantity of dilution air required, one must also consider mixing characteristics of the work area in addition to the quantity (mass or volume) of contaminant to be diluted. Thus, the amount of air required in a dilution ventilation system is much higher than the amount required in a local exhaust system. In addition, if the replacement air requires heating or cooling to maintain an acceptable workplace temperature, then the operating cost of a dilution ventilation system may further exceed the cost of a local exhaust system. The amount of dilution air required in a dilution ventilation system, q, can be estimated using the following expression:

q = K(qc/Ca)

(24.1)

where K = an empirical dimensionless mixing factor; qc = flowrate of pure contaminant vapor; and Ca = acceptable contaminant concentration. Infiltration and exfiltration refer to the uncontrolled leakage of air into or out of a building through cracks and other unintended openings in the outer shell of the building. In addition to leakage around windows and doors, infiltration and exfiltration can also occur at openings for pipes, wires, and ducts. The rate of infiltration and exfiltration can vary greatly depending on such factors as wind temperature differences between indoors and outdoors, as well as the operation of stacks and exhaust fans. Wind effects result from wind striking one side of a building causing positive pressure on that side and lower pressure on the opposite side (the leeward side). Air is forced into the building on the windward side and out the leeward side. However, some buildings may be somewhat protected from wind effects by terrain, trees, and other buildings. The tendency of warm air to rise in a room or through the levels of a multilevel building result in what is known as stack effects. In winter, when there is a large temperature difference between indoor and outdoor air, rising warm air escapes through openings at the top of the building and outdoor air is drawn in at the bottom of the building. The effect is usually less pronounced in the summer because of smaller temperature differences and the direction of the flow may be reversed. Combustion effects often arise from fires in fireplaces, stoves, and heating systems. Combustion uses up indoor air (oxygen), which can cause the pressure in the building to drop. Outdoor air is then drawn in. This effect can double infiltration rates. Use of outdoor air in a heating system or fireplace substantially reduces this effect. Forced ventilation refers to drawing air into a building through fans and ducts. The effectiveness in removing contaminants from indoor air by the use of forced ventilation can vary widely. Fans used to exhaust specific sources of pollutants (such as the kitchen stove) can be very effective. Most forced ventilation systems are used to circulate air-conditioned air. Whole house fans and the forced ducted ventilation systems of large buildings must be carefully balanced by the air supply to prevent backdrafts of contaminants from stacks and heating plants.

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24.5 Describing Equations Theodore [5–6] and Reynolds et al. [7] have developed mathematical models describing the concentration of a chemical in a medium-sized ventilated laboratory room. The following information/data (SI units) were provided: = volume of room, m3 = volumetric flow rate of ventilation air, m3/min = concentration of the chemical in ventilation air, gmol/m3 = concentration of the chemical leaving ventilated room, gmol/m3 = concentration of the chemical initially present in ventilated room, gmol/m3 r = rate of disappearance of the chemical in the room due to chemical reaction and/or other effects, gmol/m3-min

V qo co c c1

Using the laboratory room as the control volume, one may apply the conservation law for mass to the chemical to yield:

rate of mass in

rate of mass out

rate of mass generated

rate of mass accumulated (24.2)

Employing the notation specified above gives:

{rate of mass in} = qoco

(24.3)

{rate of mass out} = qoc

(24.4)

{rate of mass generated} = rV

(24.5)

{rate of mass accumulated}

Vdc dt

(24.6)

Substituting also gives

qo co qo c rV

Vdc dt

(24.7)

Dividing both sides by V and rearranging leads to:

q0 (co c) r V

dc dt

(24.8)

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The term V/q0 represents the average residence time that the chemicals reside in the room and is usually designated as . The above equation may then be rewritten as:

co c

dc dt

r

(24.9)

Thus if r = 0,

dc dt

co c

(24.10)

If the reaction rate is zero order, r = –k, then

dc dt

co c

k

co

k

c

co k

c

(24.11)

If the reaction rate is first order, r = –kc, then

dc dt

co c

kc

c0

c

kc

co

c

k

1

(24.12)

Integrating Equation 24.12 and solving for c yields:

c ci e

(t / )(1 k )

co [1 e 1 k

(t / )(1 k )

]

(24.13)

If k = 0

c co (ci co )e

(t / )

(24.14)

Theodore [5–6] and Reynolds et al. [7] provide a host of illustrative examples demonstrating the application of the above equations to real systems.

References 1. Theodore, L., and Kunz, R., Nanotechnology: Environmental Implication and Solutions, John Wiley & Sons, Hoboken, NJ, 2005. 2. Heinsohn, R., Industrial Ventilation: Engineering Principles, John Wiley & Sons, Hoboken, NJ, 1991 3. Dupont, R., Baxter, T., and Theodore, L., Environmental Management: Problems and Solutions, CRC-Lewis Publishers, Boca Raton, FL, 1998.

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4. Ganesan, K., Theodore, L., and Dupont, R., Air Toxins: Problems and Solutions, Gordon and Beach, New York City, NY, 1996. 5. Theodore, L., Nanotechnology: Basic Calculations for Engineers and Scientists, John Wiley & Sons, Hoboken, NJ, 2007. 6. Theodore, L., Chemical Reaction Kinetics, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by the U.S. EPA/APTI, RTP, NC, 1992. 7. Reynolds, J., Jeris, J., and Theodore, L., Handbook of Chemical and Environmental Engineering Calculations, John Wiley & Sons, Hoboken, NJ, 2004.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

25 Mixing

25.1 Introduction The subject of mixing has not received the attention it deserves in the literature. Although mixing plays an important role in many environmental engineering applications, little to no advances have occurred in the past half century. The material in this chapter has been primarily drawn from the work of Rich [1] and Perry and Green [2]. Although dated, the development presented in these references still applies today. Interestingly, the terms “mixing” and “agitation” are often used interchangeably. “Mixing,” however, describes more precisely an operation in which two or more materials are intermingled to attain a desired degree of uniformity. On the other hand, “agitation” applies to those operations, the primary purpose of which is to promote turbulence. For example, agitation for the purpose of promoting floc growth in suspension is commonly called “flocculation.” Mixing finds application for various combinations of phases, i.e., gas, liquid, and solid. These combinations include: Gas-liquid Gas-solid Liquid-liquid 247

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The bulk of the applications in environmental engineering are with gas-liquid systems related to reactor aeration, water disinfection, or gas stripping; and liquidsolid systems as in anaerobic sludge digestion in wastewater treatment. The usual objective of a mixer is to obtain uniformity within a reactor/equipment/system with respect to composition and/or temperature, or to reduce mass transfer limitations between phases. Statistical methods are available for describing uniformity of the degree of mixing and for controlling such operations. Methods of sampling, calculation, and interpretation of data in mixing operations (including particulate solids, and slurries), are available in the literature [3]. These complex methods are often simplified in practice. For example, a range or spread in a particular property may be used to describe uniformity in terms of standard deviation. In a batch reactor, the time to bring its composition or properties within a specified range or spread in values is often used as a measure of mixing effectiveness. Mixing and agitation also find wide application in environmental engineering treatment processes. The suspension of solids in water treatment and wastewater sludge systems, the dispersion of gases in water and wastewater treatment systems, and the flocculation of precipitates from liquids constitute the bulk of such operations. An analysis of basic principles permits the rational development of design criteria for these varied mixing systems. The material to follow applies to both batch and continuous operation; it also primarily examines the “mixing” element as opposed to the constraining vessel.

25.2 Mixing Impellers Many types of devices are used in mixing and agitation operations. These include rotating impellers, air agitators, mixing jets, and pumps. The following discussion is confined to rotating impellers, the type most commonly used in environmental engineering. Impellers fall into one of three groups: paddles, turbines, and propellers. One or more impellers in each group can be mounted on a shaft, and more than one shaft may be used in a given reactor. 25.2.1

Paddles

Paddle impellers range in design from a single, flat paddle on a vertical shaft to a battery of multi-blade flocculators mounted on a long, horizontal shaft. Paddles turn at slow to moderate speeds (2 to 150 rpm), and are used primarily as agitators of dilute suspensions or as mixers in high-viscosity applications. The principal currents generated are radial and tangential to the rotating paddles.

Mixing 25.2.2

249

Turbines

The “turbine impeller” is a term applied to a wide variety of impeller shapes. The turbine impeller is used in many mixing applications. This type of impeller consists of several straight blades mounted vertically on a flat plate. The blades of some turbine impellers are curved or tilted from the vertical. Rotation is at moderate speeds, and fluid flow is generated in a radial and tangential direction. 25.2.3 Propellers The marine-type propeller is a relatively small, high-speed impeller widely used in low-viscosity applications. It has a high rate of flow displacement and generates strong currents in an axial direction. Speed of rotation varies from 400 rpm for large-diameter propellers to 1750 rpm for those having smaller diameters. Each impeller type has its own group of services for which it is particularly suited. Although there are situations where two impeller types will perform equally well, in general, a given service will be satisfied best by only one propeller type.

25.3 Baffling The tangential component of flow induced by a rotating impeller promotes rotational movement or vortexing about the impeller shaft. Vortexing impedes the mixing operation by reducing the velocity of the impeller relative to the liquid. Furthermore, impeller power consumption is more difficult to calculate when vortexing conditions occur. Vortexing can be reduced by proper baffling. Vertical strips placed along the walls of the tank serve to break up rotational movement by deflecting the liquid back toward the impeller shaft. For turbine mixing operations, the width of the baffles may be as small as one-tenth to one-twelfth the tank diameter; for propeller operations, even smaller widths can be used. Seldom are more than four baffles necessary for these types of impellers. To reduce rotational movement in flocculation operations, stationary strips called stator blades are arranged so as to intermesh with the rotating blades of the flocculation paddles. Where propellers are used, vortexing can be reduced by inclining the impeller shaft in some position off-center of the tank or by hsving the fluid enter the tank horizontally through the side at some angle with the tank radius. A more detailed discussion of baffling and flow patterns is available in the literature [1–3].

25.4 Fluid Regimes A rotating impeller establishes within a fluid mass a flow pattern governed not only by the shape, size, and speed of the impeller but also by the characteristics

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of the fluid container and the presence of baffling. When flow is laminar, there is no mixing within the fluid other than that due to diffusion. When flow is turbulent, however, fluid “particles” move in all directions, and mixing results primarily from convective displacement. The momentum transfer associated with such displacement generates strong shear stresses within the fluid. The term “fluid regime” refers to the flow pattern and the over-all summation of the mass flowshear relationships existing in a fluid in motion. Ordinarily, both mass flow and turbulence – or their result, fluid shear – are important in mixing operations. Most turbulence is produced as the result of highvelocity flow streams coming into contact with those of lower velocity. Flow along the sides of the container, the impeller blades, and across baffles contributes turbulence to a much lesser degree [4]. The relative importance of mass flow and fluid shear will depend upon the application, as there is a direct correlation between the fluid regime and the results desired. For this reason, the design of a mixing operation involves, first, an identification of the particular fluid regime required and, second, the design of a system for producing this regime. A fluid regime can be identified either in terms of the relationship existing between the forces involved in sustaining and resisting the regime or by some index such as power input per unit volume of liquid necessary to produce a given process result. Identification by the first method is complete in as much as the method provides for geometric, kinematic, and dynamic similarity in scaling-up operations. The second method is less complete, providing only for geometric and kinematic similarity at best.

25.5 Power Curves When a fluid regime is established by a rotating impeller, the major forces generated in the fluid are the inertial forces characterized by the power number, the viscous forces represented by the Reynolds number, and the gravitational forces described by the Froude number. These numbers consist of the following dimensionless groups:

Power number

NP

Reynolds number [3] N Re

Froude number

N Fr

Pg c n3 D 5 D 2n

Dn2 g

(25.1)

(25.2)

(25.3)

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where P = power, (ft)(lbforce)/(sec); gc = Newton’s law conversion factor, 32.17 ft-lbmass/ (sec2-lbforce); = mass density of fluid, lbmass/ft3; n = rps; D = diameter of impeller, ft; = absolute viscosity of fluid, lbmass/(s-ft); and g = acceleration of gravity, ft/sec2. The general relationship existing among the groups was shown [5] to be:

NP

K (N Re ) p (N Fr )q

(25.4)

where K is an empirical constant; and p and q are exponents, the values of which depend upon the mixing situation. The gravitational forces represented by the Froude number become effective only when flow is turbulent, and then only when a vortex is formed around the axis of the impeller.

25.6

Scale-up

Identification of the optimum fluid regime to achieve the desired process result must be made from information derived from laboratory or pilot-plant investigations. Once the optimum regime has been identified, a method for scaling up the small scale operations can be used to design dynamically similar operations of the desired size. Two systems are geometrically similar when the ratios of the dimensions in one system are equal to the corresponding ratios in the other. Two other types of similitude are important in scale-up. Kinematic similarity is achieved when the fluid motion is the same in two geometrically similar systems. Systems have dynamic similarity when, in addition to being geometrically and kinematically similar, the force ratios are equal at corresponding points in the system. Precise scale-up is achieved only in systems which are dynamically similar. For a given power expenditure, the mass-flow-shear-intensity ratio can be varied by using impellers of different sizes in what otherwise would be geometrically similar systems. Therefore, one of the first considerations at the laboratory or pilot-scale is to determine the impeller-tank-diameter ratio giving the optimum process result. Inasmuch as the ratio of mass flow to shear intensity can be varied at equal power input by using geometrically similar impellers of different sizes, it was found that there is little justification for experimenting with a large variety of impeller shapes [6]. The Reynolds number is related to the shear intensity existing in a turbulent fluid. It is not surprising, therefore, that data from reactions with rates depending upon the thickness of liquid films can be correlated with the Reynolds number. This correlation, as well as its use in scaling-up operations, was demonstrated by Rushton [7]. Power requirements which must be met in the scale-up can be determined from the following relationship

P2 P1

D2 D1

( 3 m )/m

(25.5)

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The value of m is dependent upon the particular geometry of the tank, and the shape, size, and location of the impeller and other fittings in the tank. Laboratory and pilot-scale investigations should be directed not only toward an evaluation of m but also toward determining the arrangement of the tank and impeller that yields the highest value of the exponent.

25.7 Design of Mixing Equipment The principal factors which influence mixing-equipment choice are: Process requirements Flow properties of the process fluids Equipment costs Construction materials provided Environmental considerations Ideally, the equipment chosen should be that of the lowest total cost which meets all process requirements. The total cost includes depreciation on investment, operating cost such as power, and maintenance costs. Rarely is any more than a superficial evaluation based on this principle justified, however, because the cost of such an evaluation often exceeds the potential savings that can be realized. Usually optimization is based on experience with similar mixing operations. Process requirements can often be matched with those of a similar operation, but tests are sometimes necessary to identify a satisfactory design and to find the minimum rotational speed and power to meet the desired process outcome. There are no satisfactory specific guides for selecting mixing equipment because the ranges of application of the various types of equipment overlap and the effects of flow properties on process performance have not been adequately defined.

References 1. Rich, L., Unit Operations of Sanitary Engineering, John Wiley & Sons, Hoboken, NJ, 1961. 2. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 3. Metcalf and Eddy, AECOM, Wastewater Engineering, Treatment and Resource Recovery, 5th Edition, McGraw-Hill Education, New York, NY, 2013. 4. Rushton, J.H., Mixing of Liquids in Chemical Processing, Ind. Engr. Chem., 44, 2931–2936, 1952. 5. Rushton, J.H., Costich, E.W., and Evert, H.J., Power Characteristics of Mixing Impellers, Part I, Chem. Engr. Progress, 46, 395–404, 1950. 6. Rushton, J.H., How to Make Use of Recent Mixing Developments, Chem. Engr. Progress, 50, 587–589, 1954. 7. Rushton, J.H., The Use of Pilot Plant Mixing Data, Chem. Engr. Progress, 485, 1951.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

26 Biomedical Engineering

26.1 Introduction Biomedical Engineering (BME) is a relatively new discipline in the engineering profession. Terms such as biomedical engineering, biochemical engineering, bioengineering, biotechnology, biological engineering, genetic engineering, and so on, have been used interchangeably by many in the environmental engineering community. To date, standard definitions have not been created to distinguish these genres. Consequently, some have lumped them all together using the term BME for the sake of simplicity. What one may conclude from all of the above is that BME involves applying the concepts, knowledge, basic fundamentals, and approaches of virtually all engineering disciplines (not only chemical and environmental engineering) to solve specific health and health-care related problems; the opportunities for interaction between engineers and health-care professionals are therefore many and varied. On a personal note, the authors view BME as the application of engineering, mathematics, and physical sciences to principles in biology and medicine. The terms biophysics and bioengineering either involve the interaction of physics or engineering with either biology or medicine.

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Table 26.1 Fluid flow analogies in biomedical engineering. Topic

Fluid flow

Biomedical engineering

Fluid flow

Fluid

Blood

Conduit

Pipe

Blood vessels

Prime mover

Pump

Heart

Two-phase flow

Fluid-particle dynamics

Plasma-cell flow

Because of the broad nature of this subject, this introductory chapter addresses only the application of BME to the anatomy of humans, particularly the cardiovascular system, and attempts to relate four key anatomy topics to fluid flow. The reader is referred to three excellent references in the literature for an extensive and comprehensive treatment of this new discipline [1–3]. The four key cardiovascular “parts” to be discussed in subsequent sections are: 1. 2. 3. 4.

Blood Blood vessels Heart Plasma-cell flow

The relation of these topics to the general subject of fluid flow is provided in Table 26.1. Thus, from a fluid flow and engineering perspective, the cardiovascular system is comprised of a pump (heart) that generates a pressure difference driving force that involves the flow of a fluid (blood), where the fluid involves the transport of a two-phase medium (plasma and cells), through a complex network of pipes (blood vessels). As noted earlier, BME may therefore be viewed as an interdisciplinary branch of technology that is based on both engineering and the sciences. Following a section on definitions, each of the aforementioned topics receives a qualitative treatment from a fluid flow perspective.

26.2 Definitions The definition of a host of BME and BME-related terms is provided below. A onesentence (in most instances) description/explanation of each word/phase is provided. As one might suppose, the decision of what to include (as well as what to omit) was somewhat difficult. 1. Anatomy: The structure of an organism or body. 2. Aorta: The key artery of the body that carries blood from the left ventricle of the heart to organs.

Biomedical Engineering 255 3. Artery: Any one of the thick-walled tubes that carry blood from the heart to the principal parts of the body. 4. Atrium: Either the left or right upper chamber of the heart 5. Autonomic nervous system: The functional division of the nervous system that innervates most glands, the heart, and smooth muscle tissue in order to maintain the internal environment of the body. 6. Capillary: Any of the extremely small blood vessels connecting the arteries with veins. 7. Cardiac muscle: Involuntary muscle possessing much of the anatomic attributes of skeletal voluntary muscle and some of the physiologic attributes of involuntary smooth muscle tissue; sinoatrial node induced contraction of its interconnected network of fibers allows the heart to expel blood during systole. 8. Cell: An extremely small complex unit of protoplasm, usually with a nucleus, cytoplasm, and an enclosing membrane; the semielastic, selectively permeable cell membrane controls the transport of molecules into and out of a cell. 9. Chronotropic: Affecting the periodicity of a recurring action such as the slowing (bradycardia) or speeding up (tachycardia) of the heartbeat that results from extrinsic control of the sinoatrial node. 10. Circulatory system: The course taken by the blood through the arteries, capillaries, and veins and back to the heart. 11. Clot: A clot consists primarily of red corpuscles enmeshed in a network of fine fibrils or threads, composed of a substance called fibrin. 12. Corpuscle: An extremely small particle, especially any of the erythrocytes or leukocytes that are carried and/or float in the blood. 13. Cytoplasm: The protoplasm outside the nucleus of a cell. 14. Endocrine system: The system of ductless glands and organs secreting substances directly into the blood to produce a specific response from another “target” organ or body part. 15. Endothelium: Flat cells that line the innermost surfaces of blood and lymphatic vessels and the heart. 16. Erythrocytes: Red corpuscles. 17. Gland: Any organ or group of cells that separates certain elements from the blood and secretes them in a form for the body to use or discard. 18. Heart: The organ that receives blood from the veins and pumps it through the arteries by alternate dilation and contraction. 19. Homeostasis: A tendency to uniformity or stability in an organism by maintaining within narrow limits certain variables that are critical to life.

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Unit Operations in Environmental Engineering 20. Inotropic: Affecting the contractility of muscular tissue such as the increase in cardiac power that results from extrinsic control of the myocardial musculature. 21. Leukocytes: White corpuscles. 22. Nucleoplasm: The protoplasm that composes the nucleus of a cell. 23. Plasma: The fluid part of blood, as distinguished from corpuscles; its principal component is water. 24. Precapillary sphincters: Rings of smooth muscle surrounding the entrance to capillaries where they branch off from upstream metarterioles. Contraction and release of these sphincters close and open the access to downstream blood vessels, thus controlling the irrigation of different capillary networks. 25. Protoplasm: A semifluid viscous colloidal that is the living matter of humans and is differentiated into nucleoplasm and cytoplasm. 26. Pulse: The expansion-and contraction of the arterial walls that can be felt in all the arteries near the surface of the skin. 27. Stem cells: A generalized parent cell spawning descendants that become individually specialized. 28. Vein: Any blood vessel that carries blood from some part of the body back to the heart; in a loose sense any blood vessel. 29. Ventricle: Either of the two lower chambers of the heart that receive blood from the atria and pump it into the arteries.

Since the biomedical engineering and environmental engineering/biotechnology are somewhat overlapping, several additional definitions in the latter subject are listed below [4]: 1. Algae: Algae are a very diverse group of photosynthetic organisms that range from microscopic size to lengths that can exceed the height of a human. 2. Bacteria: The bacteria are tiny single-cell organisms ranging from 0.5 to 5 μm in size although some may be smaller, and a few can exceed 100 μm in length. 3. Fungi: As a group, fungi are characterized by simple vegetative bodies from which reproductive structures are elaborated; all fungal cells possess distinct nuclei and produce spores in specialized fruiting bodies at some stage in their life cycle. 4. Microorganism: A class of organisms that includes bacteria, protozoa, viruses, and so on. 5. Photosynthesis: All living cells synthesize Adenosine 5 -triphosphate (ATP), but only green plants and a few photosynthetic (or phototrophic) microorganisms can drive biochemical reactions to form ATP with radiant energy though the process of photosynthesis.

Biomedical Engineering 257 6. Protozoa: A unicellular animal existing singly or aggregating into colonies, that are usually non-photosynthetic, and are often classified further according to their capacity for and means of motility, as by pseudopods, flagella, or cilia, living primarily in water. 7. Viruses: Viruses are particles of a size below the resolution of the light microscope and are composed mainly of nucleic acid, either DNA or RNA, surrounded by a protein sheath. 8. Yeasts: Yeasts are a single-celled fungi capable of fermenting carbohydrates into alcohol and carbon dioxide.

26.3

Blood

Blood. It has been justifiably described as the “river of life.” It is the transport medium that serves as a dispenser and collector of nutrients, gases, and waste that allows life to be sustained. In terms of composition, blood is primarily composed of water. The average human contains about 5 L of blood. This 5 L volume of blood contains in numbers (approximately): 5 × 1016 red corpuscles (erythrocytes) 1 × 1014 white corpuscles (leukocytes) 2.5 × 1015 platelets (thrombocytes)

Red blood cells are typically round disks, concave on two sides, and approximately 7.5 μm in diameter. They are made in the bone marrow and have a relatively robust lifespan of 120 days. These cells take up oxygen as blood passes through the lungs, and subsequently release the oxygen in the capillaries of tissues. From a mechanics aspect, red blood cells have the ability to deform. This is important because erythrocytes often have to navigate through irregular shapes within the vasculature, as well as squeeze through small diameters such as those encountered in capillaries. In contrast, white blood cells serve a wider variety of functions. These cells can be classified into two categories based on the type of granule within their cytoplasm, and the shape of their nucleus. More specifically, the two categories are: 1) granulocytes or polymorphonuclear (PMN) leukocytes, and 2) mononuclear leukocytes. Examples of PMNs include neutrophils, eosinophils, and basophils, while those of mononuclear leukocytes include lymphocytes and monocytes. The final component of blood is plasma, and is the fluid in which the aforementioned cells remain in suspension. It is comprised primarily of water (90%), and the rest plasma protein (7%), inorganic salts (1%), and organic molecules such as amino acids, hormones, and lipoproteins (2%). The primary functions of plasma are to allow exchange of chemical messages between distant parts of the body, and maintenance of body temperature and osmotic balance.

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In terms of physical properties, the density of blood is approximately that of water, that is, 1.0 g/cm3 or 62.4 lb/ft3. The other key property is viscosity and it is approximately 50% greater than that for water. The reader is no doubt familiar with the expression “blood is thicker than water.” Well, it turns out not to be that much “thicker” than water since its viscosity is only moderately (≈50%) higher. In addition, the viscosity is a strong function of temperature; it has been reported that the viscosity increases by 50% over the 20 to 40 °C range. Although plasma (the main constituent of blood) is Newtonian, blood (with the added blood cells) is non-Newtonian.

26.4 Blood Vessels The study of the motion of blood is defined as hemodynamics. Part of the cardiovascular system involves the flow of blood through a complex network of blood vessels. Blood flows through organs and tissues either to nourish and sanitize them or to be itself processed in some sense, e.g., to be oxygenated (pulmonary circulation), filtered of dilapidated red blood cells (splenic circulation), and so on [3]. The aforementioned “river of life” flows through the piping network, which is made up of blood vessels, by the action of two pump stations arranged in series. This complex network also consists of thousands of miles of blood vessels. The network also consists of various complex branching configurations. Regarding the branching, the aorta divides the discharge from the heart into a number of main branches, which in turn divide into smaller ones until the entire body is supplied by an elaborately branching series of blood vessels. The smallest arteries divide into a fine network of still more minute vessels, defined as capillaries, which have extremely thin walls; thus, the blood comes into close contact with the fluids and tissues of the body. In the capillaries, the blood performs three functions: 1. It releases oxygen to the tissues. 2. It furnishes nutrients and other essential substances that it carries to the body cells. 3. It takes up waste products from the tissues. The capillaries then unite to form small veins. The veins, in turn, unite with each other to form larger veins until the blood is finally collected into the venal cavae from where it goes to the heart, thus completing the blood vessel circuit (see next section for additional details). This complex network is designed to bring blood to within a capillary size of each and every one of the more than approximately 1014 cells of the body. Which cells receive blood at any given time, how much blood they get, the composition of the fluid flowing by them, and related physiologic considerations are all matters that are not left up to chance [1]. Information on average radius and number of each of the various vessels has been provided by LaBarbara [5]. His data/information is given in Table 26.2. The

Biomedical Engineering 259 Table 26.2 The average radii and total numbers of conventional categories of vessels of the human circulatory system. Vessel

Average radius (mm)

Aorta

Number

12.5

1

2

159

Arterioles

0.03

1.4 × 107

Capillaries

0.006

3.9 × 109

Venules

0.02

3.2 × 108

Veins

2.5

200

Vena cava

15

1

Arteries

average diameter of blood vessels in humans has not been determined since there is a wide distribution of sizes, but based on the data in Table 26.2, one may assume the value of 150 μm for the smaller vessels. Coronary arteries are very small in diameter and can become narrow as fatty deposits (plaque) build-up. As the vessels narrow, less blood can get through to the heart when it needs it. If the heart is not getting enough oxygen to meet its needs, one may experience pain (angina). When the heart muscle does not receive enough oxygen-rich blood, it can become injured since oxygen is important to help keep the heart functioning properly. If blood flow is completely cut off for more than a few minutes, muscle cells die, causing a heart attack. The reader should note that the flow of blood through blood vessels can be safely assumed to be laminar flow. Why? For flow in circular conduits, the flow is laminar if the Reynolds number (Re = Dv / ) is less than 2,100. In nearly all biomedical applications, the numerical values for D and v are extremely small while the viscosity is large, thus resulting in Reynolds numbers significantly below 2,100. From an environmental engineering perspective, one should note that since vessel flow is laminar, the equations presented in Chapter 13 may be assumed to apply. For example, the velocity profile of the blood in the vessel may be assumed to be parabolic with the average velocity equal to one-half the maximum velocity. Equation 26.1 applies so that

P

4 fLv 2 2gcD

f

16/Re

(26.1)

For laminar flow, f is equal to: (26.2)

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so that

P

32 gvL g c D2

(26.3)

This equation may also be expressed in terms of the volumetric flow rate (setting g/gc = 1.0) as:

P

128 q L D4

(26.4)

Equation 26.4 may be rearranged to solve for q as:

q

D4 ( P) 128 L

(26.5)

The blood vessel branching discussed above may be viewed as flow through a number of pipes or conduits, an application that often arises in environmental engineering practice. If flow originates from the same source and exits at the same location, the pressure drop across each conduit must be the same. Thus, for flow through Conduits 1, 2, and 3, one may write:

P1

P2

P3

(26.6)

From a fluid flow perspective, the presence of blood vessel branching enhances the performance of the cardiovascular circulatory system. When branching involves flow into smaller diameter vessels, the oxygen has a shorter distance and shorter residence time requirement to reach the tissues that require oxygen. In addition, if the velocity through the branched vessels remains unchanged, the pressure drop correspondingly decreases. This drop is linearly related to the diameter though Equation 26.3. Finally, it should be noted that the: 1. 2. 3. 4. 5.

Blood flow is pulsating. Blood is non-Newtonian. Blood vessels vary in cross-sectional area. Blood vessels vary in shape. Items (3) and (4) vary with time.

Despite the above five limitations, efforts abound to attempt to model the flow of the “river of life” in the cardiovascular complex blood vessel network.

Biomedical Engineering 261

26.5 Heart The “river of life” flows through the cardiovascular circulatory system by the action of the heart which essentially provides two pumps that are arranged in series. (See Chapter 13’s section on flow in pumps arranged in series). In simple terms, the heart is an organ that receives blood from veins and propels it into and through the arteries. It is primarily held in place by its attachment to the arteries and veins, and by its confinement, via a double-walled sac with one layer enveloping the heart and the other attached to the breastbone. Furthermore, the heart consists of two parallel independent systems, each consisting of an atrium and a ventricle that have been referred to by some as the right heart and the left heart. The heart is divided into four chambers through which blood flows. These chambers are separated by valves that help keep the blood moving in the right direction. The chambers on the right side of the heart take blood from the body and push it through the lungs to pick up oxygen. The left side takes blood from the lungs and pumps it out the aorta. The cycle is completed when the blood is returned to the heart by the superior and inferior vena cava and enters the atrium (or chambers) on the upper right side of the heart. Regarding details of the action of the heart, blood is drawn into the right ventricle by a partial vacuum when the lower chamber relaxes after a beat. On the next contraction of the heart muscle, the blood is squeezed into the pulmonary arteries that carry it to the left and right lungs. The blood receives a fresh supply of oxygen in the lungs and is pumped into the heart by way of the pulmonary veins, entering at the left atrium. The blood is first drawn into the left ventricle and then pumped out again through the aorta (the major artery), which connects with smaller arteries and capillaries reaching all parts of the body. Thus, the cardiovascular circulatory system consists of the heart (a pump), the arteries (pipes) that transport blood from the heart, and the capillaries and veins (pipes) that transport the blood back to the heart. These form a complete recycle process. A line diagram of the recycle circulatory system is given in Figure 26.1 [5]. The cycle begins at Point 1. Oxygenated blood is pumped from the left lower ventricle (LLV) at an elevated pressure through the aorta and discharges oxygen to various parts of the body. Deoxygenated blood then enters the right upper atrium (RUA), passes through the right lower ventricle (RLV) and then enters the lungs where its oxygen supply is replenished. The oxygenated discharge from the lungs enters the left upper atrium (LUA) and returns to the lower left ventricle, completing the cycle. The concepts regarding recycle, bypass and purge presented in Chapter 3, Part I of this text readily apply to the discussion above. In addition to blood being recycled through the circulatory system, it is also being reused. Bypass occurs when part of the blood is bypassed to the liver for cleansing purposes. The purging process may be viewed as occurring during the oxygen transfer to tissues as well as the aforementioned cleansing process. To summarize, from a fluid flow/environmental engineering perspective, systemic circulation carries blood to the neighborhood of each cell in the body and

262

Unit Operations in Environmental Engineering Lung O2 intake +O2

LUA

RUA

Semilunar Valves

Antrioventricular Valves

LLV

RLV

Veins

1 Aorta

Arteries

-O2 Capillary O2 Discharge

Figure 26.1 Cardiovascular circulatory system.

then returns it to the right side of the heart low in oxygen and rich in carbon dioxide. Pulmonary circulation carries the blood to the lungs where its oxygen supply is replenished and its carbon dioxide content is purged before it returns to the left side of the heart to repeat the cycle. The driving force for flow arises from the pressure difference between the high pressure left side of the heart (systemic) to the lower pressure right side (pulmonary). This pressure difference provides the impetus for the “river of life” to flow. Thus, the heart may be viewed as a pump that has many of the characteristics of a centrifugal pump. Regarding physical properties of the heart, it is approximately the size of a clenched fist. It is inverted, conically-shaped, measuring 12 to 13 cm from base (top) to apex (bottom), 7 to 8 cm at its widest point, weighing just under 0.75 lb (less than 0.5% of a human’s body weight), and located between the third and sixth ribs. It rests between the lower part of the two lungs [3].

Biomedical Engineering 263 As with a mechanical pump, when flow rate increases, the pressure increase delivered by the pump decreases. In fact, the pump’s power is given by the product of the two terms, i.e., the pressure drop and volumetric flow rate:

hp q P

(26.7)

For the heart, hp represents the power required to maintain the recycle process that constitutes the circulatory flow of the blood in the cardiovascular system.

26.6 Plasma/Cell Flow As described earlier, blood is comprised of a fluid plasma and suspended cells that primarily include erythrocytes (red blood cells), leukocytes (white blood cells), and platelets. From a fluid dynamics perspective, blood motion can be viewed as a fluid-particle application involving two-phase flow. However, a rigorous theoretical description of flow with a concentrated suspension of particles is not available. Unfortunately, such a description is necessary for a quantitatively accurate understanding of the flow of blood in blood vessels. The blood vessels through which the blood flows have dimensions that are small enough so that the effects of the particulate nature of blood should not be ignored. Strictly speaking, the describing equations and calculations presented earlier for both flow and pressure drop are valid only under restricted conditions. The equations are not strictly valid if [6]: 1. The particle is not “very” small. 2. The particle is not a smooth, rigid sphere. 3. The particle is located “near” the surrounding walls containing the fluid. 4. The particle is located “near” one or more other particles. 5. The motion of the fluid and particle is multidimensional. 6. Brownian motion effect is significant.

References 1. Enderle, J., Blanchard, S., and Bronzino, J., Introduction to Biomedical Engineering, 2nd Edition, Elsevier/Academic Press, New York, 2000. 2. Bronzino, J., (Ed.), Biomedical Engineering Fundamentals, 3rd Edition, CRC/Taylor & Francis, Boca Raton, FL, 2000. 3. Vogel, S., Life in Moving Fluids, 2nd Edition, Princeton University Press, Princeton, NJ, 1994. 4. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 5. LaBarbara, M., Principles of Design of Fluid Transport Systems in Zoology, Science, 249. 992–1000, 1990. 6. Theodore, L., Air Pollution Control Equipment Calculations, John Wiley & Sons, Hoboken, NJ, 2009.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

Part III HEAT TRANSFER

The heat transfer part of this text was a rather unique undertaking. Heat transfer is one of the three basic tenants of chemical engineering and engineering science, and contains many basic and practical concepts that are utilized in countless industrial applications. The authors therefore considered writing a series of practical chapters on heat transfer. The material would hopefully serve as a training tool for those individuals in industry and academia involved directly, or indirectly, with heat transfer applications. Although the literature is inundated with texts emphasizing theory and theoretical derivations, the goal of this part of the text is to present the subject of heat transfer from a strictly pragmatic point-of-view. The general subject of heat transfer was rarely (if ever) included in early environmental engineering curricula. For example, in Rich’s classic 1961 Unit Operations in Sanitary Engineering text, only one of 15 chapters dealt with heat transfer. That has changed today because of the environmental engineer’s interest in energy, energy conservation, combustion, hazardous waste incineration, global warming, etc. In effect, heat transfer has become the new kid on the block in the unit operations arena, and is a topic that every environmental engineer should be proficient in. Part III contains three sections and a total of 15 chapters, and each serves a unique purpose in an attempt to treat important aspects of heat transfer. 265

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Part III chapter content includes: III-A 27. 28. 29. 30. 31. 32. III-B 33. 34. 35. 36. III-C 37. 38. 39. 40. 41.

Fundamentals Steady-State Conduction Unsteady-State Conduction Forced Convection Free Convection Radiation The Heat Transfer Equation Equipment Double-Pipe Heat Exchangers Shell-and-Tube Heat Exchangers Finned Heat Exchangers Other Heat Transfer Equipment Other Considerations Insulation and Refractory Refrigeration and Cryogenics Condensation and Boiling Operation, Maintenance, and Inspection (OM&I) Design Principles

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

27 Steady-State Conduction

27.1 Introduction Environmental engineers are involved with numerous heat transfer applications. Heat exchangers (see Chapters 33 through 36) are employed to heat and cool various fluids and wastes. Heat conduction plays an important role in the many of these real-world applications. As the temperature of a solid increases, the molecules that make up the solid experience an increase in vibrational kinetic energy. Since every molecule is bonded in some way to neighboring molecules, usually by electrical force fields, this energy can be passed through the solid. Thus, the transfer of heat may be viewed as thermal energy in transit due to the presence of a temperature difference. Heating a wire at one end eventually results in raising the temperature at the other. This type of heat transfer is called conduction and is the principle mechanism by which solids transfer heat. Fluids are capable of transporting heat in a similar fashion. Conduction in a stagnant liquid, for example, occurs by the movement of not only vibrational kinetic energy but also translational kinetic energy as the molecules move throughout the body of the stationary liquid. It is also important to note that there

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is no bulk motion with conduction. The ability of a fluid to flow, mix, and form internal currents on a macroscopic level (as opposed to the molecular mixing just described) allows fluids to carry heat energy by convection as well, a topic that receives treatment in the next two chapters.

27.2 Fourier’s Law The rate of heat flow by conduction is given by Fourier’s law [1]

Q

kA

Q

dT dx

Q A

kA

k

dT dx

dT dx

(27.1)

(27.2)

where (in English units) Q is the heat flow rate, Btu/h; k is the thermal conductivity, Btu/hr – ft-°F; A is the heat transfer area perpendicular to the flow of heat, ft2; T is the temperature, °F; and x is the direction of heat flow, ft. The term ( dT/dx) is defined as the temperature gradient and carries a negative sign since T is assumed to decrease with increasing x for Q being transferred in the positive x direction. Fourier derived his law from experimental evidence and/or physical phenomena, not from basic principles (i.e., theory). The reader should also note that Equations 27.1 and 27.2 are the basis for the definition of thermal conductivity as given in Chapter 3 (i.e., the amount of heat (Btu) that flows in a unit of time (1 h) through a unit area of surface (1 ft2) of unit thickness (1 ft) by virtue of a difference in temperature (1 °F)). Refer to Figure 27.1. Once again, the negative sign in Equation 27.1 reflects the fact that heat flow is from a high to low temperature and therefore the sign of the derivative (dT/dx) is opposite to that of the heat flow. If the thermal conductivity, k, can be considered constant over a limited temperature range ( T), Equation 27.1 can be integrated to give:

Q

kA(TH TC ) ( x H xC )

kA T x

kA T L

(27.3)

where TH is the higher temperature at point xH, °F; TC is the lower temperature at point, °F; T is the difference between TH and TC; L is the distance between points xH and xC = x, ft; and A is the area across which the heat is flowing, ft2. It should be noted that the thermal conductivity, k, is a property of the material through which the heat is passing and, as such, does vary somewhat with temperature. Equation 27.3 should therefore be strictly employed only for small values of T with an average value of the conductivity used for k.

Steady-State Conduction 269

H

Q

C

k

A

L

TH xH

>

TC


TS1, T2 > TS2). The equation is valid for fluids with Prandtl numbers from about 0.6 to 100. There may be an appreciable change in the fluid properties between the wall of the tube and the central flow if wide temperature differences are present in the flow. The Seider and Tate [1–3] equation should be used to take into account these property variations:

Nu 0.027 Re 0.8 Pr 1/3 ( / s )0.14

(29.9)

All properties are evaluated at bulk temperature conditions, except s, which is evaluated at the wall surface temperature, TS. 2. Turbulent Flow in Rough Pipes The recommended equation for this heat transfer situation is the ChiltonColburn jH analogy between heat transfer and fluid flow. It relates the jH factor (= St Pr2/3) to the Darcy friction factor, f:

jH

St Pr 2 / 3

Nu Re

1

Pr

1/ 3

f /8

(29.10)

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or

Nu (f /8) Re Pr 1/3

(29.11)

where f is obtained from the Moody chart [4]. The average Stanton number (St) is based on the fluid bulk temperature, Tm, while Pr and f are evaluated at the film temperature i.e., (TS + Tav)/2. Internal flow film coefficients are also available in the literature [1–3].

29.5 Convection Across Cylinders Forced air coolers and heaters, forced air condensers, and cross-flow heat exchangers (to be discussed later) are examples of the equipment that transfers heat primarily by the forced convection of a fluid flowing across a cylinder. For environmental engineering calculations, the average heat transfer coefficient between a cylinder (at temperature TS) and a fluid flowing across the cylinder (at a temperature T ) is calculated from the Knudsen and Katz equation:

Nu D

hD k

C Re m Pr 1/ 3

(29.12)

All the fluid physical properties for this equation are evaluated at the mean film temperature (the arithmetic average temperature of the cylinder surface temperature and the bulk fluid temperature). The constants C and m depend on the Reynolds number of the flow and are available in a host of references [1–3].

References 1. Farag, I., and Reynolds, J., Heat Transfer, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1996. 2. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th edition, McGraw-Hill, New York, NY, 2008. 3. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 4. Abulencia, P., and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 5. Dittus, F.W., and Boelter, L.M.K., Heat Transfer in Automobile Radiators of the Tubular Type, University of California, Berkeley, Publications in Engineering, 2, 443, 1930.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

30 Free Convection

30.1 Introduction Convective effects, previously described as forced convection, are due to the bulk motion of the fluid. This bulk motion is caused by external forces, such as that provided by pumps, fans, compressors, etc., and is essentially independent of “thermal” effects. Free convection is another effect that occasionally develops and was briefly discussed in the previous chapter. This effect is almost always attributed to buoyant forces that arise due to density differences within a system. It is treated analytically as another external force term in the momentum equation. Momentum (velocity) and energy (temperature) effects are therefore interdependent; consequently, both equations must be solved simultaneously, and as indicated in the previous chapter, this treatment is beyond the scope of this text but is available in the literature [1]. Consider a heated body in an unbounded medium. In natural convection, the velocity is zero (no-slip boundary condition) at the heated body. The velocity increases rapidly in a thin boundary adjacent to the body, and ultimately approaches zero when significantly displaced from the body. In reality, both natural convection and forced convection effects occur simultaneously so that one may be required to determine which is predominant. Both effects may therefore be included in some analyses even though one is often tempted to attach less 289

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significance to free convection effects. However, this temptation should be resisted since free convection occasionally plays the more important role in the design and/or performance of some heated systems. As noted above, free convection fluid motion arises due to buoyant forces. Buoyancy arises due to the combined presence of a fluid density gradient and a body force that is proportional to density. The body force is usually gravity. Density gradients arise due to the presence of a temperature gradient. Furthermore, the density of gases and liquids depends on temperature, generally decreasing with increasing temperature, i.e.( / T)P < 0. There are both industrial and environmental applications of free convection. Free convection influences industrial heat transfer from and within pipes. It is also important in transferring heat from heaters or radiators to ambient air and in removing heat from the coil of a refrigeration unit to the surrounding air. It is also relevant to environmental sciences and engineering where it gives rise to atmospheric, lake and oceanic motion, a topic treated in the last section of this chapter.

30.2 Key Dimensionless Numbers If a solid surface at temperature, TS, is in contact with a gas or liquid at temperature, T , the fluid moves solely as a result of density variations in natural convection. It is the fluid motion that causes the so-called natural convection. The nature of the buoyant force is characterized by the coefficient of volumetric expansion, , an important term in natural convection theory and applications. For an ideal gas, is given by:

1 T

(30.1)

where T is the absolute temperature. Semi-theoretical equations for natural convection use the following key dimensionless numbers, some of which have been discussed earlier:

Gr

Grashof number

L3 g T v2

L3

3

g

T

(30.2)

2

Nu = Nusselt number = hL/k Ra Rayleigh number (Gr )(Pr )

Pr

L3 g v

Prandtl number

(30.3)

L3 g

T

T k

v

cp k

2

cp

(30.4)

(30.5)

Free Convection

291

where v = kinematic viscosity; = absolute viscosity; = thermal diffusivity = k/ cp; = fluid density; cp = fluid heat capacity; k = thermal conductivity; L = characteristic length of system; and T = temperature difference between the surface and the fluid = |TS – T |. The above Rayleigh number is used to classify natural convection as either laminar or turbulent based on the following range of Ra values:

Ra < 109 laminar free convection

(30.6)

Ra > 109 turbulent free convection

(30.7)

In the previous chapter on forced convection, the effects of natural convection were neglected, a valid assumption in many environmental applications characterized by moderate-to-high-velocity fluids. However, free convection may be significant with low-velocity fluids. A measure of the influence of each convection effect is provided by the ratio:

Gr Re 2

buoyancy fource inertia force

g L( T ) v2

(30.8)

where v = fluid velocity. This dimensionless number can be represented by LT so that

Gr Re 2

LT

(30.9)

For LT > 1.0, free convection is important. The regimes of these convection effects are:

Free convection predominates, i.e., LT >> 1.0 or Gr >> Re2 (30.10) Forced convection predominates, i.e., LT > Re2

(30.11)

Both effects contribute; mixed free and forced convection, i.e., LT

1.0 or Gr

Re2

(30.12)

Combining these three convection regimes with the two flow regimes – laminar and turbulent – produces six subregimes of potential interest. For example, if the Grashof and Reynolds numbers for a system involved in a heat transfer process are approximately 100 and 50, respectively, one can determine if free convection effects can be neglected. By employing Equation 30.9, one obtains

LT

Gr Re 2

100 502

0.04

Since LT T ) or upper surface of horizontal cooled (TS < T ) plates

12

0.25

0.13 0.333

12

Spheres

0

10 –10 9

Upper surface of horizontal heated plates; plate is hotter than surroundings (TS > T ) or lower surface of horizontal cooled plates (TS < T )

0.4 4

0.60 0.025

0–10 2

104 – 8

106

8

106 – 1

1011 0.15 0.333

105–1011

0.54

0.58

0.25

0.2

Free Convection

293

Another correlation that can be used to calculate the heat transfer coefficient for natural convection from spheres is Churchill’s equation,

0.589 Ra 0.25

Nu 2

0.469 Pr

1

9 /16

4 /9

(30.18)

The Rayleigh number, Ra, and the Nusselt number, Nu, in Equation 30.18 are based on the diameter of the sphere. Churchill’s equation is valid for Pr 0.7 and Ra 1011. There are also simplified correlations for natural (or free) convection in air at 1 atm. The correlations are dimensional and are based on the following SI units: h = heat transfer coefficient, W/m2 – K; T = TS – T , °C; TS= surface temperature, °C; T = surroundings temperature, °C; L = vertical or horizontal dimension, m; and D = diameter, m. These correlations are presented in Table 30.2. The average heat transfer coefficient can be calculated once the Nusselt number has been determined. Rearranging Equation 30.14 gives:

h

Nuk /L

(30.19)

The heat transfer rate, Q , is given by the standard heat transfer equation:

Q hA(TS T )

(30.20)

If the air is at a pressure other than 1 atm, the following correction may be applied to the reference value at 1 atm:

h href (P in atmospheres)n

(30.21)

Table 30.2 Free convection equation in air. Geometry

104 < Gr Pr = Ra < 109 laminar

Gr Pr = Ra > 109 turbulent

Vertical planes and cylinders

h = 1.42( T/L)0.25

h = 0.95( T/L)0.333

0.25

h = 1.24( T/L)0.333

Upper surface of horizontal heated plates; h = 1.32( T/L)0.25 plate is hotter than surroundings (TS > T ) or lower surface of horizontal cooled plates (TS < T )

h = 1.43( T/L)0.333

Horizontal cylinders

Lower surface of horizontal heated plates or upper surface of horizontal cooled plates (TS < T ) plates

h = 1.32( T/L)

h = 0.61( T/L2)0.2

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where

n n

30.4

1 for laminar cases (Ra 109 ) 2

(30.22)

2 for turbulent cases (Ra 109 ) 3

(30.23)

Environmental Applications

Two applications that involve environment engineering comprise the concluding section of this chapter. The first is concerned with lapse rates and the other with plume rise. 30.4.1

Lapse Rates

The concept behind the so-called lapse rate that has found application in environmental science and engineering can be best demonstrated with the following example. Consider the situation where a fluid is enclosed by two horizontal plates of different temperature (T1 > T2), see Figure 30.1. In case (a), the temperature of the lower plate exceeds that of the upper plate (T2 > T1)and the density decreases in the (downward) direction of the gravitational force. If the temperature difference exceeds a particular value, conditions are termed unstable and buoyancy forces become important. In Figure 30.1(a), the gravitational force of the cooler and denser fluid near the top plate exceeds that acting on the lighter hot fluid near the bottom plate and the circulation pattern as shown on the right-hand side of Figure 30.1(a) will exist. The cooler, heavier fluid will descend, being warmed in the process, while the lighter hot fluid will rise, cooling as it moves. However, for case (b) where T1 > T2, the density no longer decreases in the direction of the gravitational force. Conditions are now reversed and are defined as stable since there is no bulk fluid motion. In case (a), heat transfer occurs from the bottom to the top surface by free convection; for case (b), any heat transfer (from top to bottom) occurs by conduction. These two conditions are similar to that experienced in the environment, particularly with regard to the atmosphere. The extension of the above development is now applied to the atmosphere but could just as easily be applied to oceanographic or large lake systems. Apart from the mechanical interference of the steady flow of air caused by buildings and other obstacles, the most important factor that determines the degree of turbulence, and hence how fast diffusion in the lower atmosphere occurs, is the variation of temperature with height above the ground (i.e., the aforementioned “lapse rate”). Air is a good insulator. Therefore, heat transfer in the atmosphere is caused by radiative heating or by mixing due to turbulence. If there is no mixing, an air parcel rises adiabatically (no heat transfer) in the atmosphere.

Free Convection (a)

295

(b) T1

1

1

T Hot fluid

Cool fluid

(z)

T(z)

Unstable fluid circulation

z

Stable

z ρ(z)

T(z)

Cool fluid

Hot fluid 2

T2

T

T2

d dT > 0, 0 dz dz

Figure 30.1 A fluid contained between two horizontal plates at different temperatures. (a) Unstable temperature gradient, (b) Stable temperature gradient.

The Earth’s atmosphere is normally treated as a perfect gas mixture. If a moving air parcel is chosen as a control volume, it will contain a fixed number of molecules. The volume of such an air parcel must be inversely proportional to the density. The ideal gas law is (see also Chapter 4, Part I)

PV

RT

(30.24)

P

RT

(30.25)

or

The air pressure at a fixed point is caused by the weight of the air above that point. The pressure is highest at the Earth’s surface and decreases with altitude. This is a hydrostatic pressure distribution with the change in pressure proportional to the change in height [2].

dP

g dz

(30.26)

Since air is compressible, the density is also a decreasing function with height. An air parcel must have the same pressure as the surrounding air and so, as it rises, its pressure decreases. As the pressure drops, the parcel must expand adiabatically. The work done in the adiabatic expansion (P dV) comes from the thermal energy of the air parcel [3]. As the parcel expands, the internal energy then decreases and the temperature therefore decreases. If a parcel of air is treated as a perfect gas rising in a hydrostatic pressure distribution, the rate of cooling produced by the adiabatic expansion can be calculated. The rate of expansion with altitude is fixed by the vertical pressure variation

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described in Equation 30.26. Near the Earth’s surface, a rising air parcel’s temperature normally decreases by 0.98°C with every 100-m increase in altitude. The vertical temperature gradient in the atmosphere (the amount the temperature changes with altitude dT/dz) is defined as the lapse rate. The dry adiabatic lapse rate (DALR) is the temperature change for a rising parcel of dry air. As noted, the dry adiabatic lapse rate is approximately – l °C/100 m or dT/dz = – 10–2 °C/m or – 5.4 °F/1,000 ft. Strongly stable lapse rates are commonly referred to as inversions; dT/dz > 0. This strong stability inhibits mixing. Normally, these conditions of strong stability only extend for several hundred meters vertically. The vertical extent of an inversion is referred to as the inversion height. Thus, a positive rate is particularly important in air pollution episodes because it limits vertical motion (i.e., the inversion traps pollutants between the ground and the inversion layer). Ground-level inversions inhibit the upward mixing of pollutants emitted from automobiles, smoke stacks, etc. This increases the ground-level concentrations of pollutants. At night the ground reradiates the solar energy that it received during the day. On a clear night with low wind speeds, the air near the ground is cooled and forms a ground-level inversion. By morning, the inversion depth may be 200–300 m with a 5 to 10 °F temperature difference from bottom to top. Clouds cut down the amount of heat radiated by the ground out into space because they reflect the radiation back to the ground. Higher wind speeds tend to cause more mixing and spread the cooling effect over a larger vertical segment of the atmosphere, thus decreasing the change in lapse rate during the night [3]. Since temperature inversions arise because of solar radiation, the effects of nocturnal radiation often results in the formation of frost. When the sun is down, some thermal radiation is still received by the Earth’s surface from space, but the amount is small. Consequently, there is a net loss of radiation energy from the ground at night. If the air is relatively still, the surface temperature may drop below 32°F. Thus, frost can form, even though the air temperature is above freezing. This frost is often avoided with the presence of a slight breeze or by cloud cover. 30.4.2

Plume Rise

Smoke from a stack will usually rise above the top of the stack for a certain distance (see Figure 30.2). The distance that the plume rises above the stack is called the plume rise. It is actually calculated as the distance to the imaginary centerline of the plume rather than to the upper or lower edge of the plume. Plume rise, normally denoted h, depends on the stack’s physical characteristics and on the effluent stack gas characteristics. For example, the effluent characteristic of stack gas temperature, Ts, in relation to the surrounding air temperature, Ta, is more important than the stack height. The difference in temperature between the stack gas and ambient air determines the plume’s density, and this density affects plume rise. Therefore, smoke from a short stack could climb just as high as smoke from a taller stack. Stack characteristics are used to determine momentum, while effluent stack gas characteristics are used to determine buoyancy. The momentum of the effluent is

Free Convection

297

h = Plume rise

Ta

Ts

Figure 30.2 Stack plume rise.

initially provided by the stack. It is determined by the velocity of the effluent stack gas as it exits the stack. As momentum carries the effluent out of the stack, atmospheric conditions begin to affect the plume. The condition of the atmosphere, including the winds and temperature profile along the path of the plume, will primarily determine the plume’s rise. As the plume rises from the stack, the wind speed across the stack top begins to tilt (or bend) the plume. Wind speed usually increases with distance above the Earth’s surface. As the plume continues upward, stronger winds tilt the plume even farther. This process continues until the plume may appear to be horizontal to the ground. The point where the plume appears level may be a considerable distance downwind from the stack. Plume buoyancy is a function of temperature. When the effluent’s temperature, Ts, is warmer than the atmosphere’s temperature, Ta, the plume will be less dense than the surrounding air. In this case, the density difference between the plume and air will cause the plume to rise. The greater the temperature difference, T, the more buoyant the plume. As long as the temperature of the plume remains warmer than the atmosphere, the plume will continue to rise. The distance downwind where the plume cools to atmospheric temperature may also be quite displaced from its original release point. Buoyancy is taken out of the plume by the same mechanism that tilts the plume over – the wind. The faster the wind speed, the faster this mixing with outside air takes place. This mixing is called entrainment. Strong wind will “rob” the plume of its buoyancy rapidly and, on windy days, the plume will not climb significantly above the stack. Many individuals have studied plume rise over the years. The most popular plume rise formulas in use are those of Briggs [4]. Most plume rise equations

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are used on plumes with temperatures greater than the ambient air temperature. Plume rise formulas determine the imaginary centerline of the plume; the centerline is located where the greatest concentration of pollutant occurs at a given downward distance. Finally, plume rise is a linear measurement, usually expressed in feet or meters, and in some instances may be negative due to surrounding structures, topography, and so on.

References 1. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th edition, McGraw-Hill, New York, NY, 2008. 2. Ablencia, P., and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 3. Theodore, L., and Buonicore, A.J., Air Pollution Control Equipment: Gases, CRC Press/Taylor & Francis Group, Boca Raton, FL, 1976. 4. Briggs, G., Plume Rise, US Atomic Energy Commission Critical Review Series, Technical Information Division Report #TID-25075, Washington, D.C., 1969.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

31 Radiation

31.1 Introduction In addition to conduction and convection, heat can be transmitted by radiation. Conduction and convection both require the presence of molecules to “carry” or pass along energy. Unlike conduction or convection, radiation does not require the presence of any medium between the heat source and the heat sink since the thermal energy travels as electromagnetic waves. This radiant energy (thermal radiation) phenomena is emitted by everybody having a temperature greater than absolute zero. Quantities of radiation emitted by a body are a function of both temperature and surface conditions, details of which will be presented later in this chapter. Applications of thermal radiation include industrial heating, drying, energy conversion, solar radiation, and combustion. The amount of thermal radiation emitted is not always significant. Its importance in a heat transfer process depends on the quantity of heat being transferred simultaneously by the other aforementioned mechanisms. The reader should note that the thermal radiation of systems operating at or below room temperature is often negligible. In contrast, thermal radiation tends to be the principal mechanism for heat transfer for systems operating in excess of 1200 °F. When systems operate between room temperature and 1200 °F, the amount of heat transfer 299

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contributed by radiation depends on such variables as the convection film coefficient and the nature of the radiating surface. Radiation heat transfer in an industrial boiler from the hot gases to most solid surfaces inside the combustion chamber is considerable. However, in most heat exchangers, the contribution of radiation as a heat transfer mechanism is usually minor. In the heat transfer mechanisms of conduction and convection discussed in the four previous chapters, movement of energy in the form of heat takes place through a material medium – a fluid in the case of convection. Since a transfer medium is not required for this third mechanism, the energy is carried by electromagnetic radiation. Thus, a piece of steel plate heated in a furnace until it is glowing red and then is placed several inches away from a cold piece of steel plate will cause the temperature of the cold steel to rise, even if the process takes place in an evacuated container, e.g., a vacuum. As noted above, radiation generally becomes important as a heat transfer mechanism only when the temperature of the source is very high. As will become clear later in the chapter, the energy transfer is approximately proportional to the fourth power of the absolute temperature of the radiating body. However, the driving force for conduction and convection is simply the temperature difference (driving force) between the source and the receptor; the actual temperatures have only a minor influence. For these two mechanisms, it does not matter whether the temperatures are 120 °F and 60 °F or 520 °F and 460 °F. Radiation, on the other hand, is strongly influenced by the temperature level; as the temperature increases, the extent of radiation as a heat transfer mechanism increases rapidly. It therefore follows that, at very low temperatures, conduction and convection are the major contributors to the total heat transfer; however, at very high temperatures, radiation is often the prime contributor. Several additional examples of heat transfer mechanisms by radiation have appeared in the literature. Badger and Banchero [1] provided the following explanation of radiation: “If radiation is passing through empty space, it is not transformed to heat or any other form of energy and it is not diverted from its path. If, however, matter appears in its path, it is only the absorbed energy that appears as heat, and this transformation is quantitative. For example, fused quartz transmits practically all the radiation which strikes it; a polished opaque surface or mirror will reflect most of the radiation impinging on it; a black surface will absorb most of the radiation received by it (as one can experience on a sunny day while wearing a black shirt) and will transform such absorbed energy quantitatively into heat. The relationship between the energy transmitted, reflected, and absorbed is discussed in the next section. Bennett and Meyers [2] provide an additional example involving the operation of a steam “radiator.” Characteristic wavelengths of radiation are provided in Table 31.1. Note that light received from the Sun passes through the Earth’s atmosphere which absorbs some of the energy and thus affects the quality of visible light as it is received. The units of wavelength may be expressed in meters (m), centimeters (cm), microm. eters ( m), or Angstroms (1.0 A= 10–4 m), with the centimeter being the unit of

Radiation

301

Table 31.1 Characteristic wavelengths [3]. Type of radiation

108

Gamma Rays

0.01–0.15 cm

X-Rays

0.06–1000 cm

Ultraviolet

100–35,000 cm

Visible

3500–7800 cm

Infrared

7800–4,000,000 cm

Radio

0.01–0.15 cm

choice [3]. The speed of electromagnetic radiation is approximately 3 108 m/s in a vacuum. This velocity (c) is given by the product of wavelength ( ) and the frequency (v) of the radiation, that is:

c

v; consistent units

(31.1)

As noted earlier, the energy emitted from a “hot” surface is in the form of electromagnetic waves. One of the types of electromagnetic waves is thermal radiation. Thermal radiation is defined as electromagnetic waves falling within the following range:

0. 1 m

100 m; 1.0 m 10 6 m 10 4 cm

However, most of this energy is in the interval from 0.1 to 10 m. The visible range of thermal radiation lies within the narrow range of 0.4 m (violet) < A< 0.8 m (red).

