Mass Transfers and Physical Data Estimation 9781119663263, 1119663261

Table of contents :
Cover
Half-Title Page
Title Page
Copyright Page
Contents
Preface
Introduction
1. Determination of Physical Data
1.1. Introduction
1.2. Estimating critical properties
1.2.1. Estimating critical temperature
1.2.2. Estimating critical pressure
1.2.3. Estimating the critical volume: Benson correlation (Benson, 1948)
1.2.4. Estimating the critical compressibility factor
1.3. Methods for estimating boiling temperature
1.4. Methods for estimating density
1.4.1. Estimating liquid densities
1.5. Methods for estimating viscosity
1.5.1. Estimating viscosities of pure liquids 1.5.2. Correlations for the viscosity of liquid mixtures1.5.3. Estimating gas viscosities
1.6. Methods for estimating specific heat
1.6.1. Heat capacities of petroleum oils
1.6.2. Heat capacities of petroleum vapors
1.6.3. Estimations for anthracite and bituminous coals
1.6.4. Heat capacities for cement, mortar and sand
1.6.5. Heat capacities of organic liquids
1.7. Estimating latent heat of vaporization
1.7.1. Rapid estimations
1.7.2. Calculating latent heat from critical data
1.7.3. Chen correlation
1.7.4. Calculations at different temperatures 1.10.6. Correlation of the Prandtl number1.10.7. Correlation for calculating the expansion coefficient
1.10.8. Correlation for calculating the saturating pressure
1.10.9. Correlation for calculating latent heat
1.11. Physical properties of air
1.11.1. Correlation of density
1.11.2. Heat capacity
1.11.3. Correlation of heat conductivity
1.11.4. Correlation of viscosity
1.11.5. Correlation of thermal diffusivity
1.11.6. Correlation of the Prandtl number
1.11.7. Correlation for calculating the expansion coefficient
2. Determinants and Parameters of Mass Transfer
2.1. Introduction 2.2. Relative transfer velocities2.2.1. Velocity relating to average mass velocity
2.2.2. Velocity relative to average molar velocity
2.3. Amount of matter transferred
2.4. Expressions of flux density
2.4.1. Total flux
2.4.2. Specific fluxes
2.5. Operations on diffusion flux densities
2.5.1. Total density as a function of the specific densities
2.5.2. Sum of mass densities with respect to v
2.5.3. Sum of molar flux densities with respect to v*
2.5.4. Sum of mass flux densities with respect to a mobile reference frame at v*
2.6. Relations between flux densities fi and ji

Citation preview

Mass Transfers and Physical Data Estimation

Energy Engineering Set coordinated by Abdelhanine Benallou

Volume 5

Mass Transfers and Physical Data Estimation

Abdelhanine Benallou

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Abdelhanine Benallou to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019943764 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-285-4

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1. Determination of Physical Data . . . . . . . . . . . . . . . . . .

1

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Estimating critical properties . . . . . . . . . . . . . . 1.2.1. Estimating critical temperature . . . . . . . . . . . 1.2.2. Estimating critical pressure . . . . . . . . . . . . . 1.2.3. Estimating the critical volume: Benson correlation (Benson, 1948) . . . . . . . . . . . . 1.2.4. Estimating the critical compressibility factor . . 1.3. Methods for estimating boiling temperature . . . . . 1.4. Methods for estimating density . . . . . . . . . . . . . 1.4.1. Estimating liquid densities . . . . . . . . . . . . . 1.5. Methods for estimating viscosity . . . . . . . . . . . . 1.5.1. Estimating viscosities of pure liquids . . . . . . . 1.5.2. Correlations for the viscosity of liquid mixtures . 1.5.3. Estimating gas viscosities . . . . . . . . . . . . . . 1.6. Methods for estimating specific heat . . . . . . . . . . 1.6.1. Heat capacities of petroleum oils . . . . . . . . . . 1.6.2. Heat capacities of petroleum vapors . . . . . . . . 1.6.3. Estimations for anthracite and bituminous coals. 1.6.4. Heat capacities for cement, mortar and sand . . . 1.6.5. Heat capacities of organic liquids . . . . . . . . . 1.7. Estimating latent heat of vaporization . . . . . . . . . 1.7.1. Rapid estimations . . . . . . . . . . . . . . . . . . . 1.7.2. Calculating latent heat from critical data . . . . .

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1 2 2 5

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8 10 11 14 14 15 15 17 18 19 19 20 20 21 21 22 22 23

vi

Mass Transfers and Physical Data Estimation

1.7.3. Chen correlation . . . . . . . . . . . . . . . . . . . . . . . 1.7.4. Calculations at different temperatures . . . . . . . . . . . 1.8. Estimating expansion coefficients β . . . . . . . . . . . . . . 1.9. Methods for estimating heat conductivity . . . . . . . . . . . 1.9.1. Heat conductivity of metals and alloys . . . . . . . . . . 1.9.2. Heat conductivity of wood . . . . . . . . . . . . . . . . . 1.9.3. Conductivity of chains of liquid hydrocarbons . . . . . 1.9.4. Conductivity of gases and vapors . . . . . . . . . . . . . 1.9.5. Conductivity of monatomic gases . . . . . . . . . . . . . 1.9.6. Conductivity of non-polar gases with linear molecules 1.10. Physical properties of water . . . . . . . . . . . . . . . . . . 1.10.1. Correlation of density. . . . . . . . . . . . . . . . . . . . 1.10.2. Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.3. Correlation of heat conductivity . . . . . . . . . . . . . 1.10.4. Correlation of viscosity . . . . . . . . . . . . . . . . . . 1.10.5. Correlation of thermal diffusivity. . . . . . . . . . . . . 1.10.6. Correlation of the Prandtl number . . . . . . . . . . . . 1.10.7. Correlation for calculating the expansion coefficient . 1.10.8. Correlation for calculating the saturating pressure . . . 1.10.9. Correlation for calculating latent heat . . . . . . . . . . 1.11. Physical properties of air . . . . . . . . . . . . . . . . . . . . 1.11.1. Correlation of density. . . . . . . . . . . . . . . . . . . . 1.11.2. Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.3. Correlation of heat conductivity . . . . . . . . . . . . . 1.11.4. Correlation of viscosity . . . . . . . . . . . . . . . . . . 1.11.5. Correlation of thermal diffusivity. . . . . . . . . . . . . 1.11.6. Correlation of the Prandtl number . . . . . . . . . . . . 1.11.7. Correlation for calculating the expansion coefficient .

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23 24 24 25 25 26 26 27 28 28 29 29 29 29 29 30 30 30 30 31 31 32 32 32 33 33 33 33

Chapter 2. Determinants and Parameters of Mass Transfer . . . . . .

35

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Relative transfer velocities . . . . . . . . . . . . . . . . . . 2.2.1. Velocity relating to average mass velocity . . . . . . 2.2.2. Velocity relative to average molar velocity. . . . . . 2.3. Amount of matter transferred . . . . . . . . . . . . . . . . 2.4. Expressions of flux density . . . . . . . . . . . . . . . . . 2.4.1. Total flux . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Specific fluxes . . . . . . . . . . . . . . . . . . . . . . . 2.5. Operations on diffusion flux densities . . . . . . . . . . . 2.5.1. Total density as a function of the specific densities . 2.5.2. Sum of mass densities with respect to v. . . . . . . .

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35 36 36 37 38 39 39 41 44 44 45

Contents

2.5.3. Sum of molar flux densities with respect to v* . 2.5.4. Sum of mass flux densities with respect to a mobile reference frame at v* . . . . . . . . . . . . . . . 2.6. Relations between flux densities fi and ji. . . . . . . 2.7. Relations between flux densities Fi and Ji* . . . . .

vii

. . . . . . . . . . . .

46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 47 47

Chapter 3. Fick’s First Law: Diffusion Coefficients . . . . . . . . . . . .

49

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Fick’s first law . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Expressing the flux density vector . . . . . . . . . . . . 3.2.2. Similarities to energy and momentum transfer laws . 3.2.3. Convective analogy. . . . . . . . . . . . . . . . . . . . . 3.3. Fick’s first law in different forms . . . . . . . . . . . . . . . 3.4. Determining diffusion coefficients from tabulated data . . 3.4.1. Gaseous binary diffusion coefficients . . . . . . . . . . 3.4.2. Illustration: diffusion coefficients of CO2 in air and in water vapor. . . . . . . . . . . . . . . . . . . . . . 3.4.3. Diffusion coefficients for liquid binaries . . . . . . . . 3.5. Estimating diffusion coefficients from correlations . . . . 3.5.1. Estimating gaseous binary diffusion coefficients . . . 3.5.2. Estimating diffusion coefficients of liquid binaries . . 3.6. Diffusion coefficients for multicomponent mixtures . . . 3.6.1. Stefan–Maxwell equation . . . . . . . . . . . . . . . . . 3.6.2. Effective diffusion coefficient for complex mixtures .

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49 50 50 51 52 52 53 53

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54 58 60 60 71 81 81 82

Chapter 4. Fick’s Second Law: Macroscopic Balances . . . . . . . . .

85

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Overall continuity equation . . . . . . . . . . . . . . . . . . . . 4.2.1. The accumulation term. . . . . . . . . . . . . . . . . . . . . 4.2.2. The generation term . . . . . . . . . . . . . . . . . . . . . . 4.2.3. The term I – O . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. The balance equation . . . . . . . . . . . . . . . . . . . . . . 4.2.5. The balance equation in Cartesian coordinates . . . . . . 4.3. Particular continuity equations . . . . . . . . . . . . . . . . . . 4.3.1. The term Ii – Oi . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. The accumulation term. . . . . . . . . . . . . . . . . . . . . 4.3.3. The generation term . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Continuity equations in molar terms . . . . . . . . . . . . . 4.4. Illustration: diffusion with chemical reaction . . . . . . . . . . 4.5. Illustration: diffusion of a component in a stagnant mixture . 4.6. Reading: background to Fick’s Laws. . . . . . . . . . . . . . .

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85 85 86 86 87 87 88 88 88 89 89 90 92 94 97

viii

Mass Transfers and Physical Data Estimation

Chapter 5. Exercises and Solutions . . . . . . . . . . . . . . . . . . . . . .

101

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205

Preface

“A river cuts through rock, not because of its power, but because of its persistence” Confucius (551–479 BC) For several years, I have cherished the wish of devoting enough time to the writing of a series of books on energy engineering. The reason is simple: for having practiced for years teaching as well as consulting in different areas ranging from energy planning to rational use of energy and renewable energies, I have always noted the lack of formal documentation in these fields to constitute a complete and coherent source of reference, both as a tool for teaching to be used by engineering professors and as a source of information summarizing, for engineering students and practicing engineers, the basic principles and the founding mechanisms of energy and mass transfers leading to calculation methods and design techniques. But between the teaching and research tasks (first as a teaching assistant at the University of California and later as a professor at the École des mines in Rabat, Morocco) and the consulting and management endeavors conducted in the private and in the public sectors, this wish remained for more than twenty years in my long list of priorities, without having the possibility to make its way up to the top. Only providence was able to unleash the constraints and provide enough time to achieve a lifetime objective. This led to a series consisting of nine volumes: – Volume 1: Energy and Mass Transfers; – Volume 2: Energy Transfers by Conduction; – Volume 3: Energy Transfers by Convection;

x

Mass Transfers and Physical Data Estimation

– Volume 4: Energy Transfers by Radiation; – Volume 5: Mass Transfers and Physical Data Estimation; – Volume 6: Design and Calculation of Heat Exchangers; – Volume 7: Solar Thermal Engineering; – Volume 8: Solar Photovoltaic Energy Engineering; – Volume 9: Rational Energy Use Engineering. The present book is the fifth volume of this series. It concerns the study of mass transfer from one system to another or between different parts of the same system. This volume also features a substantial section on physical data estimation, in view of the importance of the availability of information on physical properties for the completion of calculations. The usefulness of this book is clear considering that the subject of mass transfer has always been somewhat neglected in academic scientific literature. Indeed, there are very few textbooks dedicated to this subject. The objective is thus to provide students with the founding principles and basic data enabling an understanding of the determinants of this transfer, which govern industrial equipment design and sizing equations. Particular importance has been attached to the search for physical data, given that data availability is often one of the factors that can impede the realization of design calculations. A series of exercises is presented at the end of this document, aimed at helping students to implement the calculation techniques specific to mass transfer as rapidly as possible. These exercises are designed to closely correspond to real-life situations occurring in industrial practice or everyday life. Abdelhanine BENALLOU June 2019

Introduction

As we saw in Volume 1 of this series, most industrial manufacturing processes involve the elaboration of products using, in one stage or another, mass transfer. Indeed, the making of a desired end product is actually the outcome of a certain number of handling operations or stages in which raw materials are mixed and sometimes react to form a new mixture from which the desired product would need to be extracted. The purpose of this volume is to examine the way mass transfer occurs and to present the laws which govern it, the ultimate objective being to establish design equations which enable calculations of the exchanged mass fluxes. It should, however, be emphasized that for all of the calculations required for the quantification of transferred matter fluxes, access to reliable physicochemical data will be necessary. Indeed, the availability of physical data, such as viscosity or diffusion coefficients, is essential in order to be able to perform the calculations. It is for this reason, and in order to facilitate access to such information, that a collection of physical data is presented in Appendix 1. Yet, it has to be acknowledged that no matter the extent of the data collection effort made, it would realistically never be possible to get a hold on all the properties of all the chemical compounds and materials. For this reason, data estimation tools are needed to enable the engineer to determine the missing physical data which could be required in a given situation. Chapter 1 of this book is dedicated to presenting data estimation methods. It groups together the correlations that can be used to calculate the physicochemical properties of materials from basic data with a certain degree of accuracy.

xii

Mass Transfers and Physical Data Estimation

Subsequently, Chapter 2 of this book is dedicated to presenting parameters used in different situations where mass transfer occurs, including transfers in resting mixtures as well as matter exchanged in situations where the fluids considered are in motion. The characterization of a diffusion velocity, for example, becomes linked to the type of coordinates system (fixed or mobile) in which it will be expressed. The same applies for transfer fluxes. These fluxes are expressed using Fick’s first law (see Chapter 3), similarly to the way in which Fournier’s law expresses the thermal flux exchanged in conduction (see Volumes 1 and 2). Fick’s law makes it possible to understand mass transfer at the microscopic level. It states that the flux of matter transferred is proportional to the driving force expressed by the concentration gradient. But for the analysis of transfers in real systems, the development of macroscopic balances is required, enabling deduction of the fluxes exchanged. These balances are established in Chapter 4. They enable continuity equations to be developed for macroscopic systems with or without chemical reactions. Application of these principles is presented in the form of exercises, with solutions, in Chapter 5.

1 Determination of Physical Data

1.1. Introduction In engineering calculations, the availability of reliable data on the physicochemical properties of the materials used is of crucial importance. When these materials or components are fairly common, the values of these data can be directly found, in the literature, in tables generally established from experimental data. Whilst for the most common materials these values are easily accessible, this is not the case for new materials, nor for unusual components. For this reason, when searching for physical data, we will of course give priority to tabulated values when these can be found. A collection of such data – including those most commonly used in transfer processes – is presented in Appendix 1. However more extensive data can be found, where applicable, in the specialized literature (Sherwood, 1966; Chilton, 1973, etc.). Yet, in certain cases commonly encountered in engineering calculations, the data sought are not found in the tables. Under these conditions, it will be necessary to use data estimation methods developed by various authors. These often consist of correlations that have been established from experimental data. As a result, they do not give precise values, but rather estimations that present a given degree of accuracy. Such levels of accuracy are often sufficient for calculations, provided that their limits of use are respected when these are recommended by the authors of the correlations. This chapter presents estimation methods or correlations for determining the most common physical data, specifying the accuracy limits for each relation and, where applicable, the conditions of use.

Mass Transfers and Physical Data Estimation, First Edition. Abdelhanine Benallou. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

2

Mass Transfers and Physical Data Estimation

It is important to remember that these methods are only to be used when data cannot be found directly in tables or in specialized works. In many cases, however, they represent the only means of acquiring the necessary data. Nevertheless, in order to achieve the best results, the use of these methods must meet a number of conditions that define the limits of use. These conditions are generally specified by the authors, and particular attention must be paid to the units, as several of these correlations are not given in dimensionless form. 1.2. Estimating critical properties Critical parameters are extremely useful when determining physical data. The critical properties of the usual compounds are given in the tables in Appendix 1, presented at the end of this volume. Tabulated values should of course be used when they exist. If these cannot be found in the tables, they may be estimated from the correlations presented in this section. 1.2.1. Estimating critical temperature 1.2.1.1. Eduljee correlation (Gambill, 1959) The estimation of the critical temperature is given by the following relation:

Tc =

Tb , εTci 100

∑ i

where the εTci values are the Eduljee contributions for critical temperatures, defined in Table 1.1. Atoms or structures Atoms C H O N N Cl Br F

εTci

Notes

-55.32 28.52 1.59 30.6 -26.29 29.89 31.15 29.75

In amines In a ring

Determination of Physical Data

S Si Radical groups

OH

CO COOH COO CN Contribution of bonds Carbon-carbon double bond Carbon-carbon triple bond Nitrogen-carbon single bond N-H bond Contribution of rings 5-membered rings 6-membered rings Fusion of two rings

1.31 54.00

In silanes

31.63 35.62 34.0 32.72 31.40 30.1 29 28.52 31.63 35.94 4.12 33.83

In phenols In alcohols with carbon atoms ≤3 In alcohols with carbon atoms =4 In alcohols with carbon atoms =5 In alcohols with carbon atoms =6 In alcohols with carbon atoms =7 In alcohols with carbon atoms =8 In alcohols with carbon atoms >8 Ketone group Carboxyl group In esters Nitrile group

56.61 55.21

Up to 3 carbon atoms For 4 or more carbon atoms

3

112.9 -19.17 -18.37

In amines But take 0 when the N-C bond is in a ring Amine type

54.28 53.52 0.25

Table 1.1. Contributions of Eduljee, εTci, for critical pressures: atoms, radical groups, bonds and rings

This correlation gives estimations with an average accuracy of around 99% and a maximum deviation of the order of 6%. 1.2.1.2. Nokay correlation (Nokay, 1959) For pure hydrocarbons, the Nokay relation is simpler to use and provides estimations with accuracies around 98%, within a wide range of densities. Thus, when the boiling temperature and density are known, the critical temperature can be easily deduced by: = 1.2806 + 0.2985

+ 0.62164

,

4

Mass Transfers and Physical Data Estimation

where: Tb is the boiling temperature, in °R d is the specific density of the liquid at 60 °F Tc is expressed in °R 1.2.1.3. Guldberg–Guye rule: for inorganic compounds This correlation is used to obtain the critical temperature of inorganic compounds and non-metallic elements: = 0.614 The Guldberg–Guye rule is simple to use and provides estimations with accuracies of around 95%. 1.2.1.4. Gates and Thodos correlation: for metallic elements The Gates and Thodos correlation (Thodos, 1960) provides a rapid estimation of critical temperature based on boiling temperature: = 0,4

.

,

where the temperatures are expressed in degrees Rankine. The Gates and Thodos correlation provides estimations with accuracies close to 96%. 1.2.1.5. Correlation for mixtures: pseudocritical temperatures For mixtures of several components, Kay suggests weighting the critical temperature of component i, Tci, by its mole fraction, yi (Kay, 1936). The pseudocritical temperature, Tpc, of the mixture is then estimated as follows: n

Tpc =

∑ yiTci , i =1

where: yi is the mole fraction of component i Tci is the critical temperature of the component, i n is the number of components in the mixture

Determination of Physical Data

5

1.2.1.6. Correlation for hydrocarbon mixtures For hydrocarbon mixtures, Edmister proposes to weight the critical temperatures, Tci, by the mass fractions wi, instead of using the mole fractions yi; such that (Edmister, 1949): n

Tpc =

∑ wiTci , i =1

where: wi are the mass fractions of the different components of the mixture Tci is the critical temperature of the component, i n is the number of components in the mixture 1.2.1.7. Correlation for natural-gas mixtures (Edmister, 1948) For mixtures of natural gases, the following correlation can be used to obtain the pseudocritical temperature of the mixture directly from the density of the gas with respect to air: = 190 + 283.04

,

where: dg is the density of the gaseous mixture with respect to air Tpc is expressed in degrees Rankine 1.2.2. Estimating critical pressure 1.2.2.1. Eduljee correlation (Gambill, 1959) The critical pressures of organic compounds can be estimated from the Eduljee contributions defined in Table 1.2. The estimation of the critical pressure is given by the following relation: Pc =

104 M ⎡ ⎢ ⎢⎣

∑ i

⎤ ε Pi ⎥ ⎥⎦

2

6

Mass Transfers and Physical Data Estimation

where: Pc is in atmospheres M is the molar mass (in g/mol) ∑ is the sum of the Eduljee contributions, εPci , for critical pressures, defined in Table 1.1. These contributions can be positive or negative. Note that εPci must include both the contributions of atoms and molecular bonds, and that of rings, radical groups and bonds, as specified in Table 1.2. The Eduljee correlation gives estimations with an average accuracy greater than 96% and a maximum deviation of the order of 6%. Atoms or radical groups

εPi

Notes

Atoms C

-9.35

H

16.20

O

17.20

N

0.0

Cl

48.0

Br

68.8

F

39.9

S

27.8

Si

22.4

In silanes

Radical groups OH

In alcohols or phenols

CO

30.2

Ketone group

COOH

52.5

Carboxyl group

CN

Nitrile group

Contribution of bonds

Carbon-carbon double bond

28.6

For 2 carbon atoms in the chain

27.9

For 3 carbon atoms in the chain

25.2

For 4 carbon atoms in the chain

21.2

For 5 carbon atoms in the chain

16.4

For 6 carbon atoms in the chain

11.0

For 7 carbon atoms in the chain

5.3

For 8 carbon atoms in the chain

0.0

Beyond 8 carbon atoms in the chain

Determination of Physical Data

Carbon-carbon triple bond

51.1

As in acetyles

N-H bond

-3.15

Amine type

10.5

Saturated homo and heterocycles

6-membered rings

7.2

Saturated homo and heterocycles

Benzene ring

84.5

7

Contribution of rings 5-membered rings

Table 1.2. Contributions of Eduljee, εPi, for critical pressures: atoms, radical groups, bonds and rings

1.2.2.2. Herzog correlation (Herzog, 1944) Using this correlation, the critical pressure can be obtained from the critical volume and the critical temperature. =

.

,

where: Pc is in atmospheres Tc is in K Vc is in cm3/g-mole This correlation can of course only be used when critical temperature and critical volume values are available. The accuracy of the estimations, therefore, depends on the accuracies with which the values of Pc and Vc are obtained. 1.2.2.3. Correlation for mixtures (Prausnitz, 1958) For mixtures, Prausnitz proposes the following correlation, used to obtain the pseudocritical pressure from the pseudocritical temperature of the mixture and the critical volumes of the components: ⎡ R⎢ ⎢ Ppc = ⎣

∑

⎤⎡ n ⎤ yi Tci ⎥ ⎢ yi Zci ⎥ ⎥⎦ ⎢⎣ i =1 ⎥⎦ ,

∑

∑ yi Vci

8

Mass Transfers and Physical Data Estimation

where:

Ppc is the pseudocritical pressure of the mixture R is the perfect gas constant yi is the mole fraction of the component, i Tci is the critical temperature of the component, i Zci is the critical compressibility factor of the component, i Vci is the critical molar volume of the component, i The Prausnitz correlation provides estimations with an average accuracy of the order of 95%.

1.2.2.4. Correlation for natural gas mixtures (Edmister, 1948) For mixtures of natural gases, the following correlation can be used to obtain the pseudocritical pressure of the mixture directly from the density of the gaseous mixture with respect to air:

= 710 − 60.03 where:

dg is the density of the gaseous mixture with respect to air Ppc is expressed in atmospheres 1.2.3. Estimating the critical volume: Benson correlation (Benson, 1948) When the critical pressure and the molar volumes at boiling point are known, or when they have been determined elsewhere, the critical volume can be obtained from the following relation:

= 0.422

+ 1.981 ,

where:

Vc is in cm3/g-mole Pc is the critical pressure, in atmospheres Vb is the molar volume at normal boiling point, in cm3/g-mole

Determination of Physical Data

9

The Benson correlation for the critical volume gives estimations with an average accuracy of the order of 97% and a maximum deviation of the order of 10%. In order to use this correlation, the volume at normal boiling point needs to be known, however: Vb.

Vb can be calculated based on the density, point using the following relation:

Vb =

, of the liquid at its normal boiling

M ρb

In the case where is unavailable, Vb can also be estimated from the following relation (LeBas, 1915):

∑

Vb =

Vbi ,

Contributions

where the Vbi values are the elementary structural contributions to molar volume at normal boiling point. These contributions are presented in Table 1.3. Atoms and cycles Contributions of atoms As Bi Br C Cl Cr F Ge H Hg I

N

Vbi (in cm3/g-mol) 30.5 48.0 27.0 14.8 21.6 24.6 27.4 8.7 34.5 3.7 19.0 37.0 15.6 16.2 10.5 12.0 10.8

Notes

Terminal, R-Cl Central, R-CHCl-R

Double bond Triple bond, as in nitriles In primary amines, R-NH2 In secondary amines, R2-NH In tertiary amines, R3-NH

10

Mass Transfers and Physical Data Estimation

O

7.4 9.1 9.9 11.0 12.0 8.3

Except in the cases indicated below In methyl esters In methyl ethers In esters and higher ethers In acids Bonded with S, P or N

-6.0 -8.5 -11.5

3-membered rings 4-membered rings 5-membered rings 6-membered rings, as in benzene, cyclohexane and pyridine Naphthalene Anthracene

Contribution of rings

O

-15.0 -30.0 -47.5

Table 1.3. Elementary contributions to molar volume at normal boiling point: atoms and cycles

1.2.4. Estimating the critical compressibility factor 1.2.4.1. Lydersen correlation (Chilton, 1973) The critical compressibility factor can be estimated from the following relation:

=

1 3.43 + 0.0067 ⋅

Where Δ Hv is the latent heat at boiling point, expressed in kcal/g-mole. This correlation provides estimations with an average accuracy of the order of 96%. In contrast, significant deviations (of the order of 30%) are encountered in the case of polar compounds such as organic acids and nitriles.

1.2.4.2. Edmister correlation (Edmister, 1958) In the case where the critical pressure, Pc, and the critical temperature, Tc, are known or have been estimated elsewhere, the critical compressibility factor, Zc, can be estimated from the following relation:

= 0.371 − 0.0343 −1

Determination of Physical Data

11

where:

Pc is the critical pressure, expressed in atmospheres Tc is the critical temperature, expressed in K Tb is the temperature at normal boiling point, expressed in K 1.3. Methods for estimating boiling temperature Boiling-temperature values are presented in the tables in Appendix 1 for the usual compounds. When the required boiling temperatures cannot be found in the tables, they may be estimated from the following correlation (Meissner, 1949), with an average accuracy of 98% and a maximum deviation not exceeding 7%.

=

.

637 ∑

+

∑

where:

Tb is the boiling temperature, in K Constant B is given in Table 1.4 for the different compounds

Ri is the contribution to the refractive index by atom i or structure i (see Table 1.5) Pi is the contribution to the Parachor index by atom i or structure i (see Table 1.6) Type of components

B

Notes

Acids

28,000

Monocarboxylic

Alcohols

16,500

Monohydroxy, including phenol, cresols, etc.