32.2 Energy and Intensity A body at a given temperature will emit radiation over a range of wavelengths, not a single wavelength. Information is available on the intensity of the radiant energy I (Btu/hr-ft2-μm) as a function of the wavelength, ( m). In addition, at any given temperature, a wavelength exists at which the amount of energy given off is a maximum. For the same body at a lower temperature, the maximum intensity of radiation is obviously less; however, it is also significant that the wavelength at which the maximum intensity exists is higher in value. Since the I – curve for a single temperature depicts the amount of energy emitted at a given wavelength, the sum of all the energy radiated by a body at all its wavelengths is simply the area under a plot of I versus l. This quantity of radiant energy (of all wavelengths) emitted by a body per unit area and time is defined as

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the total emissive power E (Btu/hr-ft2). Given the intensity of the radiation at any wavelength, I, one may calculate the total emissive power, E, from

E

Id

(31.2)

0

Maxwell Planck was the first to fit the I versus relationship to equation form, as 5

C1

I

e C2 /

T

(31.3)

1

where I = intensity of emission, Btu/hr-ft2-μm, at ; = wavelength, m; C1 = 1.16 × 108, dimensionless; C2 = 25,740, dimensionless; and T = temperature. °R. It was later shown that the product of the wavelength of the maximum value of the intensity of emission and the absolute temperature is a constant. This is referred to as Wien’s displacement law:

T = 2,884 μm-°R ≈ 5,200 μm-K

(31.4)

One can derive Equation 31.4 from 31.3 as follows. Since at the maximum value of the intensity

dI d

5

C1

d

e C2 /

T

1

/d

(31.5)

0

After differentiation,

( 5C1

6

)(e C2 /

T

1) C1

6

(e C2 / T )/(e C2 /

T

1)2

0

(31.6)

which can be reduced to

( 5C1

6

)(e C2 /

T

1) C1

5

(e C2 / T )

C2 2 T

0

(31.7)

This ultimately simplifies to

5

C2 C2 / e T

T

5 0; C2

25, 740

(31.8)

The reader is left with the exercise of showing that the first term equals –5 when T = 2884.

Radiation

303

Atmospheric data indicate that the maximum intensity, I, of the Sun is experienced at around 0.25 m wavelength. This accounts for the predominance of blue in the visible spectrum and the high ultraviolet content of the Sun’s rays. The reader is also left the exercise of estimating the Sun’s temperature employing Equation 31.4 (approximately 11,500 °F).

31.3 Radiant Exchange The conservation law of energy indicates that any radiant energy incident on a body will partially absorb, reflect, or transmit stored energy. An energy balance around a receiving body on which the total incident energy is assumed to be unity gives

1

(31.9)

where the absorptivity, , is the fraction absorbed, the reflectivity, , is the fraction reflected, and the transmissivity, , is the fraction transmitted. It should be noted that the majority of engineering applications involve opaque substances having transmissivities approaching zero (i.e., = 0). This topic receives additional treatment below in a later paragraph. When an ordinary body emits radiation to another body, it will have some of the emitted energy returned to itself by reflection. Equation 31.9 assumes that none of the emitted energy is returned; this is equivalent to assuming that bodies having zero transmissivity also have zero reflectivity. This introduces the concept of a perfect “black body” for which = 1. Not all substances radiate energy at the same rate at a given temperature. The theoretical substance to which most radiation discussions refer to is called a “black body.” This is defined as a body that radiates the maximum possible amount of energy at a given temperature. Much of the development to follow is based on this concept. To summarize, when radiation strikes the surface of a semi-transparent material such as a glass plate or a layer of water, three types of interactive effects occur. Some of the incident radiation is reflected off the surface, some of it is absorbed within the material, the remainder is transmitted through the material. Examining the three fates of the incident radiation, one can see that and depend on inherent properties of the material; it is for this reason that they are referred to as surface properties. The transmissivity, , on the other hand, depends on the amount of the material in question; it is therefore referred to as a volumetric property. It is appropriate to examine a few common surfaces. An opaque surface, the most commonly encountered surface type, has ≈ 0. Because of this, Equation 31.9 becomes:

1

(31.10)

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Unit Operations in Environmental Engineering

or

1

(31.11)

Equation 31.11 may also be applied to gases; this may be counterintuitive since most gases are invisible. With respect to the reflectivity term, , surfaces may have either specular reflection, in which the angle of incidence of the radiation is equal to the angle of reflection, or diffuse reflection, in which the reflected radiation scatters in all directions. In addition, a gray surface is one for which the absorptivity is the same as the emissivity, , at the temperature of the radiation source. For this case, (31.12) and

1

(31.13)

Reflectivity and transmissivity are characteristics experienced in the everyday world. Polished metallic surfaces have high reflectivities and granular surfaces have low reflectivities. Reflection from a surface depends greatly on the characteristics of the surface. If a surface is very smooth, the angles of incidence and reflection are essentially the same. However, most surfaces encountered in engineering practice are sufficiently rough so that some reflection occurs in all directions. Finally, one may state that a system in thermal equilibrium has its absorptivity equal to its emissivity.

31.4 Kirschoff ’s Law Consider a body of given size and shape placed within a hollow sphere of constant temperature, and assume that the air has been evacuated. After thermal equilibrium has been reached, the temperature of the body and that of the enclosure (the sphere) will be the same, inferring that the body is absorbing and radiating heat at equal rates. Let the total intensity of radiation falling on the body be I (now, with units of Btu/hr-ft2), the fraction absorbed, 1, and the total emissive power, E1 (Btu/hr-ft2). Kirschoff noted that the energy emitted by a body of surface A1 at thermal equilibrium is equal to that received, so that:

E1 A1

I 1 A1

(31.14)

I

(31.15)

or simply

E1

1

Radiation

305

If the body is replaced by another of identical shape, then:

E2

I

(31.16)

2

If a third body that is a blackbody is introduced, then:

Eb

I

(31.17)

Since the absorptivity, , of a black body is 1.0, one may write:

I

E1

E2

1

2

Eb

(31.18)

Thus, at thermal equilibrium, the ratio of the total emissive power to the absorptivity for all bodies is the same. This is referred to as Kirschhoff ’s law. Since = , the above equation may also be written

E1 Eb

1

1

(31.19)

E2 Eb

2

2

(31.20)

and

where the ratio of the actual emissive power to the black-body emissive power is defined as the aforementioned emissivity, . Values of for various bodies and surfaces are available in the literature [1–3]. If a black body radiates energy, the total radiation may be determined from Planck’s law as: 5

C1

I

e C2 /

T

(31.3)

1

Integration over the entire spectrum at a particular temperature yields: 5

C1

Eb 0

e C2 /

T

1

d

(31.21)

The evaluation of the previous integral can be shown to be:

Eb

0.173 10 8 T 4

T4; T

R

(31.22)

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Thus, the total radiation from a perfect black body is proportional to the fourth power of the absolute temperature of the body. This is also referred to as the Stefan-Boltzmann law. The constant 0.173 × 10–8 Btu/hr-ft2-°R [2] is known as the Stefan-Boltzmann constant, usually designated by . Its counterpart in SI units is 5.669 × 10–8 W/m2-K4. Note that this equation was derived for a perfect black body. If the body is non-black, the emissivity is given by:

E

Eb

(31.23)

Substituting Equation 31.22 into Equation 31.23 gives

E

T4

(31.24)

Q A

T4

(31.25)

or, since E Q /A,

Thus, when the law is applied to a real surface with emissivity, , the total emissive power of a real body is given by Equation 31.25. Typical values for emissivity are available in the literature [1–3]. Now consider the energy transferred between two black bodies. Assume the energy transferred from the hotter body and the colder body is EH and EC, respectively. All of the energy that each body receives is absorbed since they are black bodies. Then, the net exchange between the two bodies maintained at two constant temperatures, TH and TC, is therefore:

Q A

EH

EC

(TH4 TC4 ) 0.173

TH 100

4

TC 100

4

(31.26)

31.5 Emissivity Factors If two large walls are not black bodies, and instead (each) have an emissivity , then the net interchange of radiant energy is given by:

Q A

(TH4 TC4 )

(31.27)

This equation can be “verified” as follows [2]. If the two planes discussed above are not black bodies and have different emissivities, the net exchange of energy will be different. Some of the energy emitted from the first body will be absorbed and the remainder radiated back to the other source. For two parallel bodies of infinite

Radiation

307

size, the radiation of each body can be accounted for. If the energy emitted from the first body is EH with emissivity Hthe second body will absorb EH C and reflect 1 – C of it, i.e., EH(1 – C). The first body will then receive EH(1 – C) H and again radiate to the cold body, but also in the amount EH(1 – C)(1 – H). The exchanges of the two bodies are therefore: Hot body Radiated: EH Reflected back: EH(1 – C) Radiated: EH(1 – C)(1 – C) Reflected back: EH(1 – C)(1 – etc. Cold body Radiated: EC Reflected back: EC(1 – H) Radiated: EC(1 – H)(1 – C) Reflected back: EC(1 – H)(1 – etc.

)(1 –

C

)(1 –

H

H

C

)

)

For a non-black body, is not unity and must be included in Equation 31.22, that is,

E

T 100

0.173

4

(31.28)

When Equation 31.28 is applied to the above infinite series analysis, one can show that Equation 31.29 results:

E

Q A

1

1

H

C

(TH4 TC4 )

(31.29)

1

The radiation between a sphere and an enclosed sphere of radii RH and RC, respectively, may be treated in a manner similar to that provided above. The radiation emitted initially by the inner sphere is EHAH, all of which falls on AC. Of this total, however, (1– C)EHAH is reflected back to the hot body. If this analysis is similarly extended as above, the radiant exchange will again be represented by an infinite series whose solution may be shown to give:

E

Q AH

H

1 H

(TH4 TC4 ) AH AC

1 C

H

= 1

1 H

(TH4 TC4 ) RH RC

2

1 C

(31.30)

1

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A similar relation applies for infinitely long concentric cylinders except that AH/AC is replaced RH/RC, not RH2 /RC2 . In general, an emissivity correction factor, F , is introduced to account for the exchange of energy between different surfaces of different emissivities. The describing equation takes the form:

EH

Q AH

F (TH4 TC4 )

(31.31)

Values of F for the interchange between surface are also available in the literature. Finally, when a heat source is small compared to its enclosure, it is customary to assume that some of the heat radiated from the source is reflected back to it. Such is often the case in the loss of heat from a pipe to surrounding air. For these applications, it is convenient to represent the net radiation heat transfer in the same form employed for convection, i.e.,

Q hr A(TH TC )

(31.32)

where hr is the effective radiation heat transfer coefficient. When TH – Tc is less than 120 °C (120 K or 216 °R), one may calculate the radiation heat transfer coefficient using

hr

4

Tav3

(31.33)

where Tav = (TH + TC)/2.

31.6

View Factors

As indicated earlier, the amount of heat transfer between two surfaces depends on geometry and orientation of the two surface. Again, it is assumed that the intervening medium is non-participating. The previous analyses were concerned with sources that were situated so that every point on one surface could be “connected” with every surface on the second ... in effect possessing a perfect view. This is very rarely the case in real-world engineering applications, particularly in the design of boilers and furnaces. Here, the receiving surface, such as a bank of tubes, is cylindrical and may partially obscure some of the surfaces from “viewing” the source. These systems are difficult to evaluate. The simplest case is addressed below; however, many practical applications must resort to the use of empirical methods. To introduce the subject of view factors, the reader should note that the flow of radiant heat is analogous to the flow of light (i.e., one may follow the path of radiant heat as one may follow the path of light). If any object is placed between a hot

Radiation

309

and cold body, a light from the hot body would cast a shadow on the cold body and prevent it from receiving all the light leaving the hot one. As noted above, the computation of actual problems involving “viewing” difficulty is beyond the scope of this book. One simple problem may illustrate the nature of these complications. Contemplate two black bodies, with surfaces consisting of parallel planes of finite size and separated by a finite distance (see Figure 31.1). Then, a small differential unit of surface dA can see the colder body only through solid angle , and any radiation emitted by it through solid angles and will fall elsewhere. To evaluate the heat lost by radiation from body H, one must integrate the loss from element dA over the whole surface of A. The previously mentioned problem is sufficiently complicated. However, it can become more complicated. If the colder body is not a black body, then it will reflect some of the energy imparted upon it. Some of the reflected energy will return to the hot body. Since the hot body is black, it will absorb the reflected energy, tending to raise its temperature. One can envision even more complicated scenarios. In order to include the effect of “viewing,” Equation 31.31 is expanded to:

Q AH

Fv F (TH4 TC4 )

(31.34)

C

1

dA

H

Figure 31.1 View factor illustration.

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with the inclusion of view factor, Fv. Additional information and applications involving view factors are available in the literature [4–6].

References 1. Badger, W., and Banchero, J., Introduction to Chemical Engineering, McGraw-Hill, New York City, NY, 1955. 2. Bennett, C., and Meyers, J., Momentum, Heat and Mass Transfer, McGraw-Hill, New York City, NY, 1962. 3. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th edition, McGraw-Hill, New York, NY, 2008. 4. Holman, J., Heat Transfer, McGraw-Hill, New York City, NY, 1981. 5. Farag, I., and Reynolds, J., Heat Transfer, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1996. 6. Incropera, F., and DeWitt, D., Fundamentals of Heat Transfer, John Wiley & Sons, Hoboken, NJ, 2002.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

32 The Heat Transfer Equation

32.1 Introduction Heat exchangers are defined as equipment that effect the transfer of thermal energy in the form of heat from one fluid to another. The simplest exchangers involve the direct mixing of hot and cold fluids. Most industrial exchangers are those in which the fluids are separated by a wall. The latter type, referred to by some as a recuperator, can range from a simple plane wall between two flowing fluids to more complex configurations involving multiple passes, fins, or baffles. Conductive and convective heat transfer principles are required to describe and design these units; radiation effects are generally neglected [1]. Heat exchangers for the chemical, petrochemical, petroleum, paper, environmental, and power industries encompass a wide variety of designs that are available from many manufacturers. Equipment design practice first requires the selection of safe, operable equipment. The selection and design process must also seek a cost-effective balance between initial (capital) installation costs, operating costs, and maintenance costs. The proper application of heat exchange principles can significantly minimize both the initial cost of a plant and its daily operating and/or utility costs. Each heat exchange application may be accomplished by the use of many types of heat 311

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exchange equipment. To perform these applications, their design and materials of construction must be suitable for the desired operating conditions; the selection of materials of construction is primarily influenced by the operating temperature and the corrosive nature of the fluids being handled. The material in Part III-B examines the various heat exchanger equipment (and their classification). This chapter discusses the log mean temperature difference driving force, Tlm, and the overall heat transfer coefficient, U. The development of both Tlm and U ultimately leads to the classic heat exchanger equation. Topics addressed in this chapter include: Energy Relationships, Heat Exchange Equipment Classification, The Log Mean Temperature Difference (LMTD) Driving Force, Overall Heat Transfer Coefficients, and The Classic Heat Transfer Equation.

32.2 Energy Relationships The flow of heat from a hot fluid to a cooler fluid through a solid wall is a situation often encountered in engineering equipment. Examples of such equipment are the aforementioned heat exchangers, condensers, evaporators, boilers, and economizers. The heat absorbed by the cool fluid or given up by the hot fluid may be sensible heat, causing a temperature change in the fluid, or it may be latent heat, causing a phase change such as vaporization or condensation. In a typical waste heat boiler, for example, the hot flue gas gives up heat to water through thin metal tube walls separating the two fluids. As the flue gas loses heat, its temperature drops. As the water gains heat, its temperature quickly reaches the boiling point where it continues to absorb heat with no further temperature rise as it changes into steam. The rate of heat transfer between the two streams, assuming no heat loss due to the surroundings, may be calculated by the enthalpy change of either fluid as described by:

Q mh (hh1 hh 2 ) mc (hc1 hc 2 )

(32.1)

where Q is the rate of heat flow, Btu/hr; mh is the mass flow rate of hot fluid, lb/ hr; mc is the mass flow rate of cold fluid, lb/hr; hh1 is the enthalpy of entering hot fluid, Btu/ lb; hh2 is the enthalpy of exiting hot fluid, Btu/lb; hc1 is the enthalpy of entering cold fluid, Btu/lb; and hc2 is the enthalpy of exiting cold fluid, Btu/lb. Equation 32.1 is applicable to the heat exchange between two fluids whether a phase change is involved or not. In the above waste heat boiler example, the enthalpy change of the flue gas is calculated from its sensible temperature change:

Q mh (hh1 hh 2 ) mc c ph (Th1 Th 2 )

(32.2)

where cph is the heat capacity of the hot fluid, Btu/lb-°F; Th1 is the temperature of the entering hot fluid, °F; and Th2 is the temperature of the exiting hot fluid, °F.

The Heat Transfer Equation

313

The enthalpy change of the water involves a small amount of sensible heat to bring the water to its boiling point plus a considerable amount (usually) of latent heat to vaporize the water. Assuming all of the water is vaporized and no superheating of the steam occurs, the enthalpy change is:

Q mc (hc 2 hc1 ) mc c pc (t c 2 t c1 ) mc hvap

(32.3)

where cpc is the heat capacity of the cold fluid, Btu/lb-°F; tc1 is the temperature of the entering cold fluid, °F; tc2 is the temperature of the exiting cold fluid, °F; and hvap is the heat of vaporization of the cold fluid, Btu/lb. Note that, wherever possible, lower case t and uppercase T will be employed to represent the cooler fluid temperature and hotter fluid temperature, respectively. In addition, lowercase c/ uppercase C and lowercase h/uppercase H will be employed to represent the cold and hot fluid, respectively.

32.3 Heat Exchange Equipment Classification There is a near infinite variety of heat exchange equipment. These can vary from a simple electric heater in the home to a giant boiler in a utility power plant. A limited number of heat transfer devices likely to be encountered by the practicing engineer have been selected for description in this chapter. The size, shape, and material employed to separate the two fluids contained within a heat exchanger is of course important, as is the method of confining one or both of the two fluids involved in the heat transfer process. Heat exchangers are classified by their functions. In a general sense, heat exchanges are classified into three broad types: 1. Recuperators or through-the-wall non-storing exchangers (e.g., double-pipe heat exchangers, shell-and-tube heat exchangers) 2. Direct contact non-storing exchangers 3. Regenerators, accumulators, or heat storage exchangers Through-the-wall non-storing exchangers [1] can be further classified into: 1. Double pipe heat exchangers 2. Shell-and-tube heat exchangers 3. Cross-flow exchangers Exchangers of Type 1 and 2 receive extensive treatment in Chapters 33 and 34, respectively. The flow in a double pipe heat exchanger [1–4] may be countercurrent or parallel (co-current). In countercurrent flow, the fluid in the pipe flows in a direction

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opposite to the fluid in the annulus. In parallel/co-current flow, the two fluids flow in the same direction. The variations of fluid temperature within the heat exchanger depend on whether the flow is parallel (co-current) or countercurrent. The definitions below are employed in the development to follow: t1 t2 T1 T2

= = = =

temperature of fluid entering the inside tube temperature of fluid exiting the inside tube temperature of fluid entering the annulus (space between the two tubes) temperature of fluid exiting the annulus

The difference between the temperature of the tube side fluid and that of the annulus side is the temperature difference driving force, T. In a parallel heat exchanger, both hot and cold fluids enter on the same side and flow through the exchanger in the same direction. The temperature approach is defined as the temperature difference driving force at the heat exchanger entrance, T1 or (T1 – t1). This driving force drops as the streams approach the exit of the exchanger. At the exit, the temperature difference driving force is T2 or (T2 – t2). The heat exchanger is more effective at the entrance than at the exit because of the higher driving force that occurs there. In a countercurrent flow exchanger, the two fluids exchanging heat flow in opposite directions. The temperature approach at the tube entrance end, T1 or (T1 – t2) and at the annulus entrance end, T2 or (T2 – t1) are usually roughly the same. The thermal driving force is normally relatively constant over the length of the exchanger, and thus countercurrent flow exchangers generally have a higher transfer efficiency than parallel heat exchangers. This topic is revisited in Section 32.5.

32.4 The Log Mean Temperature Difference (LMTD) Driving Force When heat is exchanged between a surface and a fluid, or between two fluids flowing through a heat exchanger, the local temperature driving force, T, usually varies along the flow path. This effect is treated through the log mean temperature difference approach, as discussed below. The concept of a logmean temperature is first developed before proceeding to the log-mean temperature difference driving force. Consider an absolute temperature profile T(x) that is continuous and where T1 is the absolute temperature at some point x1 with T2 at x2. By definition, the mean value of the reciprocal temperatures (1/T) between x1 and x2 is given by:

1 T

T2

1 dT T T1

T2

dT T1

1 T2

T2

1 dT T1 T1 T

ln(T2 /T1 ) T2 T1

(32.4)

The Heat Transfer Equation

315

If the average of the reciprocal is a reasonable approximation to the reciprocal of the average, i.e.,

1 t

1 T

(32.5)

then one can combine the above two equations to give:

1 T

ln(T2 /T1 ) T2 T1

(32.6)

T

T2 T1 ln(T2 /T1 )

(32.7)

or

The right-hand side of Equation 32.7 is defined to be the logmean temperature between T1 and T2, i.e.,

T2 T1 ln(T2 /T1 )

Tlm

(32.8)

Note that absolute temperatures must be employed in Equation 32.8. Assuming steady-state operation and constant properties independent of the temperature, an overall energy balance is first applied to a fluid in a conduit (e.g., a pipe with inlet and exit temperatures t1 and t2, respectively), and heated by a source at temperature TS (see also Figure 32.1):

Q mc p (t 2 t1 ) mc p [(TS t1 ) (TS t 2 )] mc p ( T1

T2 )

(32.9)

where T1 or (TS – t1), is the temperature difference driving force (also termed the approach) at the fluid entrance, and T2 or (TS – t2), is the temperature driving force at the fluid exit. The differential balance is:

dQ mc p dt

mc p d( T )

(32.10)

One may also apply an energy balance to a differential fluid element of crosssectional area, A, and a thickness of dx. Set x = 0 at the pipe entrance and x = L at the pipe exit. The pipe diameter is D, so:

A

D2 4

(32.11)

316

Unit Operations in Environmental Engineering dx

Ts

. m

t1

Ts

t2

D

A

D

L Figure 32.1 Energy balance of a differential element of a pipe.

The outer surface differential area of the element is dA = D(dx). One may also apply the energy balance across the walls of the unit to yield:

dQ h(Ts t )( D)(dx ) mc p dt

(32.12)

where h is the heat transfer coefficient of the fluid defined earlier. Separation of variables in Equation 32.11 yields:

dt Ts t

h ( D)(dx ) mc p

(32.13)

Integrating from x = 0, where t = t1 to x = L, where t = t2, yields: t2

t1

dt Ts t

L T t h ( D) (dx ); ln S 1 mc p TS t 2 0

h ( D)L mc p

(32.14)

Letting T1 = TS – t1 and T2 = TS – t2 leads to:

ln

T1 T2

h ( D)L mc p

(32.15)

Thus,

mc p

h( D)L ln( T1 / T2 )

Combining Equations 32.4 and 32.16 gives:

(32.16)

The Heat Transfer Equation

Q h( D)L

T1 T2 ln( T1 / T2 )

hA Tlm

317

(32.17)

where Tlm = log mean temperature difference, or log mean temperature approach, and is defined as:

Tlm

T1 T2 ln( T1 / T2 )

(32.18)

For the special case of T1 = T2,

Tlm

T1

T2

(32.19)

Heat transfer calculations using Tlm are convenient when terminal temperatures are known. If the temperature of the fluid leaving the tube or exchanger is not known, the procedure may require trial-and-error calculation.

32.5 Temperature Profiles Consider a heated surface, as represented in Figure 32.2, where temperatures are plotted against distance along the surface. The temperature drop at the left-hand side of the figure is much greater than that at the right-hand side. Heat is being transferred more rapidly at the left-hand end since the hot fluid and the cold fluid enter at the same end of the unit and flow parallel to each other. This arrangement is known as parallel flow or co-current flow. The alternative method is to feed the hot fluid at one end of the unit and the cold fluid at the other, allowing the fluids to pass by each other in opposite directions. Such an arrangement is called counterflow or counter-current flow. The temperature gradients for this case are provided in Figure 32.3. In this case of counter-current operation, the temperature drop along the length of the unit is relatively more constant than in parallel flow.

32.6 Overall Heat Transfer Coefficients In order to design heat transfer equipment and calculate the required energy, one must know more than just the heat transfer rate calculated by the enthalpy (energy) balances described previously. The rate at which heat can travel from the hot fluid at tH, through the tube walls, into the cold fluid at tC, must also be considered in the calculation of certain design variables (e.g., the contact area). The slower this rate is, for given hot and cold fluid flow rates, the more contact area is required. The rate of heat transfer through a unit of contact area was

318

Unit Operations in Environmental Engineering T1 Temperature

Hot fluid flow T2

T2

T1

t1

t2

Cold fluid flow

Length of exchanger Figure 32.2 Parallel flow heat transfer.

T2

Hot fluid flow

Temperature

T2 t2

T2 T1 t1

Cold fluid flow

Length of exchanger Figure 32.3 Counterflow heat transfer.

referred to earlier as the heat flux and, at any point along the area or the tube length, is given by:

dQ dA

U (TH t C )

(32.20)

where dQ /dA is the local heat flux, Btu/hr-ft2; and U is defined as the local overall heat transfer coefficient, Btu/hr-ft2-°F, a term that provides a measure (inversely) of the resistance to heat transfer. The use of the above overall heat transfer coefficient, U, is a simple, yet powerful concept. In most applications, it combines both conduction and convection effects, although heat transfer by radiation can also be included. In actual practice, it is not uncommon for vendors to provide a numerical value for U. For example, a typical value for U for estimating heat losses from an incinerator is approximately

The Heat Transfer Equation

319

0.1 Btu/hr-ft2-°F. Methods for calculating the overall heat transfer coefficient are presented later in this section. With reference to Equation 32.20, the temperatures TH and tC are actually local average values. As described in the previous section, when a fluid is being heated or cooled, the temperature will vary throughout the cross-section of the stream. If the fluid is being heated, its temperature will be highest at the tube  wall and will decrease with increasing distance from the tube wall. The average temperature across the stream cross-section is therefore tC; i.e., the temperature that would be achieved if the fluid at this cross-section was suddenly mixed to a uniform temperature. If the fluid is being cooled, on the other hand, its temperature will be lowest at the tube wall and will increase with increasing distance from the wall. In order to apply Equation 32.20 to an entire heat exchanger, the equation must be integrated. This cannot be done unless the geometry of the exchanger is first defined. For simplicity, one of the simplest geometries will be assumed here, i.e., the double pipe heat exchanger to be discussed in the next chapter. This device consists of two parallel concentric pipes. The outer surface of the outer pipe is well insulated so that no heat exchange with the surroundings may be assumed. One of the fluids flows through the center pipe and the other flows through the annular channel (known as the annulus) between the pipes. The fluid flows may be either co-current where the two fluids flow in the same directions, or countercurrent where the flows are in the opposite directions; however, the countercurrent arrangement is more efficient and is more commonly used. For this heat exchanger, integration of Equation 32.20 along the exchanger area or length, and applying several simplifying assumptions, yields:

Q UA T

(32.21)

The above equation was previously applied to heat transfer across a plane wall in Chapters 29 and 30, and it was shown that:

Q

1 hi A

TH t C x 1 kA h0 A

(32.22)

where the h coefficients represent the individual heat transfer coefficients, also discussed in Chapters 29 and 30. Since Q UA T

U

Since A is a constant,

1 hi

1 x k

1 h0

(32.23)

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Unit Operations in Environmental Engineering

Q T

UA

1 hi A

1 x kA

(32.24)

1 h0 A

For a tubular unit, Q UA T still applies, but A is now a variable. Equation 32.24 is rewritten for this geometry as:

Q T

1 hi A

1 D 2kAlm

1 ho Ao

; D

Do Di

2(ro ri )

(32.25)

where the subscripts i and o refer to the inside and outside of the tube, respectively. In addition,

Ao Ai ln(Ao /Ai )

Alm

L D ln(Do /Di )

(32.26)

and L = tube length (ft). Thus,

Q T

UA

1 hi Ai

1 ln(Do /Di ) 1 2 Lk ho Ao

(32.27)

Equation 32.27 may again be written as:

Q T

UA

Ri

1 Rw

Ro

(32.28)

The term U in Equation 32.27 may be based on the inner area, Ai, or the outer area, Ao, so that

Q T

U i Ai

U o Ao

1 hi Ai

1 ln(Do /Di ) 1 2 Lk ho Ao

(32.29)

Dividing by either Ai or Ao yields an expression for Ui and Uo:

Ui

1 hi

1 Ai ln(Do /Di ) Ai ho Ao 2 Lk

(32.30)

The Heat Transfer Equation

Uo

Ao hi Ai

1 Ao ln(Do /Di ) 1 ho 2 Lk

1 Do ln(Do /Di ) 1 ho 2k

Do hi Di

321

(32.31)

Numerical values of U can range from as low as 2 Btu/hr-ft2-°F (10 W/m2-K) for gas-to-gas heat exchangers to as high as 250 Btu/hr-ft2-°F (1250 W/m2-K) for water-to-water units. Additional values are provided in the literature [1–3]. 32.6.1

Fouling Factors

During heat exchange operation with liquids and/or gases, a “dirt” film gradually builds up on the exchanger surface(s). This deposit may be rust, boiler scale, silt, coke, or any number of things. Its effect, which is referred to as fouling, is to increase the thermal resistance, R, which results in decreased performance. The nature of the rate of deposit is generally difficult to predict a priori. Therefore, only the performance of clean exchangers is usually guaranteed. The fouling resistance is often obtained from field, pilot, or lab data, or from experience [1-3]. This unknown factor enters into every design. The scale of fouling is dependent on the fluids, their temperature, velocity, and to a certain extent, the nature of the heat transfer surface and its chemical composition. Due to the unknown nature of the assumptions, these fouling factors can markedly affect the design of heat transfer equipment. The effect (resistance) of fouling, Rf, can be obtained from the following equation:

Rf

1

1

U dirty

U clean

(32.32)

where Udirty is the overall heat transfer coefficient for design (with fouling) and Uclean is the overall heat transfer coefficient for the clean condition (without fouling). Thus, in general, Equation 32.28 can be rewritten as:

UA

1 Ri

R fi

R fo R0

1 Rt

(32.33)

where Rt = total resistance. If U is based on the outside area of the tube, Ao,

Uo

1 Ao hi Ai

R fi

1 Ao Ai

Rw

R fo

1 ho

(32.34)

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Unit Operations in Environmental Engineering

where Rw is once again the wall resistance and given by (see also Equation 32.31):

Rw

Ao ln(Do /Di ) 2 Lk

(32.35)

The two fouling factors, Rfi and Rfo, are sometimes combined and treated as one resistance, Rf. For this condition:

Uo

1 Ao ln(Do /Di ) Rf 2 Lk

1 Ao hi Ai

(32.36)

1 ho

Typical values of the fouling coefficient, hf, are shown in Table 32.1. Additional values are available in the literature [1, 3]. Note that

hf

Rf 1

(32.37)

One of the main contributors to fouling is corrosion. Therefore, corrosion protection is important and the selection of equipment materials with reasonable corrosion rates must be balanced against initial and replacement costs. Materials chosen for corrosion resistance in a given process service may include carbon steels, stainless steels, aluminum, copper alloys, nickel alloys, graphite, glass, and other non-metallics. Corrosion effects can be reduced without costly materials by means of special design and fabrication techniques; among these are limitations on flow velocities and use of impingement baffles. 32.6.2 The Controlling Resistance The reader should note that the numerical value of the overall heat transfer coefficient may be primarily governed by only one of the individual coefficients Table 32.1 Fouling coefficients (english units). Fluid

Rf, hr-ft2-°F/Btu

Alcohol Vapors

0.0005

Broiler Feed Water

0.001

Fuel Oil

0.005

Industrial Air

0.002

Quench Oil

0.004

Refrigerating Liquids

0.001

Seawater

0.0008

Steam, Non Oil-Bearing

0.0005

The Heat Transfer Equation

323

(and, therefore, only one of the resistances). That coefficient is then referred to as the controlling resistance. As one might expect, the resistance due-to conduction (the wall) is often negligible relative to the convective resistances [1–4].

32.7 The Classic Heat Transfer Equation By now the reader should have concluded that

Q UA Tlm

(32.38)

is that equation. It will find use in the remainder of the text, and particularly in the remaining chapters in this Part. A summary of the above equations is presented below for a plane wall [5],

U

1 Rw

Ri

1 x k

1 hi

Ro

1 ho

(32.23)

If the area is constant, this equation may be written as:

UA

1 hi A

1 x kA

1 ho A

(32.24)

For a tubular wall, Equation 32.24 remains the same, but which A should be employed? For inside/wall/outside resistances present, and employing the radius rather than the diameter leads to:

U

1 hi

1 r k

1 ho

(32.39)

In addition (and employing radii),

UA

with

1 hi Ai

1 r kAlm

1 ho Ao

(32.40)

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Unit Operations in Environmental Engineering

Alm

Ao Ai ln(ro /ri )

(32.41)

Equation 32.40 may therefore be written as:

UA

1 hi Ai

1 rln(ro /ri ) 1 ho Ao 2 kL

(32.42)

U may once again be based on the inside, i, or outside, o, area. For UiAi:

U i Ai

1 hi Ai

1 rln(ro /ri ) 1 ho Ao 2 kL

(32.43)

or

Ui

1 hi

1 Ai ln(ro /ri ) 1 Ai ho Ao 2 kL

(32.44)

The reader should note that although the over-all coefficient can be based on any area, the area selected should be specified. Where the barrier is a thin-walled tube of large diameter, a negligible error will be introduced by using a common area A for A1, Am, and A2. Since the surface area of a tubular barrier is proportional to its diameter, diameter terms may be substituted for the corresponding area terms.

References 1. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 2. Farag, I., and Reynolds, J., Heat Transfer, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1996. 3. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 4. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2014. 5. Reynolds, J., personal notes, Bronx, NY, 1981.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

33 Double Pipe Heat Exchangers

33.1 Introduction Heat transfer operations are important in waste treatment. Applications include the cooling of liquids (usually water), the evaporation of liquids (usually water), the heating of fluids (usually air), the drying of solids, and the heating of anaerobic biological processes. Of the various types of heat exchangers that are employed in environmental engineering, perhaps the two most fundamental are the double pipe and the shell and tube heat exchanger systems. Despite the fact that shell and tube heat exchangers (see next chapter) generally provide greater surface area for heat transfer with a more compact design, greater ease of cleaning, and less possibility of leakage, the double pipe heat exchanger still finds use in practice. The double pipe unit consists of two concentric pipes. Each of the two fluids, i.e., hot and cold, flow either through the inside of the inner pipe or through the annulus formed between the outside of the inner pipe and the inside of the outer pipe. Generally, it is more economical from a heat efficiency perspective for the hot fluid to flow through the inner pipe and the cold fluid through the annulus, thereby reducing heat losses to the surroundings. In order to ensure sufficient contacting time, pipes longer than approximately 20 ft are extended by connecting 325

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Unit Operations in Environmental Engineering

them to return bends. The length of pipe is generally kept to a maximum of 20 ft because the weight of the piping may cause the pipe(s) to sag. Sagging may allow the inner pipe to touch “the outer pipe, distorting the annulus flow region and disturbing proper operation. When two pipes are connected in a “U” configuration by a return bend, the bend is referred to as a hairpin. In some instances, several hairpins may be connected in series. Double pipe heat exchangers have been used in the chemical process industry for over 90 years. The first patent on this unit appeared in 1923. The unique (at that time) design provided “the fluid to be heated or cooled and flow longitudinally and transversely around a tube containing the cooling or heating liquid, etc.” The original design has not changed significantly since that time. Although this unit is not extensively employed in industry now (the heat transfer area is small relative to other heat exchanger), it serves as an excellent starting point from an academic and/or training perspective in the treatment of all the various heat exchangers reviewed in this Part. Topics covered in this chapter include Equipment Description, Describing Equations, Effectiveness Factor and Number of Transfer Units, and Wilson’s Method.

33.2 Equipment Description As discussed above, the simplest kind of heat exchanger, which confines both the hot and cold fluids, consists of two concentric tubes, or pipes. When conditions are such that only a few tubes per pass are required, the simplest construction is the double pipe heat interchanger. This consists of special fittings that are attached to standard iron (typically) pipe so that one fluid (usually liquid) flows through the inside pipe and the second fluid (also usually a liquid) flows through the annular space between the two pipes. Such a heat interchanger will usually consist of a number of passes which are almost invariably arranged in a vertical stack. If more than one pipe per pass is required, the proper number of such stacks is connected in parallel. Although they do not provide a large surface area for heat transfer, double pipe heat exchangers are at times used in industrial settings. A schematic of this unit and flow classification(s) is provided in Figure 33.1. Recommended standard fittings for double pipe interchangers are provided in the literature [1]. Tube materials include: carbon steel, carbon alloy steels, stainless steels, brass and alloys, cupro-nickel, nickel, monel, glass, reinforced fiberglass plastic (RFP), etc. As noted in Chapter 12, calculations for flow in the annular region referred to above require the use of a characteristic or equivalent diameter. By definition, this diameter, Deq, is given by 4 times the area available for flow divided by the “wetted” perimeter. Note that there are two wetted perimeters in an annular area. This reduces to:

Double Pipe Heat Exchangers 327 Cold fluid in

Parallel (co-current) flow

Inner pipe

Outer pipe

Hot fluid in Hot fluid out

Annulus area

(a) Cold fluid out Cold fluid in

Countercurrent flow Inner pipe

Outer pipe

Hot fluid in

Hot fluid out

Annulus area

(b) Cold fluid out

Figure 33.1 Double pipe heat exchanger schematic. (a) Parallel (co-current) flow. (b) Countercurrent flow.

Deq

4

(Do2,i Di2,o )/ (Do.i

4

Di ,o )

(33.1)

where Do,i is the inside diameter of the outer pipe, and Di,o is the outside diameter of the inner pipe. This equation reduces to:

Deq

Do.i Di ,o

(33.2)

which is four times the hydraulic radius, i.e.,

Deq

4rH

(33.3)

1 (Do.i Di ,o ) 4

(33.4)

with

rH

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Unit Operations in Environmental Engineering

The flow in a double pipe heat exchanger may be countercurrent or parallel (co-current) as shown in Figure 33.1. In countercurrent flow, the fluid in the pipe flows in a direction opposite to the fluid in the annulus. In parallel flow, the two fluids flow in the same direction. The variations of fluid temperature within the heat exchanger depend on whether the flow is parallel or countercurrent. The definitions below are employed in the development to follow: T1 = temperature of the hot fluid entering the inside pipe/tube T2 = temperature of the hot fluid exiting the inside pipe/tube t1 = temperature of the cold fluid entering the annulus t2 = temperature of the cold fluid exiting the annulus The difference between the temperature of the tube side fluid and that of the annulus side is the temperature difference driving force (TDDF), T. As noted in the previous chapter, in a co-current flow heat exchanger, both hot and cold fluids enter on the same side and flow through the exchanger in the same direction. The temperature approach is defined as the TDDF at the heat exchanger entrance, T1 or (T1 – t1). This driving force drops as the streams approach the exit of the exchanger. At the exit, the TDDF is T2 or (T2 – t2). Thus, the heat exchanger is more effective at the entrance than at the exit. In a countercurrent flow exchanger, the two fluids exchange heat while flowing in opposite directions. The temperature approach at the tube entrance end T1 or (T1 – t2) and at the annular entrance end, T2 or (T2 – t1) is usually roughly the same. In addition, the thermal driving force is normally relatively constant over the length of the exchanger. The temperature profiles for both parallel and countercurrent systems are presented in Figure 33.2. Summarizing, the double pipe heat exchanger employed in practice consists of two pipes: an inner and an outer pipe. Hot fluid normally flows in the inner pipe and the cold fluid flows in the annulus between the outer diameter of the inner pipe and inner diameter of the outer pipe. Heat transfer occurs from the inner pipe to the outer pipe as the fluids flow through the piping system. By recording Countercurrent Hot fluid flow

Co-current

(a)

T1 t1

T1

Hot fluid flow Local TDDF

Cold fluid flow Length of exchanger

T2

T2 t2

Temperature

Temperature

T1

T2

Local TDDF

T2 t2

T1 t1

Cold fluid flow

(b)

Length of exchanger

Figure 33.2 (a) Co-current and (b) countercurrent flow in a double pipe heat exchanger.

Double Pipe Heat Exchangers 329 both inner and outer fluid temperatures at various points along the length of the exchanger, it is possible to calculate heat exchanger duties and heat transfer coefficients (to be discussed shortly).

33.3 Describing Equations In designing a double pipe heat exchanger, mass balances, heat balances, and the applicable heat transfer equation(s) are used. The steady-state heat balance equation is:

Q

hH

hC

mH c pH (T1 T2 ) mC c pC (t 2 t1 )

(33.5)

This equation assumes steady state, no heat loss, no viscous dissipation, no heat generation, and no latent heat effects [1]. The rate equation used to design an exchanger is the design equation, which includes the previously developed log mean temperature or global temperature difference driving force ( T1m or LMTD), and overall heat transfer coefficient, U, as:

Q UA Tlm

(33.6)

where Q is the heat load; and A is the heat transfer area. If the TDDFs are T1 and T2 at the entrance and exit of the heat exchanger, respectively, then

Tlm

T1 T2 ln( T1 / T2 )

(33.7)

If T1 = T2, then T1m = T1 = T2. The overall heat transfer coefficient, U, is usually based on the inside area of the tube; the A term in Equation (33.6) should then be based on the inside surface area. The typical values of U for new, clean double pipe exchangers are available in the literature and, as noted earlier, are often assigned in the symbol, Uclean. If the heat duties are known (or have been calculated), values for the overall heat transfer coefficient, U, can be calculated as follows:

Uo

QC Ao Tlm

(33.8)

Ui

QH Ai Tlm

(33.9)

where A = surface area for heat transfer (outside or inside), ft2; Q = average heat duty (cold or hot), Btu/h; Uo = overall heat transfer coefficient based on the outside

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area, Btu/hr-ft2-°F; and Ui = overall heat transfer coefficient based on the inside area, Btu/hr-ft2-°F. In addition, one may write

1 UA

1 U i Ai

1 U o Ao

(33.10)

As described in the previous chapter, the overall heat transfer coefficient, U, for flow in a tube is related to the individual coefficients by the following equation:

1 UA

1 hi Ai

R f ,i Ai

ln(Do /Di ) 2 kL

R f ,o Ao

1 ho Ao

(33.11)

where hi = inside heat transfer coefficient, Btu/hr-ft2-°F; ho = outside heating transfer coefficient, Btu/hr-ft2-°F; Rf,i = fouling factor based on inner tube surfaces; Rf,o = fouling factor based on outer tube surfaces; Di/Do = diameter of pipe (inside or outside), ft; k = pipe thermal conductivity, Btu/hr-ft-°F; and L = tube length, ft. If the fouling factors, Rf,i and Rf,o, and the tube wall resistance (middle term in right-hand side of Equation 33.11) are negligible, then the relationship between the overall heat transfer coefficient and the individual coefficients simplifies as follows:

1 U

Do hi Di

1 Ui

1 hi

1 ho

(33.12)

Di ho Do

(33.13)

or

Individual coefficients, hi and ho, can be calculated using empirical equations discussed in Chapters 27 to 32, some of which is repeated below. Except for the viscosity term that is evaluated at the wall temperature, all of the physical properties in the equations that follow are evaluated at bulk temperatures. For the hot stream in the inner tube, the bulk temperature is Th,bulk = (Th,i + Th,o)/2. For the cold stream in the annulus, the bulk temperature is tc,bulk = (tc,i + tc,o)/2. Employing viscosity values at the wrong temperature can lead to substantial errors; however, the density and thermal conductivity of liquids do not vary significantly with temperature. The Reynolds number for both the cold and hot process streams must be found in order to determine whether the flow rate for each stream is in the laminar, turbulent, or transition region. In all of the equations that follow, calculations for the annulus require that the aforementioned equivalent or hydraulic diameter,

Double Pipe Heat Exchangers 331 Deq = (Do,i – Di,o) replace the tubular diameter, D. Thus, the Reynolds numbers, Rei and Reo, are defined as follows:

4mi Di ,i

Inner pipe: Rei

Annulus between pipes: Reo

(33.14) i

4mo (Do.i Di ,o )

Deq mo oS

o

(33.15)

where Di,i = inside diameter of inner pipe, ft; Do,i = outside diameter of inner pipe, ft; Do,i = inner diameter of outer pipe, ft; Deq = equivalent diameter, ft; S = crosssectional annular area, ft2; and = viscosity of hot or cold fluid at bulk temperature, lb/ft-hr. Flow regimes for various Reynolds numbers appear in Table 33.1 (also see Chapter 13). Similarly, Nusselt numbers, Nui and Nuo, are defined by the following equations:

Inner pipe: Nui

hi Di ,i

(33.16)

ki

Annulus between pipes: Nuo

ho (Do.i Di ,o ) ko

(33.17)

where k = thermal conductivity at bulk temperature of hot or cold fluid, Btu/hr-ft°F; and Nu = Nusselt number, inside or outside. For laminar flow, the Nusselt number equals 4.36 for uniform surface heat flux, Q /A , or 3.66 for constant surface temperature [2, 3]. This value should only be used for Graetz numbers, Gz, less than 10. For laminar flow with Graetz numbers from 10 to 1000, the following equation applies [3]: 0.14

Nu 2.0 Gz

1/ 3

; Gz wall

mc p kL

Table 33.1 Reynolds number values versus type of flow. Reynolds number, Re

Flow region

Re < 2100

Laminar

2100 < Re < 4,000

Transitional

Re > 4,000

Turbulent

(33.18)

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where wall = viscosity at wall temperature, lb/ft-hr; L = total length of tubular exchanger, ft; and Gz = Graetz number. For turbulent flow (Re > 4,000), the Nusselt number may be calculated from the Dittus-Boelter equation if 0.7 < Pr < 160 or the Sieder-Tate equation if 0.7 ≤ Pr ≤ 16,700 where Pr is the Prandtl number. Both equations are valid for L/D greater than 10 [3]. Details of both equations follow. Dittus-Boelter equation:

Nu = 0.023 Re0.8 Prn; or St = 0.023 Re(Re) where Pr

0.2

(Pr)

0.667

(33.19)

cp

; and n = 0.4 for heating or 0.3 for cooling. k Sieder-Tate equation: 0.14

Nu 0.023 Re

0. 8

Pr

1/ 3

(33.20) wall

Note that Equation 33.19 is often written in terms of the Stanton number, St (occasionally referred to as the modified Nusselt number), where St = Nu/RePr = h/ Vcp. The Dittus-Boelter equation should only be used for small to moderate temperature differences. The Sieder-Tate equation applies for larger temperature differences [2,3]. Errors as large as 25% are associated with both equations [2]. Other empirical equations with more complicated formulas and less error are available in the literature [2-6]. After several months of use, the tubes in an exchanger can become fouled by scale or dirt. This scale adds an extra resistance to heat transfer and causes a decrease in the overall heat transfer coefficient from Uclean to Udirty. The relationship between these two values of U (see previous chapter, Equation 32.32) is given by:

1

1

U dirty

U clean

Rf

(33.21)

where Rf is the fouling or dirt factor in typical units of ft2-hr-°F/Btu, or m2-K/W. Typical values of these fouling factors are given in Table 33.2. Information is also available in the previous chapter regarding fouling factors and their effect on heat transfer. In designing a double pipe heat exchanger, it is often desirable to estimate the minimum required pipe length. This is accomplished by using the pipe conduction resistance as the only resistance to heat transfer and taking the larger of the two temperature differences at the ends of the exchanger, Tmax(i.e., either T1 – t2 or T2 – t1 for countercurrent flow and T1 – t1 for co-current flow), to be the driving force. For this condition,

Double Pipe Heat Exchangers 333 Table 33.2 Representative fouling factors. Rf, m2-K/W

Fluid Seawater and treated boiler feed water (below 50 °C)

0.0001

Seawater and treated boiler feed water (above 50 °C)

0.0002

River water (below 50 °C)

0.0002–0.0001

Fuel oil

0.0009

Quenching oil

0.0007

Refrigerating liquid

0.0002

Steam (not oil bearing)

0.001

Q

2 kLmin Tmax ln(ro /ri )

(33.22)

This finds limited use in some real world applications.

33.4 Effectiveness Factor and Number of Transfer Units The effectiveness factor (or effectiveness) and number of transfer units are two approaches that have been occasionally employed in the past in heat exchanger design and analysis. The effectiveness, , of a heat exchanger is defined as:

actual heat transfer rate maximum heat transfer rate

Q Qmax

(33.23)

A maximum rate, Qmax , is calculated for the hot side and the cold side (tube and annulus side, respectively) for the double pipe heat exchanger. For example, assuming countercurrent flow,

Qmax , tube Qmax , annulus

(mc p )tube |T1 t1 | Ctube |T1 t1|

(33.24)

(mc p )annulus |T1 t1| Cannulus |T1 t1|

(33.25)

The C term is defined as the thermal capacitance rate and is simply mc p . The lower Qmax value is used in evaluating the heat transfer effectiveness. The effectiveness factor, , is applied in simulation (predictive) studies rather than design. It enables one to calculate exit temperatures and heat transfer rates for an existing unit. The effectiveness factor is provided below for both systems (parallel, countercurrent) and for both hot and cold side. Note the development of these equations assumes that the hot fluid is flowing in the inside tube.

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Cocurrent flow:

Tube,

T1 T2 T1 t1

H

Annulus,

t 2 t1 T1 t1

C

(33.26)

(33.27)

Countercurrent flow:

Tube,

Annulus,

T1 T2 T1 t1

H

C

t 2 t1 T1 t1

(33.28)

(33.29)

Note that subscripts 1 and 2 refer to inlet and outlet conditions, respectively. Thus, if the effectiveness factor is known, one could calculate the exit temperature T2 or t2 (and the corresponding Q ). The question that remains is whether to use H or C. The fluid to employ, hot or cold, is that which undergoes the maximum temperature change (or difference) or has the corresponding minimum value of C:

Q (mc p ) T

(mc p )H (T1 T2 ) (mc p )C (t 2 t1 )

(33.30)

If the temperatures are unknown and the flow rates are known, the minimum value of mc p is obtained from flow rate data. The mc p value for the fluid in question thus requires the use of the appropriate H or C, and this equation calculates the exit temperature. This general topic will be reviewed again in the next chapter, particularly as it applies to shell and tube heat exchangers. Another term employed in heat exchanger studies is the number of transfer units, NTU, of a heat exchanger. It is defined as:

NTU

UA (mc p )min

UA Cmin

(33.31)

where Cmin is the minimum thermal capacitance rate. NTU is determined by calculating C for each fluid and choosing the lower value.

33.5

Wilson’s Method

There is a procedure for evaluating the outside film coefficient for a double pipe unit. Wilson’s method [7, 8] is a graphical technique for evaluating this coefficient.

Double Pipe Heat Exchangers 335 The inside coefficient is a function of the Reynolds and Prandtl numbers via the Dittus-Boelter equation presented in Equation 33.19, i.e.,

hi

f (Re 0.8 Pr 0.3 )

(33.32)

A series of experiments can be carried out on a double pipe exchanger where all conditions are held relatively constant except for the velocity, V, of the cooling (in this case) inner stream. Therefore, for the proposed experiment:

hi

f (Re 0.8 )

f (V 0.8 )

f (m); Re

DV /

(33.33)

or, in equation form

hi

aV 0.8

(33.34)

where a is a constant. Equation 33.34 can be substituted into the overall heat transfer coefficient equation so that:

1 U o Ao

Ro Rw

Ri

1 ho Ao

x kAlm

1 aV 0.8 Ai

(33.35)

Data can be taken at varying velocities. By plotting 1/UoAo versus 1/V0.8, a straight line should be obtained since the first two terms of Equation 33.35 are constants. The intercept of this line corresponds to an infinite velocity and an inside heat resistance of zero. Thus, the above equation may be rewritten as:

1 U o Ao

intercept

1 ho Ao

x kAlm

(33.36)

The second term on the right-hand side is known and/or can be calculated, and ho can then be evaluated from the intercept. Fouling coefficients, f, can be estimated by the Wilson method if the outside fluid coefficient, ho, can be predicted or is negligible. Note that the fouling resistance is normally included in the intercept value.

References 1. Badger, W., and Banchero, J., Introduction to Chemical Engineering, McGraw-Hill, New York City, NY, 1955. 2. Bergman, T.L., Lavine, A.S., Incropera, F.P., and De Witt, D.P., Fundamentals of Heat and Mass Transfer, 7th Edition, John Wiley & Sons, Hoboken, NJ, 2011.

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3. McCabe, W. Smith, J., and Harriott, P., Unit Operations of Chemical Engineering, 7th Edition, McGraw-Hill, New York City, NY, 2005. 4. McAdams, W., Heat Transmission, 3rd Edition, McGraw-Hill, New York City, NY. 1954. 5. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 6. Zimmerman, O., and Lavine, I., Chemical Engineering Laboratory Equipment, 2nd Edition, pp. 79–145, Industrial Research Service, Inc., Dover, NH, 1955. 7. Wilson, E.E., A basis for traditional design of heat transfer apparatus, Trans. ASME, 37, 47, 1915. 8. Theodore, L., Ricci, F., and Van Vliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

34 Shell and Tube Heat Exchangers

34.1 Introduction There are vast industrial uses of shell and tube heat exchangers. They are employed to heat or cool process fluids, either through a single-phase heat exchanger or a two-phase heat exchanger. In single-phase exchangers, both the tube side and shell side fluids remain in the same phase as they enter. In two-phase exchangers (examples include condensers and boilers), the shell side fluid is usually condensed to a liquid or heated to a gas, while the tube-side fluid remains in the same phase. Shell and tube (also referred to as tube and bundle) heat exchangers provide a large heat transfer area economically and practically. The tubes are placed in a bundle and the ends of the tubes are mounted in tube sheets. The tube bundle is enclosed in a cylindrical shell through which the second fluid flows. Most shell and tube exchangers used in practice are of welded construction. The shells are built as a piece of pipe with flanged ends and necessary branch connections. The shells are made of seamless pipe up to 24 inches in diameter; however, they are made of bent and welded steel plates if above 24 inches. Channel sections are usually of built-up construction, with welding-neck, forged-steel flanges, rolled-steel barrels and welded-in place partitions. Shell covers are either welded directly to the shell, or are built-up constructions of flanged and dished heads and welding-neck, 337

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Unit Operations in Environmental Engineering

forged-steel flanges. The tube sheets are usually nonferrous castings in which the holes for inserting the tubes have been drilled and reamed before assembly. Baffles can be employed to both control the flow of the fluid outside the tubes and provide turbulence [1]. Generally, shell and tube exchangers are employed when double pipe exchangers do not provide sufficient area for the required heat transfer. Shell and tube exchangers usually require less materials of construction and are consequently more economical when compared to double pipe and/or multiple, parallel double pipe heat exchangers. Chapter topics include Equipment Description, Describing Equations, The “F” Factor, Effectiveness Factor and Number of Transfer Units, and Design Procedure.

34.2 Equipment Description There are more than 280 different types of shell and tube heat exchangers that have been defined by the Tubular Exchanger Manufacturers Association (TEMA). The simplest shell and tube heat exchanger has a single pass through the shell and a single pass through the tubes. This is termed a 1–1 shell and tube heat exchanger. A line diagram of a 1–2 unit (one pass on the shell side and two tube passes) can be found in Figure 34.1. A side view of the tubes in a typical exchanger is shown in Figure 34.2. Fluids that flow through tubes at low velocity result in low heat transfer coefficients and low pressure drops. To increase the heat transfer rates, multi-pass operations may be used. As noted earlier, baffles are used to divert the fluid within the distribution header. An exchanger with one pass on the shell side and four tube passes is termed a 1–4 shell and tube heat exchanger. It is also possible to increase the number of passes on the shell side by using dividers. A 2–8 shell and tube heat exchanger has two passes on the shell side and eight passes on the tube side. The choice of which fluid should flow in the tubes is an important design decision. If one of the fluids is particularly corrosive, it is typically introduced on the tube side since more expensive resistant/tubes can be purchased. If one of the T1

t2

t1 T2 Figure 34.1 One shell pass, 2 tube passes (1–2 unit) schematic.

Shell and Tube Heat Exchangers 339

Figure 34.2 Side view of tube layout in the shell and tube heat exchanger.

fluids is more likely to form scale and/or deposits, it should be the tube side fluid since the inside of the tubes are much easier to clean than the outside of the tubes. In addition, viscous fluids normally flow on the shell side in order to help induce turbulence. The impact on the overall heat transfer coefficient and pressure drop should also be considered when determining the flow locations of the fluids [1].

34.3 Describing Equations When two process fluids at different temperatures pass one another as in a shell and tube exchanger, heat transfer occurs due to the temperature difference between the two streams. The energy required to accomplish this heat transfer, i.e., the heat exchanger duty, can be determined for both process streams. For a well-insulated exchanger, the two duties should equal one another. The describing equations for

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the two duties are similar to those presented for the double pipe heat exchanger (see previous chapter). However, the following equation gives the heat exchanger duty of the hot process stream when saturated steam is introduced on the shell side (enthalpy balances between steam and condensate may also be used):

Qh

ms

ms c p ,hw (Ts ,i Ts ,o ) ms

ms c p ,hw (T1 T2 )

(34.1)

where Qh = QH = rate of energy lost by steam, Btu/hr; ms = total mass flow rate of steam condensate, lb/hr; hvap = λ = heat of vaporization of steam, Btu/lb; cp,hw = average heat capacity of condensate (hot water), Btu/lb-°F; Ts,i = T1 = steam inlet temperature, °F; and Ts,o = T2 = condensate outlet temperature, °F. The cold process stream duty is calculated similarly:

Qc

mc c p ,cw (t c ,o t c ,i ) mc c p ,cw (t 2 t1 )

(34.2)

where Qc = QC = rate of energy gained by cooling water, Btu/hr; mc = total mass flow rate of the cold water (all tubes), lb/hr; cp,cw = average heat capacity of the cold process stream, Btu/lb-°F; tc,o = t2 = cold water outlet temperature, °F; and tc,i = t1 = cold water inlet temperature, °F. The reader should note that much of the material that follows was presented in the last chapter. However, as each chapter is presented on a stand-alone basis, the applicable equations for shell and tube exchangers receive treatment here. If the heat exchanger duty is known, the overall heat transfer coefficient is calculated from either of the two equations provided below:

Uo

Qh Ao Tlm

(34.3)

Ui

Qc Ai Tlm

(34.4)

or

where Uo = overall heat transfer coefficient based on hot (h) process stream, Btu/ ft2-hr-°F; Ui = overall heat transfer coefficient based on cold (c) process stream, Btu/ft2-hr-°F; Ao = exchanger transfer area based on outside tube surfaces, ft2; Ai = exchanger transfer area based on inside tube surface, ft2; and Tlm = logmean temperature difference driving force, °F. Typical values for the overall heat transfer coefficient in steam condensers with water in the tubes range from 1,000 to 6,000 W/m2-K [2,3]. Perry’s Chemical Engineers’ Handbook [4] reports coefficients ranging from 400 to 1,000 Btu/hr-ft2°R. These values are approximately the same after conversion of units.

Shell and Tube Heat Exchangers 341 For tubular heat exchangers, as with double pipe heat exchangers, the overall heat transfer coefficient is related to the individual coefficients by the following equation:

1 UA

1 U i Ai

1 U o Ao

R f ,i

1 hi Ai

Ai

ln(Do /Di ) 2 kL

R f ,o Ao

1 ho Ao

(34.5)

where U = overall heat transfer coefficient, Btu/ft2-hr-°F; Ai, Ao = inside and outside surface areas ( DiL or DoL), ft2; A = average of two areas, ft2; Ui, Uo = coefficients based on inner and outer tube surfaces, Btu/ft2-hr-°F; hi, ho = inside and outside heat transfer coefficients, Btu/ft2-hr-°F; Rf,i, Rf,o = fouling factors based on inner and outer tube surfaces, ft2-hr-°F/Btu; k = thermal conductivity, Btu/ft-hr-°F; and L = tube length, ft. If the fouling factors, Rf,i and Rf,o, and the tube wall resistance, i.e., the middle terms on the right-hand side of Equation 34.5 are negligible, then the relationship between the overall heat transfer coefficient and the individual coefficients simplifies once again to:

1 Uo

Do hi Di

1 ho

(34.6)

The overall heat transfer coefficient, U, is an average value based on the average duty or Q (Qh Qc )/2 , and an average heat transfer area or A = (Ai + Ao)/2. The individual coefficients above, hi and ho, are almost always calculated using empirical equations. First, the Reynolds number, Re, for the cold process fluid must be found in order to determine whether the flow is in the laminar, turbulent, or transition region.

Re

4mc Di c

(34.7)

where mc = mass flow rate of tube-side fluid through one tube, lb/hr; Di = inside diameter of one tube, ft; c = viscosisty of tube-side fluid at bulk temperature, lb/ ft-hr; and Re = Reynolds number. All of the physical properties of the tube-side fluid are evaluated at the average bulk temperature, tc,bulk = (tc,i + tc,o)/2. The Nusselt number, Nu, can be determined from empirical equations. For laminar flow in a circular tube (Re < 2,100), the Nusselt number equals 4.36 for uniform surface heat flux (Q /A) or 3.66 for constant surface temperature [3–5]. This value should only be used for Graetz numbers, Gz, less than 10 [5]. For laminar flow with Graetz numbers from 10 to 1000, the following equation applies [5]: 0.14

Nu 2Gz

1/ 3

c c , wall

; Nu

hi Di , Gz kc

mc C p ,c kc L

(34.8)

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where c,wall = viscosity of tube-side fluid at wall temperature, lb/ft-hr; kc = thermal conductivity of tube-side fluid at bulk temperature, Btu/ft-hr-°F; cp,c = heat capacity of tube-side fluid at bulk temperature, Btu/lb-°F; L = length of tubes, ft; Nu = Nusselt number; and Gr = Graetz number. For turbulent flow (Re > 4,000), the Nusselt number may be calculated from the Dittus-Boelter equation if 0.7 ≤ Pr ≤ 160 or the Sieder-Tate equation if 0.7 ≤ Pr ≤ 6700, where Pr is the Prandtl number. Both equations are valid for L/D greater than 10 [3, 5]. Dittus-Boelter equation:

Nu 0.023 Re 4 /5 Pr n ; Pr

c p ,c

c

, kc n 0.4 for heating or 0.3 or cooling

(34.9)

Sieder-Tate equation:

Nu 0.023 Re 4 /5 Pr 1/3

c

(34.10)

c , wall

The Dittus-Boelter equation should only be used for small to moderate temperature differences. The Sieder-Tate equation applies for larger temperature differences [3–5]. Errors as large as 25% are associated with both equations. Other empirical equations with more complicated formulas and smaller errors are available in the literature [3–5]. The above empirical equations do not apply for flows in the transition region (Re between 2,100 and 4,000). For the transition region, one should review the literature to determine the inside heat transfer coefficient. The reader is cautioned to use the correct mass flow rate; i.e., flow through a single tube rather than the total flow rate. The outside heat transfer coefficient, ho, is calculated using empirical correlations for condensation of a saturated vapor on a cold surface if steam is employed in the shell. Under normal conditions, a continuous flow of liquid is formed over the surface (film condensation) and condensate flows downward due to gravity. In most cases, the motion of the condensate is laminar and heat is transferred from the vapor-liquid interface to the surface by conduction through the film. Heat transfer coefficients for laminar film condensation on a single horizontal tube or on a vertical tier of N horizontal tubes can be calculated using appropriate equations from the literature [3–5].

34.4

The “F” Factor

Some heat exchangers use true countercurrent flow. However, these heat exchangers are not as economical as multi-pass and crossflow units. In multi-pass

Shell and Tube Heat Exchangers 343 exchangers, the flow alternates between co-current and countercurrent between the different sections of the exchanger. As a result, the driving force is not the same as a true countercurrent or a true co-current exchanger. The aforementioned Tlm (log mean temperature difference driving force or LMTD) for these exchangers is almost always less than that of countercurrent flow and greater than that of cocurrent flow. The analysis of these units requires including an F factor in the describing equation (i.e., the heat transfer equation). For those multi-pass and crossflow exchangers, the LMTD method is applicable, i.e.,

T1

Tlm

ln

T2 T1 T2

(34.11)

or

T2

Tlm

ln

T1 T2 T1

(34.12)

For the special case of

T1

T2

T

(34.13)

one may employ

Tlm

T

(34.14)

The transfer rate is still given by

Q UA Tlm

(34.15)

except that Tlm must be corrected by a geometry factor, F. This correction factor accounts for portions of multi-pass heat exchangers where the flow is not countercurrent (e.g., hairpin turns). Equation 34.15 is now written as:

Q UAF Tlm ,cp Tlm

F Tlm ,cp

(34.16) (34.17)

where Tlm,cp is the LMTD for Ts based on “ideal” countercurrent operation and F is the geometry factor, or correction factor, applied to a different flow arrangement

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with the same hot and cold fluid temperature. Bowman et al. [6] coordinated the results of earlier studies regarding the Tlm for exchangers that experience neither co-current or countercurrent flow to provide as complete a picture as possible of the various arrangements of surface conditions and flow directions. Shell and tube heat exchangers that experience any number of passes on the shell and tube side were covered in their work as well as crossflow exchangers with different pass arrangements. The results for the various classifications follow. Using the equations developed by Bowman et al. [6], Underwood [7], and Nagle [8], Theodore [1] provided a summary of their results for: 1. 2. 3. 4. 5. 6.

One Pass Shell Side; Two Pass Tube Side One Pass Shell Side; Four Pass Tube Side One Pass Shell Side; Six Pass Tube Side Two Pass Shell Side; Four Pass Tube Side Three and More Shell-Side Passes Fluids Mixed

34.5 Effectiveness Factor and Number of Transfer Units The effectiveness factor and number of transfer units method described in the previous chapter for double pipe exchangers is again preferable to use when only inlet temperatures to the shell and tube heat exchanger are known. The following notation (different from the previous chapter since the colder fluid is normally on the tube side) is used for this shell and tube exchanger development below: m1 = mass flow rate of fluid entering or leaving the tube side m2 = mass flow rate of fluid entering or leaving the shell side T1,T2 = inlet and exit temperatures of the shell side fluid, respectively t1,t2 = inlet and exit temperatures of the tube side fluid, respectively Q = heat duty, heat load, or heat transfer rate Qmax = maximum possible heat transfer rate c1,c2 = heat capacity of fluids on the tube and shell sides, respectively C1,C2 = heat capacitance rate of fluids on the tube and shell sides, respectively As noted earlier, the heat duty of a shell and tube exchanger is calculated from the energy balance on the tube side or on the shell side:

Q m1c1 (T1 T2 ) m2 c2 (T1 T2 ) C1 (t 2 t1 ) C2 (T1 T2 )

(34.18)

where m1c1 C1 , and m2 c2 C2 . This assumes that the fluid on the tube side is being heated (t2 > t1) and the shell side fluid is being cooled (T2 < T1). To determine Qmax , the maximum heat transfer rates for both the tube, Qmax , t , and shell side fluids, Qmax ,s , are first determined and the lower of the two values is used. For

Shell and Tube Heat Exchangers 345 the tube side, Qmax , t occurs when the exit fluid temperature, t2, is the same as the temperature of the incoming fluid on the shell side (i.e., t2 = T1), so that

Qmax , t

C1 (T1 t1 )

(34.19)

Similarly, the shell side fluid, Qmax ,s , occurs when T2 = t1:

Qmax ,s

C2 (T1 t1 )

(34.20)

The lower of the two heat capacitance rates, C1 and C2, determines Cmin and the permissible maximum heat load, Qmax . If the heat capacitance rate C1 < C2, then Cmin = C1 and Cmax = C2. In this case,

Qmax

C1 (T1 t1 ) Cmin (T1 t1 )

(34.21)

Using the definition of the effectiveness factor, , provided in the previous chapter, and combining Equations 34.18 and 34.21 yields:

Q Qmax

t 2 t1 T1 t1

(34.22)

If the shell side fluid has the lower heat capacitance rate (i.e., C2 < C1), then Cmin = C2, Cmax = C1, and

Q Qmax

T1 T2 T1 t1

(34.23)

For any heat exchanger, it can be shown that

f NTU,

Cmin ; Cmax

NTU

UA Cmin

(34.24)

Several graphs in which is plotted against the NTU (Number of Transfer Units), with the Cmin/Cmax ratio Cr as the third parameter, are available for six different heat exchanger geometries [3]. These are: 1. A parallel flow heat exchanger. 2. A counterflow heat exchanger. 3. A shell and tube heat exchanger with one shell pass and any even multiple of two tube passes (two, four, etc.). 4. A shell and tube heat exchanger with two shell passes and any even multiple of four tube passes (four, eight, etc.).

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Unit Operations in Environmental Engineering 5. A single pass, crossflow heat exchanger with both fluids unmixed. 6. A single pass, crossflow heat exchange with one fluid mixed and the other unmixed.

34.6 Design Procedure When designing a shell and tube heat exchanger, the layout of the inner tube bundle is the main physical design concern and includes: 1. 2. 3. 4. 5. 6.

Tube diameter Tube wall thickness Tube length Tube layout Tube corrugation Baffle design

Each of the above are detailed by Theodore [9].

References 1. Theodore, L., and Buonicore, A., adapted from Control of Gaseous Emissions, Air Pollution Training Institute (APTI), EPA 450/2–81/005, U.S. Environmental Protection Agency, Environmental Research Center, Research Triangle Park, NC, 1981. 2. Saunders, E., Heat Exchangers: Selection Design and Construction, p. 46, Longman Scientific and Technical, John Wiley & Sons, Hoboken, NJ, 1988. 3. Incropera, F., and De Witt, D., Fundamentals of Heat and Mass Transfer, 5th Edition, John Wiley & Sons, Hoboken, NJ, 2002. 4. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 5. McCabe, W. Smith, J., and Harriott, P., Unit Operations of Chemical Engineering, 7th Edition, McGraw-Hill, New York City, NY, 2005. 6. Bowman, R., Mueller, A., and Nagle, W., Mean temperature difference in design, Trans. ASME, 62, 283, 1940. 7. Underwood, A., Calculation of the mean temperature difference in multi-pass heat exchangers, J. Inst. Petroleum Technol., 20, 145–158, 1934. 8. Nagle, W., Mean temperature differences in multipass heat exchangers, Ind. Eng. Chem., 25, 604–608, 1933. 9. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

35 Finned Heat Exchangers

35.1 Introduction Consider the case of a heat exchanger that is heating air outside the tubes by means of steam inside the tubes. The steam-side heat transfer coefficient will be very high, while the air-side heat transfer coefficient will be extremely low. Therefore, the overall heat transfer coefficient will approximate that of the air side (i.e., the air is the controlling resistance). One method of increasing the heat transfer rate is to increase the surface area of a heat exchanger. Therefore, if the surface of the metal on the air side could be increased, it would increase the area term without putting more tubes in the exchanger. This can be accomplished by mounting metal fins on a tube in such a way that there is good metallic contact between the base of the fin and the wall of the tube. If this contact is secured, the temperature throughout the fins will approximate that of the temperature of the heating (cooling) medium because of the high thermal conductivity of most metals used in practice. Consequently, the heat transfer surface will be increased without more tubes [1]. Additional metal is often added to the outside of ordinary heat transfer surfaces such as pipes, tubes, walls, etc. These extended surfaces, usually referred to as fins, increase the surface available for heat flow and result in an increase of the total transmission of heat. Examples of fin usage includes automobile radiators, air 347

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conditioning units, cooling of electronic components, and heat exchangers. They are primarily employed for heat transfer to gases where film coefficients are very low. The remaining sections of this chapter include Fin Types, Describing Equations, Fin Effectiveness and Performance, and Fin Considerations.

35.2

Fin Types

Extended surfaces, or fins, are classified into longitudinal fins, transverse fins, and spine fins. Longitudinal fins (also termed straight fins) are attached continuously along the length of the surface (Figure 35.1). They are employed in cases involving gases or viscous liquids. As one might suppose, they are primarily employed with double pipe heat exchangers. Transverse or circumferential fins are positioned approximately perpendicular to the pipe or tube axis and are usually used in the cooling of gases (Figure 35.2). These fins find their major application with shell and tube exchangers. Transverse fins may be continuous or discontinuous (segmented). Annular fins are examples of continuous transverse fins. Spine or peg fins employ cones or cylinders, which extend from the heat transfer surface, and are used for either longitudinal flow or cross flow.

|t|

w

Tb

L

T∞

r0

Ti

Figure 35.1 Longitudinal fins.

Finned Heat Exchangers 349 |

t

|

L

T∞

Tb Ti ro

rf

Figure 35.2 Transverse or circumferential fins.

Table 35.1 Fin metal data.

Metal

Thermal conductivity, W/m-K

Specific gravity

Relative volume

Relative mass

Copper

400

8.9

1.00

1.00

Aluminum

210

2.7

1.83

0.556

55

7.8

7.33

0.43

Steel

Fins are constructed of highly conductive materials. The optimum fin design is one that gives the highest heat transfer for the minimum amount of metal. The metal used in their manufacture has a strong influence on fin efficiency. Table 35.1 compares the volume and mass of three different metals required to give the same amount of heat transfer for fins with identical shapes. The values in the volume and mass columns are relative to the volume and mass of a copper fin.

35.3 Describing Equations There are two problems in calculating the heat transfer coefficient for smooth finned tubes [2]: 1. The mean surface temperature of the fin is lower than the surface temperature of a smooth tube under the same conditions (due to the flow of heat through the metal of the fin). 2. There is a question as to whether or not the flow of the fluid outside the tube is as great at the bottom of the space between the fins as the unobstructed space. Both these factors depend on the size and thickness of the fins and their spacing, and the conditions of flow.

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To analyze the heat transfer in extended surfaces, the following assumptions are usually made [3]: a. b. c. d. e. f. g. h. i. j. k. l.

Steady-state operation Constant properties Constant surrounding air temperature of T Homogenous isotropic material, with thermal conductivity, k One-dimensional heat transfer by conduction in the radial direction No internal heat generation Heat transfer coefficient, h, is uniform along the fin surface Negligible thermal radiation The fin perimeter at any cross-section is P The fin cross-sectional area is Af The temperature of the heat transfer surface (exposed and unexposed) at the base of the fin is constant, Tb The maximum temperature driving force for convection is Tb T

The maximum rate of heat transfer, Q f ,max , from a fin will occur when the entire fin surface is isothermal at T = Tb. In the case of a fin with total surface area A f , Q f , max is written as:

Q f ,max

hA f (Tb T ) hA f

b

(35.1)

where b, termed the excess temperature, is defined as (Tb T ). Thus, b is the excess temperature at the base of the fin, and T is the fluid temperature [3]. Since the fin has a finite thermal conductivity, a temperature gradient will exist along the fin. The actual heat transfer rate from the fin to the outside fluid, Q f , will be less than Q f ,max . The fin efficiency, f, is a measure of how close Q f comes to Q f ,max and is defined as:

Qf f

Q f ,max

Qf hA f

(35.2) b

From Equations 35.1 and 35.2,

Qf

f

hA f

b

Tb T (1/ f hA f )

Tb T Rt , f

(35.3)

where

Rt , f

fin thermal resistance

1 f hA f

(35.4)

Finned Heat Exchangers 351 100 y ≈ x2

Lc = L Ap = Lt/3

y 80

t/2x L

(c)

nf , %

60

40 Lc = L + t/2 Ap = Lct

y≈x t/2

y

L

20

(a)

t/2 L

0 0

0.5

1.0

1.5

x

Lc = L Ap = Lt/2

(b)

2.0

2.5

Lc3/2 (h/kAp)1/2

Figure 35.3 Efficiency of straight rectangular fins (a), triangular fins (b), and parabolic profile fins (c). (Adapted from [4]).

Figure 35.3 is a plot of the efficiency of straight (longitudinal) fins ( %) versus the following dimensionless group, 1/ 2 3/2 c

L

h kAp

(35.5)

where Lc is the corrected fin length and Ap is the profile area of the fin. Table 35.2 provides expressions to calculate the corrected length, Lc, the projected or profile area, Ap, and the surface area, Af, in terms of the fin length, L, fin thickness at the base, t, and the fin width, w, for various fin types. The fin efficiency figures are valid for a fin Biot number ≤ 0.25, i.e.,

Bi f

h(t /2) 0.25 k

(35.6)

Barkwill et al. [5] converted the graphical results presented in Figure 35.3 into equation form. Their results are available in the literature [1]. Figure 35.4 shows the fin efficiency (nf% of annular fins of rectangular profile. The variables Lc and Ap used in the abscissa of the graph are related to the fin

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Table 35.2 Fin data. Variable

Rectangular fin

Triangular fin

Parabolic fin

Lc = corrected height

L + t/2

L

L

Ap = profile area

Lct

Lt/2

Lt/3

Af = fin surface area of length w

2wLc

2w L2 (t /2)2

2.05w L2 (t /2)2

100

80

1 = r2c/ro

nf , %

60

2 r2c = rf + t/2 Lc = L + t/2 Ap = Lct

40

20

0

3

5

t ro rf

0

L

0.5

1.0

Lc3/2 (h/kAp)1/2

1.5

2.0

2.5

Figure 35.4 Efficiency of annular fins of rectangular profiles. (Adapted from [4]).

height, L, pipe radius, ro, and fin outside radius, rf, where L = fin height = rf – ro; R2c = corrected outside radius = rf + (t/2); Lc = corrected height = L + t/2; Ap = profile (cross-sectional) area = Lct; and, Af = fin surface area = 2 (r2c,c ro2 ) . The parameter of the curves is the ratio r2c/ro. Barkwill et al. [5] also converted the results of Figure 35.4 into equation form. Their results are available in the literature [1]. For a straight fin (one of uniform cross-section as opposed to one that, for example, tapers down to a point), the heat transfer from the fin may be represented mathematically by:

Q

hPkAc

c

tan h(mLc )

(35.7)

Finned Heat Exchangers 353 where P is the fin cross-section perimeter, Ac is the (cross-sectional) area of the fin, and m hP /kAc .

35.4 Fin Effectiveness and Performance Another dimensionless quantity used to assess the benefit of adding fins is the fin effectiveness, f, or fin performance coefficient, FPC. It is defined as:

Qf

FPC

f

(35.8)

Qw /o , f

where Qw /o , f is the rate of heat transfer without fins, i.e.,

Qw /o , f

hAb

(35.9)

b

Fins are usually not justified unless f or FPC ≥ 2. The fin efficiency, f, and the effectiveness, f, characterize the performance of a single fin. As indicated above, arrays of fins are attached to the base surface in many applications. The distance from the center of one fin to the next along the same tube surface is termed the fin pitch, S. In this case, the total heat transfer area, At, includes contributions due to the fin surfaces and the exposed (unfinned) base surface, i.e.,

At

Abe

N f Af

(35.10)

The total heat transfer area without fins, Atw/o,f, is

Atw /o , f

Abe

N f Ab

(35.11)

where Nf = number of fins, Af = surface per fin, Abe = total exposed (unfinned) base area of the surface, and Ab = the base are of one fin. Equation 35.11 is often used to determine Abe from knowledge of the surface geometry, fin base area, and number of fins. The total heat transfer rate, without fins, Qt ,w /o , f , is

Qt ,w /o , f

hAt ,w /o , f

b

h( Abe

N f Ab )

(35.12)

b

The total heat transfer rate from the finned surface, Qt , is

Qt

Qbe Q ft h( Abe

Qbe

N f Af

N f Qf f

)

b

hAbe

b

N f hA f

f

b

(35.13)

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Unit Operations in Environmental Engineering

where Q ft is the heat transfer rate due to all (total) fins and Q f is that due to a single fin. The maximum heat transfer rate, Qt ,max , of the surface occurs when nf = 1.0, i.e., when the temperature of the base and all the fins is Tb, and is given by:

Qt ,max The overall fin efficiency,

h b ( Abe o,f

o, f

N f A f ) hAt

b

(35.14)

, is defined as:

Qt Qt ,max

Abe

N f Af

f

At

(35.15)

Substituting from Equation 35.10 into 37.15 yields:

o, f

1

N f Af

(1

At

Finally, the overall surface effectiveness,

o,f

f

)

, is defined as:

Qt o, f

(35.16)

Qt , w /o,, f

(35.17)

35.5 Fin Considerations The selection of suitable fin geometry requires an overall comparison of 1. 2. 3. 4. 5.

Economics Mass of fin Space (if available) Pressure drop (if applicable) The heat transfer characteristics of the fin

Generally, fins are effective with gases; are less effective with liquids in forced (or natural) convection; are very poor with boiling liquids; and, are extremely poor with condensing vapors. As discussed earlier, fins should be placed on the side of the heat exchanger surface where h is the lowest. Thin, closely spaced fins (subject to economic constraints) are generally superior to fewer thicker fins. Their thermal conductivity, k, should obviously be high. In summary, the basic heat exchanger equations apply for fins:

Finned Heat Exchangers 355

Q UA T

(35.18)

with

1 U

1 ho

x k

1 hi

(35.19)

For many applications, hi >> ho and x/ k >> ho so that:

1 U

1 ho

(35.20)

The outside coefficient is generally the controlling resistance and it may be large. One way to increase Q is to increase A. As noted earlier, this may be accomplished by the addition of fins to the appropriate heat transfer surface and these can be mounted longitudinally or circumferentially.

References 1. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 2. Badger, W., and Banchero, J., Introduction to Chemical Engineering, McGraw-Hill, New York City, NY, 1955. 3. Farag, I., and Reynolds, J., Heat Transfer, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1996. 4. Bergman, T.L., Lavine, A.S., Incropera, F.P., and De Witt, D.P., Fundamentals of Heat and Mass Transfer, 7th Edition, John Wiley & Sons, Hoboken, NJ, 2011. 5. Barkwill, B., Spinelli, M., and Valentine, K., project submitted to L. Theodore, Manhattan College, Bronx, NY, 2009.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

36 Other Heat Transfer Equipment

36.1 Introduction The purpose of this chapter is to extend the material presented in the three previous chapters to the design, operation, and predicative calculations of several other types of heat exchangers. The presentation will focus on: evaporators, waste heat boilers, condensers, and quenchers. Heat exchangers are defined as equipment that effect the transfer of thermal energy in the form of heat from one fluid to another. The simplest exchangers involve the direct mixing of hot and cold fluids. Most industrial exchangers are those in which the fluids arc separated by a wall. The latter type, referred to by some as a recuperator, can range from a simple plane wall between two flowing fluids to more complex configurations involving multiple passes, fins, or baffles. Conductive and convective heat transfer principles are required to describe and design these units; radiation effects are generally neglected. As described earlier, heat exchangers are devices used to transfer heat from a hot fluid to a cold fluid, and can be classified by their functions. Farag and Reynolds [1] provide an abbreviated summary of these units and their functions (see Table 36.1). A host of other exchangers can be added to this list, including

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Table 36.1 Heat exchanger equipment. Equipment

Function

Chiller

Cools a fluid to a temperature that is obtainable if only water were used as a coolant. It often uses a refrigerant such as ammonia or Freon.

Condenser

Condenses a vapor or mixture of vapors, either alone or in the presence of a non-condensable gas.

Cooler Exchanger

Performs a double function: (1) heats a cold fluid and (2) cools a hot fluid. Little or none of the transferred heat is normally lost.

Final condenser

Condenses the vapors to a final storage temperature of approximately 100 °F. It uses water cooling, which means the transferred heat is often lost to the process.

Forced-circulation reboiler

A pump is used to force liquid through the reboiler (see reboiler below).

Heater

Imparts sensible heat to a liquid or a gas by means of condensing steam or some other hot fluid (e.g., Dowtherm).

Partial condenser

Condenses vapors at a temperature high enough to provide a temperature difference sufficient to preheat a cold stream of process fluid. This saves heat and eliminates the need for providing a separate preheater.

Reboiler

Connected to the bottom of a fractioning tower, it provides the reboil heat necessary for distillation. The heating medium may be either steam (usually) or a hot process fluid.

Steam generator

Generates steam for use elsewhere in the plant by using available high-level (temperature) heat, e.g., from tar or a heavy oil.

Superheater

Heats a vapor above its saturation temperature.

Thermosiphon reboiler

Natural circulation of the boiling medium is obtained by maintaining sufficient liquid head to provide for circulation (see reboiler).

Vaporizer

A heater which vaporizes all or part of the liquid.

Waste-heat boiler

Produces steam; similar to a steam generator, except that the heating medium is a hot gas or liquid produced in a chemical reaction.

parallel corrugated plates, plate units, etc. [2]. Direct-contact coolers and specialty condensers can also be included in this list [3]. The four general classifications of direct-contact gas-liquid heat transfer operations are [4]:

Other Heat Transfer Equipment 1. 2. 3. 4.

359

Simple gas cooling Gas cooling with vaporization of coolant Gas cooling with partial condensation Gas cooling with total condensation

Most of the direct-exchange applications listed above are accomplished with the following devices: 1. 2. 3. 4. 5.

36.2

Baffle-tray columns Spray chambers Packed columns Crossflow-tray columns Pipeline contractors

Evaporators

The concentration of solutions of nonvolatile solutes through heat-induced vaporization of the solvent (generally water) is called evaporation. Evaporation is a relatively expensive operation, and its application in waste treatment is limited primarily to the recovery of valuable by-products from waste liquids and to the treatment of wastes where no alternate methods are available. However, the vaporization of a liquid for the purpose of concentrating a solution is a common operation in the chemical process industry. The simplest device is an open pan or kettle that receives heat from a coil or jacket or by direct firing underneath the pan. Perhaps the traditional unit is the horizontal-tube evaporator in which a liquid (to be concentrated) in the shell side of a closed, vertical cylindrical vessel is evaporated by passing steam or another hot gas through a bundle of horizontal tubes contained in the lower part of the vessel. The liquid level in the evaporator is usually less than half the height of the vessel; the empty space permits disengagement of entrained liquid from the vapor passing overhead [5]. The describing equation of an evaporator, like that of any heat exchanger, is given by:

Q UA Tlm

(36.1)

The term (UA) 1 is equal to the sum of the individual resistances of the steam, the walls of the tubes, the boiling liquid, and any fouling that may be present. Consider Figure 36.1 [5]. Assume F lb of feed to the evaporator per hour, whose solid content is xF. (The symbol x is employed for weight fraction). Also, assume the enthalpy of the feed per lb to be hF. L lb of thick liquor, whose composition in weight fraction of solute is xL and whose enthalpy is hL leaves from the bottom of the evaporator. V lb of vapor, having a solute concentration of yV and an enthalpy

360

Unit Operations in Environmental Engineering V yV hV

Vapor

Feed F xF hF

Steam

Condensate Thick liquor L xL hL

S hS

S hc

Figure 36.1 Material and enthalpy balance for a single-effect evaporator.

of hv, Btu/lb, leaves the unit. In most evaporators, the vapor is pure water, and therefore yV is zero. The material balance equations for this evaporator are relatively simple. A total material balance gives

F

L V

(36.2)

A componential balance leads to:

Fx F

Lx L VyV

(36.3)

In order to furnish the heat necessary for evaporation, S lb of steam is supplied to the heating surface with an enthalpy of hS Btu/lb, and S lb of condensate with an enthalpy of hC leaves the unit as condensate. One simplifying assumption usually made is that, in an evaporator, there is very little cooling of the condensate. The cooling is usually less than a few degrees in practice; the sensible heat recovered

Other Heat Transfer Equipment

361

from cooling the condensate is so small compared to the latent heat of the steam supplied to the heating surface that the condensate is assumed to leave at the condensing temperature of the steam. The enthalpy balance equation is therefore,

FhF

ShS

VhV

LhL ShC

(36.4)

Both Equations 36.1 and 36.4 are applied in tandem when designing and/or predicting the performance of an evaporator. One of the principal operating expenses of evaporators is the cost of steam for heating. A considerable reduction in those costs can be achieved by operating a battery of evaporators in which the overhead vapor from one evaporator (or “effect”) becomes the heating medium in the steam chest of the next evaporator, thus saving both the cost on condensing the vapor from the first unit and supplying heat for the second. Several evaporators may operate in a battery in this fashion [2, 5, 6]. Basically, a multiple effect evaporator may be thought of as a series of resistances to the flow of heat. The main resistances are those associated with heat transfer across the heating surface of each effect in the evaporator and across the final condenser if a surface condenser is used. The resistances of the heating surfaces are equivalent to the reciprocal of the product of the area and overall heat transfer coefficient (1/UA) for each effect. Neglecting all resistances but those noted above and assuming they are equal, it can be seen that if the number of resistances (effects) is doubled, the flow of heat (steam consumption) will be cut in half. With half as much to heat, each effect will evaporate about half as much water. But since there are twice as many effects, the total evaporation will be the same. Thus, under these simplifying assumptions, the same output would be obtained regardless of the number of effects (each of equal resistances), and the steam consumption would be inversely proportional to that number of effects or the total heating surface installed. This resistance concept is also useful in understanding the design and operation of such units. The designer places as many resistances (effects) in series as can be afforded in order to reduce the steam consumption. One can also show that the lowest total area required arises when the ratio of temperature drop to area is the same for each effect.

36.3 Waste Heat Boilers Energy has become too valuable to discard. As a result, waste heat and/or heatrecovery boilers are now common in many process plants. As the chemical processing industries become more competitive, no company can afford to waste or dump thermal energy. This increased awareness of economic considerations has made waste-heat boilers one of the more important products of the boiler industry. The term “waste heat boiler” includes units in which steam is generated

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Unit Operations in Environmental Engineering

primarily from the sensible heat of an available hot flue or hot gas stream rather than by solely firing fuel. An obvious by-product of incineration processes is thermal energy - in many cases, a large amount of thermal energy. The total heat load generated by a typical hazardous waste incinerator, for example, is in the range of 10 to 150 million Btu/ hr. While waste heat boilers are capable of recovering 60 to 70% of this energy, the effort may or may not be justified economically. In assessing the feasibility of recovery, a number of factors must be taken into account; among these are the amount of heat wasted, the fraction of that heat that is realistically recoverable, the irregularity in availability of heat, the cost of equipment to recover the heat, and the cost of energy. The last factor is particularly critical and may be the most important consideration in a decision involving whether or not to harness the energy generated by a particular incinerator. Other important considerations are the incinerator capacity and nature of the waste/fuel being handled. Generally, heat recovery on incinerators of less than 5 million Btu/hr may not be economical because of capital cost considerations. Larger capacity incinerators may also be poor candidates for heat recovery if steam is not needed at the plant site or if the combustion gases are highly corrosive; in the latter case, the maintenance cost of the heat recovery equipment may be prohibitive. The main purpose of a boiler is to convert a liquid, usually water, into a vapor. In most industrial boilers, the energy required to vaporize the liquid is provided by the direct firing of a fuel in the combustion chamber. The energy is transferred from the burning fuel in the combustion chamber by convection and radiation to the metal wall separating the liquid from the combustion chamber. Conduction then takes place through the metal wall and conduction/convection into the body of the vaporizing liquid. In a waste heat boiler, no combustion occurs in the boiler itself; the energy for vaporizing the liquids is provided by the sensible heat of hot gases which are usually product (flue) gases generated by a combustion process occurring elsewhere in the system. The waste heat boilers found at many facilities make use of the flue gases for this purpose. In a typical waste heat boiler installation, the water enters the unit after it has passed through a water treatment plant or the equivalent. This boiler feed water is sent to heaters/economizers and then into a steam drum. Steam is generated in the boiler by indirectly contacting the water with hot combustion (flue) gases. These hot gases may be around 2,000 °F. The steam, which is separated from the water in the steam drum, may pass through a superheater, and is then available for internal use or export. The required steam rate for the process or facility plus the steam temperature and pressure are the key design and operating variables on the water side. The inlet and outlet flue gas temperatures also play a role, but it is the chemical properties of the flue gas that can significantly impact boiler performance. For example, acid gases can arise due to the presence of any chlorine or sulfur in the fuel or waste. The principal combustion product of chlorine is hydrogen chloride, which is extremely corrosive to most metal heat transfer surfaces. This problem is particularly aggravated if the temperature of the flue gas is below the dew point

Other Heat Transfer Equipment

363

of HCl (i.e., the temperature at which the HCl condenses). This usually occurs at temperatures of about 300 °F. In addition to acid gases, problems may also arise from the ash of incineration processes; some can contain a fairly high concentration of alkali metal salts that have melting points below 1,500 °F. The lower melting point salts can slag and ultimately foul (and, in some cases, corrode) boiler tubes and/or heat transfer surfaces. Boilers may be either fire tube or water tube (water-wall). Both are commonly used in practice; the fire-tube variety is generally employed for smaller applications (500 °F). These gases are usually cooled either by recovering the energy in a waste heat boiler, as discussed in an earlier section of this chapter, or by quenching. Both methods may be used in tandem. For example, a waste heat boiler can reduce an exit gas temperature down to about 500 °F; a water quench can then be used to further reduce the gas temperature to around 200 °F, as well as saturate the gas with water. This secondary cooling and saturation can later eliminate the problem of water evaporation and can also alleviate other potential problems [3]. Although quenching and the use of a waste heat boiler are the most commonly used methods for gas cooling applications, there are several other techniques for cooling hot gases. All methods may be divided into two categories: direct-contact and indirect-contact cooling. The direct-contact cooling methods include: (a) dilution with ambient air, (b) quenching with water, and (c) contact with high heat capacity solids. Among the indirect contact methods are: (a) natural convection and radiation from ductwork, (b) forced-draft heat exchangers, and (c) the aforementioned waste heat boilers. With the dilution method, the hot gaseous effluent is cooled by adding sufficient ambient air that results in a mixture of gases at the desired temperature. The water quench method uses the heat of vaporization of water to cool the gases. When water is sprayed into the hot gases under conditions conducive to evaporation, the energy contained in the gases evaporates the water, and this results in a cooling of the gases. The hot exhaust gases may also be quenched using submerged exhaust quenching. This is another technique employed in some applications. In

Other Heat Transfer Equipment

367

the solids contact method, the hot gases are cooled by giving up heat to a bed of ceramic elements. The bed in turn is cooled by incoming air to be used elsewhere in the process. Natural convection and radiation occur whenever there is a temperature difference between the gases inside a duct and the atmosphere surrounding it. Cooling hot gases by this method requires only the provision of enough heat transfer area to obtain the desired amount of cooling. In forced-draft heat exchangers, the hot gases are cooled by forcing cooling fluid past the barrier separating the fluid from the hot gases.

References 1. Farag, I., and Reynolds, J., Heat Transfer, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1996. 2. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 3. Santoleri, J., Reynolds, J., and Theodore, L., Introduction to Hazardous Waste Incineration, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2000. 4. Theodore, L., and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 5. Theodore, L., Ricci, F., and Van Vliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 6. Kern, D., Process Heat Transfer, McGraw-Hill, New York City, NY, 1950. 7. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 8. Ganapathy, V., Size or check waste heat boilers quickly, Hydrocarbon Processing, 9, 169-170, 1984. 9. Connery, W., Chapter 6 – Condensers, in: Air Pollution Control Equipment, Theodore, L., and Buonicore, A.J., (Ed.), Prentice-Hall, Upper Saddle River, NJ, 1982. 10. Theodore, L., Air Pollution Control Equipment Calculations, John Wiley & Sons, Hoboken, NJ, 2008.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

37 Insulation and Refractory

37.1 Introduction The development presented in earlier chapters may be expanded to include insulation plus refractory and their effects. Industrial thermal insulation usually consists of materials of low thermal conductivity combined in a way to achieve a higher overall resistance to heat flow. Webster [1] defines insulation as: “to separate from conducting bodies by means of nonconductors so as to prevent transfer of electricity, heat, or sound.” Insulation is defined in Perry’s [2] in the following manner: “Materials or combinations of materials which have air- or gas-filled pockets or void spaces that retard the transfer of heat with reasonable effectiveness are thermal insulators. Such materials may be particulate and/or fibrous, with or without binders, or may be assembled, such as multiple heat-reflecting surfaces that incorporate air- or gas-filled void spaces.” Refractory materials also serve the chemical process industries. In addition to withstanding heat, refractory also provides resistance to corrosion, erosion, abrasion and/or deformation. This chapter contains three remaining sections: Describing Equations, Insulation, and Refractory. However, the bulk of the material keys on insulation since it has found more applications than refractories, particularly in its ability

369

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to reduce heat losses. The reader should note that there is a significant overlap of common theory, equations, and applications with Chapters 27 and 28, both of which are concerned with heat conduction.

37.2 Describing Equations When insulation is added to a surface, the heat transfer between the wall surface and the surroundings will take place by a two-step steady-state process (see Figure 37.1 for flow past a base flat plate and Figure 37.2 for flow past an insulated plate): conduction from the wall surface at T0 through the wall to T1 and through the insulation from T1 to T2, and convection from the insulation surface at T2 to the surrounding fluid at T3. The temperature drop across each part of the heat flow path in Figure 37.2 is given below. The temperature drop across the wall and insulation is:

(T0 T1 ) QR0

(37.1)

(T1 T2 ) QR1

(37.2)

The temperature drop across the fluid film is:

(T2 T3 ) QR2

(37.3)

Cold fluid TH

T1 . Q

T2

TC Hot fluid Figure 37.1 Flow past a flat plate.

x

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371

where R0 is the thermal resistance due to the conduction through the pipe wall, L0/k0A0; R1 is the thermal resistance due to the conduction through the insulation, L1/k1A1; and R2 is the thermal resistance due to the convection through the fluid, 1/h2A2. As with earlier analyses, the same heat rate, Q, flows through each thermal resistance. Therefore, Equations 37.1 through 37.3 can be combined to give:

(T0 T3 ) QRt

(37.4)

The heat transfer is then:

Q

(T0 T3 ) Rt

total thermal driving force total thermal resisstance

(37.5)

Note that for the case of plane walls, the areas A0, A1, and A2 are the same (see Figure 37.2). Dividing Equation 37.2 by Equation 37.3 yields:

(T1 T2 ) (T2 T3 )

R1 R2

Wall

h2 L1 k1

conduction insulation resistancee convection resistance

R0

T0

R1

R2

Fluid

T1

. Q

T2

L0

L1 Wall Insulation

Figure 37.2 Flow past a flat insulated plate.

T3

(37.6)

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The group h2L1/k1 is a dimensionless number defined as the Biot number, Bi.

Bi

(fluid convection coefficient )(characteristic length) (thermal conductivity of insulation surface)

hL k

(37.7)

37.3 Insulation Fiber, powder, and flake-type insulation consist of finely dispersed solids throughout an air space. The ratio of the air space to the insulator volume is called the porosity or void fraction, . In cellular insulation, a material with a rigid matrix contains entrapped air pockets. An example of such rigid insulation is foamed insulation which is made from plastic and glass material. Another type of insulation consists of multi-layered thin sheets of foil of high reflectivity. The spacing between the foil sheets is intended to restrict the motion of air. This type of insulation is referred to as reflective insulation. Most thermal insulation systems consist of the insulation and a so-called “finish.” This finish provides protection against water or other liquid entry, mechanical damage, and any possible ultraviolet degradation; it can also provide fire protection. The finish usually consists of any form of coating (e.g., polymeric paint material, etc.), a membrane (e.g., felt, plastic laminate, foil, etc.), or a sheet material (e.g., fabric, plastic, etc.). Naturally, the finish must be able to withstand any potential temperature excursion in its immediate vicinity. It would normally seem that the thicker the insulation, the less the heat loss, i.e., increasing the insulation should reduce the heat loss to the surroundings. But this is not always the case. There is a “critical insulation thickness” below which the system will experience a greater heat loss due to an increase in insulation. This situation arises for “small” diameter pipes when the increase in area is more rapid than the increase in resistance opposed by the thicker insulation. Theodore [3] provides details. The reader should note that as the thickness of the insulation is increased, the cost associated with heat lost decreases but the insulation cost increases. The optimum thickness is determined by the minimum of the total costs. Thus, as the thickness of the insulation is increased, the heat loss reaches a maximum value and then decreases with further increases in insulation. Reducing this effect can be accomplished by using an insulation of low conductivity. Two important factors that should be considered in selecting the optimum insulation for a pipe include durability and maintainability. If it is determined that cost is the number one factor, and a cheaper insulation is chosen, it would be wise to investigate these two factors for the insulation. The end result might be that the cheaper insulation does not have a long life and might have to be maintained much more often than a more expensive one. This could cause the more expensive one to be more cost effective than the cheaper insulation over the life-cycle of the insulation system.

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Temperature difference will also play a role in determining the insulation thickness. It may be imperative to keep the temperature of the material in the pipe just above freezing, or it may be that the temperature needs to be 60° above freezing. These different situations call for different insulation thicknesses. Another factor that falls under the temperature category is the location of the pipe. It would be extremely different if a pipe is insulated in New York, Alaska, or Tahiti. All of these places have different climates and it is imperative that these be investigated in order to know how thick or thin the insulation needs to be. Finally, the other factors that need to be considered, and may be as important as the first, are whether the materials that constitute the insulation are harmful to human health or the environment. First and foremost, there is asbestos, and due to some detailed studies, insulation with no or very little asbestos should be used. Another insulation material is fiberglass. A good number of insulators are made with fiberglass. When the insulation is cut, fiberglass escapes into the air and workers should not breathe this harmful material.

37.4

Refractory

Webster [4] defines refractory as: “difficult to fuse, corrode, or draw out; especially: capable of enduring high temperature.” Refractory materials must obviously be chemically and physically stable at high temperatures. Depending on the operating environment, they must also be resistant to thermal shock, be chemically inert, and be resistant to wear. Refractories normally require special warm-up periods to reduce the possibility of thermal shock and/or drying stresses. The oxides of aluminum (alumina), silicon (silica), and magnesium (magnesia) are the most common materials used in the manufacture of refractories. Another oxide usually found in refractories is the oxide of calcium (lime). Fireclays are also widely used in the manufacture of refractories. Additional details are provided in the literature [3,5]. Refractories are selected primarily on operating conditions. Some applications require special refractory materials. Zirconia is used when the material must withstand extremely high temperatures. Silicon carbide and carbon are two other refractory materials used in some very severe temperature conditions, but they cannot be used in contact with oxygen, as they will oxidize and burn. There is no single design and selection procedure for refractories. Three general rules can be followed for refractory design and selection: 1. Design for compressive loading. 2. Allow for thermal expansion. 3. Take advantage of the full range of materials, forms, and shapes. These apply to whether the design is essentially brickwork and masonry construction or whether the refractory is one that might have been made of specialty metal.

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References 1. Merriam-Webster on-line dictionary, insulating, https://www.merriam-webster.com/ dictionary/ insulating, 2017. 2. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 3. Theodore, L., Heat transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 4. Merriam-Webster on-line dictionary, refractory, https://www.merriam-webster.com/ dictionary/ refractory, 2017. 5. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2014.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

38 Refrigeration and Cryogenics

38.1 Introduction Refrigeration and cryogenics have aroused considerable interest in the work required in the extraction of heat from a body of low temperature and the rejection of this heat to a body willing to accept it. Refrigeration generally refers to operations in the temperature range of 120 to 273 K, while cryogenics usually deals with temperatures below 120 K where gases, including methane, oxygen, argon, nitrogen, hydrogen, and helium can be liquefied. In addition to being employed for domestic purposes (when a small “portable” refrigerator is required), refrigeration and cryogenic units have been used for the storage of materials such as antibiotics, other medical supplies, specialty foods, etc. Much larger cooling capacities than this are needed in air conditioning equipment. Some of these units, both small and large, are especially useful in applications that require the accurate control of temperature. Most temperature-controlled enclosures are provided with a unit that can maintain a space below ambient temperature (or at precisely ambient temperature) as required. The implementation of such devices led to the recognition that cooling units would be well suited to the refrigeration of electronic components and to applications in the field of instrumentation. Such applications usually require small, compact refrigerators, with a relatively low cooling power, where economy of operation is often unimportant. 375

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One of the main cost considerations when dealing with refrigeration and cryogenics is the cost of building and powering the equipment. This is a costly element in the process, so it is important to efficiently transfer heat so that money is not wasted in lost heat in the refrigeration and cryogenic processes. Since the cost of equipment can be expensive, there are a number of factors to be considered when choosing equipment. Equipment details are discussed in later sections. Cryogenics also plays a major role in the chemical processing industry. Its importance lies in the recovery of valuable feedstocks from natural gas streams, upgrading the heat content of fuel gas, purifying many process and waste streams, and producing ethylene, as well as other chemical processes. Cryogenic air separation provides gases (nitrogen, oxygen, and argon) used in the 1. 2. 3. 4. 5.

Manufacturing of metals such as steel Chemical processing and manufacturing industries Electronic industries Enhanced oil recovery industry Partial oxidation and coal gasification processes

Other cryogenic gases, including hydrogen and carbon monoxide, are used in chemical and metal industries while helium is used in welding, medicine, and gas chromatography. Cryogenic liquids have their own applications. Liquid nitrogen is commonly used to freeze food, while cryogenic cooling techniques are used to reclaim rubber tires and scrap metal from old cars. Cryogenic freezing and storage is essential in the preservation of biological materials that include blood, bone marrow, skin, tumor cells, tissue cultures, and animal semen. Magnetic resonance imaging (MRI) also employs cryogenics to cool the highly conductive magnets that are used for these types of non-intrusive body diagnostics [1]. As will become apparent throughout this chapter, there is a wide variety of applications, uses, and methods to produce and to utilize the systems of refrigeration and cryogenics. Multiple factors must be considered when dealing with these practices, including the choice of refrigerant or cryogen, the choice of equipment and methods of insulation, and all hazards and risks must be accounted for to ensure the safest environment possible. Topics covered in this chapter include: Background Material; Equipment; Materials of Construction; Insulation and Heat Loss; Storage and Transportation: Hazards, Risks, and Safety; and Basic Principles and Applications.

38.2

Background Material

38.2.1 Refrigeration The development of refrigeration systems was rapid and continuous at the turn of the 20th century, leading to a history of steady growth. The purpose of

Refrigeration and Cryogenics 377 refrigeration, in a general sense, is to make materials colder by extracting heat from the material. As described in earlier chapters, heat moves in the direction of decreasing temperature (i.e., it is transferred from a region of high temperature to one of a lower temperature). When the opposite process needs to occur, it cannot do so by itself, and a refrigeration system (or its equivalent) is required [2]. Refrigeration, in a commercial setting, usually refers to food preservation and air conditioning. When food is kept at colder temperatures, the growth of bacteria and the accompanying spoiling of food is reduced and possibly prevented. People learned early on that certain foods had to be kept cold to maintain freshness and many kept these foods in ice boxes where melting ice usually absorbed the heat from the foods and the surrounding atmosphere. Household refrigerators became popular in the early 1900s and only the wealthy could afford them at the time. Freezers did not become a staple part of the refrigerator until after World War II when frozen foods became popular. The equipment necessary in refrigeration is dependent upon many factors, including the substances and fluids working in the system. One very important part of refrigeration is the choice of refrigerant being employed and the refrigerant choice obviously depends on the system in which it will be used. The following criteria are usually considered in refrigerant selection: 1. 2. 3. 4. 5.

Practical evaporation and condensation pressures High critical and low freezing temperatures Low liquid and vapor densities High latent heat of evaporation High vapor heat capacity

Ideally, a refrigerant should also have a low viscosity and a high coefficient of performance (to be defined later in this chapter). Practically, a refrigerant should have: 1. 2. 3. 4. 5.

Low cost Chemical and physical inertness at operating conditions No corrosiveness toward materials of construction Low explosion hazard Non-poisonous and non-irritating characteristics

Solid refrigerants are not impossible to use but liquid refrigerants are most often used in practice. These liquid refrigerants include hydrocarbon and nonhydrocarbon refrigerants. The most commonly used hydrocarbon refrigerants include: 1. 2. 3. 4.

Propane Ethane Propylene Ethylene

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Non-hydrocarbon liquid refrigerants include: 1. 2. 3. 4. 5.

Nitrogen Oxygen Neon Hydrogen Helium

38.2.2 Cryogenics Cryogenics is not, in itself, an integral field. It is merely the extension of many other fields of science that delve into the realm of the thermodynamic variable of temperature. When compared to room temperature, the properties of most substances change dramatically at extremely low temperatures. From a molecular perspective, the atoms in any substance at a lower temperature, while still vibrating, are compressed closer and closer together. Depending on the phase of the substance, various phenomena and changes to physical and chemical characteristics occur at these lower temperatures. There are many accepted definitions of cryogenics. Some classify it simply as a “temperature range below 240 °F.” Another more elaborate explanation defines it as: “the unusual and unexpected property variations appearing at low temperatures and which make extrapolations from ambient to low temperature reliable [3].” Webster’s dictionary [4] defines cryogenics as “the branch of physics that deals with the production and effects of very low temperatures.” Cryogenics has also been referred to as: “all phenomena, processes, techniques, or apparatus occurring or using temperatures below 120 K [5].” Combining all of these definitions, the contributing author of this chapter [6] has provided the all-purpose definition that “cryogenics is the study of the production and effects of materials at low temperatures.” There have been uses for cryogenic technologies as far back as the latter part of the 19th Century. It became common knowledge in the 1840s that in order to store food at low temperatures for long periods of time, it needed to be frozen, a technology that is still utilized today. At the beginning of the 20th Century, the scientist Carl von Linde produced a double distillation column process that separated air into pure streams of its basic components of 78% nitrogen, 21% oxygen, and 1% argon. By 1912, it was discovered that minor modifications to the double distillation column process could separate many other gases from the input stream. Since there are trace amounts of neon, krypton, and xenon in air, the aforementioned distillation process, with minor modifications, was found capable of separating these gases into relatively pure streams. In the 1930s, the development of the sieve tray brought additional changes in cryogenic technology. A sieve tray is a plate, utilized in distillation columns, with perforated holes about 5 to 6 mm in diameter, which enhances mass transfer [7]. These trays were highly popular in cryogenics due to their simplicity, versatility, capacity, and cost effectiveness.

Refrigeration and Cryogenics 379 When the “Space Race” hit the United States in the 1960s, cryogenic technologies were utilized to develop a process known as cryopumping which is based on the freezing of gases on a cold surface. This process helped produce an ultrahigh vacuum here on Earth that would be similar to what was to be experienced in outer space. This led to many other discoveries, including rocket propulsion technologies which enabled astronauts to better prepare for their voyage(s) into space. The cryopreservation process is based on the same principles as food storage (i.e., using extremely low temperatures to preserve a perishable item). Cryopreservation has become more and more popular because of its appeal in preserving living cells. Whole cells and tissue can be preserved by this technique by stopping biological activity at extremely low temperatures. The preservation of organs by cryogenics has been a stepping stone for cryosurgery which relies on cold temperatures to insure clean and precise incisions. More recently, cryobiology has been applied well below freezing temperatures to living organisms to observe how they “react.” Cryobiology has also provided developing technologies in order to help cure such fatal illnesses as Parkinson’s disease [8]. Most recently, incorporating electronic systems with cryo-technologies has provided valuable information on superconductivity. Extremely low temperatures have also led to systems that contain nearzero resistance through wires.

38.2.3 Liquefaction Liquefaction is the process for converting a gaseous substance to a liquid. Depending on the liquefied material, various steps are employed in an industrial process. Common to each is the use of the Joule-Thomson effect [9] (where the temperature changes as a fluid flows through a valve), heat exchangers, and refrigerants to achieve the cryogenic temperatures. Generally, the methods of refrigeration and liquefaction used include: 1. Vaporization of a liquid 2. Application of the Joule-Thomson effect (a throttling process) 3. Expansion of a gas in a work producing engine Liquid nitrogen is the best refrigerant for hydrogen and neon liquefaction systems while liquid hydrogen is usually used for helium liquefaction. The largest and most commonly used liquefaction process involves the separation of air into nitrogen and oxygen. The process starts by taking air compressed initially to 1500 psia through a four-stage compressor with intercoolers [10]. The air is then compressed again to 2000 psia and cooled down to about the freezing point of water in a pre-cooler. The high-pressure air is then further cooled by ammonia to about 70°F. The air is then split into two streams after this cooling stage. One stream leads to heat exchangers that cool most of the air by recycled cold gaseous nitrogen. This proceeds to an expansion valve which condenses most

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of the air, absorbing heat in the process. The other stream goes to a booster expansion engine which compresses the air and then allows for expansion to further cool and then condense it (by means of the aforementioned Joule-Thomson effect). This liquid air is mixed and filtered and then introduced to the first of two fractional distillation columns. One hundred percent nitrogen exits the top of the first column, and when condensed, contains less than 7 ppm oxygen. The oxygen rich mixture is pumped out of the bottom of the first column and is introduced to the second column. The oxygen leaving the bottom of the second column is usually 99.6% oxygen or higher. The remaining 0.4% is argon and requires a subsequent separation process. The columns employed are similar to distillation columns and function to separate the nitrogen and oxygen [7]. The major application of these liquefied gases is that they can act as refrigerants for other substances.

38.3 Equipment Highly specialized equipment is used to achieve the extremely low temperatures necessary for refrigeration and cryogenic processes. Compressors, expanders, heat exchangers, storage containers, and transportation devices (as a means of moving materials and end products) are just a few of the specialty pieces of equipment that are required in these processes. Since the drastically cold temperatures that are reached can damage most equipment, it is important for all devices to be made of materials that are durable and that can withstand any large temperature and pressure excursions. In addition, it is also essential to incorporate temperature, pressure, and density measurement devices/gauges into these systems to monitor their temperature/pressure status. Compressor power makes up approximately 80% of the total energy used in processes involving the production of industrial gases and the liquefaction of natural gases. Therefore, in order to operate a cryogenic facility at optimum efficiency, the compressor choice is an important factor. Selection of a compressor relies on the capital cost and the cost of installation of the equipment, the energy and fuel costs associated with operating the equipment, and the cost of inspection and maintenance. There are several different types of compressors, including reciprocating compressors and centrifugal compressors. Reciprocating compressors can adapt to a wide range of volumes and pressures, and operate with high efficiency. Centrifugal compressors are ideal for high-speed compression; these compressors are highly efficient and reliable, especially when dealing with low pressure cryogenics [10]. In refrigeration and cryogenics, expansion valves, often referred to as expanders, serve to reduce the temperature of a gas being expanded to provide refrigeration. Fluid expansion to produce refrigeration is performed by two unique methods: in an expansion valve where work is produced, or by a Joule-Thomson valve where no work is produced. Mechanical expansion valves generally work very much like a reciprocating compressor, while a Joule-Thomson valve provides constant enthalpy (isenthalpic) cooling of the flowing gas [2].

Refrigeration and Cryogenics 381 Low-temperature operation has varied effects on equipment; therefore, sophisticated heat exchangers must be implemented for optimum efficiency in heat transfer. The following guidelines should be followed when designing low-temperature heat exchangers: 1. 2. 3. 4. 5. 6. 7. 8.

Small temperature differences between the inlet and exit streams Large surface area to volume ratio High heat transfer Low mass flow rates Multichannel capability High pressure potential Minimal pressure drop Minimal maintenance

Some of the most common heat exchanger designs used for cryogenic processes include: 1. 2. 3. 4.

Coil tube exchangers Plate-fin exchangers Reversing exchangers Regenerators

Again, these units are primarily used because of their high efficiency at extremely low temperatures. A coiled tube heat exchanger is especially important to the cryogenic process because of its unique abilities. The large number of tubes that are wound in helices around a core tube can have varying spacing patterns that allow for equalized pressure drops in any stream. Systems that desire simultaneous heat transfer between multiple streams employ the coiled tube heat exchanger. The coiled tube heat exchanger is specifically useful in cryogenic processes because of the typically high demand of heat transfer and high operating pressures that are required. Plate-fin exchangers, reversing exchangers, and regenerators are all second in popularity to the coiled tube heat exchanger.

38.4 Materials of Construction It is common for basic construction materials to contract and become distorted as temperature decreases. This can result in unnecessary stresses on accompanying equipment including expanders, pumps, piping, etc. Many materials have temperature limitations and because of this, when it comes to materials use in cryogenic applications, more exotic materials must be considered and possibly implemented. It is important to know the properties and behavior of the various materials for proper design considerations. Some combinations of materials, with each other as well as with the refrigerant or cryogen, can be hazardous to the fluid in question or the outside environment.

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Equipment considerations that are taken into account when choosing a construction material are thermal conductivity, thermal expansivity, and density. Some materials exhibit the effect of superconductivity at very low temperatures. This phenomenon affects the heat capacity, thermal conductivity, electrical resistance, magnetic permeability, and thermoelectric effect of the material. These superconductive materials need to be strictly analyzed before use in cryogenic systems because high temperature superconductors usually have a brittle ceramic structure. Approximately 9% nickel steels are often utilized for high boiling cryogens (>75 K), while many aluminum alloys and austenitic steels are structurally suitable for the entire cryogenic temperature range. While aluminum alloys are acceptable, pure aluminum is not recommended across the insulation space because of its high thermal conductivity [11].

38.5 Insulation and Heat Loss Insulation must certainly be considered an integral part of any refrigeration or cryogenic unit. The extent of the problem of keeping heat out of a storage vessel containing a liquid refrigerant or a cryogenic liquid varies widely. Generally, one must decide on the permissible and/or allowable heat losses (leaks) since insulation costs money, and an economic analysis must be performed. Thus, the main purpose of insulation is to minimize radiative and convective heat transfer and to use as little material as possible in providing the optimal insulation. When choosing appropriate insulation, the following factors are taken into consideration: 1. 2. 3. 4. 5. 6.