Amines

6,500

Primary

2,000

Secondary

-3,000

Tertiary

15,000

Derived from monocarboxylic acids and monohydroxy alcohols

30,000

Derived from dibasic acids and monohydroxy alcohols

Esters

Ethers and mercaptans

4,000

12

Mass Transfers and Physical Data Estimation

Hydrocarbons

-500

Acetylenic

-2,500

Aromatic

-2,500

Paraffinic and naphthenic

-4,500

Olefinic

Ketones

15,000

Normal monochlorinated paraffins

4,000

Nitriles

20,000 Table 1.4. Values of constant B for different types of component (Gambill, 1957)

Atoms and structures Contributions of atoms C H -CH2O O2 F Cl Br I N

S Contribution of cycles All cycles Contribution of bonds Double bonds Triple bonds

Ri 2.418 1.100 4.618 1.525 1.643 2.211 3.376 0.950 5.967 8.865 13.900 2.322 2.502 2.840 5.516 7.69 7.97 7.91 8.11

Notes

Hydroxyl In ethers and esters Carboxyl Esters

Primary amines Secondary amines Tertiary amines Nitrile As in S-H As in R-S-R As in R-CN-S As in R-SS-R

0 1.733 2.398

Table 1.5. Contributions to the refractive index: atoms and structures (Gambill, 1957)

Determination of Physical Data

Atoms and structures

Pi

Contributions of atoms C

47.6

H

24.7

O

36.2

N

41.9

S

67.7

Cl

62.0

Si

79.2

Be

59.1

Contributions of atoms B

53.4

He

19.0

F

30.5

Ne

24.8

P

73.5

A

56.3

Ge

93.3

As

87.6

Se

81.9

Br

76.1

Kr

70.3

Sn

116.0

Sb

110.3

Te

104.6

I

98.9

Xe

93.2

Pb

113.8

Bi

108.1

Po

102.4

Rn

90.9

Al

96.7

Contribution of bonds For each bond* *

-19.0

Multiply the number of bonds by -19

Table 1.6. Contributions to the Parachor index: atoms and structures (Gambill, 1952)

13

14

Mass Transfers and Physical Data Estimation

1.4. Methods for estimating density The density values for the usual compounds are presented in the tables in Appendix 1. When the required densities cannot be found in the tables, they may be estimated from the correlations presented in this section.

1.4.1. Estimating liquid densities 1.4.1.1. Benson correlation (Benson, 1948) At normal boiling point, the density of a liquid can be estimated from the critical data as follows:

=

1.981 + 0.422

(

)

where:

Pc is the critical pressure, in atmospheres ρc is the critical density ρb is the density at normal boiling point

1.4.1.2. Goyal correlation (Doraiswamy, 1966) =

0.0653 .

− 0.09

where:

M is the molar mass, in g Pc is the critical pressure, in atmospheres Tc is the critical temperature, in K T R is the reduced temperature,

=

ρL is expressed in g/ml

1.4.1.3. Estimating gas densities The density of a gas can be deduced from the equation giving its molar volume:

=

hence:

=

Determination of Physical Data

15

where:

M is the molar mass, in g P is the pressure, in atmospheres T is the temperature, in K R is the perfect gas constant z is the compressibility function 1.5. Methods for estimating viscosity Viscosity data for the usual compounds are presented in the tables in Appendix 1. When the required viscosities are not available in the tables, they may be estimated from the following correlations.

1.5.1. Estimating viscosities of pure liquids 1.5.1.1. Modified Arrhenius relation (Partington, 1949) The viscosity of an organic or non-organic liquid, at normal boiling point, can be estimated as follows:

= 0.324

,

where:

ρ b is the density of the liquid considered at boiling temperature, in g/cm3 μ b is the viscosity at normal boiling point, expressed in centipoises This correlation gives estimations with an average accuracy of around 83% and a maximum deviation of the order of 39%.

1.5.1.2. Thomas correlation (Thomas, 1946) At temperatures other than boiling temperature, viscosity can be estimated from the following relation:

= 0.1167

.

10

,

16

Mass Transfers and Physical Data Estimation

where: 3 ρ is the density of the liquid considered, in g/cm

μ is the viscosity, expressed in centipoises

A is the constant, which depends on the atomic structure of the liquid considered Tr is the reduced temperature:

=

, where Tc is the critical temperature

Constant A is calculated by adding together all of the structural contributions corresponding to the atoms or bonds that constitute the component considered:

A=

∑

Ai

Contributions

The Ai contributions are presented in Table 1.7 for the different elements making up the molecular structure of the component concerned. Atoms or radical groups

Ai

Notes

Atoms C

-0.462

H

0.249

O

0.054

Cl

0.340

Br

0.326

I

0.335

S

0.043

Radical groups C5H6

0.385

CO

0.105

Ketones and esters

CN

0.381

Cyanides

Contributions of bonds Double bond

0.478

Table 1.7. Structural contributions for calculation of A: atoms, radical groups and rings

Determination of Physical Data

17

For most liquids, this correlation gives estimations with an accuracy of the order of 95%.

1.5.2. Correlations for the viscosity of liquid mixtures 1.5.2.1. For miscible liquid mixtures For a binary mixture of known composition, viscosity can be estimated from the following relation (Monroe, 1917):

=

+

where: is the viscosity mass of the mixture considered and

are the viscosities of components 1 and 2

xi is the mole fraction of component i (i = 1 or 2) This correlation gives the best representations for non-electrolyte mixtures. In this case, the average accuracy is 97% and the maximum deviation recorded is 9%.

1.5.2.2. For immiscible liquid mixtures For an immiscible mixture with known continuous and dispersed viscosities, it is possible to use Taylor’s relation (Taylor, 1932):

= 1 + 2.5

phase

+ 0.4 +

where: Subscripts c and d designate the continuous and dispersed phases, respectively is the volume fraction of the dispersed phase. This correlation gives the best results for mixtures with fairly low dispersed-phase volume fractions: ≤ 3% For mixtures with > 3%, the following correlation, developed by Olney and Carlson (Carlson, 1947), is more suitable:

=

+

18

Mass Transfers and Physical Data Estimation

1.5.3. Estimating gas viscosities 1.5.3.1. Bromley and Wilke relation When the critical volume and temperature are known or have been determined elsewhere, the viscosity of gases can be estimated as follows (Wilke, 1951):

=

33.3

( )

where:

( ) is the reduced temperature function, defined by: ( ) = 1,058

,

−(

, ,

( ,

) ,

)

μg is the gas viscosity, expressed in micropoises

Vc is the critical volume, in cm3/g-mole Tc is the critical temperature, in K Tr is the reduced temperature:

=

This correlation gives estimations with an average accuracy of around 97% and a maximum deviation of the order of 13%.

1.5.3.2. Arnold relation When the critical volume and/or the critical temperature are not known, but the boiling temperature is, or can be determined elsewhere, the viscosity of the gases at temperature T can be estimated as follows (Arnold, 1933):

=

27.0 + 1.47

where: μg is the gas viscosity, expressed in micropoises

T and Tb are expressed in K Vb is the molar volume at normal boiling point, expressed in cm3/g-mole

Determination of Physical Data

19

This correlation gives estimations with an average accuracy of around 94% and a maximum deviation of the order of 15%.

1.5.3.3. Relations for gaseous mixtures Herning and Zipperer propose the following relation (Zipperer, 1936):

=

∑ ∑

. .

where:

n is the number of components

μi is the viscosity of component i yi is the mole fraction of component i Mi is the molar mass of component i This correlation gives the best representations for mixtures containing no hydrogen or with a hydrogen composition of less than 25%. In this case, the average accuracy is 98.5% and the maximum deviation recorded is below 5%. In the case of mixtures containing more than 25% hydrogen, we refer to Wilke’s relation (Wilke, 1950), even if it is much more complicated to use, or to Brokaw’s relation (Brokaw, 1968), which requires knowledge of dipole moments for polar compounds.

1.6. Methods for estimating specific heat Data on specific heats (also known as sensible heat or heat capacities) are presented in the tables in Appendix 1 for the usual compounds. The following sections present estimation methods to be used when no tabulated experimental value can be found.

1.6.1. Heat capacities of petroleum oils The Cragoe correlation provides estimations of the heat capacity of petroleum oils as a function of the temperature and density of the liquid considered (Cragoe, 1929):

=

38.8 ⋅ 10

+ 45 ⋅ 10 .

20

Mass Transfers and Physical Data Estimation

where:

Cp is expressed in Btu/lb °F d is the specific density of liquid oil at 60 °F The Cragoe correlation for petroleum oils gives very good results in fairly wide temperature and density areas: temperatures between 32 °F and 400 °F and densities between 0.75 and 0.96. Estimations reach accuracies of the order 95% for the extended temperature range of 400 °F to 750 °F.

1.6.2. Heat capacities of petroleum vapors The Bahlke and Kay correlation provides estimations of the heat capacity of petroleum vapors, as a function of temperature and density of the corresponding liquid (Kay, 1929):

Cp =

[ 4 − d] [T + 670] 6450

,

where:

Cp is expressed in Btu/lb °F t is in °F d is the specific density, at 60 °F, of the liquid corresponding to the vapors considered The Bahlke and Kay correlation for petroleum oils gives very good results in fairly wide temperature and density areas: dew-point temperatures up to 660 °F and densities between 0.68 and 0.90. The average accuracy of the estimations is of the order of 98%.

1.6.3. Estimations for anthracite and bituminous coals The Clendenin correlation provides estimations of the heat capacity as a function of the temperature and mass fraction of the volatile products contained in the anthracite or bituminous coal considered (Clendenin, 1949):

= 0.2 + 8.8 ⋅ 10

+ 1.5 ⋅ 10

Determination of Physical Data

21

where:

w is the mass fraction of the volatile products contained in anthracite or in bituminous coal, expressed in %. Cp is expressed in Btu/lb °F T is expressed in °C The Clendenin correlation provides estimations with an accuracy of around 80%.

1.6.4. Heat capacities for cement, mortar and sand For this type of material, the Cragoe correlation provides estimations of heat capacity as a function of temperature (Cragoe, 1929):

= 0.18 + 6 ⋅ 10 where:

Cp is expressed in Btu/lb °F T is expressed in °F The Cragoe correlation provides estimations with an accuracy of around 85%.

1.6.5. Heat capacities of organic liquids The Johnson and Huang correlation provides estimations of heat capacity and organic liquids at 20 °C, based on the sum of the individual contributions, which are presented in Table 1.8. This method provides estimations with an average accuracy of the order of 95%, and a maximum deviation of 16%. Thus, the heat capacity of an organic liquid is obtained from the following relation (Huang, 1955):

Cp =

∑

Cpi

Contributions

where Cp is expressed in cal/g-mole °C. The Cpi values are given in Table 1.8 for different organic groups.

22

Mass Transfers and Physical Data Estimation

Organic groups

Cpi in cal/g-mole °C

Notes

Contributions of atoms C-H

5.4

-CH2-

6.3

CH3-

9.9

-COOH

19.1

-COO

14.5

Ester type

CO

14.7

Ketone type

-CN

13.9

-OH

11.0

-NH2

15.2

-Cl

8.6

-Br

3.7

-NO2

15.3

-O-

8.4

-S-

10.6

C6H5-

30.5

Amine type

Table 1.8. Elementary contributions of the different organic groups to heat capacity

1.7. Estimating latent heat of vaporization Latent heats of vaporization at boiling point (often noted ΔHv or Lv) are presented in the tables in Appendix 1 for the usual compounds. The following sections present estimation methods to be used when no tabulated experimental value can be found.

1.7.1. Rapid estimations When rapid estimations are required for the calculation of orders of magnitude, and when the boiling temperature is known, the following relation can be used:

= 21

Determination of Physical Data

23

where:

Lv is expressed in cal/g-mole Tb is in K This correlation presents the definite advantage of being simple to use. However it gives fairly approximate results: the average accuracy of the estimations can be as low as 70%.

1.7.2. Calculating latent heat from critical data When the critical data are available or have been estimated elsewhere, the following relation may be used (Giacalone, 1951):

=

⋅ −

( )

where:

Tc and Pc are the critical temperature and pressure, respectively Tb is the boiling temperature R is the perfect gas constant This correlation gives very good results: an average estimation accuracy of the order of 97% and a maximum deviation not exceeding 10%.

1.7.3. Chen correlation The Chen correlation also requires knowledge of the critical temperature and pressure. It gives the latent heat as a function of the boiling temperature, as follows (Chen, 1965):

=

7.9

− 7.82 + 7.11 ⋅ 1.07 −

( )

where: Tbr is the reduced boiling temperature, defined by:

Tb is the boiling temperature, in K

=

24

Mass Transfers and Physical Data Estimation

Tc is the critical temperature, in K Pc is the critical pressure, atm This correlation gives excellent results with an average estimation accuracy of the order of 98.2% and a maximum deviation not exceeding 6%. Its use nevertheless requires knowledge of the boiling temperature and critical temperature, as well as the critical pressure.

1.7.4. Calculations at different temperatures When a latent heat is known at a given temperature, T1, its extrapolation at other temperatures, T2, can be obtained by the following relation (Watson, 1943):

=

− −

.

where: ΔHv2 is the latent heat at T2 ΔHv1 is the latent heat at T1

Tc is the critical temperature 1.8. Estimating expansion coefficients β By definition:

β=

1 ⎛ ∂V ⎞ ⎜ ⎟ V ⎝ ∂T ⎠P

The expansion coefficient may be estimated within a given temperature interval, , , by the following relation:

β=

ρ12 − ρ22 2ρ1ρ2 [ T2 − T1 ]

This relation gives an estimation of β on density to be available: ρ1 and ρ2:

,

, but it requires data on the

Determination of Physical Data

25

When this type of data cannot be acquired, the Smith relation can be used (Smith, 1954):

=

0.04314 . −

where:

Tc is the critical temperature, in K T is also expressed in K The Smith correlation gives estimations with an average accuracy of 95% for organic liquids. The maximum deviations are encountered for polar liquids (35%).

1.9. Methods for estimating heat conductivity The heat conductivities (also known as heat conductibilities, conduction coefficients or k factors) are presented in the tables in Appendix 1 for the usual compounds. More extensive data have been tabulated by Powell (Powell, 1965) and by Grosse (Grosse, 1966) for metals in solid or liquid state. The following sections present estimation methods to be used when no tabulated experimental value can be found.

1.9.1. Heat conductivity of metals and alloys Both in liquid and solid state, the heat conductivity of metals and alloys can be estimated by the Ewing relation (Ewing, 1957):

2 ⋅ 10 = 2.61 ⋅ 10

−

+

97

where: λ is in W/cm °C

T is in K ρe is the electrical resistivity, in Ω/cm

Cp is in cal/g °C ρ is in g/cm3

M is in g/g-atom or in g/g-mole: for alloys, use the average of the atomic masses

26

Mass Transfers and Physical Data Estimation

For metals and alloys, this correlation gives an average accuracy of 90%, both for liquid state and for solids.

1.9.2. Heat conductivity of wood Several varieties of wood have been analyzed to establish correlations enabling calculation of the heat conductibilities at different humidity levels (Forest Products Lab, 1952).