Ruggedness Convenience Volume and weight Ease of handling Thermal effectiveness Cost

The thermal conductivity (k) of a material is a major consideration in determining the thermal effectiveness of the insulation material. Different types of insulation obviously have different k values and there are five categories of insulation. These include: 1. Vacuum insulation which employs an evacuated space that reduces radiant heat transfer 2. Multilayer insulation, referred to by some as superinsulation, which consists of alternating layers of highly reflective material and low conductivity insulation in a high vacuum

Refrigeration and Cryogenics 383 3. Powder insulation which utilizes finely divided particulate material packed between surfaces 4. Foam insulation which employs non-homogeneous foam whose thermal conductivity depends on the amount of insulation 5. Special insulation which includes composite insulation that incorporates many of the advantageous qualities of the other types of insulation It should also be noted that multilayer insulation has revolutionized the design of cryogenic refrigerant vessels. In a double-walled vessel, typical of cryogen storage, heat is usually transferred to the inner vessel by three methods: 1. Conduction through the vessel’s “jacket” by gases present in this space 2. Conduction along solid materials touching both the inner and outer containers 3. Radiation from the outer vessel It was discovered in 1898, that the optimum material to place inside the space created by the double-walled vessel is “nothing” (i.e., a vacuum). This Dewar vessel, named after Sir James Dewar, is still one of the most widely used insulation techniques for cryogenic purposes.

38.6 Storage and Transportation Storage and transportation needs must also be considered in order to use cryogenic processes in food production. As noted in the previous section, the storage and transfer of these products is dependent on the storage tanks and how they are insulated; the heat gain in a storage tank could jeopardize the cryogenic state and could potentially ruin any product that was contained in the tank. Therefore, to safely store a product in a tank, the structure of the tank needs to be protected against any heat gain in the form of radiation and conduction through the insulation itself, as well as conduction though the inner shell, the supports, and the other openings and valves in the storage tank. Furthermore, the material that the storage tank is made out of needs to be non-reactive to the material being stored. The choice of storage vessel(s) is dependent on the material being kept in the vessel, and these vessels range in type from low performance containers with minimal insulation to high performance vessels with multilayer insulation. When storing liquid refrigerants, caution must be exercised. It is crucial to avoid overpressure inside the vessel, which manufacturers try to account for by providing

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“bursting disks” that prevent actual disasters from occurring (see the following Hazards, Risks, and Safety section) [12]. Super-insulated vessels normally have an inner stainless steel vessel and are kept at the temperature of the stored material. The outer vessel, or vacuum jacket, for these units maintain a vacuum necessary for effective insulation and prevents the condensation of water or other materials on the inner vessel’s cold surface. Three major processes are employed to transport a cryogen, i.e., a cryogenic liquid: 1. A self-pressurized container 2. External gas pressurization 3. A mechanical pumping system Details of each process are provided below. Self-pressurization is a process in which some of the fluid is removed from a container, vaporized, and then the vapor is reintroduced to the excess space and displaces the contents. External gas pressurization uses an external gas to displace the contents of a vessel. In mechanical pumping, a cryogenic pump, located at the liquid drain line, removes the contents of a container. Trucks, railroad cars, and airplanes have been utilized to transport cryogens. More recently, there have been barges used to transport cryogens via waterways. Specialized equipment is also used to measure the temperature and pressure of the cryogen while being stored in these tanks and transporting units. For pressure measurements, a gauge line is run from the point of interest to a point with ambient pressure, and thus the two pressures are compared using a device such as a Bourdon gauge [10]. This procedure is not void of thermal oscillations which make it less than perfect; however, these problems can be avoided by insulating pressure transducers at the point of measurement.

38.7 Health and Hazard Risks, and Safety While there are many health and hazard risks associated with refrigerant and cryogenic fluids, time and experience has proven that under proper conditions, these materials can be used safely in industrial and laboratory applications. In these environments, all facilities and equipment, as previously discussed, must be properly designed and maintained; and, the personnel in these areas must be sufficiently trained and supervised. The primary health hazards associated with cryogenic fluids are those dealing with human response to the fluids, as well as hazard risks related to the interaction of cryogenic fluids and their surroundings. Constant attention and care must be exercised in order to avoid most, if not all, conceivable hazards that may be encountered in this field [13].

Refrigeration and Cryogenics 385

38.7.1 Physiological Hazards There are a few sources of personal hazard in the field of cryogenics. If the human body were to come in contact with a cryogenic fluid or a surface cooled by a cryogenic fluid, severe “cold burns” could result. Cold burns inflict damage similar to a regular burn, causing stinging sensations and accompanying pain. But, with cryogenics, the skin and/or tissue is essentially frozen, significantly damaging or destroying it. As with any typical burn or injury, the extent and “brutality” of a cold burn depends on the area and time of contact; medical assistance is strongly advised when one receives a burn of this type. Protective, insulated clothing should be worn during work with low temperature atmospheres to prevent “frost bite” when dealing with cryogenic liquids. Safety goggles (or in some cases, face shields), gloves and boots are integral parts of this protective clothing. The objective of these precautions is to prevent any direct contact of the skin with the cryogenic fluid itself or with surfaces in contact with the cryogenic fluid. All areas in which cryogenic liquids are either stored or used should be clean and organized in a manner to prevent any avoidable accidents or fires and explosions that could result; this is especially true when working with systems using oxygen.

38.7.2 Physical Hazards There are multiple possible hazard risks associated with high pressure gases in cryogenic situations because their stored energy may be considerable. During gas compression in liquefaction and refrigeration, liquids are pumped to high pressure and then evaporated. These high pressure liquids and gases are eventually stored. When these materials are stored, breaks or ruptures in transfer lines can occur that can cause significant force upon storage vessels and their environments, or spills that could have disastrous effects. If a spill of a cryogenic fluid occurs, the heat in a room will readily vaporize the fluid into a gas. The primary hazard that occurs when dealing with non-oxygen cryogenic fluids is asphyxiation. The rapidly expanding gas can fill a room or area and displace the oxygen that was in the room. With a lack of oxygen in the room, the environment is extremely dangerous for humans.

38.7.3 Chemical Hazards Hazards associated with the chemical properties of cryogenic fluids can give rise to fires or explosions. In order for a fire or explosion to occur, there must be a fuel and/or an oxidant, and an ignition source. Because oxygen and air are prime candidates for cryogenic fluids, and are present in high concentrations, the chances of disasters occurring dramatically increases, as oxygen will obviously act as the oxidizer. A source of fuel can range from a non-compatible material to a flammable gas, or even a compatible material under extreme heat. An ignition source could

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be any electrical or mechanical spark or flame, any undesired thermodynamic event, or even a chemical reaction [14–16].

38.8 Basic Principles and Applications As noted earlier, refrigeration systems are cyclic and operate using the following main components: a compressor, condenser, expander, and evaporator (see Figure 38.1). If the object to be refrigerated is a reservoir of heat at some low temperature and the object or reservoir where the heat is rejected is at a higher temperature, the continuous refrigeration cycle depicted in Figure 38.1 is a simplistic representation of the process. The details of a basic refrigeration cycle are seen in Figure 38.1. The cycle beings when a refrigerant enters the compressor as a low pressure gas (1). Once compressed, it leaves as a hot high pressure gas. Upon entering the condenser (2), the gas condenses to a liquid and releases heat to the outside environment, which may be air or water. The cool liquid then enters the expansion valve at a high pressure (3); the flow is restricted and the pressure is lowered. In the evaporator (4), heat from the source to be cooled is absorbed and the liquid becomes a gas. The refrigerant then repeats the process (i.e., the cycle continues) [2]. In a refrigerator, the working fluid enters the evaporator in a wet condition and leaves dry and saturated (or slightly superheated). The heat absorbed, QC , by the evaporator can therefore be estimated by multiplying the change of the fluid’s entropy, S, as it passes through the evaporator by the fluid’s temperature, TS, at

QH

Condenser

(2)

(3) Wout

Expansion Valve

Compressor (1) Evaporator

(4) QC

Figure 38.1 Basic Components of a Refrigeration System.

Win

Refrigeration and Cryogenics 387 the evaporator pressure since the fluid’s temperature will be constant while it is in a wet condition at constant pressure. Thus [2],

QC

TS S

(38.1)

where QC = heat absorbed by the evaporator, kJ/kg; TS = fluid saturation temperature at evaporator temperature, K; and S = fluid entropy change, kJ/kg-K. The performance and energy saving ability of a refrigerator is measured in terms of the system’s Coefficient of Performance (C.O.P.). This is defined as the heat removed at a low temperature, i.e., the cooling effect, QC, divided by the work input, Win, into the system:

C.O.P.

QC Win

(38.2)

where C.O.P. = coefficient of performance, dimensionless; QC = cooling effect, kJ/ kg; and Win = work input, kJ/kg. The traditional system of units used in refrigeration are English units (i.e., Btu, etc.) but SI units are also acceptable. The cooling effect, QC, is equal to the change in enthalpy of the working fluid as it passes through the evaporator, and the work input (Win) is equal to the increase in the working fluid’s enthalpy as it passes through the compressor [2]. The performance of a steam power plant process can be measured in a manner somewhat analogous to the C.O.P. for a refrigeration system. The thermal efficiency, th, of a work-producing cycle is defined as the ratio of work produced to heat added. Thus,

th

Wnet Qin

(38.3)

where th = thermal efficiency, dimensionless; Wnet = net work produced by the cycle, J/kg; and Qin = heat added to the cycle, J/kg. This can be rewritten as:

th

Wout Win Qin

(38.4)

where Wout = work produced by the cycle, J/kg; and Win = work consumed by the cycle, J/kg. For this type of cycle, the compressor, evaporator, and expansion value in Figure 38.1 are replaced by a turbine, boiler, and pump, respectively, with both QC and QH as well as Win and Wout reversed (Figure 38.2). When no velocity information

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Unit Operations in Environmental Engineering QC

Condenser

(3)

(4) Win

Pump

Turbine

Wout

(2) Boiler

(1) Q

Figure 38.2 Basic components of a steam power system.

is provided, velocity effects can be neglected and this equation can be expressed in terms of enthalpies at points on entry and exit to the boiler, turbine, and pump, which for a simple power cycle is:

th

(h2 h3 ) (h1 h4 ) (h2 h1 )

(38.5)

where h1 = enthalpy on entry to the boiler, J/kg; h2 = enthalpy on exit from the boiler, on entry to the turbine, J/kg); h3 = enthalpy on exit from the turbine, J/kg); and h4 = enthalpy on entry to the pump, J/kg. Note that the change in enthalpy across the pump is often neglected since it is close to zero relative to the other enthalpy changes [2]. In effect, h1 h4 0.

References 1. Kirk, R., and Othmer, D., Encyclopedia of Chemical Technology, 4th Edition, Vol. 7, “Cryogenics,” p. 659, John Wiley & Sons, Hoboken, NJ, 2001. 2. Theodore, L. Ricci, F., and VanVliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 3. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008. 4. Merriam-Webster on-line dictionary, cryogenics, https://www.merriam-webster.com/ dictionary/cryogenics, 2017. 5. McKetta, J., and Anthony, R., Encyclopedia of Chemical Processing and Design, Vol. 13, p. 261, Marcel Dekker, New York City, NY, 2005. 6. Mockler, C., personal notes, Manhattan College, Bronx, NY, 2010. 7. Theodore, L., and Ricci, F., Mass Transfer for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010.

Refrigeration and Cryogenics 389 8. Brock, J., Frisbie, C., and Jersey, A., Cryogenics, Term Project submitted to L. Theodore, Manhattan College, Bronx, NY, 2009. 9. Smith, J., Van Ness, H., and Abbott, M., Introduction to Chemical Engineering Thermodynamics, 7th Edition, McGraw-Hill, New York City, NY, 2005. 10. Abulencia, P., and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 11. Croft, A.J., Cryogenic Laboratory Equipment, Plenum, New York City, NY, 1970. 12. Flynn, A.M., and Theodore, L., Health, Safety, and Accident Management in the Chemical Process Industries, Marcel Dekker, acquired by CRC Press/Taylor & Francis Group, Boca Raton, FL, 2002. 13. Theodore, L., and Dupont, R., Environmental Health and Hazard Risk Assessment: Principles and Calculations, CRC Press/Taylor & Francis Group, Boca Raton, FL, 2012. 14. Reynolds, J. Jeris, J., and Theodore, L., Handbook of Chemical and Environmental Engineering Calculations, John Wiley & Sons, Hoboken, NJ, 2002. 15. Theodore, L., Nanotechnology: Basic Calculations for Engineers and Scientists, John Wiley & Sons, Hoboken, NJ, 2006. 16. Cengel, Y., and Boles, M., Thermodynamics: An Engineering Approach, 5th Edition, McGraw-Hill, New York City, NY, 2006.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

39 Condensation and Boiling

39.1 Introduction It should be noted that phase-change processes involve changes (sometimes significantly) in density, viscosity, heat capacity, and thermal conductivity of the fluid in question. The heat transfer process and the applicable heat transfer coefficients for boiling and condensation are more involved and complicated than that for a single-phase process. It is therefore not surprising that most real-world environmental applications involving boiling and condensation require the use of empirical correlations. The transfer of heat, which accompanies a change of phase, is often characterized by high rates. Heat fluxes as high as 50 million Btu/hr-ft2 have been obtained in some boiling systems. This mechanism of transferring heat has become important in rocket technology and nuclear-reactor design where large quantities of heat are usually produced in confined spaces. Although condensation rates have not reached a similar magnitude, heat transfer coefficients for condensation as high as 20,000 Btu/hr-ft2-°F have been reported in the literature [1]. Due to the somewhat complex nature of these two phenomena, simple pragmatic calculations and numerical details are provided later in this chapter.

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This chapter addresses phenomena associated with the change in phase of a fluid. The processes almost always occur at a solid-liquid interface and are referred to as boiling and condensation. The change from liquid to vapor due to boiling occurs because of heat transfer from the solid surface; alternatively, condensation of vapor to liquid occurs due to heat transfer to the solid surface. Phase changes of substances can only occur if heat transfer is involved in the process. The phase change processes that arise include: 1. 2. 3. 4. 5.

Boiling (or evaporation) Condensation Melting (or thawing) Freezing (or fusion) Sublimation

The corresponding heat of transformation arising during these processes are: 1. Enthalpy of vaporization (or condensation) 2. Enthalpy of fusion (or melting) 3. Enthalpy of sublimation The applications of phase change involving heat transfer are numerous and include utility units where water is boiled, evaporators in refrigeration systems where a refrigerant may be either vaporized or boiled, or both, and condensers that are used to cool vapors to liquids. For example, in a power cycle, pressurized liquid is converted to vapor in a boiler. After expansion in a turbine, the vapor is restored to its liquid state in a condenser, it is then pumped to the boiler to repeat the cycle. Evaporators, in which the boiling process occurs, and condensers are also essential components in vapor-compression refrigeration cycles. Thus, the practicing engineer needs to be familiar with phase change processes. Applications involving the solidification or melting of materials are also important. Typical examples include the making of ice, freezing of foods, freeze-drying processes, solidification and melting of metals, and so on. The freezing of food and other biological matter usually involves the removal of energy in the form of both sensible heat (enthalpy) and latent heat (enthalpy) of freezing. A large part of biological matter is liquid water, which has a latent enthalpy of freezing, hsf, of approximately 335 kJ/kg (144 Btu/lb or 80 cal/g). When meat is frozen from room temperature, it is typically placed in a freezer at 30 °C, which is considerably lower than the freezing point. The sensible heat to cool any liquids from the initial temperature to the freezing point is first removed, followed by the latent heat, hsf, to accomplish the actual freezing. Once frozen, the substance is often cooled further by removing some sensible heat of the solid. The objectives of this chapter are to develop an understanding of the physical conditions associated with boiling and condensation, and to provide a basis for performing related heat transfer calculations. The remaining chapter contents

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are highlighted with four sections: Condensation Fundamentals, Condensation Principles, Boiling Fundamentals, and Boiling Principles.

39.2 Condensation Fundamentals Although several earlier chapters dealt with situations in which the fluid medium remained in a single phase, a significant number of real-world environmental engineering applications involve a phase change that occurs simultaneously with the heat transfer process. As discussed earlier, the process of condensation of a vapor is usually accomplished by allowing it to come into contact with a surface where the temperature is maintained at a value lower than the saturation temperature of the vapor for the pressure at which it exists. The removal of thermal energy from the vapor causes it to lose its latent heat of vaporization and, hence, to condense onto the surface. The appearance of the liquid phase on the cooling surface, either in the form of individual drops or in the form of a continuous film, offers resistance to the removal of heat from the vapor. In most applications, the condensate is removed by the action of gravity. As one would expect, the rate of removal of condensate (and the rate of heat removal from the vapor) is greater for vertical surfaces than for horizontal surfaces. Most condensing equipment consists of an assembly of tubes around which the vapor to be condensed is allowed to flow. The cool temperature of the outer tube surface is maintained by circulating a colder medium, often water, through the inside of the tube. There are primarily three types of condensation processes: 1. Surface Condensation. This type of condensation occurs when vapor is in contact with a cool surface. This process is common in industrial applications and is discussed below. 2. Homogenous Condensation. Homogeneous condensation occurs when the vapor condenses out as droplets in the gas phase. 3. Direct Contact Condensation. This process occurs when vapor is in contact with a cold liquid. Surface condensation may occur in one of two modes depending upon the conditions of the surface. 1. Film Condensation. When the surface is clean and uncontaminated, the condensed vapor forms a liquid film that covers the entire condensing surface; this film contributes an additional resistance to heat transfer. 2. Dropwise Condensation. When the surface is coated with a substance that inhibits wetting, the condensed vapor forms drops in cracks and cavities on the surface. The drop often grow and coalesce

394

Unit Operations in Environmental Engineering and “drop” from the surface available for heat transfer. Thus, the additional resistance to the heat transfer can be either reduced or eliminated. Up to 90% of the surface can be covered by drops. This topic receives superficial treatment later in the chapter, with film condensation receiving the bulk of the analysis.

Liquid condensate provides resistance to heat transfer between the vapor and the surface. This resistance naturally increases as the thickness of the condensate later increases. For design purposes, it is usually desirable to have the condensation occur on surfaces that discourage the formation of thick liquid layers (e.g., short vertical surfaces or horizontal cylinders, or tube bundles, through which a coolant liquid flows). Dropwise condensation has a lower thermal resistance (and therefore a higher heat transfer rate) than film condensation. In industrial practice, surface coatings that inhibit wetting are often used. Examples of such coatings include Teflon, silicones, waxes, and fatty acids. The drawback is that these coatings lose their effectiveness over time, which results in the condensation mode eventually changing from dropwise to film condensation. Obviously, all other things being equal, dropwise condensation is preferred to film condensation. In fact, steps are often put in place to induce this “dropwise” effect on heat transfer surfaces. Finally, it should be noted that the local heat transfer coefficient varies along a flat surface and along the length of a vertical tube and around the perimeter of a horizontal tube (i.e., the local coefficient of heat transfer for a vapor condensing in a horizontal tube is a function of position, just like a vertical tube). As one would suppose, the highest coefficient is located at the top of a tube where the condensate film is thinnest. The condensing temperature in the application of the equations to be presented in the next section involving film coefficients is taken as the temperature at the vapor-liquid interface. From a thermodynamic point-of-view, condensation of a condensable vapor in a condensable vapor-noncondensable gas mixture can be induced by either increasing the pressure or decreasing the temperature, or both. Condensation most often occurs when a vapor mixture contacts a surface at a temperature lower than the saturation (dew point) temperature of the mixture. The dew point [2] of a vapor mixture is the temperature at which the vapor pressure exerted by the condensable component(s) is equal to the(ir) partial pressure in the vapor. For example, consider an air-water mixture at 75°F that is 80% saturated with water (or has a relative humidity, RH, of 80%):

% Sat % RH

pH2O p

(100)

(39.1)

where pH2O = partial pressure of water; and p = vapor pressure of water = 0.43 psia at 75°F (see Steam Tables, Appendix C). For this condition, pH2O = (0.8)(0.43) = 0.344 psia.

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Pure vapors condense (or vaporize) at their vapor pressure at a given temperature. For example, water at 212 °F and 1 atm will vaporize to steam (or steam will condense to water). A vapor or a mixture of vapors in a noncondensable gas is more difficult to analyze. A typical example is steam (water) in air or a high molecular weight organic in air. There are two key vapor-liquid mixtures of interest to the practicing engineer: air-water and steam-water (liquid). Information on the former is available on a psychometric chart (see Part IV) while steam tables (see Appendix C) provide information on the later. A discussion on both follows.

39.2.1 Psychrometric Chart A vapor-liquid phase equilibrium example involving raw data is the psychrometric or humidity chart [2]. A humidity chart is used to determine the properties of moist air and to calculate moisture content in air. The ordinate of the chart is the absolute humidity %, which is defined as the mass of water vapor per mass of bone-dry air. (Some charts base the ordinate on moles instead of mass). Based on this definition, Equation 39.2 gives (the humidity) in terms of moles and also in terms of partial pressure:

18nH2O

18 pH2O

2(nT nH2O )

29(P pH2O )

(39.2)

where nH2O = number of molecules of water vapor; nT = total number of moles in gas; pH2O = partial pressure of water vapor; and P = total system pressure. Curves showing the relative humidity (ratio of the mass of the water vapor in the air to the maximum mass of water vapor that the air could hold at that temperature, i.e., if the air were saturated) of humid air also appear on the charts. The curve for 100% relative humidity is also referred to as the saturation curve. The abscissa of the humidity chart is air temperature, also known as the dry-bulb temperature (TDB). The wet-bulb temperature (TWB) is another measure of humidity; it is the temperature at which a thermometer with a wet wick wrapped around the bulb stabilizes. As water evaporates from the wick to the ambient air, the bulb is cooled; the rate of cooling depends on how humid the air is. No evaporation occurs if the air is saturated with water; hence, TWB and TDB are then the same. The lower the humidity, the greater the difference between these two temperatures. On the psychrometric chart, constant wet-bulb temperature lines are straight with negative slopes. The value of TWB corresponds to the value of the abscissa at the point of intersection of this line with the saturation curve. Given the dry bulb and wet bulb temperatures, the relative humidity (along with any other quantity on the chart) may be determined by finding the point of intersection between the dry bulb abscissca and wet bulb ordinate. The point of intersection describes all humidity properties of the system. Psychrometric charts are available in the literature [2, 3]. (See also Part IV).

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39.2.2 Steam Tables The steam tables comprise a tabular representation of the thermodynamic properties of water. These tables are divided into three separate categories: 1. The saturated-steam tables provide the value of the enthalpy, specific volume, and entropy of saturated steam and saturated water as functions of pressures and/or temperatures (condensation or boiling points). Changes in these extensive properties during the evaporation of 1 lb of the saturated liquid are also tabulated. The tables normally extend from 32 to 705 °F, the temperature range where saturated liquid and vapor can coexist. 2. The superheat tables list the same properties in the superheatedvapor region. Degrees superheat or number of degrees above the boiling point temperature (at the pressure in question) are also listed. 3. The Mollier chart (enthalpy-entropy diagram) for water is so frequently used in engineering practice that it also deserves mention. This diagram is useful since the entropy function stays constant during any reversible adiabatic expansion or compression [2]. The steam tables are available in the literature [2, 3]. (See also Appendix C).

39.3 Condensation Principles As noted earlier, condensation occurs when the temperature of a vapor is reduced below its saturation temperature and results from contact between the vapor and a cooler surface. The latent enthalpy of the vapor is released, heat is transferred to the surface, and the condensate forms. As noted in the Introduction to this chapter, there are three basic types of condensation: surface condensation, homogeneous condensation, and direct contact condensation. The corresponding modes are: dropwise condensation and film condensation. Although it is desirable to achieve dropwise condensation in real-world applications, it is often difficult to maintain this condition. Although convection coefficients for film condensation are smaller than those for the dropwise case, condenser design calculations may be and usually are based on the assumption of film condensation. It is for this reason that the paragraphs to follow in this section focus on film condensation. Film condensation may be laminar or turbulent, depending on the Reynolds number of the condensate. The condensate Reynolds number is defined as:

Re

4

v

4m

L

L

L mL

4m W L

(39.3)

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397

where L is the density of the liquid condensate; vmL is the mean velocity of the condense liquid film; is the average thickness of the condensed liquid film; L is the absolute viscosity of the liquid condensate; m is the mass flow rate of condensate per unit width of the surface; m is the mass flow rate of condensate; and W is the width of the condensing surface. Laminar condensation occurs when Re < 1800, while turbulent condensation occurs when the Re > 1800. Since the flow may not occur in a circular conduit, the hydraulic diameter, Dh, must be used. The hydraulic diameter, Dh, is defined as:

Dh

4 AC PW

4m L v mL PW

(39.4)

where AC is the area of the conduit and PW is the wetted perimeter. Assuming that the vapor is at its saturation temperature, Tsat, and letting TS represent the surface temperature, the rate of heat transfer, Q , can be related to the rate of condensation, m , through an energy balance, as shown in the following equations:

Q hA(Tsat TS ) mhvap

(39.5)

with A representing the heat transfer area. Rearranging Equation 39.5 leads to:

m

m PW

hA(Tsat TS ) PW hvap

(39.6)

Substituting Equation 39.6 for m in the condensate Reynolds number definition in Equation 39.3 gives:

Re

4m L

4hA(Tsat TS ) PW hvap L

(39.7)

The values of A, PW and the ratio of A/PW for several geometries as well as heat transfer coefficients are available in the literature [2, 3]. A superheated vapor occurs when the temperature of the vapor is higher than that of the boiling point at the corresponding pressure. When a superheated vapor enters a condenser, the sensible heat of superheat and the latent heat of condensation must be transferred through the cooling surface. The condensation mechanism is therefore somewhat different if the condensing vapor is superheated rather than saturated. Experimental results have shown that, in most cases, the effect of superheat may be ignored and the equations for saturated vapors may be used with negligible error. It should be noted that

Tsat T

T

(39.8)

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is still the temperature difference driving force, and that the actual superheated vapor temperature does not enter into the calculations. For mixed vapors (i.e., if the vapor contains two or more volatile components), the condensation temperature is no longer constant at a given pressure unless the mixture is azeotropic [2]. If the cooling surface temperature is low, the vapor may condense and the condensate may be assumed to be the same as that of the original vapor. If the cooling surface temperature is straddled by the condensation temperatures of the components, part of the vapor will condense and some vapor must be vented from the condenser. Regarding dropwise condensation, heat transfer coefficients are an order of magnitude larger than those for film condensation. If other thermal resistances in the system are significantly larger than that due to condensation, the condensation resistance may be neglected. The following expressions are recommended for estimation purposes for dropwise condensation:

h

51, 000 2000Tsat

h

255, 000

22 C Tsat

100 C Tsat

100 C

(39.9) (39.10)

where the heat transfer coefficient h has units of W/m2-K.

39.4 Boiling Fundamentals The process of converting a liquid into a vapor is also of importance to practicing engineers. The production of steam for electrical power generation is a prime example. Many other processes, particularly in the refining of petroleum and the manufacture of chemicals, require the vaporization of a liquid. Boiling is the opposite of condensation. Boiling occurs when a liquid at its saturation temperature, Tsat, is in contact with a solid surface at a temperature, TS, which is above Tsat. The excess temperature, Te, is defined as:

Te

TS Tsat

(39.11)

The Te driving force causes heat to flow from the surface into the liquid, which results in the formation of vapor bubbles that move up through the liquid. Because of bubble formation, the surface tension of the liquid has an impact on the rate of heat transfer. Boiling may be classified as pool boiling or forced convection boiling. In pool boiling, the liquid forms a “pool” in a container while submerged surfaces supply the heat. The liquid motion is induced by the formation of bubbles as well as density variations. In forced convection, the liquid motion is induced by external means (e.g., pumping of liquids through a heated tube).

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399

Another common classification for types of boiling is based on the relationship of the liquid temperature to its saturated temperature. Boiling is subcooled (also known as local) when the liquid temperature is below the saturation temperature (i.e., T < Tsat). When the liquid is at the saturation temperature, the boiling is saturated boiling. The explanation of the strange behavior of boiling systems lies in the fact that boiling heat transfer occurs by several different mechanisms and the mechanism is often more important in determining the heat rate than the temperature-difference driving force. Consider, for example, the heating of water in an open pot. Heat is initially transferred within the water by natural convection. As the heat rate is further increased, the surface temperature at the base of the pot increases to and above 212 °F. Bubbles begin to form and then rise in columns from the heating surface creating a condition favorable to heat transfer. As the temperature increases further, more sites become available until the liquid can no longer reach the heating surface at a sufficient rate to form the required amount of vapor. This ends the nucleate boiling stage (to be discussed later). At this point, the mechanism changes to film boiling since the heating surface is now covered with a film of vapor. The temperature of the base of the pot rises (even though the heat rate is constant). Vaporization takes place at the liquid-vapor interface and the vapors disengage from the film in irregularly-shaped bubbles at random locations. Although the phenomena of superheating occurs in most boiling systems, the temperature of the boiling liquid, measured some distance from the heated surface, is higher than the temperature of the vapor above the liquid (which is at the saturation temperature). The liquid superheat temperature adjacent to the heated surface may be as high as 25 to 50 °F. Most of the temperature change occurs in a narrow thin film from the surface. This superheat occurs because the internal pressure in the vapor bubble is higher due to surface tension effects. The formation, growth, and release of bubbles is an extremely rapid sequence of events. The rapid growth and departure of vapor bubbles causes turbulence in the liquid, especially in the aforementioned zone of superheat near the heated surface. This turbulence assists the transport of heat from the heated surface to the liquid evaporating at the bubble surface. The rapid growth of bubbles and the turbulence in the liquid complement each other, resulting in high heat transfer coefficients. As the superheat of a boiling liquid is further increased, the concentration of active centers on the heating surface increases and the heat rate correspondingly increases. The mass rate of the vapor rising from the surface must be equal to the mass rate of liquid proceeding toward the surface if steady-state conditions prevail. As the boiling rate increases, the rate of liquid influx must increase since the area available for flow decreases with the increasing number of bubble columns; in addition, the liquid velocity must also increase. At this limiting condition, the liquid flow toward the heated surface cannot increase and the surface becomes largely blanketed with vapor. If the heat rate to the surface is held constant, the surface temperature will rise to a high value at which point heat is transmitted to the fluid by the mechanism of film boiling. Film boiling occurs when the superheat

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is sufficiently high to keep the heated surface completely blanketed with vapor. Heat may then be transmitted through the gas film by conduction, convection, and radiation.

39.5 Boiling Principles It has been found that the heat transfer coefficient, h, in boiling systems depends on the excess temperature, Te. For free convection boiling ( Te < 5 °C):

h

( Te )n

1 laminar free convection conditions, and n 4 convection conditions. The boiling heat flux Qs , is calculated as: where n

Qs

(39.12) 1 turbulent free 3

h(TS Tsat )

(29.13)

TS Tsat

(39.14)

and

Te

The mechanism of pool boiling heat transfer depends on the excess temperature, Te, where, once again, Te is the difference between surface temperature and the fluid saturation temperature. A plot of the boiling heat transfer flux, Qs , versus the excess temperature, is shown in Figure 39.1. When Te > 5 °C, bubbles start to form at the onset of nucleate boiling (ONB). As noted earlier, the bubbles rise and are dissipated in the liquid; the resulting stirring causes an increase in the heat flux. This regime was termed nucleate boiling. The heat transfer coefficient in this regime varies with Te as follows:

h

( Te )n

(39.12)

where n varies between 3 or 4. As Te increases, the rate of bubble formation increases. This causes h to increase, but at higher rates of bubble formation some bubbles cannot diffuse quickly enough and form a blanket around the heating element. This blanket increases the heat transfer resistance and slows down the increase in both h and the heat flux. These two factors balance out at approximately Te = 30 °C where the flux (introduced below) reaches its maximum. The peak flux is termed the critical heat flux, QS ,c . For boiling water, QS ,c is greater than 1 MW/ m2. As Te increases still further, the gas film formed by the coalescing bubbles

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Boiling regimes Free convection

Nucleate

Transition

Film

Isolated Jets and bubbles columns 107

106 . Q’S (W/m2)

. . Critical heat flux, Q’S, max= Q’S, c

C

. Q’max P

B

105

D

. Q’ min 104

103

. Leidenfrost point, Q’S, min

A ONB

1

Te,A

Te,B

Te,C

5

10

30

Te,D 120

1000

Te = Ts – Tsat (°C) Figure 39.1 Boiling curve. (Adapted from [4]).

around the element prevents the liquid from coming in and causes a further increase in the heat transfer resistance. This mechanism is known as film boiling. For the approximate 30 °C < Te< 120 °C range, the mechanism becomes unstable film boiling, also known as transition film boiling, or partial film boiling. The heat transfer coefficient decreases with increasing Te until a stable value is reached. At these high surface temperatures, thermal radiation contributes significantly to the heat transfer. This contribution causes the heat flux to increase again. The minimum in the curve is referred to as the Leidenfrost point and the heat flux is given by QS,min . At the point of peak (or critical) flux, QS ,c , a small increase in Te causes QS to decrease. However, with the lower heat flux, the energy from the heating surface cannot be completely dissipated, causing a further increase in Te. Eventually, the fluid temperature can exceed the melting point of the heating medium. The critical heat flux (at point C on Figure 39.1) is therefore also termed the boiling crisis point or the burnout point. A host of empirical equations describing QS for these various regimes is provided by Theodore [2, 3].

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References 1. Bennett, C., and Meyers, J., Momentum, Heat and Mass Transfer, McGraw-Hill, New York City, NY, 1962. 2. Theodore, L., Ricci, F., and Van Vliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 3. Theodore L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 4. Incoprera, J., and DeWitt, R., Fundamentals of Heat and Mass Transfer, John Wiley & sons, Hoboken, NJ, 2011.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

40 Operation, Maintenance, and Inspection (OM&I)

40.1 Introduction For the purposes of this chapter, the presentation will primarily address condensers since most of the heat exchangers reviewed up to this point in the text can be employed for condensation operations. However, this material can be applied virtually to all heat exchangers which for decades have been used in process operations, i.e., shell and tube, double pipe, air-cooled, flat plate, spiral plate, barometric jet, spray, etc. This chapter is primarily concerned with operation, maintenance, and inspection issues as they apply to heat exchangers. These issues obviously vary with the type of heat exchanger under consideration. Chapter contents include: Installation Procedures, Operation, Maintenance and Inspection, Testing, and Improving Operation and Performance. Note that the bulk of the material for this chapter has been drawn from the literature [1, 2].

40.2

Installation Procedures

The “preparation” of a condenser or heat exchanger for installation begins upon receipt of the unit from the manufacturer. Condensers are shipped domestically 403

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using skids for complete units, and boxes or crates for bare tube bundles. Units are normally removed from trucks using a crane or forklift. Lifting devices should be attached to lugs provided for that purpose (i.e., for lifting of the complete unit as opposed to individual parts), or used with slings wrapped around the main shell. Shell supports are acceptable lugs for lifting, provided that the complete set of supports are used together; nozzles should not be used for attachment of lifting cables. Upon receipt of the unit, the general condition should be noted to determine any damage sustained during transit. Any dents or cracks should be reported to the manufacturer prior to attempting to install the unit. Hanged connections are usually blanked with plywood, masonite, or equivalent covers, and threaded connections are blanked with suitable pipe plugs. These closures are to avoid entry of debris into the unit during shipping and handling, and should remain in place until actual piping connections are made.

40.2.1 Clearance Provisions Sufficient clearance is required for at least inspection of the unit or in-place maintenance. Inspection of heat exchangers requires minimal clearances for the following: access to inspection parts if provided, removal of channel or bonnet covers, and inspection of tube sheets and tube-to-tube sheet joints. If the removal of tubes or tube bundles in place is anticipated, provision should be provided in the equipment layout. Actual clearance requirements can be determined from the condenser setting plan.

40.2.2

Foundations

Heat exchangers must be supported on structures of sufficient rigidity to avoid imposing excessive strains due to settling. Horizontal units with saddle-type shell supports are normally supplied with slotted holes in one support to allow for expansion. Foundation bolts in these supports should be loose enough to allow movement.

40.2.3 Leveling Heat exchangers should be carefully leveled and squared to ensure proper drainage, venting, and alignment with piping. On occasions, these units are purposely angled to facilitate venting and drainage, and alignment with piping becomes the prime concern.

40.2.4 Piping Considerations The following guidelines for piping are necessary to avoid excessive strains, mechanical vibration, and access for regular inspection/maintenance.

Operation, Maintenance, and Inspection (OM&I) 405 1. Sufficient support devices are required to prevent the weight of piping and fittings from being imposed on the unit. 2. Piping should have sufficient expansion joints or bends to minimize expansion stresses. 3. Forcing the alignment of piping should be avoided so that residual strains will not be imposed on any nozzles, if applicable. 4. If external forces and moments are unavoidable, their magnitude should be determined and made known to the manufacturer so that a necessary stress analysis can be performed. 5. Surge drums or sufficient length of piping to the exchanger should be provided to minimize pulsations and mechanical vibrations. 6. Valves and bypasses should be provided to permit inspection or maintenance in order to isolate the exchanger during periods other than complete system shutdown. 7. Plugged drains and vents are normally provided at low and high points of shell-tube sides not otherwise drained or vented. These connections are functional during startup, operation, and shutdown, and should be piped up for either continuous or periodic use and never left plugged. 8. Instrument connections should be provided either on condenser nozzles or in the piping close to the exchanger. Pressure and temperature indicators should be installed to validate the initial performance of the unit as well as to demonstrate the need for inspection or maintenance.

40.3

Operation

The maximum allowable working pressures and temperatures are normally indicated on the heat exchanger’s nameplate. These values must not be exceeded. Special precautions should be taken if any individual part of the unit is designed for a maximum temperature lower than the unit as a whole. The most common example is some copper-alloy tubing with a maximum allowable temperature lower than the actual inlet gas temperature. This is required to compensate for the low strength of some brasses or other copper alloys at elevated temperatures. In addition, an adequate flow of the cooling medium must be maintained at all times. Exchangers are designed for a particular fluid throughput. Generally, a reasonable overload can be tolerated without causing damage. If operated at excessive flow rates, erosion or destructive vibration could result. Erosion could occur at normally acceptable flow rates if other conditions, such as entrained liquids or particulates in a gas stream or abrasive solids in a liquid stream, are present. Evidence of erosion should be investigated to determine the cause. Vibration can be propagated by other than flow overloads (e.g., improper design, fluid maldistribution, or corrosion/erosion of internal flow-directing devices such as baffles).

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Considerable study and research have been conducted in recent years to develop a reliable vibration analysis procedure to predict or correct damaging vibration. At this point in time, the developed correlations are considered “state of the art,” yet most manufacturers have the capability of applying some type of vibration check when designing an exchanger. Vibrations can produce severe mechanical damage, and operation should not be continued when an audible vibration disturbance is evident.

40.3.1 Startup Exchangers should be warmed up slowly and uniformly; the higher the temperature ranges, the slower the warm-up should be. This is generally accomplished by introducing the coolant and bringing the flow rate to the design level and gradually adding the vapor. For fixed-tube-sheet units with different shell-and-tube material, consideration should be given to differential expansion of shell and tubes. As fluids are added, the respective areas should be vented to ensure complete distribution. A procedure other than this could cause large differences in temperature between adjacent parts of the unit and result in leaks or other damage. It is recommended that gasketed joints be inspected after continuous full-flow operation has been established. Handling, temperature fluctuations, and yielding of gaskets or bolting may necessitate retightening of the bolting.

40.3.2 Shut Down Cooling down is generally accomplished by first shutting off the vapor stream and then the cooling stream. Again, fixed-tube-sheet units require consideration of differential expansion of the shell and tubes. Exchangers containing flammable, corrosive, or high-freezing-point fluids should be thoroughly drained for prolonged outages.

40.4 Maintenance and Inspection Recommended maintenance of exchangers requires regular inspection to ensure the mechanical soundness of the unit and a level of performance consistent with the original design criteria. A brief general inspection should be performed on a regular basis while the unit is operating. Vibratory disturbance, leaking gasketed joints, excessive pressure drop, decreased efficiency indicated by higher gas outlet temperatures or lower condensate rates, and intermixing of fluids are all signs that thorough inspection and maintenance procedure are required. Complete inspection requires a shutdown of the exhanger for access to internals and both pressure testing and cleaning. Scheduling can only be determined from experience and general inspections. Tube internals and exteriors, where accessible, should be visually inspected for fouling, corrosion, or damage. The nature of any

Operation, Maintenance, and Inspection (OM&I) 407 metal deterioration should be investigated to properly determine the anticipated life of the equipment or possible corrective action. Possible causes of deterioration include general corrosion, intergranular corrosion, stress cracking, galvanic corrosion, impingement, or erosion attack. Fouling of exchangers occurs because of the deposition of foreign material on the interior or exterior of tubes. Evidence of fouling during operation is increased pressure drop and a general decrease in performance. Fouling can be so severe that tubes can become completely plugged, resulting in thermal stresses and the subsequent mechanical damage of equipment. The nature of the deposited fouling determines the method of cleaning. Soft deposits can be removed by steam, hot water, various chemical solvents, or brushing. Cooling water is sometimes treated with four parts of chlorine per million to prevent algae growth and the consequent reduction in the overall heat transfer coefficient of the exchanger. Plant experience usually determines the cleaning method to be used. Chemical cleaning should be performed by contractors specialized in the field who will consider the deposit to be removed and the materials of construction. If the cleaning method involves elevated temperatures, consideration should be given to thermal stresses induced in the tubes; steaming-out individual tubes can loosen the tube-to-tube sheet joints. Mechanical methods of cleaning are useful for both soft and hard deposits. There are numerous tools for cleaning tube interiors: brushes, scrapers, and various rotating cutter-type devices. The exchanger manufacturer or suppliers of tube tools can be consulted in the selection of the correct tool for the particular deposit. When cutting or scraping deposits, care should definitely be exercised to avoid damaging tubes. Cleaning of tube exteriors is generally performed using chemicals, steam, or other suitable fluids. Mechanical cleaning is performed but requires that the tubes be exposed, as in a typical air-cooled condenser, or capable of being exposed, as in a removable bundle in a shell-and-tube condenser. The layout pattern of the tubes must provide sufficient intersecting empty lanes (void areas) between the tubes, as in a square pitch. Mechanical cleaning of tube bundles, if necessary, (once again) requires the utmost care to avoid damaging tubes or fins.

40.5 Testing Proper maintenance requires testing of an exchanger to check the integrity of the following: tubes, tube-to-tube sheet joints, welds, and gasketed joints. The normal procedure consists of pressuring the shell with water or air at the nameplatespecified test pressure and viewing the shell welds and the face of the tube sheet for leaks in the tube sheet joints or tubes. Water should be at ambient temperature to avoid false indications due to condensation. Pneumatic testing requires extra care because of the destructive nature of a rupture or explosion, or fire hazards when residual flammable materials are present [3]. Condensers of the straight-tube

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floating-head construction require a test gland to perform the test. Tube bundles without shells are tested by pressurizing the tubes and viewing the length of the tubes and back face of the tube sheets. Corrective action for leaking tube-to-tube sheet joints requires expanding the tube end with a suitable roller-type tube expander. Good practice calls for an approximate 8% reduction in wall thickness after metal-to-metal contact between the tube and tube hole. Tube expanding should not extend beyond 1/8 inch of the inner tube-sheet face to avoid cutting the tube. Care should be exercised to avoid over-rolling the tube, which can cause work-hardening of the material, an insecure seal, and/or stress-corrosion cracking of the tube. Defective tubes can either be replaced or plugged. Replacing tubes requires special tools and equipment. The user should contact the manufacturer or a qualified repair contractor. Plugging of tubes, although a temporary solution, is acceptable provided that the percentage of the total number of tubes per tube pass to be plugged is not excessive. The type of plug to be used is a tapered one-piece or twopiece metal plug suitable for the tube material and inside diameter. Care should be exercised in seating plugs to avoid damaging the tube sheets. If a significant number of tube or tube joint failures are clustered in a given area of the tube layout, their location should be noted and reported to the manufacturer. A concentration of failures is usually caused by something other than corrosion (e.g., impingement, erosion, or vibration) [3, 4].

40.6 Improving Operation and Performance Within the constraints of an existing system, improving operation and performance refers to maintaining operation and original or consistent performance. There are several factors previously mentioned which are critical to the design and performance of an exchanger: operating pressure, amount of non-condensable gases in the vapor stream, coolant temperature and flow rate, fouling resistance, and mechanical soundness. Any pressure drop in the vapor line upstream of the unit should ordinarily be minimized. Deaerators or similar devices should be operational where necessary to remove gases in solution with liquids. Proper and regular venting of equipment and leak proof, gasketed joints in vacuum systems are all necessary to prevent gas binding and alteration of the condensing equilibrium [4, 5]. Coolant flow rate and temperatures should be checked regularly to ensure that they are in accordance with the original design criteria. The importance of this can be illustrated merely by comparing the winter and summer performance of a condenser using cooling-tower or river water. Decreased performance due to fouling will generally be exhibited by a gradual decrease in efficiency and should be corrected as soon as possible. Mechanical malfunctions can also be gradual, but will eventually be evidenced by a near total lack of performance. Fouling and mechanical soundness can only be controlled by regular and complete maintenance. In some cases, fouling is much worse than predicted and

Operation, Maintenance, and Inspection (OM&I) 409 requires frequent cleaning regardless of the precautions taken in the original design. These cases require special designs to alleviate the problems associated with fouling. For example, a leading PVC manufacturer found that carryover of polymer reduced the efficiency of its monomer condenser and caused frequent downtime. The solution was to provide polished internals and high condensate loading in a vertical down flow shell-and-tube exchanger. In another example, a major pharmaceutical intermediate manufacturer had catalyst carryover to a vertical down flow shell-and-tube condenser which accumulated on the tube internals. The solution was to recirculate condensate to the top of the unit and spray it over the tube-sheet face to create a film descending down the tubes to rinse the tubes clean. Most heat exchanger manufacturers will provide designs for alternate conditions as a guide to estimating the cost of improving efficiency via other coolant flow rates and temperature as well as alternate configurations (i.e., vertical, horizontal, shell side, or tube side).

References 1. Connery, W., Kafes, N., Buonicore, A.J., and Theodore, L., Evaluating surface condensers for air pollution control applications, in: Energy and the Environment, Proceedings of the Third National Conference, E.J. Rolinski, A.J. Buonicore, D.E. Earley, L. Theodore, R.F. Rolsten, and R.A. Servais (Eds.), pp. 276-282, AIChE Dayton and Ohio Valley Sections, Dayton, OH, 1975. 2. Connery, W., Chapter 6 Operation, maintenance, and inspection, in: Air Pollution Control Equipment, Theodore, L., and Buonicore, A.J. (Ed.), Theodore Tutorials, East Williston, NY, 1992. 3. Theodore, L., and Dupont, R., Environmental Health and Hazard Risk Assessment: Principles and Calculations, CRC Press/Taylor & Francis Group, Boca Raton, FL, 2012. 4. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 5. Theodore, L. Ricci, F., and VanVliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

41 Design Principles

41.1 Introduction As described earlier, heat exchangers are devices used to transfer heat from a hot fluid to a cold fluid. They can be classified by their function. An abbreviated summary of these functions was provided earlier. The design of these exchangers is concerned with the quantities of heat to be transferred, the rates at which they are transferred, the temperature driving force, the extent and arrangement of the surface(s) separating the heat source and receiver, the area requirements, and the amount of mechanical energy (pressure drop) that is required to facilitate the heat transfer. Since heat transfer involves an exchange in a system, the loss of heat by one body (solid or fluid) will equal (for steady processes) the heat absorbed by another within the confines of the same system. These issues are addressed in this chapter. Current design practices for some heat exchangers usually fall into the categories of state-of-the-art and pure empiricism. Past experience with similar applications is commonly used as the sole basis for the design procedure. Vendors maintain proprietary files on past exchanger installations; these files are periodically revised and expanded as new orders are evaluated. In designing a new exchanger, the files are consulted for similar applications and old designs are heavily relied on.

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By contrast, the engineering profession in general, and the chemical engineering profession in particular, has developed well-defined procedures for most “standard” heat exchangers (e.g., double pipe, tube and bundle, boilers, etc.). These techniques, tested and refined for nearly a century, are routinely used by today’s practicing environmental engineers. The purpose of this chapter is to introduce the reader to some heat exchange process design principles and industrial applications. Such an introduction, however sketchy, can provide the reader with a better understanding of the major engineering aspects of a heat exchanger, including some of the operational, economics, controls and instrumentation for safety requirements, and any potential environmental factors associated with the unit. No attempt is made in the sections that follow to provide extensive coverage of this topic; only general procedures and concepts are presented and discussed. Chapter contents include General Design Procedures, Other Design Considerations, Process Schematics, and Purchasing a Heat Exchanger.

41.2 General Design Procedures There are usually five conceptual steps to be considered in the design of any equipment and they naturally apply to heat exchangers. These are: 1. Identification of the parameters that must be specified. 2. Application of the fundamentals underlying theoretical equations or concepts. 3. Enumeration, explanation, and application of simplifying assumptions. 4. Possible use of correction factors for non-ideal behavior. 5. Identification of other factors that must be considered for adequate equipment specification. Calculation procedures for most of these have been presented earlier in this Part. Since design calculations are generally based on the maximum throughput capacity for the heat exchanger, these calculations are never completely accurate. It is usually necessary to apply reasonable safety factors when deciding on the final design. Safety factors vary widely and are a strong function of the accuracy of the data involved, calculational procedures, and past experience. Attempting to justify these is a difficult task [1, 2]. Unlike many of the problems encountered and solved by practicing engineers, there is no absolutely correct solution to a design problem; however, there is usually a better solution. Many alternative exchanger designs when properly implemented will function satisfactorily, but one alternative will usually prove to be economically more efficient and/or attractive than the others.

Design Principles 413 Overall and componential material balances have already been described in rather extensive detail. Material balances may be based on mass, moles, or volume, usually on a rate (time rate of change) basis. However, the material balance calculation is usually based on mass or volume rates since both play an important role in equipment sizing calculations. Some design calculations in the chemical process industry today include transient effects that can account for process upsets, startups, shutdowns, and so on. The describing equations for these time-varying (unsteady-state) systems are differential. The equations usually take the form of a first-order derivative with respect to time, where time is the independent variable. However, design calculations for almost all heat exchangers assume steady-state conditions, with the ultimate design based on worst-case or maximum flow conditions. This greatly simplifies these calculations since the describing equations are no longer differential, but rather algebraic. The safe operation of an exchanger requires that controls keep the system operating within a safe operating envelope. The envelope is based on many of the design, process, and (if applicable) regulatory constraints. The control system should also be designed to vary one or more of the process variables to maintain the appropriate conditions for the exchanger. These variations are often programmed into the system based on past experience with a specific unit. The operational parameters that may vary include the flow rates, temperatures, and system pressure. The control system may be subjected to (extensive) analysis on operational problems and items that could go wrong. A hazard and operational (HAZOP) analysis [3] can be conducted on the system to examine and identify all possible failure mechanisms. It is important that all of the failure mechanisms have appropriate response reactions by the control system. Several of the failure mechanisms that must be addressed within the appropriate control system response are excess (excursion) or minimal temperature, excessive or subnormal flow rate, equipment failure, component failure, and broken circuits [3]. In the case of heat exchanger design, data on similar existing units are normally available and cost estimates and/or process feasibility are determined from these data. It should be pointed out again that most heat exchangers in real practice are designed by duplicating or mimicking similar existing systems. Simple algebraic correlations that are based on past experience are the rule rather than the exception. This stark reality is often disappointing and depressing to students and novice engineers involved in design, but emphasizes the importance of experience in engineering practice.

41.3 Other Design Considerations Heating or cooling of liquids in batch processes is used in a number of commercial applications. Some reasons for using a batch rather than a continuous heat transfer operation include: liquid is not continuously available, liquid cleaning and

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regeneration is a significant part of the operation, or batch operation is simpler and cheaper. Consider the following example. Vapor degreasers are widely used for cleaning metal parts. A degreaser consists of a tank partially filled with a solvent. The tank is equipped with a heating coil to heat the solvent close to its boiling point. The vapor of the solvent occupies the remaining volume of the tank, forming the “solvent vapor zone.” When a metal part is placed in the solvent vapor zone, the solvent condenses on the metal part and then drips off, taking contaminants with it. For ease of use, vapor degreasers are often open to the atmosphere. This makes it easier to introduce and remove the metal parts. It has been a common practice to use a halogenated hydrocarbon for such cleaning since they are excellent solvents, being volatile and non-flammable; however, they can be toxic and the open tank of a degreaser can be a significant source of solvent emissions or volatile organic components (VOCs). Walas [4] has also provided some simple “rules of thumb” for selecting and designing heat exchangers. An outline of his suggestions follows: 1. Take true countercurrent flow in a shell-and-tube exchanger as the basis for comparison. 2. Standard tubes are 3/4-in OD, 1-in triangular spacing, 16 ft long; a shell 1 ft in diameter accommodates 100 ft2; 2 ft diameter, 450 ft2; 3 ft diameter, 1100 ft2. 3. Tube side is for corrosive, fouling, scaling, and high-pressure fluids. 4. Shell side is for viscous and condensing fluids. 5. Pressure drops are approximately 1.5 psi for boiling liquids and 3 to 9 psi for other services. 6. Minimum temperature approach is 20 °F with normal coolants, 10°F or less with refrigerants. 7. Water inlet temperature is 90 °F, maximum outlet 120 °F. 8. For estimating, use the following heat-transfer coefficients (Btu/ hr-ft2-°F): water-to-liquid, 150; condensers, 150; liquid-to-liquid, 50; liquid-to-gas, 5.0; gas-to-gas, 5.0; reboiler, 200. For the maximum flux in reboilers, use 10,000 Btu/hr-ft2. 9. Double-pipe exchangers are competitive at duties requiring 100 to 200 ft2. 10. Compact (plate or finned-tube) exchangers have 350 ft2 of surface area/ft3 of volume and about four times the heat transfer per cubic foot of shell-and-tube units. 11. Plate-and-frame exchangers are suited for environmental service and, in stainless steel, are 25–50% cheaper than shell and tube units. 12. For air coolers: tubes are 0.75 to 1.00-in OD; total finned surface is 15 to 20 ft2/ft2 of bare surface; the overall heat-transfer coefficient,

Design Principles 415 U = 80–100 Btu/hr-ft2 bare surface-°F; fan power-input is 2–5 hp/ million Btu-hr; the approach is 50°F or more. 13. For fired heaters: radiant rate is 12,000 Btu/hr-ft2; convection rate, 4000 Btu/hr-ft2; cold oil-tube velocity, 6ft/s; thermal efficiency, 70–75%.

41.4 Process Schematics To the practicing engineer, particularly the process engineer, the process flow sheet is the key instrument for defining, refining, and documenting a process unit. The process flow diagram is the authorized process blueprint and the framework for specifications used in equipment designation and design. It is the single, authoritative document employed to define, construct, and operate the unit and/or process [5]. Beyond equipment symbols and process stream flow lines, there are several essential constituents contributing to a detailed process flow sheet. These include equipment identification numbers and names, temperature and pressure designations, utility designations, flow rates for each process stream, and a material balance table pertaining to process flow lines. The process flow diagram may also contain additional information such as major instrumentation and physical properties of the process streams. When properly assembled and employed, a process schematic provides a coherent picture of the process. It can point out some deficiencies in the process that may have been overlooked earlier in a study. Basically, the flow sheet symbolically and pictorially represents the interrelation among the various flow streams and the exchanger (or any other equipment), and permits easy calculations of material and energy balances. There are a number of symbols that are universally employed to represent equipment, equipment parts, valves, piping, and so on. Some of these are depicted in the schematic in Figure 41.1. Although there are a significant number of these symbols, only a few are needed for even the most complex heat exchanger unit. These symbols obviously reduce, and in some instances, replace detailed written descriptions of the unit or process. Note that many of the symbols are pictorial, which helps in better describing process components and information. The degree of sophistication and details of an exchanger flow sheet usually vary with time. The flow sheet may initially consist of a simple free-hand block diagram with limited information that includes only the equipment. Later versions may include line drawings with pertinent process data such as overall and componential flow rates, temperatures, pressures, and instrumentation. During the later stages, the flow sheet can consist of a highly detailed P&I (piping and instrumentation) diagram; this aspect of the design procedure is beyond the scope of this text; the reader is referred to the literature [5–8] for information on P&I diagrams. In summary, industrial plant flow sheets are the international language of the engineer, particularly the chemical and environmental engineer. Chemical engineers conceptually view a (chemical) plant as consisting of a series of interrelated

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Centrifugal pump

Process fluid on tube side

Process fluid on shell side Counter flow

Process fluid on tube side

Process fluid on shell side

Process fluid on tube side

Process fluid on shell side Parallel flow

Figure 41.1 Selected flow sheet symbols.

building blocks that are defined as units or unit operations (the heat exchanger is one such unit) [8]. The plant essentially ties together the various pieces of equipment that make up the process. Flow schematics follow the successive steps of a process by indicating where the pieces of equipment are located and the material streams entering and leaving each unit.

41.5 Purchasing a Heat Exchanger [7–8] Prior to the purchase of a heat exchanger, experience has shown that the following points should be emphasized: 1. Refrain from purchasing any heat exchanger without reviewing certified independent test data on its performance under a similar application. Request the manufacturer to provide performance information and design specifications. 2. In the event that sufficient performance data are unavailable, request that the equipment supplier provide a small pilot model for evaluation under existing conditions. 3. Prepare a good set of specifications. Include a strong performance guarantee from the manufacturer to ensure that the heat exchanger will meet all design criteria and specific process conditions.

Design Principles 417 4. Closely review the overall process, other equipment, and economic fundamentals. 5. Make a careful material balance study. 6. Refrain from purchasing any heat exchanger until firm installation cost estimates have been added to the cost. Escalating installation costs are the rule rather than the exception. 7. Give operation and maintenance costs high priority on the list of exchanger selection factors. 8. Refrain from purchasing any heat exchanger until a solid commitment from the vendor is obtained. Make every effort to ensure that the exchanger is compatible with the (plant) process. 9. The specification should include written assurance of prompt technical assistance from the supplier. This, together with a completely understandable operating manual (with parts list, full schematics, consistent units and notations, etc.), is essential and is too often forgotten in the rush to get the heat exchanger operating. 10. Schedules can be critical. In such cases, delivery guarantees should be obtained from the manufacturers and penalties identified. 11. The heat exchanger should be of fail-safe design [3] with built-in indicators to show when performance is deteriorating. 12. Perhaps most importantly, withhold 10 to 15% of the purchase price until satisfactory operation is clearly demonstrated. The usual design, procurement, installation, and/or startup problems can be further compounded by any one or a combination of the following: 1. 2. 3. 4. 5. 6. 7. 8.

Unfamiliarity of process engineers with heat exchangers New suppliers, frequently with unproven heat exchanger equipment Lack of industry standards with some designs Compliance schedules that are too tight Vague specifications Weak guarantees Unreliable delivery schedules Process reliability problems

Proper selection of a particular heat exchanger for a specific application can be extremely difficult and complicated. The final choice in heat exchanger selection is usually dictated by that unit capable of achieving the aforementioned design criteria and required process conditions at the lowest uniform annual cost (amortized capital investment plus operation and maintenance costs) [8]. In order to compare specific exchanger alternatives, knowledge of the particular application and site is also essential. A preliminary screening, however, may be performed by reviewing the advantages and disadvantages of each type or class of unit.

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However, there are many other situations where knowledge of the capabilities of the various options, combined with common sense, will simplify the selection process. Four construction considerations/measures that should be followed during the design of a process/plant heat exchanger include: 1. Keep hot stream lines away from workers. 2. Insulate the entire exchanger so it is not hot to the touch. 3. Make sure all the flows entering the exchanger are turned off when it is being serviced. 4. Without compromising efficiency, allow adequate space inside the exchanger so that it can be easily cleaned.

References 1. Santoleri, J. Reynolds, J., and Theodore, L., Introduction to Hazardous Waste Incineration, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2000. 2. Reynolds, J., Jeris, J., and Theodore, L., Handbook of Chemical and Environmental Engineering Calculations, John Wiley & Sons, Hoboken, NJ, 2004. 3. Theodore, L., and Dupont, R., Environmental Health and Hazard Risk Assessment: Principles and Calculations, CRC Press/Taylor & Francis Group, Boca Raton, FL, 2012. 4. Walas, M., Designing Heat Exchangers, Chem. Eng., March 16, 1987. 5. Ulrich, G., A Guide to Chemical Engineering Process Design and Economics, John Wiley & Sons, Hoboken, NJ, 1984. 6. Kern, D., Process Heat Transfer, McGraw-Hill, New York City, NY, 1950. 7. Theodore, Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 8. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2014.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

Part IV MASS TRANSFER

Mass transfer operations are generally applicable to processes that essentially involve either componential or physical change, or both. A substantial number of these operations are concerned with changing the composition of mixtures through methods that do not involve chemical reactions. These are defined as componential separation processes. It is also often desirable to separate the original substance into its component parts by phase. Such separations may be entirely mechanical, such as the separation of a solid from a liquid during filtration, or the classification of granular solid into fractions of different particle size by screening. On the other hand, if the operations involve the aforementioned changes in the composition of solutions, they are defined as componential separation operations, and it is with these processes that the bulk of this Part of the book is primarily concerned. Three important general topics are reviewed in this part of the book. They are: Classification of Mass Transfer Operations, Mass Transfer Equipment, and Characteristics of Mass Transfer Operations. Part IV contains three sections and a total of 17 chapters, and each serves a unique purpose in an attempt to treat important aspects of mass transfer.

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Part IV chapter content includes: IV-A Fundamentals 42. Equilibrium Principles 43. Phase Equilibrium Relationships 44. Rate Principles 45. Mass Transfer Coefficients 46. Classification of Mass Transfer Operations 47. Characteristics of Mass Transfer Operations IV-B Equipment 48. Absorption and Stripping 49. Distillation 50. Adsorption 51. Liquid-Liquid and Solid-Liquid Extraction 52. Humidification 53. Drying IV-C Other Considerations 54. Absorber Design and Performance Equations 55. Distillation Design and Performance Equations 56. Adsorber Design and Performance Equations 57. Crystallization 58. Other Novel Separation Processes

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

42 Equilibrium Principles

42.1 Introduction There are various mass transfer operations. The three that are most encountered in practice are distillation, absorption, and adsorption. As such, they receive the bulk of the treatment in this Mass Transfer section of this text. Other operations reviewed include, liquid-liquid and liquid-solid extraction, humidification and drying, crystallization, membrane separation processes, and phase separation processes. A brief introduction to each topic is provided below. In distillation (Chapter 49), a liquid mixture at its boiling point is brought into contact with a saturated vapor mixture of the same components at a different concentration. The components are transferred between the phases until equilibrium is established or the phases are separated. In gas absorption (Chapter 48), a component in the gas phase is dissolved by a liquid phase in contact with it. The opposite of gas absorption is stripping, where a component of the liquid phase is transferred to the gas phase. In adsorption (Chapter 50), a component of a gas or liquid is retained on the surface of a solid adsorbent such as activated carbon. In extraction (Chapter 51), one makes use of solubility differences in different liquid phases. In humidification (Chapter 52) and drying (Chapter 53), water or another liquid is vaporized, and the required heat of vaporization must be transferred to 421

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the liquid. Dehumidification is the opposite of humidification, and is defined as the condensation of water from air, or, in general, the condensation of any vapor from a non-condensable gas. In crystallization (Chapter 57), a solution of dissolved solids is supersaturated and allowed to crystallize out the excess solute, thereby forming a purer crystalline solid. And what do all of these mass transfer operations have in common? Both equilibrium and rate play an important role – two topics addressed in these introductory chapters. This chapter addresses equilibrium principles. Equilibrium principles play an important role in designing and predicting the performance of many mass transfer processes. In fact, several mass transfer calculations are based primarily on the application of equilibrium principles and equilibrium data. Distillation is a prime example; virtually every calculational procedure is based on vapor-liquid equilibrium data for a system, e.g., acetonewater. Similarly, adsorption engineering applications almost always utilize vapor (gas)-solid equilibrium data, e.g., acetone-activated carbon. It is for this reason that phase equilibrium principles in general, and specific equilibrium systems in particular, are reviewed in this and the next chapter. The term phase, for a pure substance, indicates a state of matter, i.e., solid, liquid, or gas. For mixtures, however, a more stringent connotation must be used, since a totally liquid or solid system may contain more than one phase (e.g., a mixture of oil and water). A phase is characterized by uniformity or homogeneity, which means that the same composition and properties must exist throughout the phase region. At most temperatures and pressures, a pure substance normally exists as a single phase. At certain temperatures and pressures, two or perhaps even three phases can coexist in equilibrium. This is shown on the phase diagram for water in Figure 42.1. Regarding the interpretation of this diagram, the following points should be noted: 1. The line between the gas and liquid phase regions is the boiling point and dew point line and represents equilibrium between the gas and liquid. 2. The boiling point of a liquid is the temperature at which its vapor pressure is equal to the external pressure. The temperature at which the vapor pressure is equal to 1 atm is the normal boiling point. 3. The line between the solid and gas phase regions is the sublimation point and deposition point line and represents equilibrium between the solid and gas. 4. The line between the solid and liquid phase regions is the melting point and freezing point line and represents equilibrium between the liquid and solid. 5. The point at which all three equilibrium lines meet (i.e., the one pressure and temperature where solid, liquid, and gas phases can all coexist) is the triple point. 6. The liquid-gas equilibrium line is bounded on one end by the triple point and the other end by the critical point. The critical

Equilibrium Principles

Pressure, atm

220

1.0

423

Critical point

Solid

Liquid

0.006

Triple point

0

Vapor

100 0.0098 Temperature, °C

374

Figure 42.1 Phase diagram for water.

temperature (the temperature coordinate of the critical point) is defined as the temperature above which a gas or vapor cannot be liquefied by the application of pressure alone. The term vapor, strictly speaking, should only be used for a condensable gas, i.e., a gas below its critical temperature, and should not be applied to a noncondensable gas. It should also be pointed out that the phase diagram for water (Figure 42.1) differs from that of other substances in one respect, i.e., the freezing point line is negatively sloped. For other substances, the slope of this line is positive. This is a consequence of the fact that liquid water is more dense than ice [1]. Mass transfer calculations rarely involve single (pure) components. Phase equilibria for multicomponent systems are considerably more complex, mainly because of the addition of composition variables. For example, in a ternary (threecomponent) system, the mole fractions of two of the components are pertinent variables along with temperature and pressure. In a single-component system, dynamic equilibrium between two phases is achieved when the rate of molecular transfer from one phase to the second equals that in the opposite direction. In multicomponent systems, the equilibrium requirement is more stringent, i.e., the rate of transfer of each and all components must be the same in both directions [2]. Relationships governing the equilibrium distribution of a substance between two phases, particularly gas and liquid phases, are the principal subject matter of phase-equilibrium thermodynamics. As noted above, these relationships form the basis of calculational procedures that are employed in the design and the prediction of performance of several mass transfer processes [1].

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Table 42.1 Mass transfer scenarios. Scenario

Phase undergoing treatment

Phase performing treatment

1

Gas

Gas

2

Gas

Liquid

3

Gas

Solid

4

Liquid

Gas

5

Liquid

Liquid

6

Liquid

Solid

7

Solid

Gas

8

Solid

Liquid

9

Solid

Solid

For many, mass transfer has come to be defined as the tendency of a component in a mixture to travel from a region of high concentration to one of low concentration. For example, if an open beaker with some alcohol in it is placed in a room in which the air is relevantly dry, alcohol vapor will diffuse out through the column of air in the beaker. Thus, there is mass transfer of alcohol from a location where its concentration is high (just above the liquid surface) to a place where its concentration is low (at the top of the beaker). If the gas mixture in the beaker is stagnant (void of any motion), the transfer is said to occur by molecular diffusion. If there is a bulk mixing of the layers of gas in the beaker, mass transfer is said to occur by the mechanism of either forced or natural convection, or both. These two mechanisms are analogous to the transfer of heat by conduction and convection [3]. Analogies with fluid flow are not nearly as obvious [4]. In discussing the principles and applications of mass transfer, the presentation will primarily consider binary mixtures, although some multicomponent mixture issues are addressed. Binary mixtures serve as an excellent starting point for training and educational purposes. However, real world applications often involve more than two components. As mentioned above, the problem of transferring materials from one phase to another is encountered in many engineering operations. There are generally three classes of phases encountered in practice: gas, liquid, and solid. If mass transfer occurs between two phases, with one phase being treated and the other performing the treatment, a total of (3)2 or nine possible combinations of operations with two phases is possible. These mini-scenarios are listed in Table 42.1.

42.2

Gibb’s Phase Rule

The general subject of phases and equilibrium is only one of several topics highlighted in this chapter. Equilibrium in a multiphase system is subject to certain

Equilibrium Principles

425

restrictions. These restrictions can be expressed in an equation form, defined as the Phase Rule. The state of a P-V-T system is established when its temperature and pressure and the compositions of all phases are fixed. However, these variables are not all independent for equilibrium states, and fixing a limited number of them automatically establishes the others. This number of independent variables is given by Gibb’s Phase Rule and is defined as the number of degrees of freedom of the system. This number of variables must be specified in order to fix the state of a system at equilibrium. The following nomenclature is employed in the discussion to follow: P = number of phases C = number of chemical components r = number of independent chemical reactions; components minus elements F = number of degrees freedom The phase-rule variables are temperature, pressure, and C 1 mole fractions in each phase. The number of these variables is therefore 2 + (C 1)P. The masses of the phases are not phase-rule variables because they have no effect on the state of the system. It should also be noted that the temperature and pressure have been assumed to be uniform throughout the system for an equilibrium state. The total number of independent equations (by a componential balance) is (P 1)C + r. These equations are functions of temperature, pressure, and composition; therefore, they represent relations connecting the phase-rule variables. Since F is the difference between the number of variables and the number of equations:

F

2 (C 1)P (P 1)C r

2 P C r

(42.1)

The variable r can be determined by subtracting the number of elements in the system from the number of components. (Note that r = 0 in a non-reacting system.) Thus, if a gaseous system at equilibrium consists of CO2, H2, H2O, CH4, and C2H6 the degrees of freedom may be calculated by noting the number of phases, P, and the number of components, C, are 1 and 5, respectively, and the number of elements (C, H, and O) are 3. Thus,

r

5 3 2

(42.2)

From Equation 42.1,

F

2 P C r

2 1 5 2 4

(42.3)

Since F = 4, one is free to specify T, P and two mole fractions in an equilibrium mixture of these five chemical species, provided nothing else is arbitrarily set.

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42.3 Important Phase Considerations In mass transfer applications, the most important equilibrium phase relationship (as noted above) is that between liquid and vapor. Raoult’s law and Henry’s law theoretically describe liquid-vapor behavior, and under certain conditions are applicable in practice. Raoult’s law and Henry’s law are the two equations most often used in the introductory study of phase equilibrium, specifically within the boundaries of ideal vapor-liquid equilibrium (VLE). Both Raoult’s and Henry’s laws help one understand the equilibrium properties of liquid mixtures. The next two chapters review these laws. Phase equilibrium examines the physical properties of various classes of mixtures, and analyzes how different components affect each other within those mixtures. There are four key classes of mixtures (see Table 42.1): 1. 2. 3. 4.

Vapor-liquid Vapor-solid Liquid-solid Liquid-liquid

It is the authors’ opinion that the two most important traditional mass transfer operations are absorption and distillation. As such, the presentation to follow will primarily address principles that directly apply to these two mass transfer processes. For example, material on Raoult’s law and equilibrium relationships will be employed in Chapter 49, Distillation. Material on Henry’s law and mass transfer coefficients will be directly applied to absorption processes in Chapter 48.

References 1. Theodore, L., Ricci, F., and VanßVliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 2. Theodore, L., and Barden, J., Mass Transfer Operations, A Theodore Tutorial, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1996. 3. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 4. Abulencia, P. and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

43 Phase Equilibrium Relationships

43.1 Introduction As noted in the previous chapter, the two most important gas-liquid relationships are Raoult’s Law and Henry’s Law. Both laws are addressed in this chapter; a brief discussion of non-ideal behavior is also included. The two laws. Is there a difference? The reader will soon realize that there are some similarities but also some differences. The basic difference between Raoult’s and Henry’s laws is that Raoult’s law applies to the solvent, while Henry’s law applies to the solute. In ideal solutions, both the solute and the solvent obey Raoult’s law. However, in ideal dilute solutions, the solute obeys Henry’s law whereas the solvent obeys Raoult’s law. This will be expanded upon later in this chapter. Finally, the reader should note that all the phase equilibrium calculations in this chapter are based on the assumption that the vapor and liquid are in equilibrium. Although the above presentation is limited to two-component systems, an approach to multiphase (two or more coexisting liquid phases) VLE is available in the literature [1].

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43.2 Raoult’s Law Raoult’s law states that the partial pressure of each component, pi, in a solution is proportional to the mole fraction, xi, of that component in the liquid mixture being studied. The “proportionality constant” is its vapor pressure, Pi . Therefore, for component i in a mixture, Raoult’s law can be expressed as:

pi

Pi xi

(43.1)

where pi is the partial pressure of component i in the vapor, Pi is the vapor pressure of pure i at the same temperature, xt is the mole fraction of component i in the liquid. This expression may be applied to all components so that the total pressure, P, is given by the sum of all the partial pressures. If the gas phase is ideal, Dalton’s law, pi = yiP, applies, and therefore, Equation 43.1 can be written as follows, equating Equation 34.1 to Dalton’s law and rearranging:

yi

xi

Pi P

(43.2)

where yi is the mole fraction of component i in the vapor. For example, the mole fraction of water vapor in air that is saturated, and in equilibrium contact with pure water (x = 1.0), is simply given by the ratio of the vapor pressure of water at the system temperature divided by the system pressure. While it is true that some of the air dissolves in the water (making xw < 1.0), one usually neglects this to simplify calculations. These equations primarily find application in distillation, and to a lesser extent, absorption and stripping [2]. The basis of Raoult’s law can be best understood in molecular terms by considering the rates at which molecules leave and return to the liquid. Raoult’s law illustrates how the presence of a second component, say B, reduces the rate at which component A molecules leave the surface of the liquid, but it does not inhibit the rate at which they return [3, 4]. In order to finally deduce Raoult’s law, one must draw on additional experimental information about the relation between the vapor pressures and the composition of the liquid. Raoult himself obtained these data from experiments on mixtures of closely related liquids in order to develop his law. Most mixtures obey Raoult’s law to some extent, however small it may be. The mixtures that closely obey Raoult’s law are those whose components are structurally similar, and therefore Raoult’s law is most useful when dealing with these types of solutions. Mixtures that obey Raoult’s law for the entire composition range are called ideal solutions. A graphical representation of this behavior can be seen in Figure 43.1. Many solutions do deviate significantly from Raoult’s law. However, the law is obeyed even in these cases for the component in excess (which, in this case, would be the solvent) as it approaches purity. Therefore, if the solution of solute is dilute,

Phase Equilibrium Relationships 429

Total pressure, P

Partial pressure

p’A

p’B

Partial pressure of A, pA

Partial pressure of B, pB 0

Mole fraction of A

1

Figure 43.1 Graphical representation of the vapor pressure of an ideal binary solution, i.e., one that obeys Raoult’s law for the entire composition range.

the properties of the solvent can be approximated using Raoult’s law. Raoult’s law, however, does not have universal applications. There are several aspects of this law which give it limitations. First, it assumes that the vapor is an ideal gas, which is not necessarily the case. Second, Raoult’s law only applies to ideal solutions, and, in reality, there are no ideal solutions. Also, a third problem to be aware of is that Raoult’s law only really works for solutes which do not change their nature when they dissolve (i.e., they do not ionize or associate). A correction factor which accounts for liquid-phase deviations from Raoult’s law is developed later in this chapter. Raoult law applications require vapor pressure information. Vapor pressure data are available in the literature [1–4]. However, there are two equations that can be used in lieu of actual vapor pressure information, the Clapeyron equation and the Antoine equation. The Clapeyron equation is given by:

ln p

A (B /T )

(43.4)

where p and T are the vapor pressure and temperature, respectively. The Antoine equation is given by:

ln p

A B /(T C )

(43.5)

Note that for both equations, the units of p and T must be specified for given values of A and B and/or C. Values for the Clapeyron equation coefficients, A and

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B, are provided in Table 43.1 for some compounds. Some Antoine equation coefficients, A, B, and C, are listed in Table 43.2. Additional values for these coefficients for both equations are available in the literature [1, 4]. The reader should note that the Clapeyron equation generally overpredicts the vapor pressure at or near ambient conditions. The Antoine equation is widely used in industry and usually provides excellent results. Also note that, contrary to statements appearing in the Federal Register and some Environmental Protection Agency (EPA) publications, vapor pressure is not a function of pressure [1, 2]. It is important to note that vapor mixtures do not condense at one temperature as a pure vapor does. The temperature at which a vapor begins to condense as the temperature is lowered is defined as the dew point. It is determined by calculating the temperature at which a given vapor mixture is saturated. The bubble point Table 43.1 Approximate Clapeyron equation coefficients*. A

B

Acetaldehyde

18.0

3.32

103

Acetic anhydride

20.0

5.47

103

Ammonium chloride

23.0

10.0

103

Ammonium cyanide

22.9

11.5

103

Benzyl alcohol

21.9

7.19

103

Hydrogen peroxide

20.4

5.85

103

Nitrobenzene

18.8

5.87

103

Nitromethane

18.5

3.32

103

Phenol

19.8

5.96

103

Tetrachloroethane

17.5

4.38

103

*T in K, p in mm Hg.

Table 43.2 Antoine equation coefficients*. A

B

C

Acetone

14.3916

2795.82

230.00

Benzene

13.8594

2773.78

220.07

Ethanol

16.6758

3674.49

226.45

n-Heptane

13.8587

2911.32

216.64

Methanol

16.5938

3644.30

239.76

Toluene

14.0098

3103.01

219.79

Water

16.2620

3799.89

226.35

*T in °C, p in kPa.

Phase Equilibrium Relationships 431 of a liquid mixture is defined as the temperature at which it begins to vaporize. The bubble point may also be viewed as the temperature at which the last vapor condenses, while the dew point is the temperature at which the last liquid vaporizes [1]. These examples are based on a given and constant pressure, and are thus referred to as the dew point and bubble point temperatures. Calculations based on holding the temperature constant lead to the dew point and bubble point pressures. Dew point and bubble point calculations enable one to obtain vapor-liquid equilibrium (VLE) relationships for a binary mixture; these are often provided as a P x, y diagram (with the temperature constant) or a T x, y diagram (with the pressure constant), or both. VLE data can be generated assuming Raoult’s law applies. Additional details and procedures for obtaining these graphs are provided in the literature [1]; algorithms are also available to generate these diagrams. An important application of interest in environmental engineering arises in gas absorption operations (see Chapter 48). The equilibrium of interest is that between a nonvolatile absorbing liquid (solvent) and a solute gas. The solute is ordinarily removed from its mixture in a relatively large amount of a carrier gas which does not dissolve in the absorbing liquid. Therefore, it is often possible, and frequently the case when considering the removal and/or recovery of a gaseous component by absorption, to assume that only the component in question is transferred between phases. Both the solubility of the non-diffusing (inert) gas in the liquid and the presence of vapor from the liquid in the gas phase are usually neglected. The important variables to be considered then are the pressure, temperature, and the concentration of the components in the liquid and the gas phases. The temperature and pressure may be fixed and the concentration(s) of the component(s) in the various phases are defined from phase equilibrium relationships. An equation that may be employed to relate to the equilibrium concentration of the absorbed species in the liquid phase is Henry’s law [5]. A discussion on the law follows.

43.3

Henry’s Law

Unfortunately, relatively few mixtures follow Raoult’s law. Henry’s law is another empirical relationship used for representing data on many systems. Henry’s law states that the partial pressure of the solute is proportional to its mole fraction, but the proportionality constant is not the vapor pressure of the pure substance (as it is in Raoult’s law). Instead, the proportionality constant is an empirical constant, denoted H. Therefore, for some component B of a solution, Henry’s Law can be written as:

pB

HB xB

(43.6)

The value of the Henry’s law constant, HB, has been found to be temperature dependent. The value generally increases with increasing temperature. As a

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consequence, the solubility of gases generally decreases with increasing temperature. In other words, the dissolved gases in a liquid can be driven off by heating the liquid. Mixtures which obey Henry’s law are known as ideal-dilute solutions. The above equation has also been written, for component A in this instance, as:

yA

mA x A

(43.7)

where mA is once again an empirical constant. In order to fully understand the concepts behind Henry’s law, one can once again examine the basic physical properties of a dilute solution on a molecular level. In a dilute solution, the solvent molecules are in an environment which is not that much different from the one they experience in a pure liquid. The solute molecules, however, are in a completely different environment than that of the pure solute state. Because of this, the solvent behaves as a slightly modified pure liquid, whereas the solute behaves entirely differently from its pure state. In this case, the rate of escape of solute molecules will be proportional to their concentration in the solution, and the solute will accumulate in the gas until the return rate is equal to the rate of the escape rate. This return rate will be proportional to the partial pressure of solute with a very dilute gas. It is important to remember that if the solute and solvent are very similar in structure, the solute obeys Raoult’s law. It may be rigorously proven that all non-ionic binary solutions (at relatively low pressures such that the vapor phase is ideal) obey Equation 43.6 as component “B” approaches infinite dilution. Moreover, when the dilute component B follows Henry’s law, the other component, A, must obey Raoult’s law over the same range of composition. The results obtained from Henry’s law for the mole fraction of dissolved gas is valid for the liquid layer just beneath the interface, but not necessarily the entire liquid. The latter will be the case only when thermodynamic equilibrium is established throughout the entire liquid body. Therefore, the use of Henry’s law is limited to dilute gas-liquid solutions, i.e., liquids with a small amount of gas dissolved in them. The linear relationship of Henry’s law does not apply in the case when the gas is highly soluble in the liquid. Henry’s law constants are available in the literature [1]. However, equations are also available to estimate this constant [1]. Henry’s law has been found to hold experimentally for all dilute solutions in which the molecular species are the same in the solution as in the gas. One of the most conspicuous and apparent exceptions to this is the class of electrolytic solutions, or solutions in which the solute has ionized or dissociated. As in the case of Raoult’s law, Henry’s law in this particular case does not hold. Since gases usually liberate heat when they dissolve in liquids, thermodynamics reveals [4] that an increase of temperature will result in a decrease in solubility. This is why gases may be readily removed from solution by heating. Another important factor influencing the solubility of a gas is pressure. As is to be expected from kinetic considerations, compression of a gas will tend to increase its solubility.

Phase Equilibrium Relationships 433

Air

NH3

Air - NH3

H2 O (a)

(c)

t t* > 0 yNH

(e)

Air - NH3

Air - NH3

H2O - NH3

H2O - NH3

t=∞

(f)

t=∞

3

xNH

3

Figure 43.2 Ammonia absorption system at T and P.

To illustrate the application of Henry’s law to the aforementioned absorption process, consider the air-water-ammonia system at the T and P pictured in Figure 43.2a to f [1]. In this system, NH3 is added to the air at time t = 0 (Figure 43.2b). The NH3 slowly proceeds to distribute itself (Figure 43.2c, d) until “equilibrium” is reached in Figure 43.2e. The mole fractions in both phases are measured in Figure 43.2f. These data are represented as Point (1) in Figure 43.3. If the process in Figure 43.2 is repeated several times with additional quantities of NH3, additional equilibrium points will be generated. These points appear in Figure 43.4. Although the plot in Figure 43.4 curves upwards, the data approach a straight line of slope m at low values of x. It is this region where it is assumed that Henry’s law applies. One of the coauthors of this text believes that as x 0, m 0. This conclusion, and its ramifications, has received further attention in the literature [6].

43.4 Raoult’s Law versus Henry’s Law [7] As noted in the introduction, a basic difference between Raoult’s and Henry’s laws is that Raoult’s law applies to the solvent, while Henry’s law applies to the solute.

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y

1 x Figure 43.3 x-y equilibrium point.

y

m

x

Figure 43.4 x-y equilibrium diagram showing multiple equilibrium points.

In ideal solutions, both the solute and the solvent obey Raoult’s law. Hover, in ideal dilute solutions, the solute obeys Henry’s law, whereas the solvent obeys Raoult’s law. In Figure 43.5 one can see the differences between Raoult’s law and Henry’s law in graphical form, in addition to the behavior of a real solution. It provides information on how two components are distributed between the vapor and liquid phase, e.g., acetone-water, while Henry’s law describes how a component, e.g., acetone will be distributed between a gas and liquid phase, e.g., air and water. There are several other significant differences between these two laws which must be recognized. Raoult’s law is much more theory-based, since it only applies to ideal situations. Henry’s law, on the other hand, is more empirically based, making it a more general and practical law. In addition to providing a link between the mole fraction of the solute and its partial pressure, Henry’s law constants may also be used to calculate gas solubilities. This plays an important role in biological functions, such as the transport of gases in the bloodstream. A knowledge of Henry’s law constants for gases in fats

Phase Equilibrium Relationships 435

olu

tio n

(H e

nr

Pressure

y’s La w)

HB

Ide al dil u

te s

Real solution

’s ult

p’B

w)

La

o

0

(Ra

lu

l so

a Ide

n tio

Mole fraction of B, xB

1

Figure 43.5 Graphical representation of the differences between Raoult’s and Henry’s law; Component B represents the solute.

and lipids is important in discussing respiration [4,7]. For example, consider scuba diving. In this recreational activity, air is supplied at a higher pressure so as to allow the pressure within the diver’s chest to match the pressure exerted by the surrounding water. This water pressure of the ocean increases at approximately 1 atm per 10 m of depth. However, air inhaled at higher pressures makes nitrogen more soluble in fats and lipids rather than water, causing nitrogen to enter the central nervous system, bone marrow, and fat reserves of the body. The nitrogen then bubbles out of its lipid solution if the diver rises to the surface too quickly, causing a condition called decompression sickness, also known as the bends. This condition can be fatal. The nitrogen gas bubbles can block arteries and cause unconsciousness as they rise to the brain. Another interesting application of Henry’s law is in the treatment of carbon monoxide poisoning. In a hyperbaric oxygen chamber, oxygen is raised to an elevated partial pressure. When an individual with carbon monoxide poisoning steps inside this chamber, there is a steep pressure gradient between the partial pressure of arterial blood oxygen, and the partial pressure of the oxygen in freshly inhaled air. In this way, oxygen floods quickly into the arterial blood, allowing a rapid resupply of oxygen to the bloodstream.

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Henry’s law and Raoult’s law have several similarities and differences, and each has their restrictions. The knowledge of both of these laws by environmental engineers is invaluable not only academically, but also practically as well.

43.5 Vapor-Liquid Equilibrium in Non-Ideal Solutions [1] In the case where liquid solutions cannot be considered ideal, Raoult’s law will give highly inaccurate results. For these non-ideal liquid solutions, various alternatives are available. These are considered in most standard thermodynamic texts and will be treated to some extent below. One important case will be mentioned because it is frequently encountered in distillation, namely, where the liquid phase is not an ideal solution, but the pressure is low enough so that the vapor phase behaves as an ideal gas. In this case, the deviations from ideality are localized in the liquid and treatment is possible by quantitatively considering deviations from Raoult’s law. These deviations are taken into account by incorporating a correction factor, , into Raoult’s Law. The purpose of , defined as the activity coefficient, is to account for the departure of the liquid phase from ideal solution behavior. It is introduced into Raoult’s law equation (for Component A) as follows:

yAP

pA

A

x A PA

(43.8)

The activity coefficient is a function of the liquid phase composition and temperature. Phase-equilibria problems of the above type are often effectively reduced to evaluating . Vapor-liquid equilibrium calculations performed with Equation 43.8 are slightly more complex than those made with Raoult’s law. The key equation then becomes:

P

pi

( yi P )

i

xi pi

(43.9)

When applied to a two-component, A-B, system, Equation 43.9 becomes:

P

yAP

yB P

A

x A PA

B

x B PB

(43.10)

so that

yAP

pA

A

x A PA

(43.11)

yB P

pB

B

x B PB

(43.12)

Qualitative information on methods to determine the activity coefficient(s) follows.

Phase Equilibrium Relationships 437 Theoretical developments in the molecular thermodynamics of non-ideal liquid solution behavior are often based on the concept of local composition. Within a liquid solution, local compositions, different from the overall mixture composition, are presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. The concept was introduced by G. M. Wilson in 1964 with the publication of a model of solution behavior, since known as the Wilson equation [8]. The success of this equation in the correlation of vapor-liquid equilibrium data prompted the development of several alternative local-composition models. Perhaps the most notable of these is the Non-Random-Two-Liquid (NRTL) equation of Renon and Prausnitz [9]. The application of both approaches has received extensive treatment in the literature [1, 10].

43.6 Vapor-Solid and Liquid-Solid Equilibrium This final section is concerned with a discussion of vapor-solid and liquid-solid equilibria. The relation between the amount of substance adsorbed by an adsorbent (solid) and the equilibrium partial pressure or concentration of the substance at constant temperature is called an adsorption isotherm. The adsorption isotherm is the most important and by far the most often used of the various vapor-solid or liquid-solid equilibria data which can be measured. Note that in vapor-solid equilibria, if the substance being adsorbed (adsorbate) is above its critical temperature, it is technically considered a “gas.” Under such conditions, the term “gas-solid” equilibrium is more appropriate. Most available data on adsorption systems are determined at equilibrium conditions. Adsorption equilibrium is the set of conditions at which the number of molecules arriving on the surface of the adsorbent equals the number of molecules that are leaving. At equilibrium, an adsorbent bed is said to be “saturated with vapors” and can remove no more vapors from the exhaust stream. Equilibrium determines the maximum amount of vapor that may be adsorbed on a solid at a given set of operating conditions. Although a number of variables affect adsorption, the two most important ones in determining equilibrium for a given system are temperature and pressure. Three types of equilibrium graphs and/or data are used to describe adsorption systems: isotherm at constant temperature, isobar at constant pressure, and isostere at constant mass of vapor adsorbed. As noted above, the most common and useful adsorption equilibrium data come from the adsorption isotherm. The isotherm is a plot of the adsorbent capacity versus the partial pressure of the adsorbate at a constant temperature. Adsorbent capacity is usually given in weight fraction (or percent) expressed as grams of adsorbate per 100 g of adsorbent. Figure 43.5 shows a typical example of an adsorption isotherm for carbon tetrachloride on activated carbon. Graphs of this type are used to estimate the size of adsorption systems required for a given treatment application [1, 6].

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Capacity weight, % (1 lb CCl4/100 lb C)

100

32°F

77°F

F

140°

F 212° F

300°

10

3.0 2.0 1.5 1.0 0.0001

0.001

0.01

0.1

1.0

Partial pressure, psia

Figure 43.6 Adsorption isotherms for carbon tetrachloride on activated carbon.

Attempts have been made to develop generalized equations that can predict adsorption equilibrium from physical data. This is very difficult because adsorption isotherms take many shapes depending on the forces involved. Isotherms may be concave upward, concave downward, or “S”-shaped. To date, most of the theories agree with data only for specific adsorbate-systems and are valid over limited concentration ranges. Two additional adsorption equilibrium relationships are the aforementioned isostere and the isobar. The isostere is a plot of the ln p versus 1/T at a constant amount of vapor adsorbed. Adsorption isostere lines are usually straight for most adsorbate-adsorbent systems. The isostere is important in that the slope of the isostere (approximately) corresponds to the heat (enthalpy) of adsorption. The isobar is a plot of the mass of vapors adsorbed versus temperature at a constant partial pressure. However, in the design of most engineering systems, the adsorption isotherm is by far the most commonly used equilibrium relationship. Several models have been proposed to describe this equilibrium phenomena. Theodore provides a detailed review of these equations [1, 6]. The simplest system in liquid-solid equilibria is one in which the components are completely miscible in the liquid state and the solid phase consists of a pure component. These systems find application in water purification processes. The equilibrium equations and relationships presented in the vapor-solid equilibrium section generally apply to these systems as well. Details are available in the literature [11].

References 1. Theodore, L., Ricci, F., and VanVliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009.

Phase Equilibrium Relationships 439 2. Theodore, L., and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 3. Theodore, L., and Barden, J., Mass Transfer Operations, A Theodore Tutorial, Theodore Tutorials, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1996. 4. Atkins, P., and De Paoula, J., Atkins’ Physical Chemistry, 8th Edition, W. H. Freeman and Co., San Francisco, CA, 2006. 5. Henry, W., Experiments on the quantity of gases absorbed by water, at different temperatures, and under different pressures, Phil. Trans. R. Soc. London, 93, 29–274, 1803. 6. Theodore, L., Air Pollution Control Equipment Calculations, John Wiley & Sons, Hoboken, NJ, 2009. 7. Mohan, A., adapted from homework assignment submitted to L. Theodore, 2007. 8. Wilson, G., Vapor-liquid equilibrium. xi. a new expression for the excess free energy of mixing, J. Am. Chem. Soc., 86, 127–130, 1964. 9. Renon, H., and Prausnitz, J.M., Local compositions in thermodynamic excess functions for liquid mixtures, AIChE J., 14, 135–144, 1968. 10. Smith, J., Van Ness, H., and Abbott, M., Chemical Engineering Thermodynamics, 6th Edition, McGraw-Hill, New York City, NY, 2001. 11. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York, NY, 2008.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

44 Rate Principles

44.1 Introduction Rate principles may be applied at the molecular, microscopic, or macroscopic levels. These three approaches were previously discussed in a generic sense in Chapter 1. For mass transfer operations, the molecular rate process employs the diffusivity in Fick’s law [1], while the microscopic approach employs overall and individual mass transfer coefficients. The rate transfer process can be described by the product of three terms: 1. The area available for transfer 2. The driving force for transfer 3. The (reciprocal of the) resistance to the transfer process In effect, the rate process in equation form is:

rate

(area )(driving force) (resistance)

441

(44.1)

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Unit Operations in Environmental Engineering

For mass transfer (MT) applications, Equation 44.1 becomes:

(rate of MT)

(area availabe for MT)(driving force for MT ) (resistance to MT)

(44.2)

Both the molecular and microscopic approaches are reviewed in this chapter and every attempt has been made to relate the molecular and microscopic approaches to each other. Relating the microscopic treatment of mass transfer operations to the macroscopic treatment is not as simple, but information is available in the literature on this topic [2,3]. As noted in Chapter 1, macroscopic approaches often produce algebraic equations that have withstood the test of time. For example, the macroscopic equation for calculating the height of an absorber (see Chapter 48) is simply given by the product of two terms: HOG and NOG. HOG is related to the rate process while NOG is related to equilibrium conditions within the absorber.

44.2

Fick’s Law

Molecular diffusion results from the motion of molecules. At any instant, the individual molecules in a fluid are moving in random directions at speeds varying from low to high. The molecules move at random, frequently colliding with one another. Because of the frequent collisions, the molecular velocities are continually changing in both direction and magnitude. Diffusion is more rapid at higher temperatures due to greater molecular velocities at elevated temperatures. For gases, diffusion is slower at low pressures because the average distance between the molecules is greater and the collisions are less frequent. If a solution is not uniform in concentration, the solution is gradually brought into uniformity by diffusion; the molecules from an area of high concentration migrate to one of low concentration. The rate at which a solute travels depends on the concentration gradient which exists in the solution. This gradient applies across adjacent regions of high and low concentrations. However, a quantitative measure of rate is needed to describe what is occurring. The rate of diffusion can be described in terms of a molar flux term, with units of moles/(area-time), with the area being measured as that which the solute diffuses through. In a non-uniform solution containing only two components, both must diffuse if uniformity is to occur. This leads to the use of two fluxes to describe the motion of one of the components N, the flux relative to a fixed location, and J, the flux of a component relative to the average molar velocity of all components. The first of these is of importance in the design of equipment, but the second is more characteristic of the nature of the component. For example, the rate at which a fish swims upstream against the flowing current is analogous to N, while the velocity of the fish relative to the stream is more characteristic of the swimming ability of the fish and is analogous to J.

Rate Principles

443

The diffusivity, or diffusion coefficient, DAB, of component A in solution B, which is a measure of its diffusive mobility, is defined as the ratio of its flux, JA, to its concentration gradient and is given by:

JA

DAB

CA z

(44.3)

This is Fick’s first law [1] written for the z direction. The concentration gradient term represents the variation of the concentration, CA, in the z direction. The negative sign accounts for diffusion occurring from high to low concentrations. The diffusivity is a characteristic of the component and its environment (temperature, pressure, concentration, etc.). This equation is analogous to the flux equations [2,3] defined for momentum transfer (in terms of the previously defined viscosity) and for heat transfer (in terms of the thermal conductivity). The diffusivity is usually expressed with units of length2/time or moles/time-area. Imagine if two fluids are placed side by side in a container separated by a partition [2]. As pictured in Figure 44.1, Fluid A is on the left-hand side and Fluid B is on the right-hand side. When the partition is removed, the two fluids begin to diffuse (A towards B and B towards A). The diffusion process occurs because of a finite concentration driving force, which is the concentration gradient between the two containers. Diffusion stops when the concentration is uniform throughout the total mixture, i.e., there is no concentration gradient or driving force. However, imagine that there has been a net mass movement to the right. If the direction to the right is taken as positive, the flux of A (noted as NA), relative to a fixed position is positive, while the flux of B, NB, is negative. At steady state, the net flux is:

NA NB

N

(44.4)

The movement of A is made up of two parts, namely, that resulting from the bulk movement of A in N (i.e., xAN), and that resulting from the diffusion of A through B. This latter effect is defined as JA. The sum of these two effects is then:

NA

A

B

Before Figure 44.1 Diffusion process.

xAN

A

JA

B

During

(44.5)

A +B

A +B

After

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Table 44.1 Diffusion coefficients. Gas Air

a

Air

CO2

H2

N2

O2

TC0, °C

b

–

–

0.611

–

0.178

0

1.75

CO

–

0.137

0.651

0.192

0.185

0

1.75

CO2

0.138

–

–

–

0.139

0

2.00

H2

0.611

0.550

–

0.674

0.697a

0

1.75

N2

–

0.144

0.674

-

0.181

0

1.75

Temperature (TC0) for H2 in O2 is 20 °C.

Employing Fick’s first law leads to:

NA

CA z

x A (N A N B ) DAB

(44.6)

There are three different cases of steady-state molecular diffusion in gases [2, 3]. 1. Diffusion of A through non-diffusing B 2. Steady-state, equimolar, counter diffusion 3. Steady-state diffusion in multicomponent mixtures For addition analyses of these systems, the interested reader is referred to the original works of Treybal [2] and Geankopolis [4]. The diffusivity, or diffusion coefficient, D, was defined previously as the proportionality constant in the rate equation for mass transfer (Fick’s law) and it is a property of the system that is dependent on temperature, pressure, and the nature of the components. Reliable diffusion data are difficult to obtain, particularly over a wide range of temperatures. Table 44.1 lists diffusion coefficients for a few pairs of gases that have been investigated. The diffusion coefficient, DG (cm2/sec), at a temperature TC (°C) and pressure P (atm), may be determined from the data in Table 44.1 at state “0” using the following equation:

DG

DG 0

TC 273.2 TC 0 273.2

b

1 P

(44.7)

Additional values for diffusivities may be found in the literature [5]. When experimentally determined diffusivity data are not readily available, several estimation techniques are available [6]. No purely theoretical generalized correlation of liquid phase diffusivities has yet been found, but certain empirical equations are available. This probably reflects the inadequacy of kinetic theory when applied to liquids. It is therefore preferable

Rate Principles

445

Table 44.2 Liquid diffusivities at atmospheric pressure. Temperature, °C

Diffusivity, m2/sec 109

Water

5

1.24

NaCl

Water

18

1.26

Ethanol

Water

10

0.5

Acetic Acid

Water

12.5

0.82

Solute

Solvent

NH3

to use experimental data for liquid phase diffusivities. Table 44.2 provides a number of typical values for liquid phase diffusivities. Additional liquid diffusivities are available in the literature [2, 5]. Investigations on diffusion in liquids are not as extensive as those on diffusion in gases, since less experimental data are available. The rate of diffusion in liquids may take a long time to reach equilibrium unless agitated. This is, in part, explained by the fact that there is a much closer spacing of the molecules in a liquid, thereby retarding the movement of solute. Thus, molecular attractions become more important. Also note that diffusivity values in liquids are therefore significantly smaller than in gases. In the absence of an adequate theory for diffusion in liquids, it is usually assumed that Fick’s law is obeyed and that the equations developed in the subsection for diffusion in gases can also be applied to diffusion in liquids. Two situations can arise. 1. Steady-state diffusion of A through non-diffusing B 2. Steady-state, equimolar, counter diffusion When no experimental data are available, an estimation of the liquid diffusivity, DL, can be obtained using various empirical approaches [6].

44.3 Early Rate Transfer Theories When a liquid containing dissolved gases is brought into contact with an atmosphere of gas other than that with which it is in equilibrium, an exchange of gases takes place between the atmosphere and the solution. Two early theories were advanced to explain the mechanism of the exchange. These were called the penetration theory and the two-film theory. According to the penetration theory, eddies originating in the turbulent bulk of the liquid migrate to the gas-liquid interface where they are exposed briefly to the gas before being displaced by other eddies arriving at the interface. During their brief residence at the interface, the eddies absorb molecules from the gas. Upon return of the eddies to the bulk liquid, the molecules are distributed by turbulence.

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The rate of transfer is considered to be a function of the diffusivity, of a concentration gradient, and of a surface renewal factor [5, 7]. The two-film theory, although more questionable from a theoretical standpoint, at present is of greater practical value than is the penetration theory. The two-film theory is based on a physical model in which two fictitious stagnant films exist on either side of the gas-liquid interface, one liquid film and one gas film. These films are considered to be stagnant and furnish all the resistance to gas transfer. They are thought of as persisting regardless of how much turbulence is present in the bulk gas and liquid phases, the turbulence serving only to reduce the thickness of the stagnant film. A schematic of the imaginary transfer mechanism for the two-film theory is shown in Figure 44.2. Suppose, for example, gas is being transferred to a liquid which is unsaturated with respect to the gas being transferred. First, through a combined process of mixing and diffusion, the gas molecules are transported to the outer surface of the gas film. The molecules then diffuse across the stagnant gas film to the gasliquid interface where they dissolve into the liquid film. The dissolved gas then diffuses through this stagnant liquid film to the boundary between the liquid film and the bulk liquid phase, from where it is transported throughout the bulk liquid phase by turbulent mixing. The same mechanism in reverse prevails when gases are released from a supersaturated solution. Since transfer by diffusion is Bulk gas phase

Mixing

Gas film

Liquid film

Diffusion

Bulk liquid phase

DiffusionDiffusion Mixing

p

pi ci

c Figure 44.2 Schematic sketch of gas transfer mechanism.

Rate Principles

447

slow compared to that by turbulent mixing, the rate of transfer from one phase to another is considered to be controlled by the stagnant films.

44.4

The Operating Line

The NH3-air-H2O discussion presented in Chapter 43 is now revisited in an attempt to shed some light on rate considerations. Refer first to Figure 43.4 in the previous chapter. The initial state and final (equilibrium) state of the system pictured in Figure 44.3 are designated with a square point and a circular point, respectively. If V and L represent the moles of flowing air and water, respectively, and y and x the mole fraction of NH3 in the air and H2O, respectively, an NH3 mole balance written between the initial (0) and final (1) conditions gives:

Lx0 Vy0

Lx1 Vy1

(44.8)

(The reader should note that the terms V and G for gas flow are used interchangeably in this text as well as in the literature.) This assumes that if the mole fraction of NH3 is small in both phases (L and V are essentially constant) the above equation may be rearranged to give:

L(x1 x0 ) V ( y0

y1 );

L V

y0 y1 x1 x0

(44.9)

or

L V

y1 y0 x1 x0

0 y

a L V

b

c

1 x

Figure 44.3 Equilibrium-operating line plot, liquid/gas mole ratio = 2.

(44.10)

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Unit Operations in Environmental Engineering

0 a

1 y

c

b

x Figure 44.4 Equilibrium operating line plot, liquid/gas mole ratio = 0.5.

There are two important points to be made regarding Figure 44.3. 1. The dashed line may be thought of as an “operating line” since it describes the operating x, y values during the system’s transition from Point 0 to 1. 2. The vertical displacement of any point from the equilibrium line provides a direct measure of the driving force (and thus the rate) of the system’s attempt to achieve equilibrium. In effect, the driving force is maximum at Point 0 and zero at Point 1. Equation 44.9 plots as a straight line as shown with dashes in Figure 44.3. If ab/bc = 2.0, the mole ratio of liquid to gas is correspondingly 2.0. If the initial state is as shown in Figure 44.4, and ab/bc = 0.5, then the liquid to gas ratio is also 0.5. The important point to be made in the above analysis is that the rate of the NH3 transfer process is linearly related to the vertical displacement of the point representing the state of the system from the equilibrium point directly below and, as indicated above, the rate ultimately becomes zero when the operating point reaches the equilibrium point. While this type of macroscopic rate/equilibrium approach receives treatment in Part IV B – Applications, what follows in this Part of the book keys on the aforementioned molecular and microscopic considerations.

References 1. Fick, A., On diffusion, Pogg. Ann. (Ann. Der Phys. Und Chem.), 94, 59–86, 1855. 2. Treybal, R., Mass Transfer Operations, 1st Edition, McGraw-Hill, New York City, NY, 1955.

Rate Principles

449

3. Bennett, C., and Meyers, J., Momentum, Heat, and Mass Transfer, McGraw-Hill, New York City, NY, 1962. 4. Geankoplis, C., Transport Processes and Unit Operations, Allyn and Bacon, Boston, MA, 1978. 5. Higbie, R., The rate of absorption of a pure gas into a still liquid during short periods of exposure, Trans. Am. Inst. Chem. Engr., 31, 365–389, 1935. 6. Danckwerts, P.V., Significance of liquid-film coefficients in gas absorption, Ind. Engr. Chem., 43, 1460–1467, 1951. 7. Hanratty, T., J., Turbulent exchange of mass and momentum with a boundary, J. Am. Inst. Chem. Engr., 2, 359–362, 1956.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

45 Mass Transfer Coefficients

45.1 Introduction In the discussion of diffusion in the previous chapter, the emphasis was placed on the molecular transport in fluids that were stagnant or in laminar flow. However, in many cases, these diffusion processes are too slow, and more rapid diffusion or transport is required. Quite often, to speed up this diffusion, the fluid velocity is increased so that turbulent transport occurs. When a fluid flows past a surface under such conditions that the fluid is in turbulent flow, the actual velocity of small parcels or lumps of fluid cannot be described as simply as in laminar flow. Since fluid flows in smooth streamlines in laminar flow, its behavior can usually be described mathematically. However, there are no orderly streamlines or equations to describe fluid behavior under turbulent motion. Since there are large eddies or “chunks” of fluid which move rapidly in a seemingly random fashion. This eddy transfer, or turbulent diffusion, is very fast in comparison to the relatively slow process of molecular diffusion, where each solute molecule must move by random motion through the fluid. When a fluid flows past a surface under conditions such that turbulence generally prevails, a thin laminar-type sublayer film exists adjacent to the surface. The mass transfer in this region occurs by molecular diffusion since little or no eddies 451

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are present. Since this is a slow process, a large concentration gradient or decrease in concentration across this laminar film occurs. Adjacent to this is the transition or buffer region. Here, some eddy activity exists and the transfer occurs by the sum of both molecular and turbulent diffusion. In this region, there is a gradual and non-abrupt transition from the total transfer occurring by almost pure molecular diffusion at one end to mainly turbulence at the other end. The concentration decrease is much less in this region. Although most of the transfer is by turbulent or eddy diffusion, molecular diffusion still occurs, but it contributes little to the overall mass transfer. The concentration decrease is very small here since the rapid eddy movement evens out any gradients tending to exist. Many approaches to the turbulent (convective) mass transfer problem exit: film theory, combined film-surface-renewal theory, boundary layer theory, empirical approaches, etc. One former theory has somehow managed to survive the test of time, having been successful in interpreting the results of most two-phase mass transfer operations of industrial importance. The two film theory [1] (as applied by Whitman) postulates the existence of an imaginary stagnant film next to the interface between two fluid phases, whose resistance to mass transfer is equal to the total mass transfer resistance of the system. The difficulty with this theory is in the calculation of the effective film thickness. Other theories are briefly detailed below. 1. Surface-renewal theory assumes that a clump of fluid far from the interface moves to the interface without transferring mass, sits there stagnant, transferring mass by molecular diffusion for a time short enough such that little change in the concentration profile occurs in the clump, and then moves away from the interface without transferring mass en route and mixes with the bulk fluid instantly. This theory is somewhat more satisfactory in general than the two-film theory. 2. Boundary layer theory rests on the solution of a set of simplified differential equations which are approximations to a more nearly correct set of differential equations [2]. 3. Empirical approaches, which are merely data correlations, serve for specific cases, but give little information about extrapolation to more general cases. Further details regarding these approaches are available in the literature [2].

45.2 Individual Mass Transfer Coefficients In the mass transfer operation of gas absorption, two insoluble phases are brought into contact in order to permit the transfer of a solute from one phase to another (e.g., ammonia can be absorbed from an air-ammonia mixture into a water stream

Mass Transfer Coefficients

453

without air dissolving appreciably in the water). Concern for this application is with the simultaneous application of the diffusion mechanism for each phase to the combined system. It has already been shown that the rate of diffusion within each phase is dependent on the concentration gradient existing within it. At the same time, the concentration gradients of a two-phase system are indicative of the departure from equilibrium which exists between the phases. Since this departure from equilibrium provides the driving force for diffusion, the rates of diffusion in terms of the driving forces may now be studied. In the view of Whitman’s two-film theory, it is assumed that at the gas-liquid interface, the principal diffusion resistances occur in a thin film of gas and a thin film of liquid. The rates of diffusion in these two films will describe the mass transfer operation. The diffusion coefficient, D, in Fick’s law is inversely proportional to the concentration of the inert material, cB, in the liquid film through which material A must diffuse. Replacing D with (k/cB) yields:

NA

k (c AI c AL ) zc BM

(45.1)

where cBM is the log mean concentration difference of the inert material across the film, k is a proportionality constant, and N is the amount of material transferred per unit area per unit time, or mol/area-time. The practical application of Equation 45.1 is based on the assumption that z, the film thickness, is a constant that represents an effective average value throughout the length of the contact path. Also, cBM is considered to be constant since many mass transfer processes usually involve fairly dilute mixtures and solutions. Equation 45.1 can therefore be written as:

NA

kL (c AI c AL )

NA

kG ( pAG

(45.2)

or

pAI )

(45.3)

where kL is the liquid phase mass transfer coefficient based on concentration, kG is the gas phase mass transfer coefficient based on partial pressure, cAI is the interfacial (surface) concentration of Component A, cAL is the bulk liquid concentration of Component A, pAI is the interfacial partial pressure of component A, and pAGis the partial pressure of Component A. Equation 45.2 expresses the transfer of N molecules of Solute A through the liquid film under a concentration driving force, and Equation 45.3, the transfer of the same number of molecules of solute through the gas film under a partial pressure difference driving force.

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For certain simplified cases of molecular diffusion, an equation can be derived to determine precisely the rate at which mass is being transferred.

NA

DG , AB P RTzpB , M

( pA1

pA 2 ) .

(45.4)

The bracketed term above is an exact definition for the individual mass transfer coefficient corresponding to the steady-state situation of one component diffusing through a non-diffusing second component. In principle, it is then not necessary to calculate any other mass transfer coefficient for laminar flow since molecular diffusion prevails and exact equations are available. However, in general, obtaining such analytically-derived expressions are difficult; in most cases encountered in practice, it is impossible since turbulent mass transfer, which becomes quite complex, usually prevails. Only the more frequently encountered situations of diffusion will now be discussed, namely, equimolar counter diffusion and diffusion of one component through another non-diffusing component [3, 4].

45.2.1 Equimolar Counter Diffusion In absorption operations, the absorbing medium may evaporate into the gas being treated, resulting in the simultaneous diffusion of both gases in opposite directions. The diffusion of each gas is affected by the presence of the molecules of the other gas, and hindered if the gases are diffusing in opposite directions. When such a situation exists, the diffusion of equal moles per unit area per unit time occurs in opposite directions. This is referred to as equimolar counter diffusion. For the case of equimolar counter diffusion, the concentration profile shown in Figure 45.1 can be plotted graphically as in Figure 45.2. In this latter figure, point P represents the bulk phase compositions yAG and xAL, and point M represents the concentrations yAI, and xAI, at the interface. The equations for the flux of Component A, when A is diffusing from a gas to a liquid and there is equimolar counter diffusion of Component B from the liquid, are given by:

NA

k y ( y AG

y AI ) kx (x AI

x AL )

(45.5)

where kx is the liquid mass transfer coefficient based on mole fraction, k y is the gas mass transfer coefficient based on mole fraction, xAI and yAI are the bulk liquid and gas interfacial mole fractions, respectively, and xAL and yAG are the bulk liquid and gas mole fractions, respectively. Note that the prime with the individual mass transfer coefficient, i.e., k y , is a reminder that the transfer process involves equimolar counter diffusion.

Mass Transfer Coefficients Liquid phase solution of A in liquid L

455

Gas phase mixture of A in Gas G

yAG yAi

NA xAi Interface

xAL

Distance from interface

Mole fraction of solute in the gas

Figure 45.1 Concentration (mole fraction) profile of Solute A diffusing from one phase to another. Equilibrium curve

P

yAG

D

Slope = k’x /k’y Slope = m” yAl

M Slope = m’

yA*

C

xAl

xAL

xA*

Mole fraction of solute in the liquid Figure 45.2 Overall concentration differences for diffusion profile in Figure 45.1.

The values (yAG yAI) and (xAI xAL) are the differences in concentration, or driving forces, in each phase. For example, (yAG yAI) is the driving force in the vapor phases since yAG represents the average vapor concentration at a distance from the liquid vapor interface which has a composition yAI. Rearranging Equation 45.5 gives:

kx ky

( y AG y AI ) (x AI x AL )

(45.6)

Hence, the slope of PM in Figure 45.2 is (kx /k y ) . This means that if the two mass transfer coefficients are known, then the interfacial compositions can be determined by the line PM. While the bulk concentrations yAG and xAL can

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ordinarily be determined experimentally (from equilibrium relationships), the concentrations at the interface cannot, and Equation 45.6 can instead be used.

45.2.2 Diffusion of Component A Through Non-diffusing Component B In the case of Component A diffusing through non-diffusing Component B, the concentrations can also be plotted as in Figure 45.2 where P represents the bulk phase compositions and M, the interface compositions. The equations for A diffusing through a stagnant gas and through a stagnant liquid can be shown to be [2–4]:

NA

ky (1 y A )IM

( y AG

y AI )

kx (x AI (1 x A )IM

x AL )

(45.7)

where the subscript “IM” represents the bulk flow correction factor for non-dilute liquid and gas phases. The correction terms can be calculated from the following two equations:

(1 y A )IM

(1 y AI ) (1 y AG ) (1 y AI ) ln (1 y AG )

(45.8)

or,

(1 x A )IM

(1 x AL ) (1 x AI ) (1 x AL ) ln (1 x AI )

(45.9)

Equation 45.7 may be combined with Equations 45.8 and 45.9 to give:

kx /(1 x A )IM k y /(1 y A )IM

( y AG y AI ) (x AL x AI )

(45.10)

The slope of the line PM for the case of A diffusing through stagnant B is given by the left-hand side of Equation 45.10. The slope of Equation 45.10 differs from that of Equation 45.6 for equimolar counter diffusion by the bulk flow correction terms, (1 yA)IM and (1 xA)IM. When A is diffusing through non-diffusing B and the solutions are dilute, the bulk flow correction terms are approximately unity, and Equation 45.6 can be used instead of Equation 45.10. It is for this reason that Equation 45.6 is often employed even if the transfer process involves A diffusing through non-diffusing B. Also, note that the subscripts L and G, e.g., kL and kG,

Mass Transfer Coefficients

457

are employed when the rate is expressed in terms of the concentration and partial pressure, respectively. The use of Equation 45.10 to obtain the slope is, by necessity, trial-and-error because the left-hand side contains the interfacial concentrations yAI and xAI which are being sought. A first trial estimate can be obtained using Equation 45.6. With these estimates for yAI and xAI, a value of the left-hand side of Equation 45.10 can be computed and a new slope drawn to obtain new values of yAI and xAI (read off the equilibrium line). The second trial is repeated until the values of yAI and xAI do not change significantly with successive trials. Three trials will usually suffice.

45.3 Overall Mass Transfer Coefficients It is generally more convenient to utilize an overall coefficient for the gas and liquid phases rather than the individual coefficients since it is not possible to measure the partial pressure and concentration at the interface (pA1 or yA1, and cA1 or xA1, respectively). The preferred procedure is to express the overall coefficient in terms of the individual coefficients. For this approach, it is common to employ overall mass transfer coefficients based on the overall driving force between pAG(or yAG) and cAL (or xAL). The overall coefficients may be defined on the basis of the gas film, KG, or the liquid film KL, by the equations:

NA NA

K G ( pAG

pA* ) K y ( y AG

y A* )

K L (c A* c AL ) K x (x A* x AL )

(45.11) (45.12)

* * where pA is the partial pressure in equilibrium with cAL, and c A is the concentration in equilibrium with pAG. Consider again the situation shown in Figures 45.1 and 45.2 using the concentration (mole fraction) driving force. The equilibrium-distribution curve for the * system is unique at a fixed temperature and pressure. Then, y A (since it is in equilibrium with xAL) is as good a measure of xAL as xAL itself, and, moreover, it is on the same basis as yAG. In this situation, the entire two-phase mass transfer effect may be determined from expressions such as those given in Equations 45.11 and 45.12. In this manner, the ratio of the resistance of either phase to the total resistance is given by the ratio of the driving force through that phase to the “total” driving force across both films. In effect,

1/k y 1/K y

y AG y AI y AG y A*

(45.13)

with a similar equation on the liquid side. This analysis is further expanded in the next three subsections.

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45.3.1 Equimolar Counter Diffusion and/ or Diffusion in Dilute Solutions When equimolar counter diffusion is occurring, or when the solutions are quite dilute, the following equation applies [2-4]:

NA

k y ( y AG

y AI ) kx (x AI

x AL )

(45.14)

y A* )

(45.15)

From the geometry of Figure 45.2,

( y AG

y A* ) (y AG

y AI ) (y AI

or

( y AG

y A* ) ( y AG

y AI ) m (x AI

x AL )

(45.16)

where m is the slope of the chord CM. Substituting for the concentration differences, one obtains:

NA ky

NA ky

N Am kx

(45.17)

m kx

(45.18)

or

1 ky

1 ky

Equation 45.18 demonstrates the relationship between the individual mass transfer coefficients and the overall mass transfer coefficient. The left-hand side of Equation 45.18 can be looked upon as the total resistance based on the overall gas driving force, which is equal to the sum of the gas film resistance (1/k y ) and the liquid film resistance (m /kx ) . In a similar manner, from the geometry of Figure 45.2,

(x A* x AL ) (x A* x AI ) (x AI

x AL )

(45.19)

The slope between the points M and D is

m

y AG y AI x A* x AI

(45.20)

Mass Transfer Coefficients

459

Then,

y AG y AI m

x A* x AI

(45.21)

and it can be readily shown that:

1 kx

1 m ky

1 kx

(45.22)

As before, the left-hand side of Equation 45.22 is the total resistance and is equal to the sum of the individual resistances.