1.9.2.1. Wood with humidity levels below 40% = 13.75 ⋅ 10

+

11.59 ⋅ 10

+ 2.33 ⋅ 10

where: λ is in Btu/(h)(ft)(°F) ρ is in g/cm3 τH is the average humidity level

1.9.2.2. Wood with humidity levels above 40% = 13.75 ⋅ 10

+

11.59 ⋅ 10

+ 3.16 ⋅ 10

where: λ is in Btu/(h)(ft)(°F) ρ is in g/cm3 τH is the average humidity level The two previous correlations are valid for fluxes perpendicular to the directions of the wood veins. It should be noted that, for fluxes in the direction of the veins, it would be necessary to multiply by 4 the heat conductivity values given by the above correlations.

1.9.3. Conductivity of chains of liquid hydrocarbons Quick, but fairly rough estimations (accuracies of the order of 87% and maximum deviation of 45%) can be obtained from the Weber relation (Weber, 1880):

Determination of Physical Data

=

27

0.869

where: λ is in Btu/(h)(ft)(°F)

Cp is in Btu/(lb)(°F) ρ is in g/cm3

M is in g/g-mole More accurate estimations can be obtained as a function of reduced temperature (Kingrea, 1962):

=

2.4 ⋅ 10

where:

Tr is the reduced temperature λ is in Btu/(h)(ft)(°F)

Cp is in Btu/(lb)(°F) ρ is in g/cm3

M is in g/g-mole 1.9.4. Conductivity of gases and vapors Estimations of the heat conductivity of gases and vapors and low pressure may be obtained using the Eucken correlation (Eucken, 1913):

=

+

2.48

where: λ is in Btu/(h)(ft)(°F) μ is in Lb/(h)(ft)

28

Mass Transfers and Physical Data Estimation

Cp is in Btu/(lb)(°F) M is in g/g-mole The Eucken correlation gives an average accuracy of 87% with a maximum deviation of 25%.

1.9.5. Conductivity of monatomic gases The heat conductivity of monatomic gases at low pressures can be estimated from the Reid and Sherwood correlation (Sherwood, 1966):

=

2,4 + 0.016

where: λ is in Btu/(h)(ft)(°F) μ is in Lb/(h)(ft)

Cv is in Btu/(lb)(°F) M is in g/g-mole 1.9.6. Conductivity of non-polar gases with linear molecules At low pressures, this conductivity can be estimated from the following correlation:

=

1.3

+ 3.40 −

where: λ is in Btu/(h)(ft)(°F) μ is in Lb/(h)(ft)

Cv is in Btu/(lb)(°F) M is in g/g-mole

0.70

Determination of Physical Data

29

1.10. Physical properties of water Extensive data for densities, heat capacities, heat conductivities, thermal diffusivities and Prandtl numbers between 0 °C and 300°C are presented in Appendix 1. When analytical representations are required, the following correlations may be used within temperature ranges of 0 °C to 100 °C, unless a different indication is specified.

1.10.1. Correlation of density ρ = -0.00380 θ2 – 0.0505 θ + 1002.6, where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C ρ is in kg m-3

1.10.2. Heat capacity Cp = 4,180 J kg-1 °C-1 Cp can be considered constant for temperatures 0 °C ≤ θ ≤ 100 °C. 1.10.3. Correlation of heat conductivity λ = -9.87 10-6 θ2 + 2.238 10-3θ + 0.5536, where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C λ is in W m-1 °C-1

1.10.4. Correlation of viscosity = 10

17.9 − 7.377 10 1 + 3.032 10

+ 3.354 10 + 8.765 10

30

Mass Transfers and Physical Data Estimation

where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C μ is in Pa.s

1.10.5. Correlation of thermal diffusivity For temperatures 0 °C ≤ θ ≤ 100 °C: α = 10-7θ - 0.00360θ + 1.340, where – θ is in °C and α is in m2 s-1.

1.10.6. Correlation of the Prandtl number =

13.06 + 1.387 + 3,7 ⋅ 10 1 + 12.407 ⋅ 10 + 52.97 ⋅ 10

where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C

Pr is dimensionless 1.10.7. Correlation for calculating the expansion coefficient = 10.5 ⋅ 10

+ 47.7 ⋅ 10

− 36.3 ⋅ 10

where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C the group

is in °C-1m-3

1.10.8. Correlation for calculating the saturating pressure = 20.3182 −

2,795 − 3,868 + 273

+ 273

Determination of Physical Data

31

where:

ps is in mmHg θ is the temperature in °C This correlation is valid for temperatures θ/5 °C < θ < 200 °C.

1.10.9. Correlation for calculating latent heat Lv = 2,495 -2.346 θ, where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C

Lv is the latent heat, expressed in kJ.kg-1 1.11. Physical properties of air The densities, heat capacities, heat conductivities, thermal diffusivities and Prandtl numbers for air are presented in Table 1.9, for temperatures between 0 °C and 300 °C. θ

ρ

-1

104μ

λ

Cp -3

-1

-1

-1

-1

107α

Pr

°C

kg.m

J.kg .°C

W.m .°C

Pa.s

m2.s-1

dimensionless

0

1.292

1,006

0.0242

1.72

1.86

0.72

20

1.204

1,006

0.0257

1.81

2.12

0.71

40

1.127

1,007

0.0272

1.90

2.40

0.70

60

1.059

1,008

0.0287

1.99

2.69

0.70

80

0.999

1,010

0.0302

2.09

3.00

0.70

100

0.946

1,012

0.0318

2.18

3.32

0.69

120

0.898

1,014

0.0333

2.27

3.66

0.69

140

0.854

1,016

0.0345

2.34

3.98

0.69

160

0.815

1,019

0.0359

2.42

4.32

0.69

180

0.779

1,022

0.0372

2.50

4.67

0.69

200

0.746

1,025

0.0386

2.57

5.05

0.68

32

Mass Transfers and Physical Data Estimation

220

0.700

1,028

0.0399

2.64

5.43

0.68

240

0.688

1,032

0.0412

2.72

5.80

0.68

260

0.662

1,036

0.0425

2.79

6.20

0.68

280

0.638

1,040

0.0437

2.86

6.59

0.68

300

0.616

1,045

0.0450

2.93

6.99

0.68

Table 1.9. Physical properties of air at different temperatures

When analytical representations are required, the following correlations may be used within temperature ranges of 0 °C to 100 °C, unless a different indication is specified.

1.11.1. Correlation of density =

353 + 273

where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C ρ is in kg m-3

1.11.2. Heat capacity Cp = 1008 J kg-1 °C-1 Cp can be considered constant for temperatures 0 °C ≤ θ ≤ 100 °C. 1.11.3. Correlation of heat conductivity λ = 7.57.10-5θ + 0.0242, where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C λ is in W m-1 °C-1

Determination of Physical Data

1.11.4. Correlation of viscosity μ = 10-5 (0.0046 θ + 1.7176), where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C μ is in Pa.s

1.11.5. Correlation of thermal diffusivity For temperatures 0 °C ≤ θ ≤ 100 °C: α = 10-5 (0.0146 θ + 1.8343), where θ is in °C and α is in m2 s-1.

1.11.6. Correlation of the Prandtl number Pr = -2.54.10-4 θ+ 0.7147, where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °C

Pr is dimensionless 1.11.7. Correlation for calculating the expansion coefficient

β≅

1 θ + 273

where: θ is the temperature in °C; 0 °C ≤ θ ≤ 100 °Cβ is in °-1

33

2 Determinants and Parameters of Mass Transfer

2.1. Introduction As we saw in Volume 1 of this series, the transfer of energy and mass is conditioned by the existence of a driving potential difference. Indeed, just as heat transfer requires a temperature gradient, matter transfer needs a mass potential difference expressed by a concentration gradient. This implies the existence of two areas within the system considered, which coexist at different concentrations. In this case, a chemical-species flux will travel from the area of high concentration to the area of lower concentration. Thus, mass or concentration potential difference is the driving force for mass transfer. It is one of the determinants which enables the transfer of mass. It is not the only one, however: unlike with heat transfer, it is to be noted that matter is transferred in solutions that can themselves be in motion, and where several species can be in migration. This implies that there is an overall motion, determined by the average velocity of the fluid, and specific motions due to the relative velocities of the different species. Therefore, another determinant of matter transfer will be the average transfer velocity and the relative velocities of the different species. This specific aspect requires precise definitions, given that it is necessary to differentiate the velocities and the fluxes expressed with respect to a fixed coordinates system from those relative to a moving coordinates system, which are expressed using relative velocities.

Mass Transfers and Physical Data Estimation, First Edition. Abdelhanine Benallou. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

36

Mass Transfers and Physical Data Estimation

Consequently, this specific aspect will need to be taken care of in the transferred flux calculations. It is therefore important to make sure to specify whether fluxes of matter are to be expressed with respect to a fixed coordinates system or relative to a system of coordinates moving at the average velocity. This chapter aims to specify the basis for calculating transfer velocities and fluxes, with a view to then presenting, in Chapter 3, the laws enabling the quantification of these transfers. 2.2. Relative transfer velocities In a mixture with several components, the latter are able to migrate with different velocities, owing on the one hand to the overall motion of the fluid and on the other hand to the mass potential differences. The particles of components in migration will then have differentiated motions. The differences in transfer velocities can be justified by the different sizes of the molecules in migration and by the potential gradients. Yet, in all cases, the motion of species i is characterized by its transfer velocity with respect to a fixed coordinates system, designated by vi. However, the motion of a species with respect to the rest of the fluid can only be captured using a relativist approach to express its migration with respect to the rest of the fluid. In this perspective, two types of relative velocities can be defined: the diffusion velocity of species i, with respect to average mass velocity, and the diffusion velocity of species i, with respect to average molar velocity. 2.2.1. Velocity relating to average mass velocity This is defined as the velocity with respect to a coordinates system that moves at average mass velocity, v, defined as the average velocity of the different species, weighted by their mass concentrations: n

∑ ρ iv i

v=

i=1 n

∑ρi

i=1

where: n is the number of components vi is the transfer velocity of species i with respect to a fixed coordinates system

Determinants and Parameters of Mass Transfer

37

ρi is the mass concentration of component i, that is to say, the mass, mi, of component i, divided by the total volume of the system: ρ i =

mi V

DEFINITION.– The diffusion velocity of species i with respect to the average mass velocity is given by: υi = vi – v, where vi is the transfer velocity of species i with respect to a fixed reference frame. Note that: mi = Ni Mi, where Ni is the number of moles of i and Mi is the molar mass of i. If n is the number of components, we have: n

n

mi

i=1

i=1

V

∑ρi = ∑

=

1

n

∑m i =

V i=1

m V

I.e.: n

∑ρi = ρ

i=1

2.2.2. Velocity relative to average molar velocity The diffusion velocity of species i with respect to the average molar velocity is given by: υi* = vi - v* The average molar velocity is defined by: n

*

∑ ci vi

v = i =1

n

∑ ci i =1

where: n is the number of components

38

Mass Transfers and Physical Data Estimation

vi is the transfer velocity of species i with respect to a fixed reference frame ci is the molar concentration of component i, that is, the number of moles of i, N divided by the total system volume: c i = i V Note that, if C designates the total concentration of the system, we have: n

∑ci =

i=1

1

n

∑Ni =

V i=1

N V

=C

It can also be noted that: ci =

Ni V

=

Ni Mi V Mi

=

mi VM i

=

ρi Mi

2.3. Amount of matter transferred The ultimate purpose of an analysis is to be able to determine the amount of matter crossing a given area. In practice, the passage rate of the entirety of the matter may be examined, without differentiating between its different components. In this case, the amount sought is the flux φ defined below. DEFINITION.– The flux of matter, φ, crossing area S per unit time is defined as follows, from the flux density, ϕ, that crosses this surface:

JG G φ = ∫∫ ϕ ⋅ dS S

Yet very often, the value sought is the flux of a given component migrating across a transfer area, S. This involves determining the specific flux, φi, of species i, defined below. DEFINITION.– The amount, φi, of species i crossing surface S per unit time is defined from the flux density, ϕi, that crosses this surface as follows:

JJG G φ i = ∫∫ ϕ i ⋅ dS S

Determinants and Parameters of Mass Transfer

39

Thus, in order to know the matter fluxes, the flux densities need to be determined. However, given the diverse nature of the benchmarks (fixed reference frame or relative velocities) and of the possible measurements (molar or mass), particular rigor needs to be applied with regards to the notations used and their definitions. The following sections place great importance in these definitions with a view to facilitating subsequent processing and avoiding any confusion. 2.4. Expressions of flux density In the developments presented below, particular emphasis is placed on the expressions of the flux densities, considering that when multiplied by the transfer areas, the latter provide the flux values. Moreover, the flux densities may either be related to a fixed reference frame, or expressed as a function of the average molar or mass velocities. 2.4.1. Total flux Generally speaking, the value of interest is the flux of a determined component, i. This is known as the flux specific to component i (see section 2.4.2). Yet, the flux in itself provides information on the overall transition rate of matter across a transfer area. The overall or total flux does not differentiate between the different components: it gives all of the transferred matter, for all components at once. The amount of matter per unit time and per unit transfer area is expressed by the total flux density. The latter can be sought in mass terms (g/h/m2) or in molar terms (mol/h/m2). It can be expressed with respect to a fixed reference frame or with respect to the average velocity. To facilitate presentations and help the reader become accustomed to the numerous notations and references that may be useful, depending on the scenario, Table 2.1 presents the different notations, which are defined more precisely in the following sections.