45.3.2 Gas Phase Resistance Controlling Assuming that the numerical values of kx and k y are roughly equal, the importance of the slope of the equilibrium curve chords can be readily demonstrated. However, if m is very small, so that the equilibrium curve in Figure 45.2 is nearly flat, then only a very small concentration of yA in the gas will give a relatively large value of xA in equilibrium with the liquid. This indicates that gas Solute A is very soluble in the liquid phase, and hence, the term m /kx in Equation 45.18 becomes very small or negligible. Then:

1 ky

1 ky

(45.23)

and the major resistance is said to be in the gas phases, or the “gas phase is controlling.” Also,

y AG

y A*

y AG

y AI

(45.24)

Under such circumstances, even fairly large percentage changes in kx will not significantly affect K y , and efforts to increase the rate of mass transfer would best be directed toward decreasing the gas-phase resistance, e.g., by increasing the gas phase turbulence or using equipment that specifically will have a high turbulence in the gas phase.

45.3.3 Liquid Phase Resistance Controlling In a similar manner, when m is very large or the Solute A is very insoluble in the liquid, with kx and k y again very roughly equal, the term 1/(m k y ) becomes very small and

1 kx

1 kx

(45.25)

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Table 45.1 Controlling films for various systems. Gas film 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Absorption of ammonia in water Absorption of ammonia in aqueous ammonia Stripping of ammonia from aqueous ammonia Absorption of water vapor in strong acids Absorption of sulfur trioxide in strong sulfuric acid Absorption of hydrogen chloride in water Absorption of hydrogen chloride in weak hydrochloric acid Absorption of 5 vol% ammonia in acids Absorption of sulfur dioxide in alkali solutions Absorption of sulfur dioxide in ammonia solutions Absorption of hydrogen sulfide in weak caustic solution Evaporation of liquids Condensation of liquids

Liquid film 1. Absorption of carbon dioxide in water 2. Absorption of oxygen in water 3. Absorption of hydrogen in water 4. Absorption of carbon dioxide in weak alkali solution 5. Absorption of chlorine in water Both gas and liquid film 1. Absorption of sulfur dioxide in water 2. Absorption of acetone in water 3. Absorption of nitrogen oxide in strong sulfuric acid solution

The major resistance to mass transfer is then in the liquid, the “liquid phase is controlling,” and

x A* x AL

x AI

x AL

(45.26)

In such cases, efforts to affect large changes in the rate of mass transfer are best directed toward conditions influencing the liquid coefficient, kx , i.e., increasing the turbulence in the liquid phase. For cases where kx and k y are not nearly equal, Figure 45.2 shows that it will be the relative size of the ratio kx /k y and of m (or m ) which will determine the location of the controlling mass transfer resistance. Table 45.1 lists some cases of specific films controlling a particular mass transfer operation [5,6].

45.4 Experimental Mass Transfer Coefficients In many instances, investigators have found that their data correlate well on the basis of some empirical relationship quite differently from the foregoing

Mass Transfer Coefficients

461

formalized treatment. These relationships are provided in the literature along with other importation data pertaining to the reported experimental studies [7]. Experimental data are often correlated in terms of dimensionless numbers such as the Schmidt number (Sc = / DAB) and the Reynolds’s number (Re = Lv / ). In the absence of experimental mass transfer data, many correlations are available which may be used to estimate the mass transfer coefficient for the system being studied. In practice, when choosing a correlation, one should make every effort to match as closely as possible the system conditions under which the correlation was formulated. Some of the correlations, primarily applicable in gas absorption mass transfer processes, are available in the literature [2, 4, 5]. In general, they apply to absorption columns loaded with various types of packing. These various types of absorption columns and the different packings utilized are discussed in Chapter 48.

References 1. Whitman, W., A preliminary experimental confirmation of the Two-Film Theory of gas absorption, it seems to explain satisfactorily the well-recognized differences of absorption rate for varying concentrations, Chem. Met. Eng., 29, 146, 1923. 2. Bird, R., Stewart, W., and Lightfoot, E., Transport Phenomena, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2004. 3. Treybal, R., Mass Transfer Operations, 1st Edition, McGraw-Hill, New York City, NY, 1955. 4. Geankoplis, C., Transport Processes and Unit Operations, Allyn and Bacon, Boston, MA, 1978. 5. Dwyer, O.E., and Dodge, B.F., Rate of absorption of ammonia by water in a packed tower, Ind. Eng. Chem., 33, 485, 1941. 6. Lara, M., Tower Packings and Packed Tower Design, 2nd Edition, U.S. Stoneware Co., Akron, OH, 1953. 7. F. Zenz, personal communication, L. Theodore, 1971.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

46 Classification of Mass Transfer Operations

46.1 Introduction Mass transfer operations are generally applicable to processes that essentially involve either componential or physical change, or both. A substantial number of these operations are concerned with changing the composition of solutions and mixtures through methods that do not involve chemical reactions; these are defined as componential separation processes. It is also often desirable to separate the original substance into its component parts by phase. Such separations may be entirely mechanical, such as the separation of a solid from a liquid during filtration or the classification of a granular solid into fractions of different particle size by screening. On the other hand, if the operation(s) involve the aforementioned changes in the composition of solutions, they are defined as componential separation operations, and it is with these process that the bulk of Part IV-B of the book is primarily concerned. It should also be noted that certain unit operations are grouped together because they involve a flow of mass between two or/more phases in contact with each other. Those operations of principle interest to environmental engineers are gas absorption and stripping (Chapter 48) in which molecules are transferred between gas and liquid phases, adsorption (Chapter 50) in which a flow of soluble materials 463

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takes place between solids and liquids, liquid-liquid extraction (Chapter 51), airwater contacts (Chapter 52, Humidification), and drying (Chapter 53), in which vapor is transported between water and air. Gas absorption and stripping have wide applications in environmental engineering treatment processes. In water treatment, these operations are employed to remove and replace carbon dioxide, and to remove hydrogen sulfide, methane, and various volatile organic compounds responsible for tastes and odors. In waste treatment, gas absorption is used primarily to provide oxygen for the activated sludge process. It is useful to classify mass transfer operations and to cite examples of each. These separations may be brought about by three principle mechanisms. There are nine different phase combinations (see Table 42.1 provided earlier). The individual mass transfer operations falling under these nine classifications are numerous and exceedingly diversified. However, since the underlying principles in all these operations and the general methods of applying them are often the same, these cases will be first grouped together for a study of the factors common to all. Later chapters in this Part of the book will provide specific details on these processes. These nine classifications are now considered in relation to the three mechanisms below [1–3]. 1. Contact of two immiscible phases, with mass transfer or diffusion through the interface between the phases. 2. Indirect contact of miscible phases separated by the permeable or semipermeable membrane, with diffusion through the membrane. 3. Direct contact of miscible phases.

46.2 Contact of Immiscible Phases The bulk of real-world industrial mass transfer operations of environmental interest reside in this category of immiscible phases. It is primarily these mass transfer processes that are addressed in the next part of this book. The various categories are briefly described below.

46.2.1

Gas-Gas

Since all gases are completely soluble in each other, this category of operation cannot often be practically realized. However, a brief discussion is presented in the next section.

46.2.2

Gas-Liquid

If all components of the system are present in appreciable amounts in both the gas and liquid phases, an operation known as distillation may be employed. In this instance, a gas phase can be formed from the liquid phase by the application of

Classification of Mass Transfer Operations 465 heat. For example, if a liquid solution of acetic acid and water is partially vaporized by heating, it is found ( as discussed in Chapter 42) that the newly created vapor phase and the residual liquid both contain acetic acid and water, but in proportions that are different for the two phases and different from those in the original solution. If the vapor and residual liquid are separated physically from one another and the vapor is condensed, two liquid solutions, one “richer” and the other “poorer” in acetic acid, are obtained. In this way, a certain degree of separation of the original components is accomplished. Conversely, should a vapor mixture of the two substances be partially condensed, the newly formed liquid phase and the residual vapor will differ in composition. An inter-diffusion of both components between the phases eventually establishes their final composition in both instances. It should be noted; however, that distillation is only feasible when the components to be separated have appreciably differing vapor pressures. All the components of the solutions discussed above may not be present in appreciable amounts in both the gas and liquid phases. If the liquid phase is a pure liquid containing one component and the gas contains two or more, the operation can be classified as humidification or dehumidification, depending upon the direction of the transfer. For example, the contact of dry air with liquid water results in evaporation of some of the water into the air (humidification of the air). Conversely, the contact of very moist air with pure liquid water may result in the condensation of part of the moisture in the air (dehumidification). Relatively little air dissolves in the water in both cases, and for most practical purposes it is assumed that only water vapor diffuses from one phase to the other. Both phases may also be solutions, each containing only one common component that is distributed between phases. For example, if a mixture of ammonia and air is brought into contact with liquid water (see Chapter 42), a large portion of the ammonia, but relatively little air, will dissolve in the liquid, and in this way the airammonia mixture may be separated. This operation is known as gas absorption. On the other hand, if air is brought into contact with an ammonia-water solution, some of the ammonia leaves the liquid and enters the gas phase. This operation is known as desorption, or stripping, a common water and wastewater treatment method. To complete this classification, consider the case where the gas phase contains one component and the liquid several, as in evaporation of a saltwater solution by boiling. Here the gas phase contains only water vapor, since the salt is essentially nonvolatile. Such operations do not depend on concentration gradients, but rather on the rate of heat transfer to the boiling solution, and are consequently not considered diffusional separations. Should the salt solution be separated by diffusion of the water into an air stream, however, the operation then becomes one of desorption, or stripping.

46.2.3 Gas-Solid It is convenient to classify mass transfer operations in the gas-solid category according to the number of components that appear in the two phases. If a solid solution

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were to be partially vaporized without the appearance of a liquid phase, the newly formed vapor phase and the residual solid would each contain all the original components but in different proportions. The operation is then fractional sublimation. As in distillation, the final compositions are established by inter-diffusion of the components between the phases. However, all components may be present in both phases. Although such an operation is theoretically possible, it is usually not practical because of the inconvenience of dealing with solid phases in this manner. If a solid that is moistened with a volatile liquid is exposed to a relatively dry gas, the liquid leaves the solid and diffuses into the gas, an operation generally known as drying, or sometimes as desorption. An example of this process is the drying of laundry by exposure to air. There are many industrial counterparts such as the drying of lumber or the removal of moisture from a wet filter cake by exposure to a dry gas. In these cases, the transfer is, of course, from the solid to the gas phase. If diffusion takes place in the opposite direction, the operation is known as gaseous adsorption. For example, if a mixture of water vapor and air is brought into contact with silica gel, the water vapor diffuses to the solid, which retains it, and the air is thus dried. In other instances, a gas mixture may contain several components, each of which is adsorbed on a solid but to different extents (fractional adsorption). For example, if a mixture of propane and propylene gases is brought into contact with a molecular sieve, the two hydrocarbons are both adsorbed, but to different degrees, thus leading to a separation of the gas mixture. In a case in which the gas phase is a pure vapor, such as in the sublimation of a volatile solid from a mixture with one that is non-volatile, the operation is dependent more on the rate of application of heat than on the concentration gradient. This process is essentially a non-diffusional separation. The same is true of the condensation of a vapor to a pure solid, where the rate of condensation depends on the rate of heat removal. Consider the following example. The “doughboys” in World War I employed gas masks to prevent problems with poisonous gas releases. The gas masks caused the ambient air being drawn in for breathing purposes to pass through a canister filled with activated carbon, which is a highly porous, granular or pelleted form of carbon. The activated carbon readily adsorbed the organic molecules that constituted the poisonous gases released but essentially did not adsorb oxygen and nitrogen, which passed through the canister freely. Thus, the gas mask can be thought of as a small-scale version of an industrial gas adsorption unit. During the war, these charcoal and other solid adsorbents were employed in chemical warfare. The use of these materials for adsorption of gases became widely introduced in industrial plants as a result of improvements made during the war in the manufacture of activated charcoal, silica gel and other highly active adsorbents.

46.2.4

Liquid-Liquid

Separations involving the contact of two insoluble liquid phases are known as liquid-extraction operations. A simple example is a familiar laboratory procedure:

Classification of Mass Transfer Operations 467 if an acetone-water solution is shaken in a separatory funnel with carbon tetrachloride and the liquids are allowed to settle, a large portion of the acetone will be found in the carbon-tetrachloride-rich phase and will thus have been separated from the water. A small amount of the water will also have been dissolved by the carbon tetrachloride, and a small amount of the latter will have entered the water layer, but these effects are relatively minor and can usually be neglected. In another example, a solution of acetic acid and acetone may be separated by adding the solution to an insoluble mixture of water and carbon tetrachloride. After shaking and settling, both acetone and acetic acid will be found in both liquid phases, but in different proportions. Such an operation is known as fractional extraction.

46.2.5 Liquid-Solid Fractional solidification of a liquid, where the solid and liquid phases are both of variable composition containing all components but in different proportions, is theoretically possible but is not ordinarily carried out because of practical difficulties in handling the solid phase and the very slow transfer rates in the solid. Cases involving the distribution of a substance between the solid and liquid phases are common, however. Dissolution of a component from a solid mixture by a liquid solvent is known as leaching (sometimes called solvent extraction). The leaching of gold from ore by cyanide solutions and the leaching of cottonseed oil from cottonseeds by hexane are two examples of this class of leaching process. Diffusion is, of course, from the solid to the liquid phase. If the concentration gradient driving diffusion is in the opposite direction, the operation is known as liquid adsorption. Thus, colored material that contaminates impure cane sugar solutions may be removed by contacting the liquid solutions with activated carbon, whereupon the colored substances are retained on the surface of the solid carbon and removed from solution. When the solid phase is a pure substance and the liquid solution is being separated, the operation is called crystallization; but as ordinarily carried out, crystallization rates are more dependent on heat-transfer rates than on solution concentrations. The reverse operations is dissolution. No known operation is included in the category involving a pure liquid phase.

46.2.6 Solid-Solid Because of the extraordinarily slow rates of diffusion within solid phases, there is at present no major practical engineering separation operation in this (solid-solid) category.

46.3 Miscible Phases Separated by a Membrane In operations involving miscible phases separated by a membrane, the membrane is necessary to prevent intermingling of the phases. It must be permeable differently

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to the components of the solutions if diffusional separations are to be possible. Three different phase combinations are briefly discussed below. Additional information can be found in Part II.

46.3.1

Gas-Gas

The operation in the gas-gas category is known as gaseous diffusion, gas permeation, or effusion. If a gas mixture whose components are of different molecular weights is brought into contact with a porous diaphragm, the various components of the gas will diffuse through the pores at different rates. This leads to different compositions on opposite sides of the diaphragm and, consequently, to separation of the gas mixture. In this manner, large-scale separation of the isotopes of uranium, in the form of uranium hexafluoride, can be carried out. Membrane production of pure oxygen from atmospheric air is another application of this mass transfer operation.

46.3.2

Liquid-Liquid

The separation of a crystalline substance from a colloid, by contact of the solution with a liquid solvent with an intervening membrane permeable only to the solvent and the dissolved crystalline substance, is known as dialysis. For example, aqueous beet sugar solutions containing undesired colloidal material are freed of the latter by contact with water with an intervening semipermeable membrane. Sugar and water diffuse through the membrane, but the larger colloidal particles cannot. Fractional dialysis for separating two crystalline substances in solution makes use of the difference in membrane permeability of the substances. If an electromotive force is applied across the membrane to assist in the diffusion of charged particles, the operation is electrodialysis. If a solution is separated from the pure solvent by a membrane that is permeable only to the solvent, the solvent diffuses into the solution; an operation that has come to be defined as osmosis. This is not a separation operation, of course, but if the flow of solvent is reversed by superimposing a pressure to oppose the osmotic pressure, the process is labeled reverse osmosis.

46.3.3 Solid-Solid The operation in the solid-solid category has found little, if any, practical application in the chemical process or environmental industries.

46.4 Direct Contact of Miscible Phases Operations involving direct contact of miscible phases are not generally considered practical industrially except in unusual circumstances because of the difficulty in maintaining concentration gradients without mixing of the fluid.

Classification of Mass Transfer Operations 469 Thermal diffusion involves the formation of a concentration difference within a single liquid or gaseous phase by the imposition of a temperature gradient upon the fluid, thus making possible a separation of the components of the solution. This process has been used, for example, in the separation of uranium isotopes in the aforementioned form of uranium hexafluoride. If a condensable vapor, such as steam, is allowed to diffuse through a gas mixture, it will preferentially carry one of the components along with it, thus making a separation by the operation known as sweep diffusion. If the two zones within the gas phase, where the concentrations are different, are separated by a screen containing relatively large-size openings, the operation is called atmolysis. If a gas mixture is subjected to very rapid centrifugation, the components will be separated because of the slightly different forces acting on the various molecules, owing to their different masses. The heavier molecules thus tend to accumulate at the periphery of the centrifuge.

46.5 Mass Transfer Operations Selection The selection decision is generally not an easy one. The environmental engineer or scientist faced with the problem of separating components of a solution must ordinarily choose among several of the aforementioned operations. Although the choice is usually limited because of the particular physical characteristics of the materials to be handled, the necessity for making this decision nevertheless almost always exists. Until the equipment and the fundamentals of the various operations are clearly understood, no basis for such a decision is available. Therefore, it would be prudent to establish the nature of the alternatives at this time. The individual may sometimes choose between using a diffusional operation of the sort discussed in the chapters that follow or a purely mechanical separation method discussed later in Part IV-B. For example, in the separation of a desired mineral from its ore, it may be possible to use either the diffusional operation of leaching with a solvent or the purely mechanical method of flotation. Vegetable oils may be separated from the seeds in which they occur by extraction or by leaching with a solvent. A vapor may be removed from a mixture with a noncondensable gas by the mechanical operation of compression or by the diffusional operations of gas absorption or adsorption. Sometimes, both mechanical and diffusional operations are used, especially where the former are incomplete, as in processes for recovering vegetable oils wherein extraction is followed by leaching. A more commonplace example is the wringing of water from wet laundry followed by air drying. It is characteristic that at the end of a solely mechanical operation, the substance removed is pure, whereas if removed by diffusional separation methods, it is associated with/transferred to another substance. Frequently, a choice may be made between a diffusional operation and a chemical reaction to bring about a separation. For example, water may be separated from other gases either by absorption in a liquid solvent or by chemical reaction

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with ferric oxide. Chemical methods ordinarily destroy the substance removed, whereas diffusional methods permit the eventual recovery of a substance in an unaltered form without great difficulty. There are also choices to be made within diffusional operations. For example, a gaseous mixture of oxygen and nitrogen may be separated by preferential adsorption of the oxygen on activated carbon, by absorption, by distillation, or by gaseous effusion. A liquid solution of acetic acid may be separated by distillation, by liquid extraction with a suitable solvent, by absorption with a suitable solvent, or by adsorption with a suitable adsorbent. The principal basis for choice in most cases is economics: the method that costs least (on an annual basis) is usually the one selected. Other factors, including legal and/or environmental constraints, also influence the decision. The simplest operation, although it may not be the least costly, is sometimes desired because it will be trouble-free. Sometimes, a method will be discarded because imperfect or incomplete knowledge of design methods or the unavailability of data for design will not permit results to be guaranteed. Favorable previous experience with a particular method is often given strong consideration. Cost, however, remains one of the prime factors in environmental engineering studies.

References 1. Dupont, R., Baxter, T., and Theodore, L., Environmental Management, CRC Press, Boca Raton, FL, 1998. 2. Dupont, R., Theodore, L., and Ganesan, K., Pollution Prevention: The Waste Management Approach for the 21st Century, CRC Press, Boca Raton, FL, 2000. 3. Treybal, R., Mass Transfer Operations, 3rd Edition, McGraw-Hill, New York City, NY, 1980.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

47 Characteristics of Mass Transfer Operations

47.1 Overview This last chapter of Part IV-A of this book is concerned with some of the characteristics of mass transfer operations. Although the topics of unsteady-state versus steady-state operation receives treatment, it is the general topic of flow patterns in mass transfer operations and the subject of stage-wise versus continuous operation that most completely characterizes a mass transfer operation. Superimposed on these three topics is the concept of concurrent versus countercurrent systems, and the concept of a theoretical or ideal stage. These concepts are briefly addressed below. Treybal [1] has discussed the two major characteristic methods of carrying out mass transfer operations. One consideration is the nature of the flow of the phases (whether in steady- or unsteady- state). However, the flow pattern, i.e., whether the flow direction of the phases is parallel, countercurrent, or cross current, must also be considered in the analysis [1]. As important is the method of contacting the phases, i.e., whether it is stage-wise or continuous contact [1]. These considerations are described in this chapter.

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Unit Operations in Environmental Engineering

47.2 Unsteady-State versus Steady-State Operation A steady-state process is one in which there is no change in conditions (pressure, temperature, composition, etc.) or rates of flow with time at any given point in the system. All other processes are unsteady-state. In a batch process, a given quantity of matter is placed in a container, and a change occurs by chemical and/or physical means. At the end of the process, the container (or adjacent containers to which material may have been transferred) holds the products. In a continuous process, materials are continuously fed to a piece of equipment or to several pieces in series, and products are continuously removed from one or more points. A continuous process may or may not be steady-state. A coal-fired power plant, for example, operates continuously. However, because of the wide variation in power demand between peak and slack periods, there is an equally wide variation in the rate at which the coal is fired. For this reason, power plant problems may require the use of average data over long periods of time. If one examines batch operations, all the phases are stationary from a point of view external to the system or on a “forward flow” basis, although internally there may be relative motion of the phases. A batch reactor, with no flow into or out of the reactor unit during the course of the reaction, is one such example [2]. Perhaps a more appropriate mass transfer example is the familiar laboratory extraction procedure involving contact of a solution with an immiscible solvent in a separatory funnel. This a batch operation since, once the liquids are in place, there is no further flow of liquid into or out of the vessel until the operation is completed. The solute diffuses from the solution into the solvent during the course of the extraction and the concentrations in both phases must therefore change with time. Provided the time of contact is sufficient, the maximum change in concentration which is possible occurs when equilibrium exists between the phases, although in practice the operation may be stopped before this occurs. The entire operation is then said to be equivalent to one stage (or step). Many mass transfer operations may be carried out in this general fashion. In a semi-batch operation, one phase is stationary while the other flows continuously into and/or out of the system. A semi-batch reactor, with either (but not both) flow into and out of the unit, is a simple example [2]. A mass transfer application is the case of a drier where a quantity of wet solid is placed in an air stream which flows continuously into and out of the drier, carrying away the vaporized moisture. The concentration of moisture in the solid and in the exiting air stream must, of course, change with time. Ultimately, if sufficient time is permitted, the stationary phase will come to equilibrium with the influent phase. It is a characteristic of steady-state operation that system variables at any position in the system remain constant with the passage of time. This requires continuous, invariable flow of all phases into and out of the system and a maintenance of the flow regime within the system. An example here would be a continuous steady-state tubular flow reactor [2]. It should be noted that most mass transfer

Characteristics of Mass Transfer Operations 473 operations are (or are assumed to be) steady-state and continuous, and unless otherwise noted, the reader should assume this to be the case.

47.3 Flow Pattern Flow options arise irrespective of whether the unit is operated in the steady-or unsteady-state mode. There are the three basic flow patterns that arise in practice: parallel, countercurrent, and cross current. Each of these flow mechanisms are discussed qualitatively below with details provided by Kelly [3].

47.3.1 Parallel Flow In parallel flow, the phases move through the unit in the same direction, entering and leaving together. The net effect, insofar as concentrations are concerned, is ultimately the same as that for a batch operation: if the phases are in contact long enough, the maximum concentration change will correspond to a state of equilibrium between the effluent phases. Consider the single-stage concurrent contracting process represented in Figure 47.1. A steady-state material balance can be used to monitor the separation taking place in the process. Only the transfer of a single component will be considered here. Treating the average compositions of the flows into and out of the system, one can write both an overall (Equation 47.1) and componential balance (Equation 47.2).

V0 L0 V0 y0 L0 x0

V1 L1

(47.1)

V1 y1 L1 x1

(47.2)

y1 , V1

y0 , V0 Single-

Stage

Process x0 , L0

Figure 47.1 Single stage concurrent contacting process.

x1 , L1

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y, mole or mass fraction

The terms V (gas) and L (liquid) refer to the molar flow rates entering and leaving the stage, and x and y are the mole fractions of the transferring component in V and L. Note that these equations may also be written on a mass basis and that the notation V and G may be used interchangeably. The separation taking place in a single stage process can be represented on an operating diagram which can be drawn in conjunction with phase equilibrium information available for the system. In this example, and for any case where the inlet streams to the process are mixed together, the contacting pattern is said to be co-current. An operating diagram for the steady-state mass transfer of a single component between these two phases is provided in Figure 47.2 where it is assumed V0 = V1 = V and L0 = L1 = L. This diagram shows the transfer of a single component from phase V to phase L. For transfer in the other direction, the equilibrium line remains the same, but the operating line would be located and appear below the equilibrium relationship from the case shown above. As the end of the operating line representing the exit stream from the process approaches the equilibrium curve, the single-stage process approaches what has come to be defined as an ideal or theoretical stage. If the operating line reaches the equilibrium curve, the single-stage device is a theoretical stage. It is also important to note that the displacement of an operating point on the operating line from the equilibrium line provides a direct measure of both the driving force and the rate of transfer. Thus, the rate of transfer is highest at x0, y0 and zero at x1, y1. The operating line connecting points x0, y0 and x1, y1 describes the actual operating state or condition during the transfer process from Point 0 to 1. The shape of the equilibrium curve on the operating diagram arises from the phase equilibria of the system which must reflect changes in temperature, pressure, ionic strength, etc., that occur in the single-stage process. The shape of the operating line reflects changes in the quantity of material in streams V and L as mass (often referred to as the solute) is transferred from one phase to another.

Slope =

y0

L V

Operating line

Equilibrium curve y1 x0

x1 x, mole or mass fraction

Figure 47.2 Operating diagram: single-stage device.

Characteristics of Mass Transfer Operations 475 To reduce the degree of curvature of the operating line, the use of mole ratios on a solute free basis may be employed instead of mole fractions. This conversion can be obtained by dividing the number of moles of a transferring species (the solute) by the number of moles of those components that are not transferred. Thus;

V0 y0

Vs y0 1 y0

Vs y0

(47.3)

where Vs is the gas non-transferring (solute-free) portion of V0 and

y0

moles of transferring component moles of nontransferring component

(47.4)

A similar expression can be written for the liquid stream, L. A component balance can be used to derive an operating line expression in terms of these solutefree mole or mass ratios:

Vs (Y0 Y1 ) Ls ( X1 X0 )

(47.5)

or

Vs (Y0 Y1 )

Ls ( X0

X1 )

(47.6)

An X-Y operating line based on Equation 47.6 would therefore be a straight line passing through points (X0, Y0) and (X1, Y1) with a slope of –Ls/Vs. However, for most environmental engineering applications, the solute-free mole fraction is not employed, particularly when mole fractions are low.

47.3.2 Countercurrent Flow In countercurrent flow, the contacted phases flow in opposite directions through the equipment. For example, the gas to be “washed” in gas absorption may flow upward through a tower while the “washing” liquid flows downward through the gas. The reverse transfer occurs in gas stripping, where the gas does the “washing.” For the maximum possible transfer, one of the effluent phases will come to equilibrium with the other influent phase, although in practice this condition is never met. If the exiting streams from a single-stage device are in equilibrium, the singlestage is defined as a theoretical stage. It is usually desirable to use several or multiple stages for a given separation. When multiple stages are used, some thought must be given to the pattern of contacting the two phases. Co-current contacting (see Figure 47.1), where the inlet stream from one phase is mixed with the inlet stream of another phase, can provide, at best, the equivalent of only one theoretical

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stage no matter how many actual stages are used. To maximize driving forces throughout a particular process, countercurrent contacting is often used. In this contacting mode, the inlet stream for one phase is contacted with the outlet stream of the other phase. A two-stage contacting system for both concurrent and countercurrent is pictured in Figure 47.3 [3]. The following componential mole balance on a solute free basis is applied to the overall countercurrent process pictured in Figure 47.3 b.

VsY0 Vs X3

VsY2 Ls X1

(47.7)

The operating line for countercurrent contacting is shown in Figure 47.4. Here, the diagram shows the transfer from phase V to phase L. If transfer were in the opposite direction, as in a stripping operation, the operating line would, Y1 , VS

Y0 , VS

Stage 1

X1 , LS

Y2 , VS

Stage 2

X3 , LS

X2 , LS

(a)

Y1 , VS

Y0 , VS

Stage 1

X1 , LS

Y2 , VS

Stage 2

X3 , LS

X2 , LS

(b)

Figure 47.3 Two-state contacting process. (a) Co-current operation. (b) Countercurrent operation.

Y, mole or mass ratio

Y0

Slope =

Operating line

LS VS

Y2 Equilibrium curve

X3

X, mole or mass ratio

Figure 47.4 Operating diagram: countercurrent contacting operation.

X1

Characteristics of Mass Transfer Operations 477 as indicated earlier, be located below the equilibrium line. In general, for a given number of stages, countercurrent contacting yields the highest mass transfer efficiency. The reason for this is that the average mass transfer driving force across the device is greater than would be the case with co-current contacting.

47.3.3 Cross Current Flow In cross current flow, the phases flow at right angles to each other, as in the case of the air and water in some atmospheric water-cooling towers. The maximum possible concentration change occurs if one of the effluent streams comes to equilibrium with the other influent stream. Cross current contacting, which is intermediate in mass transfer efficiency to co-current and countercurrent contacting, is shown in Figure 47.5 for a two-stage system [3]. Although the liquid, L, feed to both stages is the same, this need not be the case; the feed rate to each stage can be different as can the composition of that feed stream. To draw the operating line, the following solute free balances can be written:

Vs (Y0 Y1 ) Vs ( X1 X0 ); Stage 1

(47.8)

Vs (Y1 Y2 ) Vs ( X2

(47.9)

X0 ); Stage 2

The componential balance for each stage is essentially similar to that for the co-current process. However the stages are coupled. For this case, the operating diagram is presented in Figure 47.6 Cross current contacting is not used as commonly as co-current and countercurrent contacting but is employed in extraction, leaching, drying, and air pollution control applications. To illustrate the relative efficiencies of the various contacting modes of operation discussed above, Kelly [3] considered the following example. Suppose there are two discrete stages that can mix and separate phases, and that the stages can be connected co-currently, countercurrently, or cross currently for gas-liquid contacting. Find the contacting mode that will provide the maximum removal of a single transferring component from a gas phase, V, if a fixed amount of solvent, L, is to be used. Assume the inlet liquid flow, L = 20 units/time, inlet liquid X0 , LS

X0 , LS

Y0 , VS

Stage 1

Y1 , VS

X1, LS Figure 47.5 Two-stage crossflow contacting process.

Stage 2

X2 , LS

Y2 , VS

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Operating line Slope =

Y, mole or mass ratio

Y0

LS VS

Y1

Equilibrium curve

Y2

X2

X0

X1

X, mole or mass ratio Figure 47.6 Operating diagram: cross current contacting.

composition = 0.0 of the transferring component, inlet gas flow, V = 20 units/time, inlet gas composition = Y0 of the transferring (solute) component on a fractional basis, only one component is transferred between the gas and liquid, each stage can be considered to be an ideal stage, and equilibrium data in terms of mole ratios on a solute free basis is given by Y = mX, where X and Y are the mole ratios of the transferring component to the non-transferring component. Consider first a single concurrent stage, as pictured in Figure 47.3a. The ratio of flows of non-transferring components (the liquid to gas ratio on a solute-free basis) is:

Ls Vs

Ls V VY0

Ls V (1 Y0 )

(47.10)

The outlet mole ratios, X1 and Y1, can be found through graphical means on an operating diagram by constructing a line of slope - Ls/Vs from (X0, Y0) to the equilibrium line since this is a theoretical stage. Since the inlet mole ratio, X0 = 0.0, an analytical expression can also be developed to find X1 and Y1. By material balance:

VsY0 Ls X0

VsY1 Ls X1

(47.11)

But, X1 = Y1/m from the given equilibrium relationship. Noting that X0 = 0.0 and rearranging, one obtains:

Y1

Y0 1 (Ls /Vs m)

Y0 1 A

(47.12)

Characteristics of Mass Transfer Operations 479 where A is defined as the absorption factor, A = Ls/Vsm. Co-current contacting, even for a cascade of stages, yields at best one theoretical stage. Since the streams leaving the first stage are in equilibrium, the addition of a second co-current stage will not result in any further mass transfer. Thus, the addition of another co-current stage does nothing to enhance the transfer of solute as will be the case no matter how many stages are added. Consider countercurrent contacting with two stages, as pictured in Figure 47.3b. A solute balance around Stage 1 yields the following result:

Y0 (Ls /Vs ) X2 1 A

Y1

(47.13)

A balance around Stage 2 gives a similar result:

Y1 (Ls /Vs ) X3 1 A

Y2

(47.14)

Recognizing that X3 = 0.0 and that (Ls/Vs)X2 is the same as AY2, leads to the following expression:

Y2

Y0 1 A A2

(47.15)

For cross current contacting with two stages (see Figure 47.5), a material balance around State 1, with X0 = 0.0, leads to the following expression for Y1:

Y1

Y0 1 A/2

(47.16)

Y1 1 A/2

(47.17)

Similarly, for Stage 2, with X0 = 0.0

Y2

By combining the two equations, Y2 can be expressed as a function of Y0:

Y2

Y0 (1 A/2)2

(47.18)

One may now summarize the results for the outlet concentration from each process as:

Co-current, single stage: Y

Y0 1 A

(47.19)

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Crossflow, two stages: Y

Y0 (1 A/2)2

Countercurrent, two stages: Y

Y0 A2 1 A 4 Y0 1 A A2

(47.20)

(47.21)

The results clearly indicate that maximum separation (or the minimum Y) is achieved with countercurrent flow. A similar analysis of these flow modes is available for chemical reactors [2, 4, 5].

47.4 Stage Wise versus Continuous Operation Stage wise operation is considered first. If two insoluble phases are allowed to come into contact so that the various diffusing components of the mixture distribute themselves between the phases, and if the phases are then mechanically separated, the entire operation, as defined earlier, is said to constitute one stage. Thus, a stage is the unit in which contacting occurs and where the phases are separated; and, a single-stage process is one where this operation is naturally conducted once. As an example, the laboratory batch extraction in a separatory funnel, which was described earlier, may be cited. However, the operation may be carried on in a continuous as well as in a batch wise fashion. Should a series of stages be arranged so that the phases are frequently contacted and separated once in each stage, the entire multistage assemblage is called a cascade and the phases may move through the cascade in parallel, countercurrent, or crossflow mode. In order to establish a standard for the measurement of performance, the ideal, or theoretical, or equilibrium stage referred to earlier is defined as one where the effluent phases are in equilibrium, so that (any) longer time of contact will bring about no additional change of composition. Thus, at equilibrium, no further net change of composition of the phases is possible for a given set of operating conditions. (In actual industrial equipment, it is usually not practical to allow sufficient time, even with thorough mixing, to attain equilibrium). Therefore, an actual stage does not accomplish as large a change in composition as an equilibrium stage. For this reason, the fractional stage efficiency is defined as the ratio of a composition change in an actual stage to that in an equilibrium stage. Stage efficiencies for industrial equipment of interest to the environmental engineer range between a few percent to that approaching 100 percent [5,6]. The approach to equilibrium realized in any stage is then defined as the fractional stage efficiency. In the case of continuous-contact operation, the phases flow through the equipment in continuous intimate contact throughout the unit, without repeated physical separation and contacting. The nature of the method requires the operation to be either semi-batch or steady-state, and the resulting change in compositions

Characteristics of Mass Transfer Operations 481 may be equivalent to that given by a fraction of an ideal stage or by more than one stage. Note that equilibrium between two phases at any position in the equipment is generally never completely established. The essential difference between stage-wise and continuous-contact operation may then be summarized. In the case of stage-wise operation, the flow of matter between the phases is allowed to reduce the concentration difference. If allowed to contact for long enough, equilibrium can be established, after which no further transfer occurs. The rate of transfer and the time (of contact) then determine the stage efficiency realized in any particular application. On the other hand, in the case of the continuous-contact operation, the departure from equilibrium is deliberately maintained and the transfer between the phases may continue without interruption. Economics play a significant role in determining the most suitable mass transfer method.

References 1. Treybal, R., Mass Transfer Operations, 3rd Edition, McGraw-Hill, New York City, NY, 1980. 2. Theodore, L., Chemical Reactor Analysis and Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011. 3. Kelly, R., General processing calculations, in: Handbook of Separation Process Technology, R. Rousseau (Ed.), John Wiley & Sons, Hoboken, NJ, 1987. 4. Theodore, L., Chemical Reaction Kinetics, A Theodore Tutorial, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1995. 5. Reynolds, J., Jeris, J., and Theodore, L., Handbook of Chemical and Environmental Engineering Calculations, John Wiley & Sons, Hoboken, NJ, 2004. 6. Theodore, L., and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

48 Absorption and Stripping

48.1 Introduction The key unit operation of interest to the environmental engineer is gas absorption and stripping (desorption). Absorption operations are employed to remove/ recover such gases as hydrogen sulfide (H2S), methane (CH4), various volatile organic compounds (VOCs), etc. In addition, absorption is used in wastewater treatment to provide oxygen for aerobic biological processes such as activated sludge. The removal of one or more selected components from a gas mixture by absorption is therefore an important operation in environmental engineering. The process of absorption conventionally refers to the intimate contacting of a mixture of gases with a liquid so that part of one or more of the constituents of the gas will dissolve in the liquid. The contact usually takes place in some type of packed or plate column. This chapter therefore deals exclusively with packed or plate equipment. Only equipment and design procedures are emphasized, as a detailed presentation of the theory, including diffusional process, mass transfer coefficients, equilibrium (lines), operating lines, etc., has already been covered in Chapter 42 through 47. Since gas absorption is concerned with the removal of one or more species from a gas stream by treatment with a liquid, necessary information includes the 483

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solubility of these constituents in the absorbing liquid. In gas absorption operations, the equilibrium of interest is that between a solvent (nonvolatile absorbing liquid) and a solute gas. The solute is ordinarily removed from its mixture in a relatively large amount of a carrier gas that does not dissolve in the absorbing liquid. Temperature, pressure, and the concentration of solute in one phase are independently variable. The graphical equilibrium relationship of importance, again, is a plot of x, the mole fraction of solute in the liquid, against y (or y*), the mole fraction in the vapor in equilibrium with x. Thus, for cases which follow Henry’s law (see Chapter 43), Henry’s law constant, m, can be defined by:

y*

mx

(48.1)

or the equivalent y = mx*. The engineering design of gas absorption equipment must be based on a sound application of the principles of diffusion, equilibrium, and mass transfer. The main requirement in equipment design is to bring the gas into intimate contact with the liquid, i.e., to provide a large interfacial area and a high intensity of interface renewal, and to minimize resistance and maximize the driving force. This contacting of the phases can be achieved in many different types of equipment, the most important of which are either packed or plate columns. The final choice often rests with the various criteria that may have to be met. For example, if the pressure drop through the column is large enough such that horsepower costs become significant, a packed column may be preferable to a plate-type column because of the lower pressure drop. Again, primary emphasis in this Chapter is placed on packed and plate columns. In most processes involving the absorption of gaseous constituents from a gas stream, the gas stream is the process fluid; hence, its inlet conditions (flow rate, composition, and temperature) are usually known. The temperature and composition of the inlet liquid and the composition of the outlet gas are also usually specified. The main objectives, then, in the design of an absorption column, are the determination of the solvent (liquid) flow rate and the calculation of the principal dimensions of the equipment (column diameter and height) [1–3]. These three topics are reviewed sequentially later in Chapter 54. Although detailed absorber calculations will be provided, a brief introduction follows. The usual operating data to be provided or estimated are the flow rates, terminal concentrations, and terminal temperatures of the phases. The flow rates and concentrations fix the operating line, while the terminal temperatures provide an indication as to what extent the operation can be considered isothermal (i.e., whether the equilibrium line needs to be corrected for changes in liquid temperature). The operating line is obtained by a mass balance, and the outlet liquid temperature is evaluated from an energy balance on the column. In applications where relatively small quantities of gaseous constituents are being absorbed, temperature effects are usually negligible, and isothermal conditions prevail.

Absorption and Stripping 485 In gas absorption operations, the choice of a particular solvent is also important. Frequently, water is used as it is inexpensive and plentiful, but the following properties must also be considered. 1. Gas solubility – a high gas solubility is desired since this increases the absorption rate and minimizes the quantity of solvent necessary; generally, a solvent of a chemical nature similar to that of the solute to be absorbed will provide good solubility. 2. Volatility – a low solvent vapor pressure is desired since the gas leaving an absorption unit is ordinarily saturated with the solvent and much will therefore be lost if its vapor pressure is high. 3. Corrosiveness – a non-corrosive solvent is highly desirable. 4. Cost (particularly for solvents other than water) – lower cost solvents are preferable to higher cost solvents unless significantly increased absorption performance can be realized. 5. Viscosity – a low viscosity solvent is preferred for reasons of rapid absorption rates, improved flooding characteristics, lower pressure drops, and good heat transfer characteristics. 6. Chemical stability – a solvent should be chemically stable and, if possible, nonflammable 7. Toxicity – likewise, the solvent of choice should be benign environmentally and pose little to no human health risk [4]. 8. Low freezing point – if possible, a low freezing point is favored since any solidification of the solvent in the column could prove disastrous.

48.2 Description of Equipment The principal types of gas absorption equipment may be classified as follows: 1. Packed columns (continuous operation) 2. Plate columns (stage operation) 3. Miscellaneous Of the three categories, the packed column is most commonly used. Noe that plate columns will also receive treatment in the next chapter (Distillation).

48.2.1 Packed Columns Packed columns are usually vertical columns that have been filled with packing or material of large surface area. The liquid is distributed over and trickles down through the packed bed, thus exposing a large surface area to contact the gas. The

486

Unit Operations in Environmental Engineering Gas outlet

Liquid intlet

Entrainment separator (Demister) Liquid distributor Packing restrainer

Shell Random packing Access manway for packing removal Liquid re-distributor

Access manway for packing removal Packing support Gas inlet

Overflow Liquid outlet

Figure 48.1 Typical countercurrent packed column.

countercurrent packed column (Figure 48.1) is the most common encountered in environmental gaseous removal or recovery systems. The gas stream moves upward through the packed bed against an absorbing or reacting liquor (solvent-scrubbing solution), which is introduced at the top of the packing. This results in the highest possible efficiency. Since the solute concentration in the gas stream decreases as it rises through the column, there is fresh

Absorption and Stripping 487 solvent constantly available for contact. This provides the maximum average driving force for the mass transfer process throughout the packed bed. Mist eliminators also play an important role in absorbers. Mist eliminators are used to remove liquid droplets entrained in the gas stream. Ease of separation depends on the size of the droplets. Droplets formed from liquids are usually large, up to hundreds of microns in diameter, and are therefore effectively removed in mist eliminators. However, drops formed due to condensation or chemical reactions may be less than 1 μm in size and much harder to separate. Entrainment removal (mist separation) is possible by a number of methods including the following [1–3]. 1. 2. 3. 4. 5. 6.

Knitted wire or plastic mesh Swirl vanes or zigzag vanes Cyclones Gravity settling chambers Units in which the gas is forced to make a 180° turn Additional packing above the packed bed

Method 6 was employed by the graduate thesis advisor of one of the authors [4]. One of the simplest and most efficient means of mist separation is to use a porous blanket of knitted wire or plastic mesh. For most processes, the pressure drop across these mist eliminators range from 0.1 to 1.0 in of water, depending on vapor and liquid flowrates and the size of the eliminators. The efficiency of separation is generally high-usually 90% or better. The packing is the heart of this type of equipment. Its proper selection entails an understanding of the operational characteristics and the effect on performance of the significant physical differences among the various packing types. The main points to be considered in choosing the column packing include: 1. Durability and corrosion resistance (the packing should be chemically inert to the fluids being processed) 2. Free space per unit volume of packed space (this controls the liquor holdup in the column as well as the pressure drop across it; ordinarily, the fractional void volume, or fraction of free space, in the packed bed should be large) 3. Wetted surface area per unit volume of packed space (this is very important since it determines the interfacial surface between the liquid and gas; it is rarely equal to the actual geometric surface since the packing is usually not completely wetted by the fluid) 4. Resistance to the flow of gas (this affects the pressure drop across the column) 5. Packing stability and structural strength to permit easy handling and installation 6. Weight per unit volume of packed space 7. Cost per unit area of packed space

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Unit Operations in Environmental Engineering

Table 48.1 illustrates some of the various types and applications of the different column packings available. One additional distinction should be made, i.e., the difference between random and stacked (structured) packings. Random packings are those that are simply dumped into the column during installation and allowed to fall at random. It is the most common method of packing installation. During installation prior to pouring the packing into the column, the column may first be filled with water. This prevents breakage of the more fragile packing by reducing the velocity of the fall. The fall should be as gentle as possible since broken packing tightens the bed and increases the pressure drop. Stacked packing, on the other hand, is specifically laid out and stacked by hand, making it a tedious operation and rather costly; it is avoided where possible except for the initial layers on supports. Liquid distributed in this latter system usually flows straight down through the packing immediately adjacent to the point of contact. The aforementioned liquid distribution also plays an important role in the efficient operation of a packed column. A good packing from a process viewpoint can

Table 48.1 Four Typical packings and applications. Packing

Application Features

Raschig rings

Originally, the most popular type, usually cheaper per unit cost but sometimes less efficient than others; available in widest variety of materials to fit service: very sound structurally; usually packed by dumping wet or dry, with larger 4 – to 6-in sizes sometimes handstacked; wall thickness varies between manufacturers; available surface changes with wall thickness; produces considerable side thrust on tower; unfortunately it usually has more internal liquid channeling and directs more liquid to walls of the column.

Berl saddles

More efficient than Raschig rings in most applications, but more costly; packing nests together and creates “tight” spots in bed that promotes channeling but not as much as Raschig rings; do not produce much side thrust and have lower unit pressure drops with higher flooding points than Raschig rings; easier to break in than Raschig rings.

Intalox saddles

One of the most efficient packings, but more costly; very little tendency or ability to nest and block areas of bed; higher flooding limits and lower pressure drop than Raschig rings or Berl saddles; easier to break in than Raschig rings.

Pall rings

Lower pressure drop, less than half of Raschig rings; higher flooding limit; good liquid distribution; high capacity; considerable side thrust on column wall; available in metal, plastic, and ceramic.

Absorption and Stripping 489 be reduced in effectiveness by poor liquid distribution across the top of its upper surface. Poor distribution reduces the effective wetted packing area and promotes liquid channeling. The final selection of the mechanism of distributing the liquid across the packing depends on the size of the column, type of packing, tendency of the packing to divert liquid to column walls, and materials of construction for distribution. For stacked packing, the liquid usually has little tendency to cross distribute and thus moves down the column in the cross-sectional area that it enters. In the dumped condition, most liquids follow a conical distribution down the column with the apex of the cone at the liquid impingement point. For uniform liquid flow and reduced channeling of gas and liquid, the introduction of the liquid onto the packed bed should be as uniform as possible. Any impingement of the liquid on the wall of a column should be redistributed after a bed depth of approximately three column diameters for Raschig rings and five to ten column diameters for saddle packings. As a guide, Raschig rings usually a have maximum of 10 to 15 ft of packing per section, while saddle packing can use a maximum of 12 to 20 ft sections. As a general rule, however, the liquid should be redistributed every 10 ft of packing height. The redistribution brings the liquid off the wall and directs it toward the center of the column for redistribution and contact in the next lower section. Packed columns are characterized by a number of features to which their widespread popularity may be attributed. 1. Minimum structure – the packed column usually needs only a packing support and liquid distributor approximately every 10 ft along its height 2. Versatility – the packing material can be changed by simply discarding it and replacing it with a type providing better efficiency 3. Corrosive-fluids handling – ceramic packing is used and may be preferable to metal or plastic because of its corrosion resistance; when packing does deteriorate, it can be quickly and easily replaced, and it is also preferred when handling hot combustion gases 4. Low pressure drop – unless operated at very high liquid rates where the liquid becomes the continuous phase as the flowing films thicken and merge, the pressure drop per lineal foot of packed height is relatively low 5. Range of operation – although efficiency varies with gas and liquid feed rates, the range of operation is relatively broad 6. Low investment – when plastic packings are satisfactory or when the columns are less than about 3 or 4 feet in diameter, cost is relatively low

48.4 Plate Columns Plate columns (also commonly referred to as “tray columns”) are essentially vertical cylinders in which the liquid and gas are contacted in stepwise fashion (staged

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Unit Operations in Environmental Engineering

operation) on plates or “traps” (Figure 48.2). The liquid enters at the top and flows downward via gravity. On the way, it flows across each plate and through a downspout to the plate below. The gas passes upward through openings of one sort or another in the plate, then bubbles through the liquid to form a froth, disengages from the froth, and passes on to the next plate above. The overall effect is a multiple, countercurrent contact of gas and liquid. Each plate of the column is a stage since the fluids on the plate are brought into intimate contact, interface diffusion occurs, and the fluids are separated. The number of theoretical plates (or stages) is dependent on the difficulty of the separation to be carried out and is determined solely from material balances, mass transfer resistances and equilibrium considerations. The diameter of the column, on the other hand, depends on the quantities of liquid and gas flowing through the column per unit time. The actual number of plates required for a given separation is greater than the theoretical number because of plate inefficiencies [1, 2]. Gas out

Mist eliminator Shell

Liquid in

Tray Downspout Bubble-cap

Tray support ring

Vapor riser

Side stream withdrawal Intermediate feed

Froth

Tray stiffener Gas path through cap Gas in

Liquid out Figure 48.2 Typical bubble-cap plate column.

Absorption and Stripping 491 The particular plate selection and its design can materially affect the performance of a given absorption operation. Each plate should be designed so as to provide as efficient a contact between the vapor and liquid as possible, within reasonable economic limits. The two principal types of plates encountered include: 1. Bubble-cap 2. Sieve (perforated)

References 1. Theodore, L., and Barden, J., Mass Transfer Operations, A Theodore Tutorial, East Williston, NY, originally published by the USEPA/APTI, RTP, NC, 1995. 2. Theodore, L., and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 3. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2016. 4. Treybal, R.E., graduate thesis advisor to L. Theodore, New York University, Bronx, NY, 1955–1957.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

49 Distillation

49.1 Introduction This second mass transfer operation equipment chapter receives both preferential and unique treatment relative to the other chapters in Part IV-B. It is no secret that the subject matter of distillation is solely located in the domain of the chemical engineer. In many ways, it is a topic which separates (no pun intended) chemical engineering from other engineering and applied science disciplines, including environmental engineering. As such, the authors made a conscious decision to provide preferential treatment to distillation and to include developmental material which may be lacking in many environmental engineering texts. It was also decided to expand upon the concepts of binary distillation with separate chapters on the McCabe-Thiele design procedure in Part IV-C. Distillation may be defined as the separation of the components of a liquid feed mixture by a process involving partial vaporization through the application of heat. In general, the vapor evolved is recovered in liquid form by condensation. The more volatile (lighter) components of the liquid mixture are obtained in the vapor discharge at a higher concentration. The extent of the separation is governed by two important factors: the properties of the components involved, and by the physical arrangement of the unit used for distillation. 493

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Unit Operations in Environmental Engineering

In continuous distillation, a feed mixture is introduced to a column where vapor rising up the column is contacted with liquid flowing downward (which is provided by condensing the vapor at the top of the column). This process removes or absorbs the less volatile (heavier) components from the vapor, thus effectively enriching the vapor with the more volatile (lighter) components. This occurs in the section above the feed stream which is referred to as the enriching or rectification section of the column. The product (liquid or vapor) removed from the top of the column is rich in the more volatile components and is defined as the distillate. The section below the feed stream is referred to as the stripping section of the column. In this section, the liquid is stripped of the lighter components by the vapor produced in a reboiler that provides heat at the bottom of the column. The liquid that is removed from the bottom of the column is called the bottoms, which is richer in the heavier components. Distillation columns are used throughout the chemical and environmental engineering industries when mixtures (primarily in liquid form) must be separated. One such example is the petroleum industry. In such an application, crude oil is fed into a large distillation column and different fractions (oil mixtures of varying composition and volatility) are taken out at different heights in the column. Each fraction, such as jet fuel, home heating oil, gasoline, etc., is used by both industry and the consumer in a variety of ways. The separation achieved in a distillation column depends primarily on the relative volatilities of the components to be separated, the number of contacting trays (plates) or packing height, and the ratio of liquid and vapor flow rates. Distillation columns are rarely designed with packing in large scale production because of the liquid distribution problems that arise with large diameter units and the enormity of the height of many columns. However, where applicable, towers filled with packing are competitive in cost, and are particularly useful in cases where the pressure drop must be low and/or the liquid holdup must be small. Packed towers, an overview of which is available at the end of this chapter, are occasionally used for bench-scale or pilot plant work. In contrast, use of trayed towers extends to many chemical industries. The sections to be covered in this chapter are as follows: Flash Distillation, Batch Distillation, and Continuous Distillation with Reflux,

49.2 Flash Distillation The separation of a volatile component from a liquid can be achieved by means of flash distillation. This operation is referred to as a “flash” since the more volatile component of a saturated liquid mixture rapidly vaporizes upon entering a tank or drum which is at a lower pressure and/or higher temperature than the incoming feed. If the feed is a sub-cooled liquid, a pump and heater may be required to elevate the pressure and temperature, respectively, to achieve an effective flash (see Figure 49.1). As the feed enters the flash drum, it normally impinges against

Distillation

495

V {yi } TV PV Flash drum F {zi} TF PF

Feed pre-heater

T, P

F = feed molar flow rate V = vapor molar flow rate L = liquid molar flow rate

L {xi } TL PL

Figure 49.1 Flash distillation system.

an internal deflector plate which promotes liquid-vapor separation of the feed mixture. The composition of the feed stream, F, is given by the mole fractions belonging to the set {zi}. Similarly, the compositions of the vapor and liquid product streams are given by {yi} and {xi}, respectively. Alternatively, some processes require that a vapor stream be cooled, so as to partially condense the least volatile components in the stream. This process is referred to as a partial condensation, and the following development may be adapted to apply to such processes. As a result of the flash, the vapor phase will be mostly composed of the more volatile components. Typically, flash distillation is not an efficient means of separation when used only once. However, when several flash units are placed in series, much purer products may be achieved. Moreover, it can be a necessary and economical means of separating two or more components with discernible relative volatilities, as is often necessary in the petroleum industry. In a very real sense, an individual flash distillation unit may be seen as analogous to a single tray in a distillation column (as will be explained later in this chapter). Relative volatility can be described as the ratio of a particular compound to vaporize, relative to another compound. As is expected, a more volatile component is more likely to vaporize from a mixture (as compared to a compound of lesser volatility) when the mixture’s temperature is raised or the pressure is lowered. Relative volatility serves as a quantitative comparison of the volatility difference between two compounds of interest. When a liquid solution may be considered ideal, Raoult’s law applies, and hence the relative volatility ( ) of Component A in an A/B binary mixture may be defined in terms of each component’s vapor pressure ( pi )

AB

KA KB

pA pB

(49.1)

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Unit Operations in Environmental Engineering

where AB = relative volatility of A with respect to B; dimensionless; and Ki = yi/xi = phase equilibrium constant of component i, dimensionless. Note that the phase equilibrium constant, Ki is a function of several thermodynamic variables, namely the system’s temperature, pressure, and the compositions of each phase. For non-ideal conditions, all three of these variables must be taken into account, and K values must be determined via fugacity calculations. However, within the scope of this text, one may assume that K is solely a function of temperature and pressure [1]. The concept of relative volatility is of the utmost importance in an operation such as distillation, where two or more components are to be separated based on their differences in boiling point (which is directly correlated to differences in volatility). The subsequent development refers to the graphical solution of flash distilling a binary liquid mixture. As with many process operations, an overall material balance can be written to describe the system illustrated in Figure 49.1. The overall mole balance yields:

F

L V

(49.2)

Based on the above, a componential mole balance can be written for Component i, as shown in Equation 49.3:

Fz i

Lxi Vyi

(49.3)

where zi = mole fraction of component i in the feed stream, dimensionless; xi = mole fraction of component i in the liquid stream, dimensionless; and yi = mole fraction of component i in the vapor stream, dimensionless. Since only two components are present in the liquid feed, the subscript i can be omitted, assuming that all mole fractions refer to the lighter component. As a matter of convenience, Equation 49.3 may be rearranged in terms of the vapor composition to represent a straight line of the form y = mx + b as follows:

y

L F x z V V

(49.4)

where – L/V = slope of the operating line, m; and (F/V)z = the y-intercept of the operating line, b. (Note that (V/F) is the fraction of the feed stream which is vaporized). The above equation defines an operating line for this system, similar to that discussed in Chapter 48. Since the liquid and vapor streams are assumed to be in thermodynamic equilibrium (as is customarily practiced in the quick-sizing of mass transfer units), the flash is defined as an equilibrium stage. The equation therefore relates the liquid and vapor composition leaving the flash drum. One should also note that the vapor molar flow is normally represented by V, not G,

Distillation

497

in standard distillation notation; however, both are used interchangeably for the study of other mass transfer operations in certain chapters to follow.

49.3 Batch Distillation Although batch distillations are generally more costly than their continuous counterparts, there are certain applications in which batch distillation is the method of choice. Batch distillation is typically chosen when it is not possible to run a continuous process due to limiting process constraints, the need to distill other process streams, or because the low frequency use of distillation does not warrant a unit devoted solely to a specific product or operation. A relatively efficient separation of two or more components may be accomplished through batch distillation in a pot or tank. Although the purity of the distilled product varies throughout the course of batch distillation, it still has its use in industry. As shown in Figure 49.2, a feed is initially charged to a tank, and the vapor generated by boiling the liquid is withdrawn and enters a condenser. The condensed product is collected as distillate, D, with composition xD and the liquid remaining in the pot, W, has composition xw Total and componential material balances around a batch distillation unit are shown below:

F W D

(49.5)

Wxw

(49.6)

Fx F

Dx D

Note that W, xw, D, and xD all vary throughout the batch distillation process. A convenient method for mathematically representing a binary batch distillation process is known as the Rayleigh equation. This equation relates the Condenser D, y

Initially (t = 0) F, xF

W, xW

Steam coil

D, xD Batch still

Figure 49.2 Batch distillation diagram.

Condensate tank

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Unit Operations in Environmental Engineering

composition and amount of material remaining in the batch to initial feed charge, F, and composition, xF, as:

ln

xF

W F

xW

dx (y

*

(49.7)

x)

where y* = mole fraction of vapor in equilibrium with liquid of composition x. At the desired distillate composition, the distillation is stopped. At this time, the moles of residue remaining in the still is denoted Wfinal, with composition xW,final. As such, Equation 49.7 may be re-written as: xF

W final

F exp xW , final

dx (y

*

x)

(49.8)

This equation may be solved numerically by plotting 1/(y* x) vs x and integrating between the limits xW, final and xF to determine the area, A, under the curve. The above can therefore be written as:

W final

F exp[ A]

(49.9)

where A = area under the curve. In some situations, the degree of separation achieved in a single equilibrium stage (such as a flash distillation column) or in a batch still is often not large enough to obtain the desired distillate and/or bottoms purities. To improve the recovery of the desired product, a multi-staged distillation column may be employed. The analysis of such multi-stage columns is the focus of the remainder of this chapter.