References

Reference frame Fixed

Mobile at velocity v

Mobile at velocity v*

Mass

f

j

j*

Molar

F

J

J

Table 2.1. Different notations of total flux according to the coordinate system

40

Mass Transfers and Physical Data Estimation

2.4.1.1. Total mass flux density with respect to a fixed reference frame: f The mass flux density with respect to a fixed reference frame is noted n. It is defined by: f = ρ v, where: v is the average mass transfer velocity with respect to a fixed reference frame n

∑ ρ iv i

v has been defined by: v = i=1n

∑ρi

i=1

ρi is the mass concentration of component i, that is to say, the mass of mi component i, divided by the total volume of the system: ρ i = with respect to a V fixed reference frame or with respect to average velocity 2.4.1.2. Total molar flux density with respect to a fixed reference frame: F The molar flux density with respect to a fixed reference frame is noted F. It is defined by: F = C v*, where: C is the overall molar concentration of the system v* is the average molar transfer velocity with respect to a fixed reference frame n

*

∑ ci vi

v has been defined above by: v = i =1 *

n

∑ ci i =1

ci is the molar concentration of component i, that is, the number of moles of N component i, divided by the total system volume: c i = i V

Determinants and Parameters of Mass Transfer

41

2.4.2. Specific fluxes Specific fluxes are those relating to each of the components of a given mixture. These fluxes are of the most interest in engineering calculations, given that they express the passage rates of a species, i, across a given transfer area. For a mixture of n components, there are n specific flux densities. In addition, these flux densities can be expressed in mass or molar terms. They can also be related to a fixed reference frame or to coordinates that move at average molar or mass velocity. Table 2.2 indicates the diversity of the notations and reference frames that can be encountered in the literature. Yet it should be pointed out that the relative mass or molar flux densities (in relation to average mass or molar velocity) are the most commonly used, given that they enable quantification of the mass transfers of a species, i, which results from the driving potential difference, rather than from the movement as a whole. Reference frame

References

Fixed

Mobile at velocity v

Mobile at velocity v*

Mass

f i; 1 ≤ i ≤ n

j i; 1 ≤ i ≤ n

j i *; 1 ≤ i ≤ n

Molar

F i; 1 ≤ i ≤ n

J i; 1 ≤ i ≤ n

J i *; 1 ≤ i ≤ n

Table 2.2. Specific fluxes in different coordinate systems

2.4.2.1. Mass flux density of species i with respect to a fixed reference frame: fi It is defined by: fi = ρivi, where: fi is the mass flux density of component i with respect to a fixed reference frame ρi is the mass concentration of component i, that is, the mass of component i, divided by the total volume of the system: ρ i =

mi V

vi is the transfer velocity of component i with respect to a fixed reference frame

42

Mass Transfers and Physical Data Estimation

2.4.2.2. Molar flux density of species i with respect to a fixed reference frame: Fi It is defined by: Fi = civi*, where: vi* is the average molar transfer velocity of species i with respect to a fixed reference frame ci is the molar concentration of component i, that is, the number of moles of N component i, divided by the total system volume: c i = i V 2.4.2.3. Mass flux density of species i with respect to v: ji This density expresses the relative motion of species i with respect to a mobile reference frame, at average mass velocity, v: it is given by: ji = ρi(vi - v), where: ji is the mass flux density of component i with respect to mass velocity, v ρi is the mass concentration of component i, that is to say, the mass of component i, divided by the total volume of the system: ρ i =

mi V

vi is the transfer velocity of component i with respect to a fixed reference frame v is the average mass transfer velocity with respect to a fixed reference frame, n

*

∑ ci vi

which has been defined above by: v = i =1

n

∑ ci i =1

2.4.2.4. Molar flux density of species i with respect to v: Ji It is defined by: Ji = ci(vi - v),

Determinants and Parameters of Mass Transfer

43

where: Ji is the molar flux of component i with respect to the average mass velocity ci is the molar concentration of component i, that is to say, the number of moles N of i, divided by the total volume of the system: c i = i V vi is the transfer velocity of component i with respect to a fixed reference frame v is the average mass transfer velocity with respect to a fixed reference frame, as defined above 2.4.2.5. Mass flux density of species i with respect to v*: ji* It is defined by: ji* = ρi(vi - v*), where: ji* is the mass flux density of component i with respect to the average molar velocity, v* ρi is the mass concentration of component i, that is to say, the mass of component i, divided by the total volume of the system: ρ i =

mi V

vi is the transfer velocity of component i with respect to a fixed reference frame v* is the average molar velocity 2.4.2.6. Molar flux density of species i with respect to v*: Ji* It is defined by: Ji* = ci(vi - v*), where: Ji* is the molar flux of component i with respect to the average molar velocity ci is the molar concentration of component i vi is the transfer velocity of component i with respect to a fixed reference frame

44

Mass Transfers and Physical Data Estimation

v* is the average molar velocity with respect to a fixed reference frame: n

*

∑ ci vi

v = i =1

n

∑ ci i =1

Table 2.3 presents a summary of the expressions of the molar or mass fluxes expressed in a fixed reference frame, or with respect to mobile coordinates at v or v*.

Fixed

Reference frame Mobile at velocity v

Mobile at velocity v*

Mass

fi = ρivi

ji = ρi(vi - v)

ji* = ρi(vi - v*)

Molar

Fi = civi*

Ji = ci(vi - v)

Ji* = ci(vi - v*)

1≤i≤n

1≤i≤n

1≤i≤n

References

1≤i≤n

1≤i≤n

1≤i≤n

Table 2.3. Specific molar or mass fluxes in different coordinate systems

2.5. Operations on diffusion flux densities 2.5.1. Total density as a function of the specific densities The total matter flux density is the sum of the specific densities:

F=

n

∑ Fi

i=1

n

*

∑ ci vi

Indeed: Φ = C v and v = i =1 *

n

∑ ci i =1

n

∑ c iv i

Hence: F = i=1n

∑ci

i=1

C.

.

Determinants and Parameters of Mass Transfer

45

n

Yet:

∑ c i = C.

i=1

n

n

i=1

i=1

Therefore: F =

∑ c iv i = ∑ F i . n

∑ ρ iv i .

NOTE.– It can similarly be demonstrated that f = cv =

i=1

2.5.2. Sum of mass densities with respect to v n

It can easily be shown that, for a mixture of n components:

∑ j i = 0.

i=1

Indeed, the mass flux density of species i with respect to average mass velocity, v, is given by: ji = ρi(vi - v). Hence:

n

n

n

i=1

i=1

i=1

⎛

n

⎞

⎝ i=1

⎠

∑ j i = ∑ ρ i ( v i − v ) = ∑ ρ iv i − ⎜ ∑ ρ i ⎟ v .

Yet, in the case of a mixture with n components, the average velocity, v, is given by: n

∑ ρ iv i

v=

i=1 n

∑ρi

i=1

n

Therefore:

I.e.:

n

n

i=1

i=1

⎛

n

∑ρ v ⎞ i=1 i i

∑ j i = ∑ ρ iv i − ⎜ ∑ ρ i ⎟ ⎝ i=1

n

n

n

i=1

i=1

i=1

⎠⎛ n ⎞ ⎜ ∑ ρ i⎟ ⎝ i=1 ⎠

.

∑ j i = ∑ ρ iv i − ∑ ρ iv i = 0 .

SPECIFIC CASE: BINARY MIXTURE.– For a binary (A, B), we will have: jB = -jA.

46

Mass Transfers and Physical Data Estimation

2.5.3. Sum of molar flux densities with respect to v* For a mixture with n components, the molar flux density with respect to v* is given by: Ji* = ci(vi - v*); 1 ≤ i ≤ n. The sum of the flux densities is then: n

∑

J*i =

∑

c i v i − v*

i=1

i=1

n

Hence:

n

∑

J*i =

i=1

n

∑ ci i=1

n

∑ Fi − v*C . i=1

n

Or:

∑ J*i = F − Cv* = 0 . i=1

2.5.4. Sum of mass flux densities with respect to a mobile reference frame at v* For a mixture with n components, the mass flux density with respect to v* is given by: ji* = ρi(vi - v*); 1 ≤ i ≤ n. The sum of the flux densities is then:

Or:

Yet:

n

n

i=1

i=1

⎛

n

⎞

∑ j *i = ∑ ρ iv i − ⎜ ∑ ρ i ⎟ v * . ⎝ i=1

⎠

n

∑ ρ i = ρ.

i=1

Hence:

n

n

i=1

i=1

∑ j *i = ∑ ρ iv i − ρv * .

n

n

i=1

i=1

(

)

∑ j *i = ∑ ρ i v i − v * .

Determinants and Parameters of Mass Transfer

Furthermore, based on the definition of v, we have:

Hence:

n

n

i=1

i=1

(

n

n

i=1

i=1

47

∑ ρ iv i = v ∑ ρ i .

)

∑ j *i = vρ − ρv * , i.e.: ∑ j *i = ρ v − v * .

2.6. Relations between flux densities fi and ji The flux density, ji, is given by: ji = ρi(vi-v). Hence: ji = ρi vi −

ρi ρ

n

∑ ρk v k . k =1

n

Or: ji = fi − ωi

ρ

∑ fk ; where ω is the mass fraction of i; ωi = ρi . i

k =1

Let this be written as follows: fi = ji + ωi f. Indeed, this relation expresses the fact that mass flux density with respect to a fixed reference frame, fi, is the result of the superposing of the following two phenomena: – the transfer of species i, generated by the overall motion, the latter defined by ωif; – the transfer of species i, occurring outside the overall motion. Moreover, this transfer is that expressed by the relative flux density, ji; that is, with respect to average velocity, v.

2.7. Relations between flux densities Fi and Ji* The development is similar to that presented in section 2.6, only here a molar reference is adopted, with the flux density, Ji*, given by:

Ji* = ci(v - v*) c Hence: J*i = ci vi − i c

n

∑ ck vk . k =1

48

Mass Transfers and Physical Data Estimation

Or: Fi = J*i + x i

n

N

∑ Fk ; where x is the molar fraction of i; xi = Ni . i

k =1

This relation shows that Fi (molar flux density of i with respect to a mobile reference frame at v*) is the sum of the following two components: – the transfer of species i, generated by the overall motion, the latter defined by n

xi

∑ Fk ; k =1

– the transfer of species i, occurring outside the overall motion. Moreover, this transfer is that expressed by the relative flux density, Ji*; that is, with respect to average velocity, v*. SPECIFIC CASE: BINARY MIXTURE.– For a binary (A, B), we will have: – in molar terms: FA = J*A + x A ( FA + FB ) ; – in mass terms: f A = jA + ωA ( f A + f B ) .

3 Fick’s First Law: Diffusion Coefficients

3.1. Introduction As we saw in Volume 1 of this series, whenever there is a mass potential difference there will be transfer of matter, whether or not the fluids are in motion. In the previous chapter we demonstrated that, generally, the flux generated by a potential difference is superposed onto transfer due to the overall motion. Thus, when a fluid is not in motion and is composed of two areas with different concentrations in a given component, we will observe the transfer of that component which will be induced by the difference in concentration only. Yet if the fluid sets in motion, a stationary observer will witness the transfer due to the overall motion together with the transfer generated by the potential difference. In this chapter, we will explore the laws governing mass transfer inside a homogeneous system, that is, composed of a mixture in a single phase, either gaseous or liquid. The system can be binary (with two components, A and B), or complex, that is, formed of several components. Transfers in such systems underlie those that take place in the osmosis and dialysis processes presented in Volume 1. Our aim in this chapter is to establish the expressions enabling calculation of the fluxes transferred, with a view to determining the concentration profiles in the systems considered.

Mass Transfers and Physical Data Estimation, First Edition. Abdelhanine Benallou. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

50

Mass Transfers and Physical Data Estimation

3.2. Fick’s first law For clarity of presentation, we will consider in this section a system containing two components A and B. We are specifically interested in the transfer of a given component, i. Fick’s first law actually states the following two simple facts related to transfer mechanisms (see Volume 1): (i) The existence of a transfer: when two areas containing components A and B in different concentrations are brought into contact, then their concentrations will tend to balance out. This means that there will be mass (A and/or B) passing from areas of high concentration to areas of low concentration. This quite simply reflects the first law of thermodynamics; (ii) the flux density, jA or JA*, of matter, with respect to a coordinates system moving at the fluid’s average velocity is proportional to the driving potential difference, i.e. to the concentration gradient. This in turn is simply analogous to the law stated by Fourier on heat conduction. Thus, as in the case of heat transfer by conduction, mass is transferred under the effect of a driving potential difference; but for mass transfer, it is the concentration difference (or gradient) that governs exchanges. 3.2.1. Expressing the flux density vector If we focus on component A, the concentration of A at point M(x, y, z) depends, in general, on x, on y, on z and on time t. Let us note it c(x, y, z, t). In molar terms, the flux density vector of transferred mass, with respect to a coordinates system that moves at v*, is three-dimensional. It is of the form:

⎡ * J A,x JJJG ⎢ ⎢ J *A = ⎢ J *A,y ⎢ * ⎢ J A,z ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

For a given velocity v*, the transferred-mass flux density vector will therefore depend on: – the nature of the medium (liquid or gas) and of molecules A and B that are interacting therein. This parameter is covered by the diffusion coefficient of A in mixture AB: DAB;

Fick’s First Law: Diffusion Coefficients

51

– the concentration gradient representing the mass driving potential difference. It is precisely this dependence that is translated mathematically by Fick’s first law in the simplest of forms, that is, a proportionality between the determining factors: the concentration gradient and the diffusion coefficient, DAB:

JJJG JJJJJJG J *A = − D AB grad c A or, when the overall concentration of the system, C, is constant: JJJG G G J *A = − D AB ∇ c A = − C D AB ∇ x A ,

(

)

(

)

where: xA is the molar fraction of A in the mixture:

=

The dimension of the diffusion coefficient of component A in mixture AB, DAB, is [L]2 [T]-1. 3.2.2. Similarities to energy and momentum transfer laws 3.2.2.1. Similarity to Fourier’s law Let us recall that the Fourier equation links thermal flux density to the temperature gradient as follows: G ϕ=− λ

G

( ∇T)

This presents total analogy with Fick’s law, where heat conductivity, λ, is the analog of the diffusion coefficient, DAB. 3.2.2.2. Similarity to Newton’s law for momentum transfer Let us recall that in fluid mechanics, the viscous stress tensor is linked to the velocity gradient by Newton’s law, which is written in the simplest form as follow:

⎛ ∂v ⎞ τxy = − μ ⎜ x ⎟ ⎝ ∂y ⎠ This represents total analogy with Fick’s law, where viscosity, μ, is the analog of the diffusion coefficient, DAB.

52

Mass Transfers and Physical Data Estimation

3.2.3. Convective analogy Let us recall here that the heat flux transferred by convection is greatly simplified by the introduction of the convection heat transfer coefficient, h. The convective heat flux density is then written in the form: ϕ = h ΔT In a similar manner, let us introduce the mass transfer convection coefficient, generally noted k. The convective mass flux is then given by: f = k Δc Coefficient k is generally linked to the diffusion coefficients, DAB, which apply to the system concerned. In problems where, in addition to dependence on time, the concentration distribution depends on a single dimension, z for example, the concentration is a function of t and z alone. The density flux is then a scalar given by: J*A,z = − D AB

∂C A ∂z

In steady state and for constant C, this expression is written: J*A,z = − CD AB Sz

dx A dz

3.3. Fick’s first law in different forms The expression of Fick’s law presented in section 3.2.1 gives the molar flux density of species A; namely JA*. Other expressions of this law can be encountered depending on the reference used (mass or molar), and the coordinates system chosen for evaluating the fluxes (fixed, mobile at v or mobile at v*). Thus, we have the following equivalent expressions:

JJJG JJJJJJG J *A = − D AB C grad x A

jA = − ρD AB∇ω A

Fick’s First Law: Diffusion Coefficients

(

FA = − C D AB∇x A + x A FA + FB

(

f A = − ρD AB∇ω A + ω A f A + f B

53

)

)

Table 3.1 summarizes these expressions of Fick’s law, specifying the references (molar or mass) and the coordinates system used. Coordinates system

Reference

Flux density

Fick’s law

Mass

fA

f A = -ρ D AB ∇ω A + ω A ( f A + f B )

Molar

FA

FA = -C D AB ∇ x A + x A ( FA + FB )

Mobile at v

Mass

jA

jA = -ρDAB∇ωA

Mobile at v*

Molar

JA*

J*A = -C DAB∇x A

Fixed

Table 3.1. Different forms of Fick’s law

3.4. Determining diffusion coefficients from tabulated data Like thermal conductivity, viscosity and density, the diffusion coefficient DAB is a physical property that depends on the binary mixture considered (A, B), on its state (gaseous or liquid), on its temperature and often on its concentration. 3.4.1. Gaseous binary diffusion coefficients Diffusion coefficients for several gaseous binaries have been determined experimentally by several authors (Cowling, 1970; Hirschfelder, 1964; LandoltBörnstein, 1968, 1969; Marrero, 1972; Sherwood, 1964). Table 3.2 presents examples of these coefficients for some gaseous binaries. Gas A

Gas B Air

Carbon dioxide

Nitrogen Oxygen

T (K)

DAB in cm2/sec

273 317.2 293 298 273 293.2

0.1380 0.1770 0.1630 0.1670 0.1390 0.1530

54

Mass Transfers and Physical Data Estimation

Air Nitrogen Water

Carbon dioxide Hydrogen Oxygen Acetone

Hydrogen

Ammonia

273 313 307.5 352.1 273 307.2 352.3 273 328.5 352.3 723 296 298 358 473 533

0.2200 0.2880 0.2560 0.3590 0.1380 0.1980 0.2450 0.7500 1.1210 0.3520 1.3000 0.4240 0.7830 1.0930 1.8600 2.1490

Table 3.2. Examples of diffusion coefficients for gaseous binaries at 1 Atm as a function of temperature

A more extensive list of gaseous binary diffusion coefficients is presented in Table A1.2 in Appendix 1. Note that the diffusion coefficients are given for A-B pairs, and the coefficient is the same for the diffusion of A in B, and for that of B in A; i.e.: DAB = DBA It is also to be noted that diffusion coefficients of different gaseous binaries depend not only on the characteristics of the components considered, but also on temperature. Consequently, when using the values given in the tables of Appendix 1, the indicated temperature ranges must be observed. Should a diffusion coefficient be sought at a temperature not featured in the table, an estimation thereof could be obtained by interpolation or regression of the data available. 3.4.2. Illustration: diffusion coefficients of CO2 in air and in water vapor Within the context of an analysis relating to environmental protection, we are led to conduct a study of carbon gas diffusion in air and in water vapor.

Fick’s First Law: Diffusion Coefficients

55

Questions 1) By conducting an interpolation, determine the diffusion coefficient of carbon dioxide in air at 30 °C. 2) By conducting a regression of the data available on the diffusion coefficient of carbon dioxide in water, develop a model that expresses the variation of DCO2-H2O as a function of temperature. 3) Using this model, determine DCO2-H2O at 273 K, 307.2 K at 352.3 K and calculate the residue. 4) Using this model, calculate DCO2-H2O at 20 °C, at 40 °C and at 60 °C. Solutions 1) Diffusion coefficient of carbon dioxide in air at 30 °C Table 3.3 shows the data available for the gaseous binary, CO2-Air: T (K)

DCO2-Air (in cm2/sec)

273

0.1380

317.2

0.1770

Table 3.3. Data available on DCO2-Air

NUMERICAL APPLICATION.– Interpolation gives:

Yet:

And:

D CO2 − Air − D CO 2 −Air (303)

D CO2 −Air − D CO 2 −Air ( 317.2 )

( 273)

317.2 − 273

D CO2 −Air − D CO2 −Air (303)

( 273)

303 − 273

Hence:

303 − 273

30

=

D CO2 − Air − D CO 2 −Air

=

D (303)

CO 2 − Air

− 0.1380 30

= 8.8235 × 10 −4.

I.e.: D CO2 −Air = 0.1380 + 30 *8.8235 × 10−4. (303)

Or: D (303)

CO 2 − Air

(317 ,2 )

( 273)

317.2 − 273

.

0.1770 − 0.1380 = 8.8235 × 10 −4. 317, 2 − 273

=

D CO2 −Air − 0.1380 (303)

( 273)

= 0.1645 cm 2 / s.

.

56

Mass Transfers and Physical Data Estimation

2) Regression of data available on DCO2-H2O For the gaseous binary CO2-H2O, Table 3.4 gives: T (K)

DCO2-H2O (in cm2/sec)

273

0.1380

307.2

0.1980

352.3

0.2450

Table 3.4. Data available for DCO2-H2O

Verification of the data presented in Table A1.2 in Appendix 1 shows that these are the only data available. Yet, these data do not correspond to the temperatures that are of interest to us, namely: T1 = 293 K; T2 = 313 K and T3 = 333 K We therefore need to carry out a regression of the data available with a view to determining a model representing the variation of DCO2-H2O as a function of temperature. Adopting a linear regression, the model sought takes the form: DCO2-H2O = a*T + b We write: x = T and: y = DCO2-H2O. The model sought then takes the form: y = a*x + b. For n available measures (xi, yi), coefficients a and b are given by (see Appendix 2): n ⎞⎛ n ⎞ n 1⎛ ⎜ xi ⎟⎜ yi ⎟ − ( x i yi ) ⎟⎜ ⎟ n⎜ a = ⎝ i =1 ⎠⎝ i =1 ⎠ i =1 2 ⎛ n ⎞ ⎜ xi ⎟ ⎜ ⎟ ⎛ n ⎞ ⎝ i =1 ⎠ − ⎜ x 2 ⎟ i ⎜ ⎟ n ⎝ i =1 ⎠

∑

∑

∑

∑

∑

Fick’s First Law: Diffusion Coefficients

57

and: b=

n n ⎤ 1⎡ ⎢ yi − a * xi ⎥ n⎢ i =1 ⎥⎦ ⎣ i =1

∑

∑

NUMERICAL APPLICATION.– For the present case, we have 3 measurements, therefore n = 3. We then have: Measurements

xi

yi

xi yi

x i2

1

273

0.138

37.674

74,529

2

307.2

0.198

60.8256

94,371.84

3

352.3

0.245

86.3135

124,115.29

Σ

932.5

0.581

184.8131

293,016.13

Table 3.5. Extrapolation calculations

. ∗ .

Hence: And: b =

=

.

.

i.e.: a = 1.33 × 10-3. .

0.581 − 1.33 × 10

∗ 932.5 i.e.:

b = -2.21 × 10-1.

Hence the model giving DCO2-H2O as a function of temperature: DCO2-H2O = 1.33 × 10-3* T -2.21 × 10-1 3) DCO2-H2O at 273 K, at 307.2 K and at 352.3 K and calculation of the residue Table 3.6 presents the measured values of DCO2-H2O and those generated by the linear model developed in the above relation: T (K)

DCO2-H2O measured

DCO2-H2O calculated

273

0.138

0.143

307.2

0.198

0.189

352.3

0.245

0.249

Table 3.6. Measured and calculated values of DCO2-H2O

58

Mass Transfers and Physical Data Estimation

Next, the squares of the deviations between the values measured and those generated by the model are calculated. The results are given in Table 3.7, which presents the residue of the deviations. (DCO2-H2O measured - DCO2-H2O calculated)2 2.72461 × 10-5 8.42359 × 10-5 1.56676 × 10-5 1.27 × 10-4

Residue

Table 3.7. Calculating the residue

4) Calculating DCO2-H2O at 20 °C, at 40 °C and at 60 °C The values calculated from the model for the temperatures considered are presented in Table 3.8. T (°C)

DCO2-H2O (in cm2/sec)

20

0.170

40

0.197

60

0.223

Table 3.8. Result of regressions

3.4.3. Diffusion coefficients for liquid binaries As with gases, published data exist on the diffusion coefficients of liquid binaries. The latter have been experimentally determined, for several liquid binaries, by several authors. The works are summarized in the Critical Tables (National Research Council, 1929), and by Chang (Chang and Wilke, 1955) and Hammond (Hammond and Stokes, 1956). Table 3.9 presents examples of values of such diffusion coefficients. A more extensive list of the diffusion coefficients for liquids is presented in Appendix 1.

Fick’s First Law: Diffusion Coefficients

Gas A

Gas B Air

Carbon dioxide

Nitrogen Oxygen Air

Water

Gas A

Water

Nitrogen

DAB in cm2/sec

273

0.1380

317.2

0.1770

293

0.1630

298

0.1670

273

0.1390

293.2

0.1530

273

0.2200

313

0.2880

307.5

0.2560

352.1

0.3590

Carbon dioxide

273

0.1380

307.2

0.1980

Gas B

T (K)

DAB in cm2/sec

Carbon dioxide

352.3

0.2450

Hydrogen Oxygen Acetone

Hydrogen

T (K)

Ammonia

59

273

0.7500

328.5

1.1210

352.3

0.3520

723

1.3000

296

0.4240

298

0.7830

358

1.0930

473

1.8600

533

2.1490

Table 3.9. Examples of diffusion coefficients for liquid binaries at 1 Atm as a function of temperature

A more extensive list of the diffusion coefficients of liquid binaries is presented in Table A1.2 in Appendix 1. Note that the diffusion coefficients are given for pairs (solvent and solute) that need to be observed. It is also to be noted that diffusion coefficients of liquid binaries are, in orders of magnitude roughly four times smaller than the gas diffusion coefficients.

60

Mass Transfers and Physical Data Estimation

NOTE.– The diffusion coefficients of the liquid binaries depend not only on the characteristics of the components considered, but also on the concentration and temperature. Consequently, when using the values given in the tables of Appendix 1, the temperature ranges indicated must be observed. While concentration dependence can be understood by distinguishing between infinite dilution solutions and real solutions (see below), dependence on temperature is taken into account using the following correlation: D AB μ = Cste T Thus, should a diffusion coefficient be sought at a temperature not featured in the table, an estimation thereof could be obtained based on knowledge of the viscosities expressed at a known temperature.

3.5. Estimating diffusion coefficients from correlations In section 3.4.2 we proceeded using interpolation or regression as we were fortunate enough to find the values of DCO2-Air and DCO2-H2O in the tables in Appendix 1. When the diffusion coefficients sought are not found in this appendix, or when the temperature of interest differs significantly from the temperature ranges tabulated in Appendix 1, the estimation methods presented in the following sections can be used.

3.5.1. Estimating gaseous binary diffusion coefficients The methods for gaseous binary diffusion coefficients are based on critical gas theory and on the different models of interaction between the molecules (Hirschfelder, 1954).

3.5.1.1. Modeling interactions between molecules To represent the interactions between the molecules of a binary gas mixture, let us use the kinetic theory of gases with the Lennard-Jones potential model, the field of which is defined by Gosse, Déroulède, and Dutheuil (1991):

⎡⎛ σ ⎞ 12 ⎛ σ ⎞ 6 ⎤ ϕ(r) = 4ε ⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎢⎝ r ⎠ ⎝ r⎠ ⎥ ⎦ ⎣

Fick’s First Law: Diffusion Coefficients

61

where: r is a magnitude characterizing the sizes of the molecules and the distances between molecules. It is known as the intermolecular distance σ characterizes the distances where probabilities for collision between molecules exist. It is known as the collision diameter ε is a characteristic of the forces between molecules. It is known as interaction energy. The interaction energies, ε, intermolecular distances, r, and collision diameters, σ, are physical data available for the different atoms and molecules. These data are tabulated in Appendix 1 for common molecules. Table 3.10 presents examples of these. Molecule

Name

σ (in Angström)

ε (K) k

He

Helium

2.551

10.22

Air

Air

3.711

78.6

CH4

Methane

3.758

148.6

CO

Carbon monoxide

3.690

91.7

CO2

Carbon dioxide

3.941

195.2

C2H2

Acetylene

4.033

231.8

C2H4

Ethylene

4.163

224.7

C2H6

Ethane

4.443

215.7

C6H6

Benzene

5.349

412.3

Cl2

Chlorine

4.217

316.0

F2

Fluorine

3.357

112.6

HCl

Hydrogen chloride

3.339

344.7

H2

Hydrogen

2.827

59.7

H2O

Water

2.641

809.1

H2S

Hydrogen sulfide

3.623

301.1

NH3

Ammonia

2.900

558.3

O2

Oxygen

3.467

106.7

SO2

Sulfur dioxide

4.112

335.4

UF6

Uranium hexafluoride

5.967

236.8

Table 3.10. Several collision diameters and interaction energies

62

Mass Transfers and Physical Data Estimation

Should they not be given in the tables, the ε values can be estimated from the critical temperature, , the temperature at normal boiling point, , or the fusion temperature, , using the following relations (Hirschfelder, Bird and Spotz, 1949):

= 0.77 or:

= 1.15 . ε

Or alternatively: = 1.92T . Either of the correlations can be used interchangeably, depending on the data available,T , T , or T . and the temperatures are in K. Likewise, to estimate σ, one of the following relations will be used, depending on the data available: σ = 0.841 Vc1/3 σ = 0.841T

/

or alternatively: σ = 2.44

/

,

where:

Vc is the critical volume (in cm3/mole), and k is the Boltzmann constant Pc is the critical pressure (in Atm). The values of Vc and Pc are presented in Appendix 1 for common molecules. When not found in the tables, the volumes and critical pressures can also be estimated using the methods presented in Chapter 1. The interaction force between two molecules, A and B, is then expressed as a function of the potential gradient: FAB =

dϕ dr

The developments carried out by Chapman and Enskog and the use of the Lennard-Jones potential have made it possible to determine the diffusion coefficient of a gas A in a gas B as a function of the parameters, σ and Ω, which characterize the force field:

D

=

.