49.4 Continuous Distillation with Reflux Note the design procedure employing the McCabee-Thiele method is detailed in Chapter 55, Part IV-C of the book.

49.4.1 Equipment and Operation For large-scale operations, continuous distillation is almost always more economical than batch, especially when a steady supply of feed is available. One of the disadvantages of both batch and flash distillation is the multiplicity of sequential distillations that are often necessary to achieve something approaching “complete” separation of the components. Moreover, it is possible to produce a very pure product by batch distillation. However, in order to obtain a high recovery, the liquid residue must be redistilled multiple times [2].

Distillation

499

A continuous distillation column is analogous to several small flash units in series and effectively circumvents the need for multiple units. Indeed, unless a mixture contains an azeotrope [3] a product stream of any desired purity may be theoretically obtained. In reality, a massive number of trays or an extremely high reflux ratio (to be defined shortly) would be required in the limit of a 100% pure product; these physical constraints on the system design limit the actual recovery possible. In a continuous distillation column (Figure 49.3), the mixture to be separated is fed into the column at some predetermined feed point between the top and bottom of the column. Vapor flows up the column and liquid flows countercurrently down the column. For the subsequent discussion regarding continuous distillation, a binary (2-component) feed is assumed. In the case of a binary feed, the more volatile (lower boiling) component is referred to as the light component, whereas the less volatile (higher boiling) component is referred to as the heavy component. As is standard practice in binary distillation, all mole fractions (i.e., xD, xB, xF) are representative of the light component only. The ascending vapor and descending liquid are brought into contact on either trays (plates) or packing. The plates/packing serves as a “widget” which allows more intimate contact between the liquid and vapor. As described earlier, the vapor at the top of the column enters a condenser. Part of the condensate is returned to Overhead vapor, V, ya Condenser

Liquid reflux L, xa

Rectification section Feed F, xF Stripping Section

L, xb

V, yb Reboiler

Bottoms B, xB

Figure 49.3 Schematic of a trayed, continuous distillation column.

Distillate D, xD

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the top of the column to provide reflux. The reflux descends counter to the rising vapors, and in the case of a total condenser, the remainder of the liquid condensate is withdrawn from the condenser as distillate product.

49.4.2 Reflux and Boil-up The molar flow ratio of reflux returned to distillate product collected is defined in this text as the reflux ratio, R. As the liquid/reflux stream descends, it is progressively stripped of the light constituent by the rising vapor. As a result of both heat and mass transfer effects, the vapor stream tends to vaporize the low-boiling constituent from the liquid and the liquid stream tends to condense the heavy constituent from the vapor. This liquid stream travels from the top tray to the bottom of the column, gradually increasing in the heavy component at each tray. The liquid is then collected by a pump at the bottom of the column, sent to a reboiler where it is partially vaporized (steam is usually employed as the heating medium) and returned to the column to provide an ascending vapor stream in the section of the column below the feed plate. The vapor generated by the reboiler is referred to as the boil-up. The other portion of the reboiler liquid is removed as the bottoms product. Analogous to the reflux ratio, the boil-up ratio, RB, is defined as the molar flow ratio of vapor generated by the reboiler to bottoms product.

49.4.3 Retrification and Stripping The vapor stream generated in the reboiler passes up through the portion of the column below the feed tray (the tray onto which the feed stream is added). The portion of the column below the feed tray (and including the feed tray itself) is known as the stripping section. Upon reaching the top of the stripping section, the rising vapor enters the portion of the column above the feed tray, referred to as the rectification section. As this rising vapor stream contacts the descending liquid stream on each plate, its concentration of the lighter component is increased. As previously indicated, the vapor exits from the top of the column and passes into the condenser. The coolant is normally water; however, other cooling fluids may be utilized when the situation warrants. (Additional information on condenser coolant will be discussed shortly). As seen in Figure 49.3, the rectification section is the portion of the column above the feed tray. In this section, there are n-trays; the first tray (Tray 1) is at the very top, onto which the reflux is introduced, and the nth tray (Tray n) is the tray above the feed tray. In reality, the liquid and vapor molar flow rates each vary from plate to plate because of inevitable differences in the molar enthalpy of vaporization between the light and heavy components. The vapor and liquid molar flow rates leaving the ith tray in the rectification section are, denoted Vi and Li with concentrations yi and xi, respectively (refer to Figure 49.4). Similarly, the stripping section is the portion of the column below the feed tray. The stripping section is said to have m-trays (including the feed tray). Thus, the

Distillation

Li – 2 xi – 2

501

Vi – 1 yi – 1 Tray i - 1

Li – 1 xi – 1

Vi yi

Tray i Li xi

Vi + 1 yi + 1

Tray i + 1 Li + 1 xi + 1

Vi + 2 yi + 2

Figure 49.4 Qualitative examination of rectification trays.

total number of trays in the column is N = n + m. The vapor and liquid molar flow rates in the stripping section are differentiated from those in the rectification section by using an overbar. For instance, the vapor and liquid molar flow rates leaving the jth tray in the stripping section are denoted Vj and L j with concentrations yj and xj, respectively. A column may consist of one or more feeds and may produce two or more product streams. Any product drawn off at various stages between the top and bottom are referred to as side streams. Multiple feeds and product streams do not alter the basic operation of a column, but they do complicate the analysis of the process to some extent. Not all distillation columns contain both a rectification and stripping section, depending on its specific use. For instance, if the process requirement is to strip a volatile component form a relatively nonvolatile solvent, the rectification section may be omitted; the unit is then referred to as a stripping column.

49.4.4

Pressure Drop and Flooding

There are numerous semi-empirical equations that are available for predicting the pressure drop across tray columns. As a preliminary estimate, one may assume the pressure drop is given by the height of liquid supported on the tray. Typically, this liquid height is in the 4 to 6 in range. Therefore, a reasonable approximation for pressure drop may be 4 to 6 in of H2O per tray, which is approximately 0.1 to

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0.2 psi per tray. The lower value applies to smaller diameter columns and the upper value applying to larger diameter units [4]. It is important that the vapor stream has the correct superficial velocity (linear average velocity, calculated as if the column conduit was empty) as it flows upwards in the column. Should the vapor flow too slowly, liquid may pass down through the tray perforations instead of over the weir, a condition known as weeping. However, if the vapor has a velocity that is too high, some liquid may be carried from the froth to the tray above by the rapidly flowing vapor. This condition is known as entrainment. Should the vapor velocity be increased further, entrainment may become excessive such that the liquid level in the downcomer will reach the plate above. At this point, liquid flow from the tray(s) in question becomes hindered, and the column’s entrainment flooding point has been reached. When calculating the allowable superficial velocity of a vapor, certain effects, such as the foaming tendency of a distillation mixture, are often taken into account, as foaming increases the likelihood of entrainment. Both excessive entrainment and weeping greatly influence tray efficiency and negatively impact overall column performance.

49.4.5 Condenser and Reboilers In some operations where the top product is required as a vapor, the liquid condensed is sufficient only to produce reflux to the column, and the condenser is referred to as a partial condenser. In a partial condenser, the reflux will be in equilibrium with the vapor leaving the condenser, and the condenser is considered to be a theoretical stage (an equilibrium stage) when estimating the column height. However, in actual practice it may be advisable not to rely on the action of a partial condenser as an extra stage, but instead to add extra plates to the column in order to affect the desired separation. In contrast, when the vapor is totally condensed, the liquid returned to the column will have the same composition as the distillate product and the condenser is not considered to be a theoretical stage. A partial reboiler is generally used at the bottom of the column in order to operate the column and produce vapor which flows upwards through the stripping section and into the rectification section. The liquid in the reboiler generally exits as bottoms product. Since both the liquid and vapor are considered to be in thermodynamic equilibrium, the partial reboiler is usually considered a theoretical stage.

References 1. Theodore, L., Ricci, F., and VanVliet, T., Thermodynamics for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2009. 2. Coates, J., and Pressburg, B., Analyze Material and Heat Balances for Continuous Distillation, Chem. Eng., 68, 131-136, 1961. 3. Treybal, R., Mass Transfer Operations, 3rd Edition, McGraw-Hill, New York City, NY, 1980. 4. Fair, T.R., Chapter 5 Distillation, in: R.W. Rousseau (Ed.), Handbook of Separation Process Technology, John Wiley & Sons, Hoboken, NJ, 1987.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

50 Adsorption

50.1 Introduction It is well established that the molecular forces at the surface of a liquid are in a state of imbalance or unsaturation. The same is true of the surface of a solid, where the molecules or ions on the surface may not have all their forces satisfied by union with other particles. As a result of this unsaturation, solid and liquid surfaces tend to satisfy their residual forces by attracting and retaining onto their surfaces gases or dissolved substances with which they come in contact. This phenomenon of the concentration of a substance on the surface of a solid (or liquid) is called adsorption. Thus, the substance attracted to a surface is said to be the adsorbed phase or adsorbate, while the substance to which it is attached is the adsorbent. Adsorption should be carefully distinguished from absorption, the later process being characterized by a substance not only being retained on a surface, but also passing through the surface to become distributed throughout the phase. Where doubt exists as to whether a process is true adsorption or absorption, the noncommittal term “sorption” is sometimes employed. Adsorption operations include dehumidification of air with desiccants, removal of obnoxious components of air and water systems with activated carbon, and the exchange of ions between water solutions and ion exchange materials. Adsorption 503

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is a surface phenomenon, and good adsorbents must have both a high surface area-to-volume ratio, and an “active” or “activated” surface, i.e., a surface relatively free of adsorbed materials. Many adsorbents employed in environmental engineering practice are highly porous and filled with fine capillaries. From an environmental perspective, adsorption is primarily employed in gas treatment/recovery, and in drinking water and groundwater treatment for the removal of dissolved organic compounds that may have adverse taste and odor characteristics, or that may represent health hazards to exposed individuals. The principle types of adsorbents include activated carbon, granular ferric hydroxide, and activated alumina, although carbon-based adsorbents are used most commonly in environmental applications because of their relatively low cost. The study of adsorption of various substances on solid surfaces has revealed that the forces operative in adsorption are not the same in all cases. Two types of adsorption are generally recognized: “physical” or van der Waals adsorption and “chemical” or chemisorption. Physical adsorption (physisorption) is the result of intermolecular forces of attraction between molecules of the solid and the substance adsorbed. When, for example, the intermolecular attractive forces between a solid and a gas (or vapor) are greater than those existing between molecules of the gas itself, the gas will condense upon the surface of the solid even though its pressure may be lower than the vapor pressure corresponding to the prevailing temperature. The adsorbed substance does not penetrate within the crystal lattice of the solid and does not dissolve in it but remains entirely upon the surface. Should the solid, however, be highly porous, containing many fine capillaries, the adsorbed substance will penetrate these interstices if it “wets” the solid. The partial pressure of the adsorbed substance at equilibrium equals that of the contacting gas phase, and by lowering the pressure of the gas phase, or by raising the temperature, the adsorbed gas on  the adsorbent is readily removed or desorbed in unchanged form. Physical adsorption is characterized by low heats of adsorption (approximately 40 Btu/ lbmol of adsorbate) and by the fact that the adsorption equilibrium is both reversible and established rapidly. This latter point allows one to simplify calculations in most real-world environmental engineering adsorption applications. Chemisorption, or activated adsorption, on the other hand, is the result of chemical interaction between the solid and the adsorbed substance. The strength of the chemical bond may vary considerably and identifiable chemical compounds in the usual sense may not form. Nevertheless, the attractive forces are generally much greater than that found in physical adsorption. Chemisorption is also accompanied by much higher enthalpy changes (ranging from 80 to as high as 400 Btu/lbmol) with the heat liberated being on the order of the enthalpy of a chemical reaction. The process is frequently irreversible, and, on desorption, the original substance will often have undergone a chemical change. Although it is probable that all solids adsorb gases (or vapors) to some extent, adsorption, as a rule, is not very pronounced unless an adsorbent possesses a large surface area for a given mass. For this reason, such adsorbents as silica gel, charcoals, and molecular sieves are particularly effective as adsorbing agents. These substances

Adsorption 505 have a very porous structure and, with their large exposed surface, can take up appreciable amounts of various gases and dissolved constituents. The extent of adsorption can be increased further by “activating” the adsorbents in various ways. For example, wood charcoal can be “activated” by heating between 350°C and 1000°C in a vacuum or in air, steam, and certain other gases to a point where the adsorption of carbon tetrachloride, (e.g., at 24°C) can be increased from 0.011 to 1.48 g/g of charcoal. The activation apparently involves a distilling out of hydrocarbon impurities and thereby leads to exposure of a larger free surface for possible adsorption. The amount of gas adsorbed by a solid depends on a host of factors, including the surface area of the adsorbent, the nature of the adsorbent and gas being adsorbed, the temperature, and the pressure of the gas. Since the adsorbent surface area cannot always be readily determined, common practice is to employ the adsorbent mass as a measure of the surface available and to express the amount of adsorption per unit mass of adsorbing agent used. A concept, which becomes especially important in determining adsorbent capacity, is that of “available” surface, i.e., surface area accessible to the adsorbate molecule. It is apparent from pore size distribution data that the major contribution to surface area is located in pores of molecular dimensions. It seems logical to assume that a molecule, because of steric effects, will not readily penetrate into a pore smaller than a certain minimum diameter. Hence, the concept that molecules are screened out. This minimum diameter is the so-called critical diameter and is a characteristic of the adsorbate and related to its molecular size. Thus, for any molecule, the effective surface area for adsorption can exist only in pores that the molecule can enter. Figure 50.1 attempts to illustrate this concept for the case in which two adsorbate molecules in a solvent (not shown) compete with each other for an adsorbent surface site. The reader should note that design and performance equations for adsorbers receive treatment in Chapter 56.

50.2 Adsorption Classification Four important adsorbents widely used industrially are briefly considered, namely, activated carbon, activated alumina, silica gel, and molecular sieves. The first three of these are amorphous adsorbents with a non-uniform internal structure. Molecular sieves, however, are crystalline and therefore have an internal structure of regularly spaced cavities with interconnecting pores of definite size. Details of the properties peculiar to the various materials are best obtained directly from the manufacturer. A description of these principle adsorbents is available in the literature [1–4].

50.3 Adsorption Equilibria The adsorption process involves three necessary steps. The fluid carrying the adsorbate must first come in contact with the adsorbent, at which time the adsorbate is preferentially, or selectively, adsorbed on the adsorbent. Next, the un-adsorbed fluid must be separated from the adsorbent-adsorbate, and, finally, the adsorbent

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Macropore

Micropore

Area available to both adsorbate and solvent

Area available only to solvent and smaller adsorbate

Area available only to solvent Figure 50.1 Concept of molecular screening in micropores (diameter range = 10 to 1,000 Å).

must be regenerated by removing the adsorbate, or discarding the used adsorbent and replacing it with fresh material. Regeneration is performed in a variety of ways, depending on the nature of the adsorbent-adsorbate complex. Gases or vapors are usually desorbed by either increasing the temperature (thermal cycle) or reducing the pressure (pressure cycle) of the adsorbent-adsorbate. Activated carbon used for water treatment applications is regenerated using thermal cycling, often with steam injection. The more popular thermal cycle is accomplished by passing hot gas through the adsorption bed in the opposite direction to the flow during the adsorption cycle. This ensures that the gas passing through the unit during the adsorption cycle always meets the most active adsorbent last and that the adsorbate concentration in the adsorbent at the outlet end of the unit is always maintained at a minimum. In the first step in the adsorption process where the molecules of the fluid come in contact with the adsorbent, an equilibrium is established between the adsorbed fluid and that remaining in the fluid phase. Although adsorption equilibrium and equilibrium relationships were reviewed in Chapter 42, material associated specifically with the adsorption process was not included. Because of the unique nature of adsorption equilibrium and the method of representation, this topic is included in the next section. Figures 50.2 through 50.4 show typical experimental equilibrium adsorption isotherm data. Consider Figure 50.2, where the concentration of adsorbed gas on the solid is plotted against the (equilibrium) partial pressure, p, of the adsorbate vapor or gas

Adsorption 507 Propane 40°C Ethylene 25°C

Propylene 25°C Propane 25°C

Propane 0°C

800

Equilibrium partial pressure, p, of adsorbate, mm Hg

700

600 P

500

400

300

200

100

0

0

0.02

0.04 0.06 0.08 lb adsorbate/lb silica gel

0.10

0.12

Figure 50.2 Vapor-solid equilibrium isotherms of some hydrocarbons on silica gel.

at constant temperature. At 40 °C, for example, pure propane vapor at a pressure of 550 mm Hg is in equilibrium with an adsorbate concentration at point P of 0.04 lb adsorbed propane/lb silica gel. Increasing the pressure of the propane will cause more propane to be adsorbed, while decreasing the pressure of the system at P will cause propane to be desorbed from the silica gel. As described earlier, adsorption is an exothermic process; hence, the concentration of adsorbed gas decreases with increased temperature at a given equilibrium pressure. This is evident from the behavior of the isotherm curves. For example, as indicated in Figure 50.2, decreasing the temperature of pure propane vapor to

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Methane 25°C

Acetone 160°C

Acetone 100°C

Equilibrium partial pressure, p, of adsorbate, mm Hg

250

Acetone 30°C

200

150

Benzene 100°C 100

50

0

0

0.1

0.2 0.3 lb adsorbate/lb carbon

0.4

Figure 50.3 Vapor-solid equilibrium isotherms of some hydrocarbons on activated carbon.

25°C will increase the equilibrium concentration to 0.055 lb adsorbed propane/lb silica gel. The process of a gas being brought into contact with a porous solid, and part of it being taken up by the solid, is always accompanied by the liberation of heat. The extent to which the process is exothermic depends on the type of sorption and the particular system. For physical adsorption, the amount of heat liberated is usually equal to the latent enthalpy of condensation of the adsorbate plus the heat of wetting of the solid by the adsorbate. The heat of wetting is usually only a small fraction of the heat of adsorption. On the other hand, in chemisorption, the heat evolved can approximate the enthalpy of chemical reaction.

50.3.1

Freundlich Equation

In isotherms of Type I, the amount of gas adsorbed per given quantity of adsorbent increases relatively rapidly with pressure and then much more slowly as the

Adsorption 509 24 –103°F

Capacity, lb CO2 adsorbate/100 lb 4A molecular sieves

22 20 -22°F

18

32°F 16

77°F

14 122°F

12

212°F

10 8 6 4

392°F

2 0 0.1

1

10

100

1000

Carbon dioxide pressure, mm Hg Figure 50.4 CO2 vapor-solid equilibrium isotherms on molecular sieves.

surface becomes covered with adsorbate molecules. To represent the variation of the amount of adsorption per unit area or unit mass with pressure, Freundlich proposed the following equation:

Y

kp1/n

(50.1)

where Y is the weight or volume of gas adsorbed per unit area or unit mass of adsorbent, p is the equilibrium partial pressure, and k and n are empirical constants dependent on the nature of solid and adsorbate, and on the temperature. Equation 50.1 may be rewritten as follows. Taking logarithms of both sides:

log(Y ) log(k)

1 log(p) n

(50.2)

If the log(Y) is now plotted against log(p), a straight line should result with the slope equal to (1/n) and an ordinate intercept equal to log(k). Although the requirements of the equation are met satisfactorily at lower pressures, the experimental points curve away from the straight line at higher pressures indicating that this equation does not have general applicability in reproducing adsorption of gases by solids.

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50.3.2 Langmuir Isotherms A much better equation for Type I isotherms was deduced by Langmuir from theoretical considerations. Langmuir postulated that gases, on being adsorbed by a solid surface, cannot form a layer more than a single molecule in depth. Further, he visualized the adsorption process as consisting of two opposing actions, a condensation of molecules from the gas phase onto the surface and an evaporation of molecules from the surface back into the body of the gas. When adsorption first begins, every molecule colliding with the surface may condense on it. However, as adsorption proceeds, only those molecules that occupy a part of the surface not already covered by adsorbed molecules may be expected to be adsorbed. The result is that the initial rate of condensation of molecules on a surface is high and then falls off as the surface area available for adsorption is decreased. On the other hand, a molecule adsorbed on a surface may, by thermal agitation, become detached from the surface and escape into the gas. The rate at which desorption will occur will depend, in turn, on the amount of surface covered by molecules and will increase as the surface becomes more fully saturated. These two rates, condensation (adsorption) and evaporation (desorption), will eventually become equal and when this happens, an adsorption equilibrium will be established. If is the fraction of the total surface covered by adsorbed molecules at any instant, then the fraction of bare surface available for adsorption is (1 – ). According to kinetic theory, since the rate at which molecules strike a unit area of a surface is proportional to the pressure of the gas, the rate of condensation of molecules should be determined both by the pressure and the fraction of bare surface, or k1(1 – )p, where k1 is a proportionality constant. If k2 is the rate at which molecules evaporate from a unit surface when the surface is fully covered, then for a fraction of a fully covered surface, the rate of evaporation will be k2 . For adsorption equilibrium, these rates must be equal. Therefore:

k1 (1

) p k2

(50.3)

or

k1 p k2 k1 p

bp 1 bp

(50.4)

where b = k1/k2. Now, the amount of gas adsorbed per unit area or per unit mass of adsorbent, Y, must obviously be proportional to the fraction of surface covered; hence:

Y

k

kbp 1 bp

ap 1 bp

(50.5)

Adsorption 511 where the constant a has been written for the product kb. Equation 50.5 is the Langmuir adsorption isotherm. The constants a and b are characteristic of the system under consideration and are evaluated from experimental data. Their magnitude also depends on temperature. The validity of the Langmuir adsorption equation at any one temperature can be verified experimentally most conveniently by first dividing both sides of Equation 50.5 by p and then taking reciprocals. The result is:

p Y

1 a

b p a

(50.6)

Since a and b are constants, a plot of (p/Y) versus p should yield a straight line with slope equal to (b/a) and an intercept equal to (1/a).

50.4 Description of Equipment Because of the high cost of maintenance of air recovery and purification systems for applications with high concentrations of organic vapors, scientists and engineers have been forced into researching and designing systems for solvent recovery. The result has been the development of three types of systems, differentiated by the manner in which the adsorbent bed is maintained or handled during both phases of the adsorption-regeneration cycle: (1) fixed or stationary bed, (2) moving bed, and (3) fluidized bed [3]. Figure 50.5 presents a flow diagram of a dual stationary-bed solvent recovery system with auxiliaries for collecting the vapor-air mixture from various point sources, then transporting it through the particulate filter and into the on-stream carbon adsorber – in this case, Bed 1. The effluent air, which is virtually free of vapors, is usually vented outdoors. The lower carbon adsorber (Bed 2) is regenerated during the service time of Bed 1. A steam generator or other source of steam is often required. The effluent steam-solvent mixture from the adsorber is directed though the condenser and the liquefied mixture then passes into the decanter and /or distillation column for separation of the solvent from the steam condensate [3]. Dual adsorber systems can also be operated with both beds on-stream simultaneously, especially when solvent concentrations are low. In this situation, regeneration is less frequent than a full work shift and may be accomplished during off-work hours. Operation in parallel almost doubles the air-handling capacity of the adsorption unit and may be an advantage in terms of operating cost. The concept of a moving-bed system is provided in the literature [1–4]. Separators (decanters) are installed following the condenser to separate the contaminant from the water. Note, however, that water is generally the heavier phase. Separators work on the principle of gravitational forces where the heavier material to be separated is removed from the bottom of the canister and the lighter material is removed through a line located at the top of the canister. Water

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Process discharges, hoods or direct connections to point-emission sources

Purified air exhaust Duct Vapor-air mixture

Bed 1 Carbon bed Particulate filter

On-stream

Bed 2

Blower

Carbon bed Regenerating Adsorber

Decanter, distillation column Steam generator

Condenser

Recovered solvent

Figure 50.5 Stationary-bed carbon system with auxiliaries for vapor collection and solvent separation from steam condensate.

separators are more effective with single solvent applications, and only when the solvent is immiscible in water.

50.5 Regeneration Adsorption processes in practice use various techniques to accomplish regeneration or desorption. The adsorption-desorption cycles are usually classified into four types, used separately or in combination, as follows [1]: 1. Thermal swing cycles using either direct heat transfer by contacting the bed with a hot fluid or indirect transfer through a surface, and reactivating the adsorbent by raising the temperature. A temperature between 300 °F and 600 °F is usually reached. The bed is then flushed with a dry purge gas or reduced in pressure and returned

Adsorption 513 to adsorption operations. High design loadings on the adsorbent can usually be obtained, but a cooling step is needed. 2. Pressure swing cycles use either a lower pressure or vacuum to desorb the bed. The cycle can be operated at nearly isothermal conditions with no heating or cooling steps. The advantages include fast cycling with reduced adsorber dimensions and adsorbent inventory, direct production of a high purity product, and the ability to utilize gas compression as the main source of energy. 3. Purge gas stripping cycles use an essentially non-adsorbent purge gas to desorb the bed by reducing the partial pressure of the adsorbed component. Such stripping is more efficient at higher operating temperatures and lower operating pressures. The use of a condensable purge gas has the advantages of reduced power requirements, which are gained by using a liquid pump instead of a blower, and an effluent stream that can be condensed to separate the desorbed material by simple distillation. 4. Displacement cycles use an adsorbable purge to displace the previously adsorbed material on the bed. The stronger the adsorption of the purge, the more completely the bed is desorbed using lower amounts of purge, but the more difficult it becomes subsequently to remove the adsorbed purge itself from the bed. When deciding whether to employ a regenerative system, several factors should be considered. The principal consideration is that of economics. It is important to establish if recovery of the adsorbate will be cost-effective or if regeneration of the adsorbent is the prime consideration. If solvent recovery is the main objective, the design should be based on past experimental data to establish the ratio of sorbent fluid to recoverable adsorbent at different working capacities. Most systems today employ steam as the regenerating medium, but some of the new systems use a hot inert gas such as nitrogen.

References 1. Theodore, L., and Buonicore, A., adapted from Control of Gaseous Emissions, Air Pollution Training Institute (APTI), EPA 450/2–81/005, U.S. Environmental Protection Agency, Environmental Research Center, Research Triangle Park, NC, 1981. 2. Theodore, L., and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 3. MSA Research Corp., Package Sorption Device System Study, EPA-R2–73-202, U.S. Environmental Protection Agency, Office of Research and Monitoring, Washington D.C., 1973. 4. VIC Manufacturing Co., Installation, Operation and Maintenance for VIC 1200F/S and 1200 Advanced Series Drycleaning Machines, VIC Manufacturing Company, Minneapolis, MN, 1992.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

51 Liquid-Liquid and Solid-Liquid Extraction

51.1 Introduction Extraction is a term that is used for any operation in which a constituent of a liquid or a solid is transferred to another liquid (the solvent). The term liquidliquid extraction describes the processes in which both phases in the mass transfer process are liquids. The term solid-liquid extraction is restricted to those situations in which a solid phase is present and includes those operations frequently referred to as leaching, lixivation, elutriation, and washing. Some of these terms are used interchangeably below [1–4]. Leaching is the reverse of liquid adsorption and the term applies to those operations in which a soluble material is transferred from a solid to a liquid. The leaching or elutriation of digested sludge solids to remove excessive concentrations of ions prior to chemical conditioning is one example of an environmental engineering application. As with several other unit operations, leaching can be carried out either in fixed-bed or dispersed-contact operations. Extraction involves the following two steps: contact of the solvent with the liquid or solid to be treated so as to transfer the soluble component (solute) to the solvent, and separation or washing of the resulting solution. The complete process may also include a separate recovery procedure involving the solute and solvent; 515

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this is normally accomplished by another operation such as evaporation, distillation, or stripping. Thus, the streams leaving the extraction system usually undergo a series of further operations before the finished product is obtained. Either one or both solutions may contain the desired material. In addition to the recovery of the desired product or products, recovery of the solvent for recycling is also often an important consideration [1-4]. In practice, the manner and equipment in which these operations are carried out is based on the difference in physical states between the solvent and the material to be extracted. Because solids are more difficult to handle and do not readily lend themselves to continuous processing, leaching of solids is commonly accomplished in a batch-wise fashion by agitating the crude mixture with the leaching agent and then separating the residual insoluble materials from the resultant solution. Liquid-liquid extraction may also be carried out in a batch operation. The ease of moving liquids, however, makes liquid extraction more amenable to continuous flow in various types of columns and/or stages. For design calculations or analysis of operations, one can apply either data on the equilibrium attained between the phases or the rate of mass transfer between phases described in earlier chapters. The usual design approach is often through the theoretical-stage concept, as discussed in the absorption and distillation chapters, and as discussed below. Two sections follow. The first is concerned with liquid-liquid extraction and the second with solid-liquid extraction. Equilibrium considerations, equipment and simple design procedures/predicative methods, although different for both processes, are included for both topics. The notation employed is that typically employed in industry.

51.2 Liquid-Liquid Extraction Liquid-liquid extraction is used for the removal and recovery of primarily organic solutes from aqueous and non-aqueous streams. Concentrations of solute in these streams range from either a few hundred parts per million in environmental engineering applications to several mole/mass percent in chemical process industrial applications. Most organic solutes may be removed by this process. Extraction has been specifically used in removal and recovery of phenols, oils, and acetic acid from aqueous streams, and in removing and recovering freons and chlorinated hydrocarbons from organic streams.

51.2.1 The Extraction Process Treybal [1] has described the liquid-liquid extraction process in the following manner. If an aqueous solution of acetic acid is agitated with a liquid such as ethyl acetate, some of the acid but relatively little water will enter the ester phase. Since the densities of the aqueous and ester layers are different at equilibrium, they will

Liquid-Liquid and Solid-Liquid Extraction 517 settle on cessation of agitation and may be decanted from each other. Since the ratio of acid to water in the ester layer is now different from that in the original solution and also different from that in the residual water solution, a certain degree of separation has occurred. This is an example of stage-wise contact and it may be carried out either in a batch or continuous fashion. The residual water may be repeatedly extracted with more ester to continue to reduce the acid content. As will be discussed shortly, one may arrange a countercurrent cascade of stages to accomplish the separation. Another possibility is to use some sort of countercurrent or cross current continuous-contact device where discrete stages are not involved. More complicated processes may use two solvents to separate the components of a feed. For example, a mixture of para- and ortho-nitrobenzoic acids may be separated by distributing them between the insoluble liquids chloroform and water. The chloroform preferentially dissolves the para isomer, and the water preferentially dissolves the ortho isomer. This is called double-solvent or fractional extraction [1]. The liquid-liquid extraction described above is a process for separating a solute from a solution based on the combination of the concentration and solubility driving force between two immiscible liquid phases. Thus, liquid extraction effectively involves the transfer of solute from one liquid phase into a second immiscible liquid phase. The simplest example involves the transfer of one component from a binary mixture into a second immiscible phase such as is the case for the extraction of an impurity from wastewater into an organic solvent. Liquid extraction is usually selected when distillation or stripping is impractical or too costly (e.g., the relative volatility for the two components falls between 1.0 and 1.2). Recovery of the solute and solvent from the product stream is often carried out by stripping or distillation. The recovered solute may be either treated, reused, resold, or disposed of. Capital investment in this type of process primarily depends on the particular feed stream to be processed. The solution whose components are to be separated is the feed to the process. The feed is composed of a diluent and solute. The liquid contacting the feed for purposes of extraction is referred to as the solvent. If the solvent consists primarily of one substance (aside from small amounts of residual feed material that may be present in a recycled or recovered solvent), it is called a single solvent. A solvent consisting of a solution of one or more substances chosen to provide special properties is a mixed solvent. The solvent-lean, residual feed solution, with one or more constituents removed by extraction, is referred to as the raffinate. The solvent-rich solution containing the extracted solute(s) is the extract. The degree of separation that arises because of the aforementioned solubility difference of the solute in the two phases may be obtained by providing multiplestage countercurrent contacting and subsequent separation of the phases, similar to a distillation operation. In distillation, large density differences between the gas-liquid phases are sufficient to permit adequate dispersion of one fluid in the other as each phase moves through the column. However, in liquid extraction,

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the density differences are significantly smaller and mechanical agitation of the liquids is frequently employed at each stage to increase contact and to increase the mass transfer rates. The minimum requirement of a liquid extraction unit is to provide intimate contact between two relatively immiscible liquids for the purposes of mass transfer of constituents from one liquid phase to the other, followed by the aforementioned physical separation of the two immiscible liquids. Any device or combination of devices that accomplishes this is defined in this text as a stage. If the effluent liquids are in equilibrium, so that no further change in concentration would have occurred within them after longer contact time, the stage is considered a theoretical or ideal stage. The approach to equilibrium attained is a measure of the stage efficiency. Thus, a theoretical or equilibrium stage provides a mechanism by which two immiscible phases intimately mix until equilibrium concentrations are reached and then physically separated into clear layers. A multi-stage cascade is a group of stages, usually arranged in a countercurrent flow between stages, for the purpose of enhancing the extent of separation [1–4].

51.2.2 Equipment There are two major categories of equipment for liquid extraction. The first is single-stage units, which provide one stage of contact in a single device or combination of devices. In such equipment, the liquids are mixed, extraction occurs, and the insoluble liquids are allowed to separate as a result of their density differences. Several separate stages may be used in an application. Second, there are multistage devices, where many stages may be incorporated into a single unit. This type is normally employed in practice. There are also two categories of operation: Batch or continuous. In addition to cocurrent flow (rarely employed) provided in Figure 51.1 for a three-stage system, cross current extraction is a series of stages in which the raffinate from one extraction stage is contacted with additional fresh solvent in a subsequent stage. Cross current extraction is usually not economically appealing for large commercial processes because of the high solvent usage and low solute concentration in the extract. It can be shown that multistage cocurrent operation only increases the residence time and therefore will not increase the separation above that obtained in a single stage, provided equilibrium is established in a single stage. cross current contact, in which fresh solvent is added at each stage, will increase the separation beyond that obtainable in a single stage. However, it can also be shown that the degree of separation enhancement is not as great as can be obtained by countercurrent operation with a given amount of solvent (see Chapter 47). The maximum separation that can be achieved between two solutes in a single equilibrium stage of the two phases is governed by equilibrium factors and the relative amounts of the two phases used, i.e., the phase ratio. A combination of the overall and component mass balances with the equilibrium data allows the compositions of the phases at equilibrium to be computed. If the separation achieved

Liquid-Liquid and Solid-Liquid Extraction 519 F

R1 1

R2 2

S

E3

S

S

R1 1

R2 2

E1

R3

R1

E3 R2

2 E2

Cross current

3

E2

1

Cocurrent

E2

S F

E1

3

E1

F

R3

R3 3

E3

Countercurrent S

Figure 51.1 Multistage extractors.

is inadequate, it can be increased by either changing the phase ratio or by the addition of more contacting stages.

51.2.3 Solvent Selection There are several principles that can be used as a guide when choosing a solvent for a liquid extraction process. These are typically conflicting and certainly no single substance would ordinarily possess every desirable characteristic for a process. Compromises are inevitable, and in what follows, an attempt is made to indicate the relative importance of the various factors that must be considered. Selectivity is the first and most important property ordinarily considered in deciding on the applicability of a solvent; selectivity refers to the ability of a solvent to extract one component of a solution in preference to another. The most desirable solvent from a solubility aspect would be one that would dissolve a maximum of one component and a minimum of the other. As in the case of vapor-liquid equilibrium, numerical data that quantify selectivity can be measured or determined. The numerical values of the selectivity, normally designated as , are available in literature [1]. Note that there are numerous possible selectivities for a three-component system. For example, it could be defined as [(xCB/xAB)/(xCA/xAA)] where xCB is the concentration of Solute C in the B rich solution, xAB is the concentration of the third component in B, xCA is the concentration of C in the A rich solution, while xAA is the concentration of A in the A rich solution. Like relative volatility, has been shown to be approximately constant for a few systems. However, in most cases, varies widely with concentrations.

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Unit Operations in Environmental Engineering

The importance of “good” selectivity for extraction processes parallels that of relative volatility for distillation. Practical processes require that exceeds unity and the more so the better. Selectivities close to unity will result in a large extraction unit, a large number of extraction stages, and in general, more capital and operation costs. As one might suppose, if = 1, the separation is impossible. Of all the desirable properties, favorable selectivity, recoverability, interfacial tension, density, and chemical reactivity are essential for the process to be carried out. The remaining properties, while not always important from a technical standpoint, must be given consideration in good engineering work and in the cost estimation of a process.

51.3 Design and Predictive Equations Fortunately, simple analytical procedures are available to perform many of the key extraction calculations [5]. The equilibrium constant, K, was defined as the ratio of the mole fraction of a solute in the gas to the mole fraction of the solute in the liquid phase. A distribution coefficient, k, can also be defined as the ratio of the weight fraction of solute in the extract phase, y, to the weight fraction of solute in the raffinate phase, x, i.e.,

y x

k

(51.1)

For shortcut methods, a distribution coefficient k (or m) is represented as the ratio of the weight ratio of solute to the extracting solvent in the extract phase, Y, to the weight ratio of solute to feed solvent in the raffinate phase, X. In effect, Y and X are weight fractions on a solute-free basis.

k

m

Y X

(51.2)

Consider first the cross current extraction process in Figure 51.2. This may be viewed as a laboratory unit since the extract and raffinate phases can be analyzed after each stage to generate equilibrium data as well as to achieve solute removal. If the distribution coefficient, as well as the ratio of extraction solvent to feed solvent (S /F ) are constant, and the fresh extraction solvent is pure, then the number of cross current stages (N) required to achieve a specified raffinate composition can be estimated from:

N

log(X F /X R ) kS log 1 F

(51.3)

Liquid-Liquid and Solid-Liquid Extraction 521 S1

F XF

S2

R1 1

R2

R3

XR

3

2

E1

S3

E2

E3

S = Mass flow rate of extracting solvent F = Mass flow rate of solvent feed Figure 51.2 Three-stage cross current extraction system.

Here, XF is the weight fraction of solute in feed, XR is the weight fraction of solute in raffinate, S is the mass flow rate of solute free extraction solvent to each stage, and F’ is the mass flow rate of the solute free feed solvent. Once again, XF and XR are weight fractions on a solute-free basis. The term k (S /F ) will later be defined as the extraction factor, , analogous to the absorption factor A discussed in Chapter 48. As noted earlier, most liquid-liquid extraction systems can be treated as having either: 1. Immiscible (mutually non-dissolving) solvents 2. Partially miscible solvents with a low solute concentration in the extract, or 3. Partially miscible solvents with a high solute concentration in the extract Only the first case is briefly addressed below. The reader is referred to the literature for further information on the second and third cases [1]. For the first case where the solvents are immiscible, the rate of solvent in the feed stream (F ) is the same as the rate of feed solvent in the raffinate stream (R ). Also, the rate of extraction solvent (S ) entering the unit is the same as the extraction solvent leaving the unit in the extract phase (F ). However, the total flow rates entering and leaving the unit will be different since the extraction solvent is removing solute from the feed. Thus, the ratio of extraction-solvent to feed-solvent flow rates (S /F ) is equivalent to (E /R ).

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Unit Operations in Environmental Engineering

51.4 Solid-Liquid Extraction (Leaching) [2] There are three key unit operations that involve the “mass transfer” between solids and liquids: 1. Crystallization 2. Solid-liquid phase separation 3. Solid-liquid extraction Crystallization receives treatment in Chapter 57, while the physical separation of solids and liquids appeared earlier in Chapters 17 and 19. This last section in this chapter addresses the important topic of solid-liquid extraction. Solid-liquid extraction involves the preferential removal of one or more components from a solid by contact with a liquid solvent. The soluble constituent may be solid or liquid, and it may be chemically or mechanically held in the pore structure of the insoluble solid material. The insoluble solid material is often particulate in nature, porous, cellular with selectively permeable cell walls, or surface-activated. In engineering practice, solid-liquid extraction is also referred to by several other names such as chemical extraction, washing extraction, diffusional extraction, lixiviation, percolation, infusion, and decantation-settling. The simplest example of a leaching process is in the preparation of a cup of tea. Water is the solvent used to extract or leach tannins and other substances from the tea leaf. A brief description/definition of terms adopted by some is provided below [4,6]. 1. Leaching – The contacting of a liquid and a solid, e.g., with the potential of imposing a chemical reaction upon one or more substances in the solid matrix so as to render them soluble. 2. Chemical extraction – This is similar to leaching but it applies to removing substances from solids other than ores. The recovery of gelatin from animal bones in the presence of alkali is an example of chemical extraction. 3. Washing extraction – The solid is crushed to break cell walls, permitting the valuable soluble product to be washed from the solid matrix. Sugar recovery from cane is a prime example of washing extraction. 4. Diffusional extraction – The soluble product diffuses across a denatured cell wall (no crushing involved) and is washed out of the solid. The recovery of beet sugar is an excellent example of diffusional extraction.

51.4.1

Process Variables

In the design of solid-liquid systems, the rate of extraction is affected by a number of independent variables. These are:

Liquid-Liquid and Solid-Liquid Extraction 523 1. 2. 3. 4. 5. 6. 7. 8.

Temperature Concentration of solvent Particle size Porosity and pore-size distribution Agitation Solvent selection Terminal stream composition and quantities Materials of construction

Details on each of the above design/system variables are provided by Theodore and Ricci [4].

51.4.2 Equipment and Operation The methods of operation of a leaching system can be specified by four characteristics: operating cycle (batch, continuous, or multi-batch intermittent), direction of streams (cocurrent, countercurrent, or hybrid flow), staging (single-stage, multistage, or differential-stage), and method of contacting. The following is a list of typical leaching systems: 1. 2. 3. 4. 5. 6. 7.

Horizontal-basket design Endless-belt percolator Kennedy extractor Dispersed solids leaching Batch stirred tanks Continuous dispersed solids leaching Screw conveyor extraction

As one might suppose, details on leaching equipment and operation is similar to that for liquid-liquid extraction systems. The simplest method of operation for a solid-liquid extraction or washing of a solid is to bring all the material to be treated and all the solvent to be used into intimate contact once and then to separate the resulting solution from the undissolved solids. This single-contact or single-stage batch operation is encountered in the laboratory and in small-scale operations but rarely in industrial operations because of the low recovery efficiency of soluble material and the relatively dilute solutions produced. If the total quantities of solvent to be used is divided into portions and the solid extracted successively with each portion of fresh solvent after draining the solids between each addition of solvent, the operation is called multiple-contact or multistage. Although recovery of the soluble constituents is improved by this method, it has the disadvantage that the solutions obtained are still relatively dilute. This procedure may be used in small-scale operations where the soluble constituent need not be recovered. If the solid and solvent are mixed continuously and the mixture fed continuously to a separating device, a continuous single-contact operation is obtained.

524

Unit Operations in Environmental Engineering Fresh solvent

Overflow solution 1

Solids underflow

Solids feed

Figure 51.3 Single-stage leaching unit. Fresh solvent

Fresh solvent

Solids feed

1

Fresh solvent

2

Overflow solution

Solids underflow

N

Overflow solution

Overflow solution

Figure 51.4 Multistage cross-flow leaching unit.

High recovery of solute with a highly concentrated product solution can be obtained only by using countercurrent operation with a number of stages. In countercurrent operation, the product solution is last in contact with fresh solid feed and the extracted solids are last in contact with fresh solvent. Details on the above three methods of operation are provided below [7]. 1. The single stage operation is shown in Figure 51.3 and represents both the complete operation of contacting the solids feed and fresh solvent and the subsequent mechanical (or equivalent) separation. 2. The second type is the multistage system shown in Figure 51.4 with the flow direction termed cross-flow. Fresh solvent and solid feeds are mixed and separated in the first stage. Underflow from the first stage is sent to the second stage where more fresh solvent is added. This is repeated in all the subsequent stages. 3. The third type of flow is the multistage countercurrent system shown in. Figure 51.5. The underflow and overflow streams flow countercurrent to each other [2]. Figure 51.6 shows a material balance for a continuous countercurrent process [2]. The stages are numbered in the direction of flow of the solid (e.g., sand). The light phase is the liquid that overflows from stage to stage in a direction opposite to that of the flow of the solid, dissolving solute as it moves from Stage N to Stage 1. The heavy phase is the solid flowing from Stage 1 to Stage N. Exhausted solids leave Stage N, while concentrated solution overflow leaves from Stage 1. For purposes of analysis, it is customary to assume that the solute-free solid is

Liquid-Liquid and Solid-Liquid Extraction 525 Fresh solvent

Overflow solution 1

2

N Solid feed

Solids underflow

Figure 51.5 Multistage countercurrent leaching unit.

V1

V3

Vn

Vn+1

VN+1

y1

y3

yn

yn+1

yN+1

1

2

n

n+1

N

Lo

L2

Ln–1

Ln

LN

xo

x2

xn–1

xn

xN

Figure 51.6 Material balance-countercurrent process.

insoluble in the solvent so that the flow rate of this solid is constant throughout the process unit.

51.4.3 Design and Predictive Equations Design and predictive equations for leaching operations can be more involved than those for liquid extraction. As before, the solute/solvent equilibrium and process throughput determine the cross-sectional area and the number of theoretical and/or actual stages required to achieve the desired separation. And, as with many of the previous unit operations discussed so far, the number of equilibrium stages and stage efficiencies can be determined under somewhat similar conditions for the countercurrent units discussed earlier. As in distillation (Chapter 49) and absorption (Chapter 48), the quantitative performance of a countercurrent system can be analyzed by utilizing an equilibrium line and the operating line concept, and, as before, the method to be employed depends on whether these lines are straight or curved. Provided sufficient solvent is present to dissolve all the solute in the entering solid and there is no adsorption of solvent by the solid, equilibrium is attained when the solid is completely “saturated” and the concentration of the solution (as formed) is uniform. Assuming these requirements are met, the concentration of the liquid retained by the solid leaving any stage is the same as that of the liquid overflow from the same stage. Therefore, an equilibrium relationship exists for this (theoretical) stage in question. The equation for the operating line is obtained by writing a material balance. From Figure 51.6:

VN

1

Lo

V1 LN (total solution, including solute)

(51.4)

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Unit Operations in Environmental Engineering

VN 1 y N

1

Lo xo

V1 y1 LN x N (componential solute only)

(51.5)

Eliminating VN+1, and solving for yN+1, gives:

yN

1 V1 Lo LN

1

1

xN

V1 y1 Lo xo LN V1 Lo

(51.6)

If the density and viscosity of the solution change considerably with solute concentration, the solids from the lower stages might retain more liquid than those in the higher stages. The slope of the operating line then varies from stage to stage. If, however, the mass of the solution retained by the solid is independent of concentration, LN is constant, and the operating line is straight. The two above mentioned conditions describe variable and constant overflow, respectively. It is usually assumed that the inerts are constant from stage to stage and insoluble in the solvent. Since no inerts are usually present in the extract (overflow) solution and the solution retained by the inerts is approximately constant, both the underflow, LN, and overflow, VN, are constant, and the equation for the operating line approaches a straight line. Since the equilibrium line is also straight, the number of stages can be shown to be (with reference to Figure 51.6):

log N

yN 1 xN y1 x1

y y log N 1 1 x N x1

(51.7)

The above equation should not be used for the entire extraction cascade if L0 differs from L1, L2 …, LN, (i.e., the underflows vary within the system). For this case, the compositions of all the streams entering and leaving the first stage should first be calculated before applying this equation to the remaining cascade [3,5].

References 1. Treybal, R., Mass Transfer Operations, 1st Edition, McGraw-Hill, New York City, NY, 1955. 2. Treybal, Liquid Extraction, McGraw-Hill, New York City, NY, 1951. 3. Theodore, L., and Barden, J., Mass Transfer Operation, A Theodore Tutorial, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1995. 4. Theodore, L., and Ricci, R., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 5. Reynolds, J., Jeris, J., and Theodore, L., Handbook of Chemical and Environmental Engineering Calculations, John Wiley & Sons, Hoboken, NJ, 2001. 6. Rickles, R.N., Liquid-solid extraction, Chem. Eng., 157–172, 1965. 7. Tagiaferri, R., report submitted to L. Theodore, Manhattan College, Bronx, NY, Dec. 1964.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

52 Humidification

52.1 Introduction In many unit operations, it is necessary to perform calculations involving the properties of mixtures of air and water vapor. Such calculations often require knowledge of: 1. The amount of water vapor carried by air under various conditions 2. The thermal properties of such mixtures 3. The changes in enthalpy content and moisture content as air containing some moisture is brought into contact with water or wet solids, and other similar processes Frequently, the environmental engineer is concerned with the change in the temperature and humidity of a quantity of air as the air undergoes prolonged contact with a free water surface under adiabatic conditions. For example, consider an adiabatic operation in which undersaturated air is brought into intimate contact with recirculated water. If the temperature of the water initially is the same as that of the incoming air, the former will drop as evaporation takes place. At the same

527

528

Unit Operations in Environmental Engineering

time, sensible heat will flow from the air to the water in response to the temperature difference developing across the stationary air film at the air-water interface. Ultimately there is established a steady-state wherein the water is at the adiabatic saturation temperature, Ts. If the contact between the water and air is sufficiently long, the temperature of the exit air will be close to Ts and its humidity close to saturation. Air with a certain temperature and humidity is often required in environmental applications. Here, adiabatic humidification is commonly carried out in a horizontal chamber, baffled at both ends and fitted with centrifugal spray nozzles. Water collected at the bottom is recirculated. As a rule, make-up water is used with a temperature such that the water temperature in the humidifier remains close to the adiabatic saturation temperature. The heating operations are accomplished by passing the air over heated coils or banks of finned tubes. Hot water can be cooled by contacting it with air in a counterflow arrangement in a cooling apparatus. It is to be noted that both heat and mass transfers are taking place between the phases. Sensible heat is transferred to the air as the result of the temperature difference existing between the interface and the bulk air. Water cooling operations can be carried out in a variety of contacting equipment. Packed towers and tray towers of the types ordinarily used in gas transfer can be employed for such operations. Of more popular use, however, are cooling towers constructed of tiers of horizontal wood slats. The tiers are arranged in such a manner that the water is intercepted repeatedly by the slats as it drops through the tower. Air circulation can be either natural or forced. In towers having natural circulation, the flow of air is either horizontal across the tower or vertical, induced by the rise of warm air from the top of the tower. This chapter reviews the properties of mixtures of air and water vapor, the mechanisms of the humidification process, and the equipment in which these processes are carried out. The remainder of this chapter focuses on five topics: Psychrometry, The Psychrometric Chart (including some key definitions), The Humidification Process, Equipment, and Describing Equations.

52.2

Psychrometry

Some key (and important) terms are introduced before proceeding to the general subject of psychrometry. In the discussion of the physical properties of mixtures of air and water vapor. Certain terms need to be defined. The definitions of several key terms follow [1, 2]. 1. The humidity, or absolute humidity, YA, is defined as the mass of water (designated with subscript A) carried by one unit of mass of bone dry air (BDA). The modal humidity, YA , can also be defined, but most of the literature on humidification and drying is in terms of YA. The two quantities are related by:

Humidification 529

YA

18 YA 29

(52.1)

The humidity at 1.0 atm is then related to the mole fraction, YA, and the partial pressure, pA, by:

YR

yA 18 29 1.0 y A

pA 18 29 1.0 pA

(52.2)

2. Air is said to be saturated with water vapor at a given temperature and pressure when it contains the maximum amount of water vapor possible at that condition. This is achieved when the air is in equilibrium with liquid water. The saturation humidity, YAs, is the value of YA corresponding to a partial pressure, pA, equal to the vapor pressure of water, pA , at the given temperature. The relative humidity, YR, is defined by:

YR

100

pA pA

(52.3)

3. The percent humidity, or percent saturation, PS, is given by:

PS 100

YA YAs

p A 1. 0 p A p A 1. 0 p A

(52.4)

where pA and pA must be expressed in atmospheres. 4. The humid heat capacity, CpH, is defined as the energy required to raise the temperature of 1 lb of the carrier gas (air, B) and its accompanying vapor (water, A) by 1°F. Thus, if CpB and CpH are the heat capacities of the carrier gas and vapor, respectively, CpH is given by:

C pH

C pB YAC pA

(52.5)

For the air-water system, the above equation becomes:

C pH

0.24 0.46YA ; [Btu/ F lb BDA]

(52.6)

Note that heat capacities on a mass basis are designated with a capital C in this field. The molal humid heat is based on moles of BDA.

530

Unit Operations in Environmental Engineering 5. The humid volume, VH, is defined as the volume, in ft3, of “moist” gas per unit mass of bone dry gas where:

VH

VR YAVA ; [ft 3 /lb BDA]

(52.7)

The terms YR and VA represent the volume, in ft3, of BDA/lb BDA and volume of A/lb A, respectively. 6. The enthalpy of moist air, Hy, at Ty is defined by Equation 52.8 in units consistent with the present discussion:

Hy

C pH (Ty To )

YA ; Btu/lb BDA

o

(52.8)

The term o is the heat of vaporization at some reference temperature, To, which is normally assumed to be either 0 °F or 32 °F. This effectively represents the sensible enthalpy of the air-water mixture relative to To plus the enthalpy of vaporization at To. 7. The adiabatic saturation temperature, Tas, of moist air is the temperature that air reaches when it is saturated adiabatically (i.e., at constant enthalpy). 8. The wet-bulb temperature, Twb, is the temperature attained by a small reservoir of water in contact with a large amount of air flowing past it. Normally, one can assume that Twb and Tas are the same for the water-air system at temperatures low enough to form a dilute gas-phase solution. For other liquid-vapor mixtures, the adiabatic saturation and wet-bulb temperatures are normally not equal. 9. The dew point temperature, Tdp, is the temperature at which a given sample of moist air becomes saturated as it is cooled at constant pressure and absolute humidity. Thus, the dew-point pressure is the (total) pressure to which moist air must be compressed at constant temperature and humidity to bring it to saturation. As an example, suppose the air in a room at 70°F is at 50% relative humidity. From the steam tables (see Appendix C), the vapor pressure of water at 70 °F is 0.3631 psi, which means that the air at 50% humidity holds water vapor with a partial pressure of (0.50) (0.3631) or 0.1812 psi. If the temperature is dropped at constant pressure to the point where 0.1812 psi equals the water vapor pressure (around 52 °F), the air becomes saturated with water and any further drop in temperature will cause condensation. The dew point of this air mixture is then 52 °F. Obviously, if the air were already saturated at 70 °F (i.e., 100% relative humidity), then the dew point would also be 70 °F.

Humidification 531

52.3 The Psychrometric Chart One vapor-liquid phase equilibrium example containing air-water raw data involves the psychrometric or humidity chart. A humidity chart is used to determine the properties of moist air and calculate the moisture content in air. The ordinate of the chart is the absolute humidity, Y (with the subscript A dropped for convenience), which was defined earlier as the mass of water vapor per mass of bone dry air. (Note that some charts base the ordinate on moles instead of mass). The previously defined concepts and definitions are normally presented graphically on the aforementioned psychrometric or humidity chart (Figure 52.1). There are also charts that apply to other single non-condensable gases and single condensable components at a fixed pressure (usually 1 atm). Curves showing the relative humidity (ratio of the partial pressure of the water vapor in the air to the vapor pressure of water at the system temperature) of humid air also appear on the charts. The curve for 100% relative humidity is also referred to as the saturation curve. The abscissa of the humidity chart is the air temperature, also known as the dry-bulb temperature, Tdb. The wet-bulb temperature, which has also already been defined, is another measure of the humidity. As described earlier, it is the temperature at which a thermometer stabilizes when it has a wet wick wrapped around the bulb. As water evaporates from the wick to the ambient air, the bulb is cooled; the rate of cooling depends on how humid the air is. No evaporation occurs if the air is saturated with water; hence, Twb and Tdb are

16

40

° 75 ° 70 ° 65

60 50° 55° ° 45°

25

60

satu

ratio

90

n lin

es

0.10

10

30

80

125°

120

s. me v volu rated erature u t a S temp e olum ific v re Spec peratu m e vs. t

0.14

0.12

70 60 50 40

hea mid

17

12

130°

Hu

Volume, ft3/lb dry air

18

13

batic

0.08

°

115°

110 °

0.06

105

80 °

85 °

80

90

10 0° 95°

°

0.04

°

0.02

100

120

140

160

180

200

220

Temperature, °F

Figure 52.1 Psychrometric chart for the air-water vapor system at a pressure of 1 atm.

0 240 250

Humidity, lb water vapor/lb dry air

mid ity Rela satu tive ratio n 100%

t vs

. hu

19

14

Adia

135 o

20

15

0.15

140 o

21

20

22

Humid heat, (Btu)/lb dry air) (°F) 0.22 0.24 0.26 0.28 0.30

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Unit Operations in Environmental Engineering

then the same. The lower the humidity, the greater the difference between these two temperatures. On the psychrometric chart, constant wet-bulb temperature lines are straight with negative slopes. The value of Twb corresponds to the value of the abscissa at the point of intersection of this line with the saturation curve. The humid volume, also defined earlier, is the volume of wet air per mass of BDA and is linearly related to the humidity. This quantity has been used as an alternate ordinate in Figure 52.1. A common experimental method for determining the humidity of air is to determine the wet-bulb and dry-bulb temperatures simultaneously. This measurement can be accomplished by rapidly passing a stream of air over two thermometers, the bulb of one which is dry. The bulb of the other is kept wet by means of a cloth sack either dipped in water or supplied with water. In a sling psychrometer, the two thermometers are fastened in a metal frame that may be whirled about a handle. The psychrometer is whirled for some seconds and the reading of the wetbulb thermometer is observed as quickly as possible. The operation is repeated until successive readings of the wet-bulb thermometer show that it has reached its minimum temperature. Thus, a psychrometer consists mainly of two thermometers for wet- and dry-bulb readings. The wet-bulb temperature is the equilibrium temperature obtained by the aforementioned action of an unsaturated vapor-gas mixture flowing past a wetted wick completely covering the bulb of the thermometer. At this equilibrium temperature, the sensible heat given to the water from the air is balanced by the loss of heat from the water by evaporation. An expression for the absolute humidity, Y, can be obtained from both a heat and material balance in terms of the dry-bulb temperature, Tdb, and the wet-bulb temperature, Twb:

Y

Yw

h k

(Tdb Twb )

(52.9)

w

where Yw = the saturated absolute humidity at the wet-bulb temperature, w = the latent heat of vaporization at Twb, h = the gas film coefficient for heat transfer by conduction and convection, and k = the mass transfer coefficient. The ratio of h to k (English units) has been experimentally determined to equal 0.236 for an air-water system. It is assumed that radiation effects are negligible in Equation 52.9. As mentioned earlier, the relative humidity, YR, is the ratio of the partial pressure of the water vapor, pA, in the mixture to the vapor pressure of liquid water, pA , at the system temperature. This was expressed in Equation 52.3 on a percentage basis. In addition, the dew point is the temperature to which a vapor-gas mixture must be cooled at constant total pressure and humidity to become saturated. This can be determined by calculating pA in the formula for YR (Equation 52.3), and looking up the temperature that corresponds to this vapor pressure of water. In an adiabatic “saturator,” evaporation cools the water to Tas, the adiabatic saturation temperature. If the contact time between the air and water is long enough,

Humidification 533 the air becomes saturated and leaves the apparatus at the steady-state temperature Tas. The following equation can be derived from an enthalpy balance, where the subscript as refers to conditions at Tas:

T Tas

o

(Yas Y ) C pH

(52.10)

For the air-water system at ordinary temperature ranges, the heat capacity may be assumed as constant and one may employ Equation 52.6 for CpH. Equations 52.9 and 52.10 for absolute humidity can be compared in the following form:

h

Yw Y

Yas Y

(T Tw ); for wet-dry bulb data

(52.11)

(T Tas ); for adiabatic saturation data

(52.12)

k

C pH

w

s

where h and k again represent the heat and mass transfer coefficients, respectively. Both equations would be identical if CpH = h/k. This is almost always true for the air-water system. Hence the adiabatic cooling line on the psychometric chart may be used for wet-bulb problems under ordinary conditions. Figures 52.1 and 52.2 contain the psychometric charts from thermodynamic properties drawn from the literature. The following are some helpful points on the use of psychometric charts: 1. Heating or cooling at temperatures above the dew point (temperature at which the vapor begins to condense) corresponds to a horizontal movement on the chart. As long as no condensation occurs, the absolute humidity remains constant. 2. If the air is cooled, the system follows the appropriate horizontal line to the left until it reaches the saturation curve and follows it thereafter. 3. In problems involving use of the humidity chart, it is convenient to choose the mass of air on a dry basis since the chart uses this basis.