T σ

Ω

Fick’s First Law: Diffusion Coefficients

63

where: DAB is the diffusion coefficient of A in the binary mixture, A-B (in cm2/sec) T is the absolute temperature (in K) MA and MB are the molar masses of A and B, respectively (in g/moles) and = σ

=

σ

σ

, where σ and σ are the collision diameters of components A and

B, respectively, expressed in Angström. σA and σB are given by Table 3.10 and Table A1.4 in Appendix 1 Ω

=f

, where f is a dimensionless function of T and of εAB, known as the

ε

collision integral (see Table 3.11 and Table A1.4 in Appendix 1): is the geometric average of: ε

and

ε

and of:

are calculated from the

ε

i.e.:

ε

=

ε

ε

;

values given in Table 3.11 and in

Appendix 1; k being the Boltzmann constant. Table 3.11 gives several values of

as a function of

.

kT ε AB

ΩAB

kT ε AB

ΩAB

kT ε AB

ΩAB

0.30

2.662

2.00

1.075

20

0.6640

0.40

2.218

3.00

0.9490

30

0.6232

0.50

2.066

4.00

0.8836

40

0.5960

0.60

1.877

5.00

0.8422

50

0.5756

0.70

1.729

6.00

0.8124

60

0.5596

0.80

1.612

7.00

0.7896

70

0.5464

0.90

1.517

8.00

0.7712

80

0.5352

1.00

1.439

9.00

0.7556

90

0.5256

10.00

0.7424

100

0.5130

200

0.4644

400

0.4170

Table 3.11. Values of ΩAB based on the Lennard-Jones potential

64

Mass Transfers and Physical Data Estimation

More detailed data are provided in Appendix 1. By smoothing these data it is possible to determine a regression of Ω as a function of (see Appendix 2). ε

This regression is obtained in the form: Ω

= 1.435

.

kT ε

The graph in Figure 3.1 shows that this regression agrees substantially with the available data.

Figure 3.1. Representation of regression of ΩAB and comparison with data. For a color version of this figure, see www.iste.co.uk/benallou/energy5.zip

It should be noted that the expression of the regression and the graph in Figure 3.1 are only to be used to estimate ΩAB if the data sought are not to be found in section A1.5 of Appendix 1.

3.5.1.2. Hirschfelder, Bird and Spotz correlation (gas) This correlation is based on the Chapman and Enskog equation. It is given by Bird, Stewart and Lightfoot (1975): 3

D AB = 1,8583 × 10

−3

1

⎡ 1 1 ⎤2 + ⎢ ⎥ 2 Pσ AB Ω AB ⎣ M A M B ⎦ T2

Fick’s First Law: Diffusion Coefficients

65

where: DAB is the diffusion coefficient of A in the binary mixture, A-B (in cm2/sec) T is the absolute temperature (in K) P is the total pressure, expressed in atmospheres σ

MA and MB are the molar masses of A and B, respectively (in g/moles) = , where: and are the collision diameters of components A and B,

σ

respectively, expressed in Angström (see Table 3.10 and Table A1.4 in Appendix 1) Ω

ε

is given as a function of

=

ε

ε

,

and

ε

ε

ε

are given by Table A1.5 in Appendix 1

3.5.1.3. Fuller, Schettler and Giddings correlation (gas) This relation has been established from experimental values (Fuller et al., 1966), based on the developments conducted by Gilliland (1934) and the kinetic theory of gases (Reid and Sherwood, 1966). It is to be used when the Lennard-Jones parameters (σ and Ω) are not available:

D

= 10

.

+

.

,

where: is the diffusion coefficient of A in B, expressed in cm2 s-1 T is the absolute temperature expressed in K and

are the molar masses of A and B expressed in g/moles

and are the “molecular diffusion volumes” of A and B. They are obtained from Table 3.12. Molecules

V (in cm3/mole)

Air

20.1

Br2

67.2

C Cl2 F2

114.8*

Cl2

37.7*

CO

18.9

66

Mass Transfers and Physical Data Estimation

*

CO2

26.9

D2

6.70

H2

7.07

H2O

12.7

N2

17.9

N2O

35.9

NH3

14.9

O2

16.6

SF6

69.7*

SO2

41.1*

Values established based on a small amount of experimental data only Table 3.12. Molecular diffusion volumes

In the event that the values of the molecular diffusion volumes are not available, these can be calculated by adding together the atomic diffusion volumes (see Table 3.12), observing the chemical formulas of the molecules concerned and taking into consideration the contributions of any aromatic rings or heterocyclic rings. Atoms or rings

*

V (in cm3/mole)

C

16.5

Cl

19.5*

H

1.98

He

2.88

Kr

22.8

N

5.69*

O

5.48

S

17.0*

Xe

37.9*

Aromatic rings

-20.2

Heterocyclic rings

-20.2

Values established based on a small amount of experimental data only Table 3.13. Atomic diffusion volumes and contributions of ring structures (in cm3/mole)

Fick’s First Law: Diffusion Coefficients

67

3.5.1.4. Illustration: calculating the diffusion volume of CS2 We have: VCS2 = VC + 2 VS NUMERICAL APPLICATION.– From Table 3.12, we have: VC = 16.5 cm3/mole VS = 17.0 cm3/mole Hence: VCS2 = 16.5 + 34. I.e.: VCS2 = 50.5 cm3/mole.

3.5.1.5. Choice of correlation to be used for gases The Herschfelder et al. correlation and that of Chapman and Enskog (see sections 3.5.1.1 and 3.5.1.2) both result from the Chapman and Enskog developments. They require knowledge of the same set of parameters: temperature, pressure and molar masses, but also collision diameters, σAB and the collision integral, ΩAB. In contrast, the Fuller et al. correlation is based on another set of parameters, essentially the diffusion volumes. The illustration presented in section 3.5.1.6 shows that these correlations give similar results. In reality, the use of one or another of the correlations will depend on which data are available.

3.5.1.6. Illustration: diffusion coefficients of CO2 in air One of the polluting gases most commonly encountered is carbon dioxide. It is one of the products of the combustion of liquid and solid fuels most used in the production of electricity and in boilers intended for industrial vapor production. The fumes discharged from the stacks of boilers and thermal power plants therefore contain large amounts of carbon dioxide. In order to determine the consequences of these gaseous discharges on air quality in the vicinity of thermal power plants and industrial units, it is necessary to study the diffusion of carbon dioxide in the air. For this purpose, knowledge of the diffusion coefficient of carbon dioxide in air under atmospheric pressure is necessary. Note that this coefficient, DCO2-Air , depends on temperature.

68

Mass Transfers and Physical Data Estimation

Questions 1) What DCO2-Air data are available? 2) Deduce therefrom by interpolation, an estimation of DCO2-Air at 25 °C. 3) We seek to evaluate the validity of the different correlations. We propose calculating DCO2-Air at 0 °C, at 25 °C and at 44.2 °C: a) using the Chapman and Enskog correlation; b) using the Herschfelder et al. correlation; c) using the Fuller et al. correlation. 4) Compare the results with each other and with the values measured and estimated by interpolation. 5) What conclusion can be drawn?

Data: MAir = 28.97 g/mol

MO = 16.00 g/mol

MC = 12.01 g/mol

Solutions 1) Data available on DCO2-Air Table A1.1 in Appendix 1 gives, for the gaseous binary, CO2-Air. T (K)

DCO2-Air (in cm2/sec)

273

0.138

317.2

0.177

Table 3.14. Diffusion coefficients for gaseous binary, CO2-Air

2) Estimation of DCO2-Air at 25 °C We have, through interpolation:

D

°

=D

°

+ 25

D

. °

−D 44.2

°

Fick’s First Law: Diffusion Coefficients

NUMERICAL APPLICATION.– °

D

= 0.138 °

Hence: D

. °

D

= 0.177

= 0.160.

3) Evaluating the validity of the different correlations a) Chapman and Enskog correlation =

where: M

.

and:

=

.

=

NUMERICAL APPLICATION.–

P = 1 Atm M

T = 298 K

M

= 44.01 g/mol

= 34.94 g/mol

From the data bank (Appendix 1), we have: σ

= 3.711 A and: σ

Hence: σ ε

=

σ

= 3.941 A σ

= 195. 2 K and:

Hence (see section 3.5.1.1):

kT ε

= 3.826. ε

= 78. 6 K. =T

ε

ε

= 2.4

Hence (see Table A1.5, Appendix 1): Hence:

ε

= 0.156

/ .

= 1.012.

MAir = 28.97 g/mol

69

70

Mass Transfers and Physical Data Estimation

b) Herschfelder et al. correlation DCO2-Air is given by:

= 1.8583 × 10

D

1

T Pσ

Ω

M

+

1 M

NUMERICAL APPLICATION.–

P = 1 Atm

T = 298 K

= 44.01

= 3.826

MAir = 28.97 g/mol

/

= 1.012

We deduce therefrom:

/ .

= 0.154

c) Fuller et al. correlation D

= 10

.

.

+

NUMERICAL APPLICATION.– P = 1 Atm

V

T = 298 K

= 26. 9 cm /mol

Hence: D

M

= 44.01 g/mol

= 20. 1

MAir = 28.97 g/mol

/

= 0.157 cm /s.

4) Comparison of results The following table summarizes, in cm2/sec, the different results obtained for the diffusion coefficient of CO2 in air at different temperatures, under 1 Atm.

T (°C)

Interpolation of measurements

Correlation of Chapman and Enskog

Correlation of Herschfelder et al.

Correlation of Fuller et al.

0

0.138

0.137

0.135

0.134

25

0.160

0.156

0.154

0.157

44.2

0.177

0.172

0.170

0.175

Table 3.15. Coefficients of CO2-Air by different correlations

Fick’s First Law: Diffusion Coefficients

71

5) Conclusions For all of the temperatures considered, the values estimated by the correlations are close to the results of the measurements given in Table A1.1 of Appendix 1 (DCO2-Air = 0.138 cm2/sec at 0 °C and DCO2-Air = 0.177 cm2/sec at 44.2 °C). The Chapman and Enskog correlation appears to give the closest results at low temperatures: 0.137 cm2/sec at 0 °C; whereas for 25 °C and 44.2 °C. The results given by the Fuller et al. correlation would appear to be the closest to the experimental values. Nevertheless, in each of the situations, the different correlations give results which are very similar to one another, and close to the experimental values.

3.5.2. Estimating diffusion coefficients of liquid binaries 3.5.2.1. Scheibel equation (liquids) For non-aqueous solutions, the Scheibel relation gives the diffusion coefficient of a solute, A, in infinite dilution in a solvent, S:

=

, with:k = 8.2 × 10

1+

where:

D is the diffusion coefficient of solute A in the solvant, S, expressed in [cm2/Sec] μs is the solvent viscosity, expressed in [Cp] V is the molar volume at normal boiling point, expressed in [cm3/mole] T is the temperature, expressed in [K]

3.5.2.2. Wilke and Chang equation (non-electrolytic liquids) For low concentrations of A in a solution, A-B, a good approximation of the diffusion coefficient, DAB, is given by Chang (1955):

D

= 7.4 ∗ 10

ψ μ

,

,

72

Mass Transfers and Physical Data Estimation

where: DAB is in cm2/sec T is the absolute temperature (in K) MB is the molar mass of the solvent (in g/mol) VA is the molar volume of solute A at its normal boiling point, in cm3/g-mol (see Chapter 1, section 1.2.3.1) μ is the viscosity of the solution (in centipoises) ψB is a parameter that characterizes the association ability of the molecules of solvent B. It is equal to 1 for non-associated liquids. Table 3.16 gives values of ψB. Solvent

Water

Methanol

Ethanol

Benzene

Ether

Heptane

ψB

2.6

1.9

1.5

1

1

1

Table 3.16. Association abilities

For liquids not listed in this table, use ψB=1, with the exception of electrolytes, for which the Nernst equation, given below, should be used. 3.5.2.3. Hayduk and Laudie correlation A simpler relation giving equivalent accuracies to the previous relation has been proposed by Hayduk and Laudie for non-electrolytic solutes in infinite dilution in water:

D

= 13.26 ∗ 10 μ

.

V

.