52.4 The Humidification Process The evaporation of water into air for the purpose of increasing the air humidity is known as humidification. Closely allied to this is the evaporation of water into air for the purpose of cooling the water. Dehumidification consists of condensing water from air to decrease the air humidity. All these processes are of considerable

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Figure 52.2 Psychrometric chart for high temperatures. Barometric pressure of 29.92 in Hg.

Humidification 535 industrial importance and involve the contacting of air and water accompanied by heat and mass transfer. These processes are discussed in this section. In terms of industrial applications of humidification, it is often necessary to employ air at a known temperature and a known humidity. This can be accomplished by bringing the air into contact with water under such conditions that a desired humidity is reached. This mixing process can occur in any of the gas-liquid contacting devices discussed in early chapters. If conditions in a humidifier are such that the air reaches complete saturation, the humidity is fixed. However, if the equipment is operated in such a manner that the exit air is not saturated, then process conditions are somewhat indeterminate. Many of these applicable calculations can be obtained directly from the psychrometric chart provided in the previous section, or from an equivalent chart. Direct contact of a condensable vapor-non-condensable gas mixture with a liquid can produce any of several results, including humidifying of the gas or cooling of the liquid. The direction of liquid transfer (either humidification or dehumidification) depends on the difference in humidity of the bulk of the gas and that at the liquid surface. If the liquid is normally a pure substance, no concentration gradient exists within it and the resistance to mass transfer lies entirely within the gas. Since evaporation and condensation of vapor simultaneously involve a latent enthalpy of vaporization or condensation, there will always be a transfer of latent heat in the direction of mass transfer. The temperature differences existing within a system additionally control the direction of any sensible heat transfer that may occur. Furthermore, since temperature gradients may reside within the liquid, within the gas, or within both, the sensible heat transfer resistance may include effects in either or both phases. The effects of latent and sensible heat transfer may be simultaneously considered in terms of the enthalpy changes which occur. The operation of an adiabatic humidifier normally involves make-up water entering the unit at the adiabatic saturation temperature. Under these conditions, the temperature of the water in the system is assumed constant at the adiabatic saturation temperature, and both the air temperature and humidity remain constant. In addition, all the heat required to vaporize the water is supplied from the sensible heat of the air. When water is present in air, it is possible to extract the water by cooling the air-water mixture below the mixture’s dew point temperature. The dew point was defined earlier as the temperature at which the air can no longer absorb more water (i.e., it is the 100% relative humidity point) and if the temperature is then reduced further, water is forced out of the air-water mixture as condensation (dew) is formed. This process is described as dehumidification. The amount of water that is removed from a mixture as a result of cooling can be determined by drawing a line on a psychrometric chart (Figure 52.l) from the mixture’s initial conditions (e.g., dry-bulb temperature and relative humidity), horizontally to the left (i.e., the cooling direction) until the 100% relative humidity line is encountered. The dehumidification process can also be accomplished by bringing moist air into contact with a spray of water, the temperature of which is lower than

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the dew point of the entering air. An example is passing the air through sprays. Furthermore, dehumidification of air may also be accomplished by passing a cold fluid through the inside of (finned) tubes arranged in banks through which the air is blown. The outside surface of the metal tubes must be below the dew point of the air so that water will condense out of the air [3]. Cooling towers also find application in industry. The same operation that is used to humidify air may also be used to cool water. There are many cases in environmental engineering practice in which warm water is discharged from condensers or other equipment and where the value of this water is such that it is more economical to cool it and reuse it than to discard it. Water shortages and thermal pollution have made the cooling tower a vital part of many plants in the chemical and environmental process industries. Cooling towers are normally employed for this purpose and they may be destined to have an increasingly important role in almost all phases of industry. Modern (newer) power-generating stations remain under construction or in the planning stage, and both water shortages and thermal pollution are serious problems that must be dealt with. The cooling of water in a cooling tower is accomplished by bringing water into contact with unsaturated air under such conditions that the air is humidified and the water is brought approximately to the wet-bulb temperature. This method is applicable only in those cases where the wet-bulb temperature of the air is below the desired temperature of the exit water [3]. Quantitatively speaking, water is cooled in cooling towers by the exchange of sensible heat, latent heat, and water vapor with a stream of relatively cool dry air. The basic relationships developed for dehumidifiers also apply to cooling towers although the transfer is in the opposite direction since the unit acts as a humidifier rather than as a dehumidifier of air [4]. Brown and Associates [5] have provided empirical correlations from the literature [6-7] for estimating (roughly) sizes and capacities of conventional cooling towers.

52.5 Equipment Any of the absorption contact equipment discussed in Chapter 48 are applicable to humidification, dehumidification, and water-cooling applications. Packed towers and plate towers are particularly effective but there are other types of specialized units. These are briefly discussed below. As mentioned above, warm water flows down cooling towers countercurrent to rising unsaturated air. Water is cooled by furnishing part of the latent heat required to vaporize some of the water into the air stream. The air is thus humidified as it rises. The calculation of the tower height for the vaporization process can be accomplished by the stepwise procedure detailed for distillation in Chapter 49. The hot water is introduced at the top of the tower and leaves at a lower temperature. The air flows countercurrent to the water. It enters at the bottom and leaves

Humidification 537 the top. The temperature of the air-water interface tends to approach the adiabatic saturation or wet-bulb temperature of the air. At the top of the tower, heat is being transferred from the inlet (hot) water to the air, since the temperature of the water is higher than that of the interface, and the interface temperature is usually higher than that of the air. This sensible heat removed from the water appears as sensible and latent heat of the air-water mixture. At the bottom of the tower, the temperature of the water and of the interface may both be lower than that of the air, with sensible heat being transferred both from the liquid and air to the interface, resulting in the vaporization of the water. Water may thus be cooled by air at a higher temperature, provided that a humidity difference driving force (which produces evaporation) is maintained [5]. Cooling towers were originally primarily constructed of redwood, a material which is very durable when in continuous contact with water; however, there are also moisture resistant polymer composite materials that are now available. The internal packing is usually in the form of horizontal wooden slats. The void volume is usually greater than 90%, leading to a pressure drop that is extremely low. The air-water interfacial surface includes not only the liquid films that wet the slats but also the surface of droplets, which settle as a “rain” from each tier of slats to the next. Natural-circulation towers consist of two types: atmospheric and natural-draft. In atmospheric towers, air circulation is dependent solely on the prevailing winds, which essentially produces a crossflow of the air and water. The towers are generally long with narrow horizontal cross-sectional areas. This leads to adequate penetration of the air into the central portions of the unit. Louvers on the sides of the tower help reduce the losses of water entrained in the gas stream. Naturaldraft towers provide a free convection effect. This ensures air movement even in calm weather; it is similar to a stack or a chimney. Large cross-sectional areas are usually required in order to maintain a low air velocity and, consequently, a low pressure drop. Both natural-circulation and natural-draft towers must be relatively tall. A pump is required for the water, but there are no fans or the accompanying power cost associated with moving the air. The emphasis in recent years has been to employ mechanical-draft equipment. These towers may be of the forced-draft type where the air is blown into the tower by a fan at the bottom, or of the induced-draft type where the air is drawn upward by a fan at the top. Since the forced-draft tower ensures the recirculation of the hot, humid discharged air, the effectiveness of the tower is somewhat compromised. However, the induced-draft tower discharges the air at a higher velocity and can lead to a more uniform air distribution in the packing. Based on these considerations, this unit is often preferred despite the fact that the fan power is higher since the air density is lower. Water is usually distributed over the packing by weirs or spray nozzles. Spray eliminators at or near the top of the unit can reduce water carryover. All these units incur losses defined as blowdown. In addition, make-up water is required for evaporation and entrainment losses as well as the blowdown.

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Unit Operations in Environmental Engineering

Spray columns (see Chapter 48) are either forced or induced-draft towers without any internal packing. Contact relies entirely on the inlet water sprays at the top to provide interfacial surface for mass transfer. Low gas pressure drop is normal for this unit. Spray chambers are essentially horizontal spray columns. They too are frequently used for adiabatic humidification-cooling operations. Dehumidification is possible by cooling the water prior to spraying or by inserting refrigerating coils in the side spray chamber. Generally, three banks of sprays in series will bring the gas to substantial equilibrium with the incoming spray liquid. Spray ponds are occasionally used for water cooling where a close approach to the air wet-bulb temperature is not required. These units essentially contain fountains from which the water is sprayed vertically upward into the air and allowed to settle by gravity into a collection basin. They are obviously subject to water loss due to any prevailing winds. Cooling ponds or cooling reservoirs are used for removing a relatively small amount of heat from water over a small temperature range. The pond required may simply be estimated from any relationship that provides the rate of evaporation of water into still air. The heat loss is then calculated from the latent enthalpy (of vaporization) of water. As one might expect, any wind will increase the capacity of a pond. The cooling capacity can be further increased if a system of spray nozzles can be installed above the surface of a cooling pond as in the spray ponds described above. Other methods of dehumidification can also include adsorption, such as those methods discussed in Chapter 50. Adsorbents employed include silica gel or alumina (commonly referred to as desiccants). In addition, treating/washing the gas with liquids can also reduce the water content of the gas.

52.6 Describing Equations For cooling or humidification, the operating line (described in Chapters 48 and 49) lies below the equilibrium line. As noted earlier, an operating line refers to the actual vapor-liquid relationship of a key component, in contrast to the true equilibrium relationship. There is, therefore, a minimum air rate that can be used to accomplish a specified water-cooling duty. In dehumidification, cool water is used to reduce both the humidity and the temperature of the air introduced at the bottom of the tower. The operating line for this case lies above the equilibrium line and there exists a minimum amount of water that can be used to dehumidify a given quantity of air. The basic relationships available for humidifiers and cooling towers apply to dehumidifiers, although the transfer occurs in the opposite direction; in effect, the tower is a dehumidifier rather than a humidifier of air. Calculational details are provided in the literature [2,5–7].

Humidification 539

References 1. Bennett, C., and Myers, J., Momentum, Heat, and Mass Transfer, 2nd Ed., McGraw-Hill, New York City, NY, 1974. 2. Theodore, L., and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010. 3. Author unknown, New York University, Chemical Engineering Unit Operations Laboratory Report, Bronx, NY, 1960. 4. Author unknown, Manhattan College, Bronx, NY, 1961. 5. G. Brown and Associates, Unit Operations, John Wiley & Sons, Hoboken, NJ, 1950. 6. Flour Corp. Ltd., Bulletin T337, 1939. 7. Kelly, K.C., paper published by the Flour Corp., presented before the California Natural Gas Association, Dec. 3, 1942.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

53 Drying

53.1 Introduction As noted in the previous chapter, it is necessary to perform calculations involving the properties of mixtures of air and water vapor. In many unit operations such calculations often require knowledge of: 1. The amount of water vapor carried by air under various conditions 2. The thermal properties of such mixtures 3. The changes in enthalpy and moisture content as air containing some moisture is brought into contact with water or wet solids, and other similar processes It should be noted that drying can be defined as the reduction of the liquid content (usually water) of solids or nearly solid materials. Moisture reduction can be accomplished through heating the solids by conduction or radiation, or by bringing the wet material into contact with unsaturated hot air or gases. Some drying operations involve only the exposure of solids to unsaturated air at normal temperature.

541

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Unit Operations in Environmental Engineering

In waste treatment, drying is applied as a terminal dewatering operation, being preceded by one or a combination of operations such as settling, thickening, stabilization, and dewatering. Heat drying is expensive and is most often used where the dried material is of commercial value. In general, air drying is employed to facilitate the disposal of worthless sludge. This chapter reviews the mechanisms of drying processes, and the equipment in which these processes are carried out. The remainder of this chapter will focus on three topics: Drying Principles, Describing Equations, and Drying Equipment.

53.2 Drying Principles In the drying process, a liquid (usually water) is separated from a wet solid by use of a hot dry gas (usually air). The drying of solids to remove moisture involves the simultaneous processes of heat and mass transfer. Heat is transferred from the gas to the solid (and liquid) in order to evaporate the liquid contained in the solid. Mass is transferred as either a liquid or vapor within the solid and then as a vapor from the surface of the solid. Additional details on this process are provided later in this section. The energy required to vaporize the liquid in a solid is almost always furnished by a hot, inert carrier gas that enters the drier. In some driers, the solid may be in contact with heated metal surfaces where the required heat of vaporization flows to the solid by conduction. In vacuum drying (where there is essentially no carrier gas), the heat of vaporization is furnished by conduction or radiation; here, the capacity of the drier is largely influenced by the heat-transfer surface available within the dryer. The curve provided in Figure 53.1 is obtained when a substance saturated with water is dried. During the drying process, a thin film of water exists on the surface of the material where water is supplied from the solid fast enough to keep the surface entirely wet. As this water is evaporated, water from the interior of the sample rises to the surface essentially by capillary action with the solid temperature approximately given by the wet-bulb temperature of the air. After drying has proceeded for some time, the surface film begins to disappear and the rate of drying decreases. This critical moisture content leads to dry patches on the surface, and as noted in Figure 53.2, the drying rate begins to fall. This corresponds to the curved portion of the graph in Figure 53.1. Ultimately, water ceases to evaporate and a final equilibrium moisture content (denoted by the dashed line in these figures) is achieved. The curve in Figure 53.1 can also be plotted as shown in Figure 53.2. In this plot, the horizontal portion corresponds to the constant rate period. The value of the mass rate of water evaporated per unit time, R, (mass/time-area), corresponding to the horizontal portion is defined as the critical drying rate, Rc, while the value of X (mass H2O evaporated/mass solid) corresponding to Rc is called the critical moisture content, Xc. Note that the drying rate may reach zero before

Mass H2O in sand

Drying 543

Time Figure 53.1 Typical drying process: moisture content versus time.

Rc

mass evaporation R= hr - ft2

Ro

Xo

Xc X=

mass H2O mass dry solid

X1

Figure 53.2 Typical drying process: drying rate versus moisture content.

the solid is completely dry. The resulting equilibrium moisture content, Xe, for the solid is a function of the drying rate. The reader is reminded that the term X is a concentration term on a water free or dry weight basis.

53.3 Describing Equations An equation for the total time of drying can be derived [1]. First note that:

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Unit Operations in Environmental Engineering

dm ; dm RAdt Adt

R

(53.1)

The term m is the mass of water, A is the area of the drying surface, and dt is the incremental time of drying. In addition,

dm 2sA s dX

(53.2)

where s is specified as one-half of the thickness of the material being dried, s is the density of the dry material, and X is the moisture content on a water-free basis. Combining Equations 53.1 and 53.2, eliminating dm, and solving for R, gives:

R

[ 2sA s ]dX Adt

2s

s

dX dt

(53.3)

The total drying time, tt, is then found by integration as: tt

X1

dt 0

tt

2s

s

dX R X2

(53.4)

where X1 and X2 are the initial and final moisture contents, respectively. During the constant rate period, R is constant and may be set equal to Rc. One may therefore integrate Equation 53.4 to give:

tc

2s s ( X1 X2 ) Rc

(53.5)

In the falling rate period, an assumption can be made that there is a linear relationship, that is, y = mx + b (as shown in Figure 53.2) between the falling rate, R, and moisture content, X. This effectively assumes that the curve approximates a straight line from the critical moisture content to the origin during the falling rate period. Thus:

R mX b

(53.6)

dR mdX ; dR/m dX

(53.7)

In addition:

For this falling rate period, one may replace dX in Equation 53.4 by Equation 53.7 to give:

Drying 545 Xc

dX 2s s R X0

tf

2s s m

Rc

dR R Ro

(53.8)

where Xo refers to the final moisture content. Upon integration, the time of drying during the falling rate period, tf, becomes:

2s s R ln c m Ro

tf

(53.9)

If the y-intercept in Figure 53.2 is assumed to be zero, i.e., Ro = Xo = 0, then

Rc Ro Xc Xo

m

Rc Xc

(53.10)

Substituting this value of m into Equation 53.9 gives:

tf

2s

s

Xc R ln c Rc Ro

(53.11)

The total time for drying is the sum of the times for the constant rate period and the falling rate period, i.e.,

tt

tc t f

2s s R ( X1 X2 ) Xc ln c Rc Ro

(53.12)

If both rates periods are present, X2 = Xc so that the above is given by:

tt

2s s R ( X1 Xc ) Xc ln c Rc Ro

(53.13)

Another term that has received attention in this field is the free moisture content, F. It is defined as the difference between the total moisture content X and the equilibrium moisture content, Xe, (the content of water in the solid that cannot be removed by the air), expressed as mass of water per mass of dry solid:

F

X Xe

(53.14)

The free moisture content is a function of the same variables as the equilibrium moisture content. As one might surmise, F is of major interest in drying calculations.

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It should be noted that drying is but one method of separating a liquid from a solid. Technically, it is the aforementioned vaporization process in which the heat rate and mass transfer rates control equipment design. In most dryers, heat is transferred by convection from a gaseous drying medium to the surface of the wet solid. (In some designs, radiation from the walls of the dryer to the wet material supplements convection). This heat vaporizes the liquid, which is usually water. The vapor that is thus formed must diffuse into the gas phase. In so doing, it passes through the same gas-phase convective resistance through which the heat passed. However, the transfer of heat is in the opposite direction to the transfer of mass. Depending on operating conditions, a small portion of the heat can act as sensible heat to raise the temperature of the wet solid. In some respects, drying is identical to humidification (see previous chapter) in that water is provided from a pure liquid while in drying it comes from liquid dispersed in a solid. Thus, drying requires that a new resistance be considered. This resistance can be thought of opposing the movement of the liquid through the solid to the gas-solid interface. Although the next section keys on rotary dryers and spray dryers, the significance of the above resistance becomes apparent if one analyzes typical drying data obtained in a batch dryer. Summarizing, one notes that the rate at which a liquid is vaporized from a solid is constant at first. This constant rate holds until the moisture content of the wet stock reaches a critical value, at which point the rate begins to decrease. It continues to decrease thereafter, ultimately falling to or approaching zero when the moisture content has been reduced to an equilibrium value that is the lowest value that it can reach with the drying conditions employed.

53.4 Drying Equipment 53.4.1 Rotary Dryers The rotary dryer is a popular device suitable for the drying of free-flowing materials that can be tumbled without concern for breaking. Moist solid is continuously fed into one end of a rotating cylinder with the simultaneous introduction of heated air. The cylinder is installed at a slight angle and with internal lifting flights so that the solid is showered through the hot air as it traverses the dryer. A rotary dryer is almost always operated in the cocurrent mode. The hot air is cooled as it is humidified; at the same time, the solid is heated and dried by contact with the hot air. A typical industrial rotary dryer consists of a cylinder, rotated upon suitable bearings and usually slightly inclined to the horizontal. The length of the cylinder may range from 4 to more than 10 ft. Solids fed into one end of the cylinder progress through it by virtue of rotation, head effect, the slope of the cylinder, and are discharged as finished product at the other end. Gases flowing through the cylinder may retard or increase the rate of solids flow, depending upon whether the gas flow is countercurrent or cocurrent with respect to solid flow.

Drying 547 Rotary dryers have been classified as direct, indirect-direct, indirect, and special types. These terms refer to the method of heat transfer: direct when heat is added to or removed from the solids by direct exchange between flowing gas and solids, and indirect when the heating medium is separated from physical contact with the solids by a metal wall or tube. Rotating equipment is applicable to batch or continuous processing solids, which are relatively free-flowing and granular when discharged as product. Materials that are not completely free-flowing in their feed condition are handled in a special manner, either by recycling a portion of the final product and premixing with the feed in an external mixer to form a uniform granular feed to the process, or by maintaining a bed of free-flowing product in the cylinder at the feed end and, in essence, performing a premixing in the cylinder itself. The method of feeding rotating equipment depends upon material characteristics and the location and type of upstream processing equipment. When the feed comes from above, a chute extending into the cylinder is usually employed. For sealing purposes, or if gravity feed is not convenient, a screw feeder is normally used. On cocurrent, direct-heat units, cold-water jacketing of the feed chute or conveyer may be desirable if it is contacted by the inlet hot gas stream. This will prevent overheating of the metal wall with resultant scaling or overheating of heat sensitive feed materials. One method of feeding direct cocurrent drying equipment utilizes dryer exhaust gases to convey, mix, and pre-dry wet feed. The latter is added to the exhaust gases from the dryer at a high velocity. The wet feed, mixed with dust entrained from the dryer, separates from the exhaust gases in a cyclone (typically) [2] and usually drops into the feed end of the cylinder. This technique combines pneumatic and rotary drying. The dust entrained in the exit-gas stream is customarily removed in the cyclone collector(s). This dust may be returned into the process or may be separately collected. For expensive materials or extremely fine particles, bag collectors (a baghouse) [2] may follow a cyclone collector, assuming fabric temperature stability is not limiting and there are assurances of no temperature excursions. Rotating equipment, with the exception of brick-lined vessels operated above ambient temperatures, are usually insulated to reduce heat losses. Other exceptions are direct-heat units of bare metal construction operating at high temperatures where heat losses from the shell are necessary to prevent overheating of the metal. Insulation is the rule with concurrent, direct-heat units. It is not unusual for product cooling or condensation on the shell to occur in the last 10 to 50 percent of the cylinder length if it is not sufficiently insulated. Countercurrent flow of gas and solids gives greater heat-transfer efficiency with a given inlet gas temperature. However, cocurrent flow is used more frequently to dry heat-sensitive materials at higher inlet gas temperature because of the rapid cooling of the gas during the initial evaporation of surface moisture. A major design variable of the rotary dryer is the drying rate. Since the solid loses moisture while the air stream is gaining moisture, the drying rate can be calculated in either of two ways. Data taken from a dryer can provide the drying

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Unit Operations in Environmental Engineering

rate based either on the moisture lost by the solid and/or the humidity gained by the air stream between the inlet and exit [1]. These two calculation methods allow checking the consistency of dryer measurement data. As noted above, the mechanism of drying involves the transfer of heat by convection for vaporizing the material. Mass is transferred as a result of a moisture concentration gradient at a rate dependent on the characteristics of the solid. The dynamic equilibrium prevailing between the rate of heat transfer to the material and rate of vapor removal from the surface during the constant drying rate period also provides a means for calculating the heat and mass transfer coefficients. The drying rate, dm/dt, can be measured by the two ways presented above, i.e., based on the loss of water by the solids or on the increase in water content of the air stream. Dry-bulb temperatures and saturated (wet-bulb) temperatures at each end of the dryer are normally measured to calculate the log mean temperature difference. This enables one to determine the heat transfer coefficient, h, based on the air and water data as follows:

dm dt

hV (T Ts )mean

(mwater )solid

(mwater )air

m

(53.15)

and from the conservation law, (53.16)

with m dm/dt = drying rate, lb water evaporated/hr; h = heat transfer coefficient based on dryer volume, Btu/hr-ft2-°F; V = dryer volume, ft3; T = temperature of drying medium, dry-bulb temperature, °F, determined at inlet and outlet conditions; Ts = surface temperature of solid, °F, wet-bulb temperature, determined at inlet and outlet conditions; and λ = latent heat of vaporization, Btu/lb. Similarly, humidities can be obtained at the inlet and outlet (including saturated humidities), and the log mean humidity difference is used to calculate the mass transfer coefficient, k, based on either the air and water data as follows since (mwater )solid (mwater )air m :

m

dm dt

kV (Yas Y )mean

(53.17)

53.4.2 Spray Dryers Spray dryers (SD) are often utilized in the chemical processing industry to obtain a dry product in a granular or powder form. Various spray-dried products include coffee, detergents, and instant beverages. In environmental engineering applications, spray dryers are used for gaseous and particulate pollutant removal in units called dry scrubbers [3].

Drying 549 Spray drying is a drying technique that involves the drying of a solid in solution via atomization of the solution. The atomized drops are then contacted with a hot air stream. The dry product is normally collected by a cyclone (or another particulate control device) at or near the bottom of the unit [2]. Spray dryers may be operated cocurrently or countercurrently. The method of operation of a spray dryer is relatively simple, requiring only two major equipment items, a spray dryer similar to those used in the chemical food-processing and mineral preparation industries, and a cyclone or fabric filter (baghouse) or electrostatic precipitator (ESP) to collect the fly ash and entrained solids [2]. In the spray dryer, the solution (or slurry) is atomized into the incoming hot gas stream to increase the liquid-gas interface and to promote the mass transfer of the gas to the slurry droplets where it is absorbed. The slurry solution may be mechanically atomized by either a rotary atomizer or by spray nozzles, and injected into the gas stream. The action of the rotary atomizer results in smaller droplet size and size distribution, and is less subject to plugging and wear than the spray nozzle; however, it is higher in cost. This atomizer propels the droplets radially outward and perpendicular to the gas flow. The droplets decelerate rapidly owing (in part) to the drag forces of the downward moving gas and eventually attain the velocity of the gas. The radial distance between the atomizer and the dryer wall must be sufficient to allow for adequate drying of the largest droplets [3]. The length to diameter (L/D) ratio of the cylindrical section of the SD is typically 0.8:1. In a SD with a dual-fluid pneumatic nozzle, atomization is in the direction of the gas flow and the L/D ratio is typically 2:1. Optimum SD performance is achieved through the proper choice of L/D, droplet size, and residence time. Spray dryers have several advantages over other types of dryers. The drying time is very short, permitting drying of highly heat sensitive materials and the creation of a solid or a hollow spherical product. Some products, such as food and detergents, require certain appearances, consistencies, and bulk densities; other drying methods may fail to produce these valuable properties. Another desirable characteristic of spray dryers is their ability to produce a dry product ready for packaging from a feed solution. This may greatly simplify the manufacturing process. Drying some solids requires removing small amounts of liquid from a solid material to reduce the liquid content of the solid to an acceptably low value. For example, a solid dissolved in a solution can be dispersed into a stream of hot gas (usually air) in the form of fine droplets. Moisture vaporizes from the droplets, leaving residual particles of dry solid behind. The inlet dry gas gains the moisture that evaporates from the droplets. A dry solids material balance requires the subtraction of the water from the product in performing the balance. This is accomplished by determining the moisture content of the feed and the product. After determining the amount of water in the feed and in the product, and subtracting the water content for both the feed and the product, one may compare the dry feed and the dry product to check the consistency of data and determine any experimental error [4].

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Unit Operations in Environmental Engineering

As atomized droplets are contacted with the hot, dry air in the drying chamber, the water evaporates and enters the air. The evaporation rate is determined from the following water mass balance equation (employing SI units in this development):

mw

dm (water ) (m1 )( X1 ) (m2 )( X2 ) dt

(53.18)

where dm/dt (water) = mw = evaporation rate, kg H2O/s; m1 = dry solids feed rate, kg dry feed/sec; X1 = feed moisture content, kg H2O/kg dry feed; m2 = dry solids product rate, kg dry feed/sec; and X2 = product moisture content, kg H2O/ kg dry feed. The moisture gain by the drying gas may be similarly calculated by:

mg

dm (gas) (m2 )(Y2 ) (m1 )(Y1 ) dt

(53.19)

where dm/dt (gas) = mg = moisture gain by drying gas, kg H2O/sec; m2 = dry air mass flow from dryer, kg dry air/sec; Y2 = air humidity at dryer outlet, kg H2O/kg dry air; m1 = dry air mass flow to dryer, kg dry air/sec; and Y1 = air humidity at dryer inlet, kg H2O/kg dry air. The air humidities in Equation 53.19 can be determined from a psychrometric chart. The moisture gain of the drying gas should agree within 10% of the evaporation rate provided in Equation 53.18. The choice of many individuals in the dryer field for collecting dry product from a spray dryer is the cyclone [2]. However, fabric filters (baghouses) and electrostatic precipitators (ESPs) may also be employed. Baghouses have an inherent advantage over ESPs in that they have high collection efficiency and simplicity in design. Two disadvantages of using a fabric filter are that the fabric is somewhat sensitive to wetting, so that a margin above the saturation temperature must be maintained to protect the bags, and there is always the possibility of catastrophic bag failure due to an elevated temperature excursion [2]. Among the inherent advantages that dryers enjoy are: 1. 2. 3. 4. 5.

Low capital cost Low pressure drop losses Reduced auxiliary power Reduced water consumption Continuous, two-stage operation, from liquid feed to dry product

A companion spray unit to a spray dryer is the spray tower. This unit can also be employed to cool a hot gas. In one spray tower design, the gas (quench) cooling is accomplished simply by spraying water at the top of the tower as the hot gases travel upwards through the tower. When the un-vaporized water reaches the pump at the bottom of the tower, it is recirculated by pumping it back up to the top. Since about 10% of the water stream is usually vaporized during contact

Drying 551 with the hot gases, makeup water must be constantly added. For efficient evaporation of the water, the gas velocity should be from 400 to 600 fpm and the entire cross-section of the gas stream should be covered with a fine spray of water. This necessitates proper location, type, and number of spray nozzles to ensure adequate coverage of the gas-side flow area. Cooling hot gases with a water spray is relatively simple and requires little space. These towers are easy to operate and, with automatic temperature controls, only that amount of water that is needed is used to maintain the desired temperature of the gases at the discharge. Their installation and operating costs are generally considered to be less than that of other cooling methods. However, quench towers require careful design review for applications where the gases to be cooled contain a significant amount of acid.

References 1. Famularo, J., Unit Operations Laboratory Manual, Manhattan College, Bronx, NY, 1996. 2. Theodore, L., Air Pollution Control Equipment Calculations, John Wiley & Sons, Hoboken, NJ, 2008. 3. Santoleri, J. Reynolds, J., and Theodore, L., Introduction to Hazardous Waste Incineration, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2000. 4. Shaefer, S., and Theodore, L., Probability and Statistics in Environmental Science, CRC Press/Taylor & Francis Group, Boca Raton, FL, 2002.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

54 Absorber Design and Performance Equations

54.1 Introduction Design and performance equations are provided for both packed and plate columns, and some overlap does exist. However, the emphasis is on packed columns since they are often the choice in absorption applications. On the other hand, plate columns, the preferred choice in distillation (Chapter 49), will receive more attention in the next chapter. For most absorption applications, sufficient information is either provided or available to enable one to completely describe the system through simple yet standard calculational procedures. These calculations generally involve the determination of three unknown system variables: the liquid rate, the column height (and corresponding pressure drop), and the column diameter. Each of the topics is treated sequentially below.

54.2 Packed Columns [1,2] 54.2.1

Liquid Rate

As described earlier, the equilibrium of interest in gas absorption is that between a relatively nonvolatile absorbing liquid (solvent) and a soluble gas (solute). 553

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Unit Operations in Environmental Engineering

For cases that follow Henry’s Law (Chapter 43), Henry’s Law constant, m, is defined by:

y*

mx

(54.1)

The usual operating data to be determined or estimated for isothermal systems are the liquid rates and the terminal concentrations or mole fractions. An operating line, which describes operating conditions in a countercurrent flow column, is obtained by a mass (or a mole) balance around the column (as shown in Figure 54.1). Note that the notation normally employed for the gas rate in absorption calculations is G, not V, as employed for the vapor rate in Chapter 47. The subscript m is often carried if the rate is based on moles. The overall mole balance is:

Gm1 Lm2

Gm2 Lm1

(54.2)

For Component A, the mass (or mole) balance becomes:

Gm1 y A1 Lm2 x A 2

Gm2 y A 2 Lm1 x A1

(54.3)

Assuming Gm1 = Gm2 = Gm, and Lm1 = Lm2 = Lm (reasonable for many applications where solute concentrations are reasonably small), then:

Gm1 y A1 Lm2 x A 2

Gm2 y A 2 Lm1 x A1

(54.4)

or, on arrangement,

Lm Gm

y A1 y A 2 x A1 x A 2

(54.5)

This is the equation of a straight line known as the operating line. It has a slope of Lm/Gm on x, y coordinates and passes through the points (xA1, yA1) and (xA2, yA2) as indicated in Figure 54.2. In the design of most absorption columns, the quantity of gas to be treated, Gm, the concentrations, yA1 and yA2, and the composition of the entering liquid, xA2, are ordinarily fixed by process requirements. However, the quantity of liquid solvent to be used is subject to some choice. This can be resolved by setting or obtaining a minimum liquid-to-gas ratio. With reference to Figure 54.2, the operating line must pass through Point A (top of column) and must terminate at the ordinate yA1. If such a quantity of liquid is used to produce Operating Line (OL) AB, the exiting liquid will have the composition xA1. If less liquid is used, the exit liquid composition will clearly be greater, as at Point C, but since the driving forces for mass transfer are less, the absorption

Absorber Design and Performance Equations

555

Treated gas Gm2

yA2

Lm2

xA2

Lean solution

Feed gas Gm1 yA1

Rich solution Lm1

xA1

Figure 54.1 Mole balance; countercurrent flow absorption column.

is more difficult. The time of contact between the gas and liquid must then be greater and the absorber must be correspondingly taller. The minimum liquid that can be used corresponds to the Operating Line AD, which has the smallest slope for any line touching the equilibrium curve (tangent to the curve at E). At Point E, the mass transfer driving force is zero, the required contact time for the concentration change desired is infinite, and an infinitely tall column results. This then represents the minimum liquid-to-gas ratio. The importance of the minimum liquid-to-gas ratio lies in the fact that column operation is frequently specified as some factor of the minimum liquid-to-gas ratio. For example, a typical situation frequently encountered is that the actual operating line, (Lm/Gm)act is 1.5(Lm/Gm)min.

54.2.2 Column Diameter Consider a packed column operating at a given liquid rate and the gas rate is then gradually increased. After a certain point, the gas rate is so high that the drag

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Unit Operations in Environmental Engineering B

yA1

C

D

Slope = (Lm/Gm)act E Equilibrium curve y* = f(x) Slope = (Lm/Gm)min

yA

yA2

A

xA2

xA1 xA

xA1

max

Figure 54.2 Operating and equilibrium lines.

on the liquid is sufficient to keep the liquid from flowing freely down the column. Liquid begins to accumulate and tends to block the entire cross-section for flow (a process referred to as loading). This, of course, increases both the pressure drop and prevents the packing from mixing the gas and liquid effectively, and ultimately some liquid is even carried back up the column. This undesirable condition, known as flooding, occurs fairly abruptly, and the superficial gas velocity at which it occurs is called the flooding velocity. The calculation of column diameter is usually based on flooding considerations, with the usual operating range being taken as 50 to 75% of the flooding rate. One of the more commonly used column diameter calculations is based on U.S. Stoneware’s [3] generalized pressure drop correlation, as presented in Figure 54.3. The procedure to determine the column diameter is as follows: 1. Calculate the abscissa, (L/G)( G/ L)0.5; mass basis for all terms 2. Proceed to the flooding line and read the ordinate (design parameter) 3. Solve the ordinate equation for Gf at flooding 4. Calculate the column cross-sectional area, S, for the fraction of flooding velocity chosen for operation, f, by the equation:

S

W fG f

(54.6)

Absorber Design and Performance Equations G = lb/ft * s F = packing factor, dimensionless = liquid density/water density = lb/ft * s = lb/ft3 G = lb/ft3 L L = lb/s G = lb/s gc = 32.2 (lb/lbf) (ft/s2) Parameter of curves in pressure drop in inches of water/foot of packed height

0.60 0.40 0.20

Floo

ding

1.50

line

0.10 1.00 0.2

0.040

0.50 0.25

G

G2F

L

g L c

0.060

0.020 0.10 0.010 0.006 0.004

.005

0.002 0.001 0.01

0.02

0.04 0.06 0.1

0.2

L G

0.4 0.6 1.0 0.5

2.0

4.0

10.0

G L

Figure 54.3 Generalized pressure drop correlation to estimate column diameter.

where W is the mass flow rate of the gas, lb/sec; and S is the area in ft2. 5. The diameter of the column is then determined by:

D

4

0. 5

S

1.13S 0.5 ; ft

(54.7)

Note that the proper units, as designated in the correlation, must be used as the plot is not dimensionless. The flooding rate is usually evaluated using total flows of the phases at the bottom of the

557

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Unit Operations in Environmental Engineering column where they are at their highest value. The pressure drop may be evaluated directly from Figure 54.3 using a revised ordinate that contains the actual, not flooding, value of G [1,2].

54.2.3 Column Height The column height may be estimated from:

Z

N OG H OG

(54.8)

where NOG is the number of overall transfer units, HOG is the height of a single transfer unit, and Z is the height of the column packing. In most design applications, the number of transfer units, NOG, is obtained experimentally or calculated using any of the methods to be explained later in this section. The height of a transfer unit, HOG, is also usually determined experimentally for the system under consideration. Information on many different systems using various types of packings has been compiled by the manufacturers of gas absorption equipment and should be consulted prior to design. The data may be in the form of graphs depicting, for a specific system and packing, the HOG versus the gas mass flux (lb/hr-ft2), with the liquid rate as a parameter. The packing height, Z, is then simply the product of the HOG and the NOG Although there are many different approaches to determine the column height, the HOG-NOG approach is the simplest and presently the most used in environmental engineering calculations, with the HOG usually being obtained from the manufacturer. Details on NOG follow. In many operations, the constituent to be absorbed (e.g., HCl) is in the very dilute range. For this condition [2,4]: y1

N OG y2

dy y y*

(54.9)

If the operating line and equilibrium line are both parallel and straight:

N OG

y1 y2 y y*

(54.10)

If the operating line and equilibrium are just straight (and not necessarily parallel):

N OG

y1 y2 ; (y y * )lm ( y y * )lm

( y y * )1 ( y y * )2 ln[( y y * )1 /(y y * )2 ]

(54.11)

Absorber Design and Performance Equations

559

However, one can show that if Henry’s Law applies, the number of transfer units is given by Coburn’s equation:

ln N OG

y1 mx2 y2 mx2

1

1 1 A

1 A

1 A

(54.12)

where

A

Lm mGm

(54.13)

and A is defined once again the absorption factor, and m is the slope of the equilibrium curve. The solution to this equation can be conveniently found graphically. However, note that the flow rates Lm and Gm are based on moles in Equation 54.13. If the gas is highly soluble in the liquid and/or reacts with the liquid, Theodore [4] has shown that:

N OG

ln

y1 y2

(54.14)

If the operating line and/or equilibrium line are curved, the integral above in Equation 54.9 should be evaluated. Qualitatively, the height of a transfer unit is a measure of the height of a contactor required to effect a standard separation, and it is a function of the gas flow rate, the liquid flow rate, the type of packing, and the chemistry of the system. As indicated above, experimental values for HOG are generally available in the literature or from vendors [1,2,4].

54.3 Plate Columns The most important design considerations for plate columns include the calculation of the column diameter, type and number of plates to be used (usually either bubble-cap or sieve plates), actual plate layout and physical design, and plate spacing; these, in turn, determine column height. To consider each of these to any great extent is beyond the scope of this chapter, particularly since it received attention in Chapter 49. The discussion that follows, therefore, will be a relatively concise presentation of some of the general absorber design techniques that will provide satisfactory results for purposes of estimation [5–8].

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Unit Operations in Environmental Engineering

The column diameter, and consequently its cross-section, must be sufficiently large to handle the gas and liquid at velocities that will not cause flooding or excessive entrainment. The superficial gas velocity for a given type of plate at flooding is given by the relation: 0.5

VF

CF

L

G

(54.15)

G

where VF (the notation usually employed for plate columns) is the gas velocity through the net column cross-sectional area for gas flow, ft3/sec-ft2; the densities are in lb/ft3; and CF is an empirical coefficient that depends on the type of plate and operating conditions. The net cross-section is the difference between the column cross-section and the area taken up by downcomers. In actual design, some percent of VF is usually used, i.e., for non-foaming liquids 80 to 85% of VF is specified, while 75% or less of VF is used for foaming liquids. Of course, the value is subject to a check of entrainment and pressure drop characteristics. The calculation of column diameter based on Equation 54.15 assumes that the gas flow rate is the controlling factor in its determination [2,6]. In the above discussion of equilibrium stages, it was assumed that the phases leaving the stage were in equilibrium. In actual countercurrent multistage equipment, it is not practical to provide the combination of residence time and intimacy of contact required to accomplish equilibrium. Hence, the concentration change for a given stage is less than that predicted by equilibrium considerations. Stage efficiencies are employed to characterize this condition. The efficiency term frequently used is the overall stage (plate) efficiency, given by the ratio of theoretical contacts required for a given separation to the actual number of contacts required for the same operation. While reliable information on the overall efficiency is most desirable and convenient to use, so many variables come into play that really reliable values for the overall stage efficiency are difficult to come by. This value is generally obtained by experiment or field test data, or may be specified by the vendor. The number of theoretical plates may be determined directly without recourse to graphical techniques for cases where both the operating line and the equilibrium curve may be considered straight (dilute solutions). This will frequently be the case for a relatively dilute gas (as usually encountered in air pollution control) and liquid solutions where, more often than not, Henry’s Law is usually applicable. Since the quantity of gas absorbed is small, the total flows of liquid and gas entering and leaving the column again essentially remain constant. Hence, the operating line will be substantially straight. For such cases, the Kremser-BrownSounders [9,10] equation applies for determining the number of theoretical plates, Np:

Absorber Design and Performance Equations

y Np Np

log

1

mx0

y1 mx0

1

log A

1 A

1 A

561

(54.16)

Note that ln may be employed rather than log in both the numerator and denominator. Here mx0 is the gas composition in equilibrium with the entering liquid (m is Henry’s Law constant = slope of the equilibrium curve). If the entering liquid contains no solute gas, then x0 = 0 and Equation 54.16 can be simplified further. The solute concentrations in the gas stream, yNp+1 and y1 represent inlet and outlet conditions, and L and V (that appear in A) are the total molar rates of liquid and gas flow per unit time per unit column cross-sectional area, respectively. Small variations in L and V may be roughly compensated for by using the geometric mean value of each taken at the top and bottom of the column. The general procedure to follow in sizing a plate tower is given below [2]. 1. Calculate the number of theoretical stages, N, using Equation 54.16. 2. Estimate the efficiency of separation, E. This may be determined at the local (across plate), plate (between plates), or overall (across column) level. The overall efficiency, E0, is generally employed. 3. Calculate the actual number of plates:

N act

N E0

(54.17)

4. Obtain the height between plates, h. This is usually in the 12- to 36-in range. Many towers use a 24-inch plate spacing. 5. The tower height, Z, is then:

Z

N act h

(54.18)

6. The diameter may be calculated directly from Equation 54.15. 7. The plate or overall pressure drop is difficult to quantify accurately. It is usually in the 2- to 6-in H2O per plate range for most columns with the lower and upper values applying to small and large diameters, respectively.

54.4 Stripping Quite often, an absorption column is followed by a liquid absorption process in which the gas solute is removed from the absorbing medium by contact with an

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Unit Operations in Environmental Engineering

insoluble gas. This operation is called “stripping” and is utilized to regenerate the solute “rich” solvent so that it can be recycled back to the absorption unit. The rich solution enters the stripping unit and the volatile solute is stripped from solution by either reducing the pressure, increasing the temperature, using a stripping gas to remove the vapor solute dissolved in the solvent, or any combination of these process options. While the concept of stripping is opposite to that of absorption, it is treated in the same manner. The operating line developed for adsorption can be applied to a stripping unit (see Figure 54.4 for Component A) [1,4]. As was developed in Equation 54.5, the operating line for a stripping unit is also given by:

Lm Gm

y A1 y A 2 x A1 x A 2

(54.5)

However, since this process is the opposite of absorption, solute is transferred from the liquid to the gas, and thus, the operating line lies below the equilibrium Rich gas Gm2 yA2

Lm2

xA2

Rich solution

Stripping gas Gm1 yA1

Lean solution Lm1 xA1 Figure 54.4 Stripping unit.

Absorber Design and Performance Equations

563

curve. When absorption was addressed, a minimum liquid to gas ratio, (Lm/Gm)min, could be set in order to determine limits on the design. However, for stripping operations, a minimum gas to liquid ratio, (Gm/Lm)min, is now used and corresponds to the minimum gas rate required to achieve the desired separation. The minimum gas to liquid ratio can be found by the following procedure. 1. As shown in Figure 54.5, a line is drawn from the point (yA1, xA1), which represents the mole fractions of solute in the stripping gas feed and the stripped liquid stream, respectively, to the intersection of x = xA2, which represents the mole fraction of the liquid stream to be stripped. If a plot of the equilibrium data results in a curve, then a tangent is drawn on the curve at the point corresponding to the value of the inlet rich solution concentration, xA2. 2. Obtain yA2 by reading or calculating the corresponding value on the y-axis. 3. Rearrange Equation 54.5 in terms of (Gm/Lm)min and insert the known quantities. 4. Multiply the results of Step 3 by the liquid flow rate to obtain the minimum stripping rate. 5. Typically, an actual value of 1.3 (Gm/Lm)min is employed to assure an efficient separation. Other calculations essentially remain the same for both packed and plate towers, except that the height of a packed tower is given by:

Z

H OL N OL

(54.19)

Equilibrium curve yA2

(Gm/Lm)min

yact y

(Gm/Lm)act yA1 xA1 Figure 54.5 Minimum gas to liquid ratio.

x

xA2

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Unit Operations in Environmental Engineering

where

ln N OL

x2 ( y1 /m) (1 A) A x1 ( y1 /m) 1 A

(54.20)

54.5 Packed versus Plate Tower Comparison Of the various types of gas absorption devices, packed columns and plate columns are the most commonly used in practice. Although packed columns are used more often, both have their special areas of usefulness, and the relative advantages and disadvantages of each are worth considering. In general: 1. The pressure drop of the gas passing through a packed column is smaller. 2. A plate column can stand an arbitrarily low liquid feed and permits a higher gas feed than a packed column. It can also be designed to handle liquid rates that would ordinarily flood a packed column. 3. If the liquid deposits a sediment, a plate column is more advisable. By fitting the column with manholes, a plate column can be cleaned of accumulated sediment that would clog many packing materials and warrant necessary costly removal and refilling of the column. Packed columns are also susceptible to plugging if the gas contains particulate contaminate(s). 4. In mass transfer processes accompanied by considerable heat effects, cooling or heating the liquid is much easier in a plate column. A system of pipes immersed in the liquid can be placed on the plates between the caps, and heat can be removed or supplied through the pipe wall directly to the area in which the process is taking place. The solution of the same problem for a packed column leads to the division of this process into a number of sections, with the cooling or heating of the liquid taking place between these sections. 5. The total weight of a plate column is usually less than a packed column designed for the same capacity. 6. A well-installed plate column avoids serious channeling difficulties insuring good, continuous contact between the gas and liquid throughout the column. 7. In highly corrosive atmospheres, a packed column is simpler and cheaper to construct than a plate column.

Absorber Design and Performance Equations

565

8. The liquid holdup in a packed column is considerably less than in a plate column. 9. Temperature changes are apt to do more damage to a packed column than to a plate column. 10. Plate columns are advantageous for absorption processes with an accompanying chemical reaction (particularly when it is not very rapid). The process is favored by a long residence time of the liquid in the column and by easier control of the reaction. 11. Packed columns are preferred for liquids with high foaming tendencies over plate columns. 12. The relative merits of a plate column versus a packed column for a specified purpose are normally determined by direct comparison of the actual cost figures resulting from a detailed design analysis for each type. Finally, the general design procedures consist of a number of steps that have to be taken into consideration. These include [1]: 1. Solvent selection 2. Equilibrium data evaluation 3. Estimation of operating data (usually obtained from a mass and energy balance, where the energy balance determines whether the absorption process can be considered isothermal or adiabatic) 4. Column selection (should the column selection not be obvious or specified, calculations must be carried out for the different types of columns, and the final selection based primarily on economic considerations) 5. Calculation of column diameter (for packed columns this is usually based on flooding conditions, and for plate columns is based on the optimum gas velocity or the liquid handling capacity of the plate) 6. Estimation of the column height or the number of plates (for a packed column, the column height is obtained by multiplying the number of transfer units, obtained from a knowledge of equilibrium and operating data by the height of a transfer unit; for plate columns, the number of theoretical plates, often determined from the plot of equilibrium and operating lines, is divided by the estimated overall efficiency to give the number of actual plates, which in turn allows the column height to be estimated from the plate spacing) 7. Determination of pressure drop through the column (for packed columns, correlations dependent on packing type, column operating data, and physical properties of the constituents involved need to be available to estimate the pressure drop through the packing; for plate columns, the pressure drop per plate is obtained and multiplied by the number of plates)

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Unit Operations in Environmental Engineering

54.6 Summary of Key Equations The key equations for absorption and stripping calculations for tower height, including a summary of earlier material, are presented below. For packed tower absorption:

ln N OG

y1 mx2 y2 mx2

1

1 A

1 A

1 1 A

(54.12)

For stripping:

ln N OL

x2 ( y1 /m) (1 A) A x1 ( y1 /m) 1 A

(54.20)

where the subscripts 1 and 2 refer to bottom and top conditions, respectively. In addition, A = Lm/mGm and S = 1.0/A. For plate tower absorption:

y Np Np

log

1

mx0

1

y1 mx0

log A

1 A

1 A

(54.16)

Note: the term ln, rather than log, may also be employed in both the numerator and denominator. If A approached unity, Equation 54.16 becomes:

N

y N 1 y1 y1 mx0

(54.21)

or

y N 1 y1 y N 1 mx0

N N 1

(54.22)

Note that the subscripts 1 and N refer to top and bottom of the column, respectively.

Absorber Design and Performance Equations

567

For stripping in plate towers:

x0 ( y Np 1 /m) Np

log

x N ( y Np 1 /m)

1

1 S

1 S

log S

(54.23)

or

x0 x Np x0 ( y Np 1 /m)

S Np 1 S S Np 1 1

(54.25)

If S is approximately 1.0, one may use either of the following equations:

Np

x0 x Np x Np ( y Np 1 /m)

(54.24)

or

x0 x Np x0 ( y Np 1 /m)

Np Np 1

(54.25)

References 1. Theodore, L., and Buonicore, A., Control of Gaseous Emissions, US Environmental Protection Agency Training Manual, Research Training Park, NC, 1982. 2. Theodore, L., and Barden, J., Mass Transfer Operations, A Theodore Tutorial, East Williston, NY, originally published by USEPA/APTI, RTP, NC, 1995. 3. U.S. Stoneware Co., Generalized Pressure Drop Correlation, Chart No. GR-109, Revision 4, U.S. Stoneware Co., Akron, OH, 1963. 4. Theodore, L., Engineering calculations: sizing packed-tower absorbers without data, Chem. Eng. Progress, 5, 18–19, 2005. 5. Chen, N., New equation gives tower diameter, Chem. Eng., 5, 1962. 6. Treybal, R., Mass Transfer Operations, 2nd Ed., McGraw-Hill, New York City, NY, 1967. 7. Henley, E., and Staffin, H., Stage Process Design, John Wiley & Sons, Hoboken, NY, 1963. 8. Sawistowski, H., and Smith, W., Mass Transfer Process Calculations, Interscience, New York City, NY, 1963. 9. Kremser, A., Theoretical analysis of absorption process, Nat’l Petrol. News, 22 (21), 42, 1930. 10. Souders, M., and Brown, G.G., Fundamental design of absorbing and stripping columns for complex vapors, Ind. Eng. Chem., 24, 519, 1932.

Unit Operations in Environmental Engineering, Louis Theodore, R. Ryan Dupont and Kumar Ganesan. © 2017 Scrivener Publishing LLC. Published 2017 by John Wiley & Sons, Inc.

55 Distillation Design and Performance Equations

55.1 Introduction In many respects, this chapter is an extension of Chapter 49. It is recommended that the reader refer to Chapter 49 before proceeding to the body of this chapter. When designing a binary distillation column, laborious tray-by-tray calculations may be circumvented by utilizing the McCabe-Thiele graphical method. The major simplifying assumptions which are generally employed with the McCabeThiele method are: 1. Each tray, a partial condenser and/or partial reboiler, all serve as theoretical stages 2. Constant molal overflow applies 3. Raoult’s law is valid While these assumptions are not necessary for completing the graphical sizing of a column, they vastly simplify quick-sizing procedures for ideal or near-ideal systems. Note that Assumption 3 is the assumption most subject to question; however, it may be circumvented by obtaining more accurate equilibrium data [1].

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The aforementioned assumptions allow one to determine several important design parameters by simply constructing a McCabe-Thiele diagram. These parameters include the number of theoretical stages, the minimum number of theoretical stages, and minimum reflux ratio [1-4]. This method is perhaps the easiest to learn and apply to binary systems since it does not require iterative calculations, nor does it include enthalpy balances (though it may be modified to include them when constant molal overflow is not assumed). In fact, this approach is fundamentally a convenient graphical representation of more drawn out calculations. Almost the entire chapter is devoted to the McCabe-Thiele binary distillation graphical method. The chapter concludes with a short discussion of packed column distillation.

55.2 Binary Distillation Design/The McCabe-Thiele Graphical Method 55.2.1

Calculation of Operating Lines

As previously discussed, an equilibrium curve may be developed to describe the equilibrium relationship between the liquid and vapor components leaving each stage of a distillation column. However, in the graphical design of a staged column, it is necessary to develop an operating line which relates the passing liquid and vapor streams between each stage. In effect, it is the operating line that provides a mathematical relationship describing the operating conditions within the column. The following analysis will develop operating lines for both the rectification section and the stripping section. To accomplish the following analysis, refer to Figure 55.1. An overall material balance is written for Envelope I in the rectification section as:

V=L+D

(55.1)

This equation relates the vapor, V, flowing up the column, the liquid reflux, L, flowing down the column, and the distillate, D, collected. A componential material balance for the light component, A, is written as

Vyn + 1 = Lxn+DxD

(55.2)

This can be rearranged to form the equation of a straight line, y = mx + b, as:

yn

1

L xn V

D xD V

where L/V = the slope of the operating line, dimensionless.

(55.3)

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571

D xD

I LC

n

III

Ln

Vn+1 n+1

F xF

m Lm

Vm+1

m+1

II

VR

LN

B xB

I - Material balance around rectification section II - Material balance around stripping section III - Material balance around column Figure 55.1 Material balances around a distillation column.

The value L/V is defined as the internal reflux ratio, which is the ratio of the liquid reflux molar flow (returned to the top of the column) to the vapor molar flow (exiting the top of the column). By substituting Equation 55.2 into Equation 55.3, then dividing both the numerator and denominator of the terms on the right-hand side by the distillate flow, D, and finally substituting in the definition of the reflux ratio, R = L/D one obtains:

yn

1

R xn R 1

xD R 1

which is the final result for the operating line of the rectification section.

(55.4)

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The corresponding operating line for the stripping section may be developed in a similar manner. The overall material balance around Envelope II in the stripping section is:

L V

B

(55.5)

A componential balance on the lighter component in the stripping section may be rearranged to form an equation analogous to that of Equation 55.3. Note once again that the overbars are reminders that the flow is occurring in the stripping section:

ym

1

L B xm xB V V

(55.6)

By performing similar substitutions, and by utilizing the definition of the boil-up ratio, RB V / B , an equation analogous to that of Equation 55.6 may be developed:

ym

1

RB 1 xm RB

xB RB

(55.7)

which is the final result for the operating line of the stripping section [1–4].

55.2.2 Thermal Condition of the Feed (q-factor) The thermal condition of the feed has a major impact on how a column operates. It has been found advantageous to represent the effect of the thermal condition of the feed graphically on a McCabe-Thiele diagram [5]. If the feed is subcooled, it will condense some of the vapor rising through the stripping section; if it is superheated, it will evaporate some of the liquid cascading down the rectifying section. In most applications, the feed is either a saturated liquid, saturated vapor, or a saturated two-phase mixture. Therefore, only these three scenarios will receive detailed treatment in this text. Figure 55.2 illustrates the operation of a standard feed tray with a saturated feed, where FL is the molar flow rate of the liquid portion of the feed and FV is the molar flow rate of the vapor portion of the feed. For the purposes of this discussion (as noted above), the feed stream may be saturated liquid (FV = 0), saturated vapor (FL = 0), or a saturated two-phase mixture (F = FL + FV). When the feed is a saturated liquid (FV = 0) and has an enthalpy equal to that of the mixture on the feed plate, the feed will flow completely into the stripping section such that V = V and L L FL . When the feed is a saturated vapor (FL = 0), the feed will completely flash into the rectification section such that L L and V V FV . When the feed is a saturated two-phase mixture (F = FL + FV), the resulting liquid and vapor streams leaving the feed tray are given by L L FL and V V FL , respectively.

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Rectification section L

V = V + FV

F = FL + FV

L = L + FL

V

Stripping section

Figure 55.2 Action on a feed tray: saturated feed.

One can now define a factor which may be used to quantitatively represent the thermal condition of the feed on a McCabe-Thiele diagram. An overall material balance around the feed tray is given by:

F V L V L

(55.8)

F (V V ) (L L)

(55.9)

Rearranging results in:

An enthalpy (H) balance, neglecting heat losses to the surroundings and any enthalpy of mixing effects, yields:

FH F VHV

LH L

VHV

LH L

(55.10)

where Hi = molar enthalpy of Stream i at the stream’s temperature and pressure about the feed tray. Since HV HV and H L H L , Equation 55.10 becomes:

FH F

HV (V V ) H L (L L)

(55.11)

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Substituting (V V ) from Equation 55.9 into Equation 55.4, one may rearrange the resulting equation to define the feed condition factor, q:

q

L L F

HV HV

HF HL

(55.12)

Note that the term q was employed earlier to represent the volumetric flowrate. The factor q is defined as the moles of liquid flowing in the stripping section which resulted from one mole of feed entering the column. For example, when the feed is a saturated liquid, each mole of feed entering the column adds directly to the stripping section’s liquid flow. Hence, the q factor is equal to unity. Conversely, when the feed is a saturated vapor, each mole of feed entering the column adds directly to the vapor flowing up the rectification section. In this case, none (of the one mole) of the feed adds to the stripping section’s liquid flow and the q factor is equal to zero. The L terms cancel by substituting L L FL into Equation 55.12, leaving the result that for any type of saturated feed, q = FL / F, which is equivalent to the fraction of the feed that is liquid. In the case of a subcooled liquid feed, the value of q is determined by:

q 1

CP , L Tb TF

(55.13)

where = the temperature of the feed, Tb = the bubble point temperature of the feed, CP , L average constant pressure heat capacity of the liquid feed, and λ= latent enthalpy of vaporization of the feed at TF. One must ensure that consistent units are employed when using Equation 55.13. Analogously, the value of q may be calculated for a superheated feed by Equation 55.14:

q

CP ,V TF Td

(55.14)

where Td = the dew point temperature of the feed, and CP ,V average constant pressure heat capacity of the vapor feed. Note that some practitioners prefer to employ a factor, f, defined as:

f=1–q

(55.15)

Values of q and f are summarized in Table 55.1. The term q also defines the point of intersection between the two operating lines on the McCabe-Thiele diagram. The operating line equations at the point of intersection are:

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Table 55.1 Values of q and f for the five general feed conditions. Feed condition

q

f

Subcooled liquid

>1

0,0,1

0,0, 1

0.9

q=1 0