,

where: DAB is in cm2/sec VA is the molar volume of solute A at its normal boiling point (in cm3/g/mol) μ is the viscosity of the solution (in centipoises)

3.5.2.4. Nernst relation (electrolytic solutions) For electrolytic solutions at infinite dilution, the diffusion coefficient of a solute, A, in water is given by (Nernst, 1888):

D

= 8.931 × 10

T

,

Fick’s First Law: Diffusion Coefficients

73

where:

D

is the diffusion coefficient of A in the solvent, S, expressed in [cm2/sec]

T is the absolute temperature expressed in [K]

C and C are, respectively, the cationic and anionic conductances, at infinite dilution, expressed in mhos/equivalent. They are given below in Table 3.17 for the most commonly encountered ions. For more data concerning these conductances, consult the reference indicated at the bottom of this table K0 = C + C is the conductance of the electrolyte at infinite dilution, expressed in mhos/equivalent

Z+ and Z- designate the valences of the cations and anions, respectively (absolute values) Cations

Anions

Monovalent

C0+

Monovalent

C 0−

Ag+

61.90

CH3COO-

40.90

CH3NH3

+

-

58.70

C6H5COO

32.40

51.90

Br-

78.40

(CH3)3NH

47.20

Cl

-

76.35

Cs+

77.30

Cl O3-

(CH3)2NH2+ +

+

-

64.60

H

349.80

F

55.4

K+

73.50

HCO3-

44.50

+

38.70

I

-

76.80

Na+

50.10

OH-

Li

NH4

+

-

198.60

73.6

NO3

71.46

Bivalent

C0+

Bivalent

C 0−

Ba++

63.60

CO3--

69.3

Be++

45.00

C2O4--

74.2

74

Mass Transfers and Physical Data Estimation

Ca++

59.50

++

55.00

Cu++

56.60

++

53.00

Zn++

52.80

Co

Mg

SO4--

80.0

Table 3.17. Conductances at infinite dilution in water, in mhos/equivalent at 25 °C (Stokes, 1955)

It is to be noted that the values of the conductances presented in Table 3.18 are given at 25 °C. In the knowledge that the latter depend on temperature, Hamed and Owen propose the following correlation to calculate C and C as a function of temperature (Owen, 1950), as a function of the value at 25 °C: 2

3

C0+ (θ) = C0+ (25°) + α ⋅ [ θ − 25] + β⋅ [ θ − 25°] − γ ⋅ [ θ − 25] , where:

C 25° is the value of C at 25 °C, taken from Table 3.17 θ is the temperature, expressed in [°C] α, β and γ are constants given in Table 3.18 for the most common anions and cations Ions +

H

α

β

γ

4.816

-1.031 × 10

-2

0.767 × 10-4

Li+

0.890

0.441 × 10-2

0.204 × 10-4

Na+

1.092

0.472 × 10-2

0.115 × 10-4

K+

1.433

0.406 × 10-2

0.318 × 10-4

Cl-

1.540

0.465 × 10-2

0.128 × 10-4

Br-

1.544

0.447 × 10-2

0.230 × 10-4

I-

1.509

0.438 × 10-2

0.217 × 10-4

Table 3.18. α, β and γ for the most common anions and cations (Owen, 1950)

Fick’s First Law: Diffusion Coefficients

75

3.5.2.5. Reid–Sherwood equation (aqueous solutions) For aqueous solutions at infinite dilution, the diffusion coefficient of a solute, A, in solvent is given by the Reid–Sherwood relation:

D

=

. μ .

.

,

where:

D

is the diffusion coefficient of A in water at infinite dilution, in cm2s-1

μe is the viscosity of water under the conditions considered, expressed in Cp VA is the solute molar volume at normal boiling point, expressed in cm3 /mole-1

3.5.2.6. Extension to real solutions The diffusion coefficient for a real solution is obtained from the diffusion coefficient at infinite dilution, using the following relation: ⎡

⎤

dLn( γ A ) , ⎥ DAB = D0AB ⎢1 + ⎢⎣ dLn(x A ) ⎥⎦

where: DAB is known as the activity-corrected diffusion coefficient γA is the activity coefficient of A xA is the molar fraction of A

3.5.2.7. Illustration: estimating the diffusion coefficient of acetic acid in water Suspicions of acetic acid pollution have been reported by an environmental organization against a riverside chemical plant. Analysis of liquid waste from this plant requires knowledge of the diffusion coefficient of acetic acid in water at the temperatures indicated in Table 3.19.

Questions 1) Give the values of DAcid-Water available in the tables. 2) Give estimations of DAcid-Water at the temperatures indicated in Table 3.19.

76

Mass Transfers and Physical Data Estimation

Data: MWater=18 g/mol

ψWater = 2.6

MAcid = 60.5 g/mol.

Density of acetic acid at its normal boiling point: ρAcid = 0.97 gcm-3. θ (°C)

μSolution (Cp)

10

1.5

12

1.3

15

1.2

21.7

1.01

27.1

0.98

29.0

0.89

30

0.74

35

0.70

40

0.64

Table 3.19. Viscosity of the solution as a function of temperature

Solutions 1) Values of DAcid-Water from the tables There is a single value of DAcid-Water available in Appendix 1: DAcid-Water = 1.24 × 10-5 cm2/sec at 20 °C

2) Estimations of DAcid-Water The Wilke and Chang equation gives (see section 3.5.2.2):

D

= 7.4 ∗ 10

ψ μ

.

With the molar volume, VAcid, at its normal boiling point given by: VAcid =

MAcid ρAcid

Fick’s First Law: Diffusion Coefficients

77

NUMERICAL APPLICATION.– MWater=18 g/mol

ψWater = 2.6

MAcid = 60.5 g/mol

ρAcid = 0.97 g cm-3

Hence: VAcid = 62.37 cm3/mole. θ (°C)

T (K)

μSolution (Cp)

10 12 15 21.7 27.1 29 30 35 40

283 285 288 294.7 300.1 302 303 308 313

1.5 1.3 1.20 1.01 0.98 0.89 0.74 0.70 0.64

DAcid-Water in cm2/sec 8 × 10-6 9.3 × 10-6 1.02 × 10-5 1.24 × 10-5 1.30 × 10-5 1.44 × 10-5 1.74 × 10-5 1.87 × 10-5 2.07 × 10-5

Table 3.20. Calculating the diffusion coefficient

3.5.2.8. Illustration: diffusion parameters of an electrodialyzer The electrodialyzer represented in Figure 3.2 consists of two compartments separated by a semi-permeable membrane (permeable to water but impermeable to salt).

Figure 3.2. Electrodialyzer. For a color version of this figure, see www.iste.co.uk/benallou/energy5.zip

78

Mass Transfers and Physical Data Estimation

Each of the two compartments initially contains 5 liters of pure water. 1 kg of salt is added to the first compartment. Only half a kilo of the same salt is added to the second compartment.

Questions Assuming that, in both compartments, the mixing is instantaneous and homogeneous: 1) calculate the mass fractions of salt and water in each of these compartments; 2) calculate the driving potential difference between the two compartments; 3) what conclusion can you draw?; 4) the semi-permeable membrane is 2.5 mm thick. Calculate the molar flux density of the water being transferred, assuming that the diffusion is unidirectional and that the diffusion coefficient is: DES = 2.7 × 10-2 m2/sec; 5) calculate the water diffusion velocity.

Data: Me = 18 g

Ms = 60 g

Solutions 1) Calculating the mass fractions Compartment 1: – total mass = 6 kg; – amount of salt = 1 kg. The salt mass fraction in compartment 1 is thus given by: ωS1 =

mS1 m1

Fick’s First Law: Diffusion Coefficients

79

NUMERICAL APPLICATION.– m S 1 = 1 kg

ω S1 =

1 6

m 1 = 6 kg

; i.e.: ω S 1 = 0.167

The water mass fraction in compartment 1 is obtained by difference (because ωs + ωe = 1): ωe1 = 1 – ω s1; i.e.: ωe1 = 0.833

Compartment 2: – total mass = 5.5 kg; – amount of salt = 0.5 kg. The salt mass fraction in compartment 1 is thus given by: ωS2 =

mS2 m2

NUMERICAL APPLICATION.–

= 0. 5 =

. .

; i.e.:

= 5. 5 = 0.091

The water mass fraction in compartment 1 is obtained by difference (because ωs + ωe = 1): ωe2 = 1 – ωs2; i.e.: ωe2 = 0.909

2) Calculating the driving potential difference The driving potential difference is given in this case by the difference in mass concentrations: Δωe = ωe2 - ωe1; i.e.: Δωe = 0.076

80

Mass Transfers and Physical Data Estimation

3) Conclusion There will therefore be water transfer from compartment 2 to compartment 1.

4) Calculating the molar flux density Assuming the diffusion is unidirectional, the water flux transferred in direction z is given by:

M = −CD S

dx dz

The molar flux density is then: F = −CD NUMERICAL APPLICATION.– Water masses: me1 = me2 = 5 kg Salt masses: ms1 = 1 kg

xe =

Ne N

N ee =

Me = 18 g

ms2 = 0.5 kg me Me

xS =

Ms = 60 g

NS N

NSe =

mS MS

Compartment 1: N

=

=

,

= 277.78

=

=

,

= 16.67

N1 = Ne1 + Ns1 = 277.78 + 16.67 = 294.45.

Hence: Therefore:

=

=

. .

= 0.94 and xs1 = 0.06.

Compartment 2: N

=

=

,

= 277.78

=

=

= 8.33

Hence:

N2 = Ne2 + Ns2 = 277.78 + 8.33 = 286.11.

Therefore:

x

0.94 − 0.97 dx = dz 2. 5 10

=

=

. .

= 0.97 and xS2 = 0.03.

Fick’s First Law: Diffusion Coefficients

I.e.:

= −12.

Furthermore: C =

=

81

.

.

.

= 0.050

.

/

Thus: Fe = -0.05 x 0.027 x (-12) = 0.0162 moles/m2 sec.

5) Calculating the diffusion velocity We have: Fe = ceve. Hence: v =

c =

.

+N V

N

NUMERICAL APPLICATION.–

c =

.

ce= 0.0483 moles/m3

.

Therefore: ve = 0.0162/0.0483

ve = 0.335 m/sec.

3.6. Diffusion coefficients for multicomponent mixtures In practice, we often encounter complex mixtures composed of several bodies. In this case, the methods for calculating or estimating the diffusion coefficients presented for binary mixtures no longer apply and the relations and parameters specific to complex mixtures must be used.

3.6.1. Stefan–Maxwell equation Diffusion in complex mixtures (with n components) is expressed by the Stefan– Maxwell law. This law gives the expression of the concentration gradient of component i as a function of the binary diffusion coefficients, diffusion fluxes and molar fractions of the components that are present: ∇x i =

1

n

x i F j − x jF i

j=1

D ij

∑ C

82

Mass Transfers and Physical Data Estimation

where: Dij is the diffusion coefficient of component i in the binary (i, j) xi is the molar fraction of component i in the mixture C is the total molar concentration Fi is the molar flux of i with respect to a fixed coordinates system The Stefan–Maxwell equation is the starting point for diffusion calculations in complex mixtures. It must be written for all components (or all but 1). This gives what is called the Stefan–Maxwell system of equations. Solving this system of equations makes it possible to determine the concentration profiles or mass fluxes.

3.6.2. Effective diffusion coefficient for complex mixtures The study of diffusion in complex mixtures is generally facilitated by introducing an effective diffusion coefficient (also known as an equivalent diffusion coefficient), which makes it possible to give expressions of the diffusion fluxes similar to those obtained for binary mixtures. Let us recall that for a binary (A, B), the molar flux density is given by: FA = J*A + x A ( FA + FB ) ,

or:

FA = −CDAB∇x A + x A ( FA + FB ) The effective or mixture diffusion coefficient, Dim, is defined in such a way that, for complex mixtures too, the expression of the diffusion flux of component i could be written in a way similar to the expression of the flux for binary mixtures, that is: n

Fi = −CDim∇xi + xi

∑ Fj j=1

The latter expression can be referred to as Fick’s law for multicomponent mixtures.

Fick’s First Law: Diffusion Coefficients

83

Thus, the gradient can be expressed as follows: n

Fi − x i

∑ Fj j=1

∇x ii =

CDim

Comparing with the expression of ∇xii given by the Stefan–Maxwell law, that is:

∇xi =

1 C

n

∑

xi Fj − x jFi Dij

j=1

We deduce therefrom the expression of the equivalent diffusion coefficient: n

Fi − x i CDim =

∑ Fj j=1

n

∑ j=1

x i Fj − x jFi CDij

Using this equation, the effective diffusion coefficient, Dim, can be determined from the binary diffusion coefficients, Dij.

4 Fick’s Second Law: Macroscopic Balances

4.1. Introduction Fick’s first law makes it possible to calculate the flux densities, mass or molar, from the driving potential difference. It thus facilitates the understanding of local matter migrations inducing the diffusion of mass within a system. It also makes it possible to interpret the physical phenomena observed in different processes, such as dialysis, which generate macroscopic mass transfer. But engineering calculations require an understanding of how microscopic interactions lead to transfers in systems as a whole, on a macroscopic scale. Indeed, on a global scale, it is necessary to analyze the relations that can be established between the expressions translating macroscopic exchanges and the laws expressing microscopic transfers. The overall mass balances are used to establish these relations. They lead to continuity equations which are necessary in order to quantify the possible transfers between a given system and its environment. 4.2. Overall continuity equation To explain the relations between the macroscopic scale and microscopic transfers, let us consider a system defined by an arbitrary volume, V, delimited by an envelope surface, S (see Figure 4.1). Let

be the vector normal to S, facing outwards.

Let us establish a balance of the mass which transfers between this system and the outside.

Mass Transfers and Physical Data Estimation, First Edition. Abdelhanine Benallou. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

86

Mass Transfers and Physical Data Estimation

V

Figure 4.1. System of arbitrary volume and of surface S

Let us recall (see Volume 1) that the overall balance equation is written quite simply as follows: Input – Output + Generation = Accumulation I–O+G=A 4.2.1. The accumulation term With the system being of volume V and of density ρ, its total mass is given by:

Thus, the accumulation of mass in the system is generally given by the differential of the integral representing the total system mass, with respect to time, i.e.: =

=

4.2.2. The generation term Given that the total mass of the system remains constant: G=0

Fick’s Second Law: Macroscopic Balances

87

4.2.3. The term I – O The oriented mass passage velocity with respect to a fixed coordinates system is noted v . The scalar product, − ρ v × n − × , then corresponds to the mass passage rate across a surface element, dS, of normal n . Integration across the entire surface enables term I – O to be obtained: I−O=

(−ρv) × n dS −

=

(−

)×

4.2.4. The balance equation The balance equation is therefore given by: I – O = A. i.e.: ×

=−

Stokes’ theorem (Green-Ostrogradsky) makes it possible to convert the surface integral into a volume integral as follows: ×

=

The balance equation thus becomes: =− Given that no scenario has been presented for volume V, we can write: ∀ :∭ (

+

)

=0

This can only be mathematically possible if the integrant is nil. Consequently:

+ ∇(ρv) = 0

+ (

) = 0.

This is the overall continuity equation, also known as Fick’s Second Law.

88

Mass Transfers and Physical Data Estimation

4.2.5. The balance equation in Cartesian coordinates In Cartesian coordinates (x, y, z), the velocity vector is written as follows: =

The continuity equation becomes: (

+

)

+

(

)

+

(

+

)

= 0.

+

(

)

+

= 0.

Or, by developing: + I.e.: where:

ρ

(

)

+

+ ρ∇v = 0.

ρ

+ +

(

)

+

+

+

= 0.

=0

is the particulate derivative of ρ, defined by: =

+

+

+

4.3. Particular continuity equations Particular continuity equations actually designate the macroscopic balances written for each of the components composing the mixture contained in a given system. To establish these balances, let us consider, as above, a system of arbitrary volume V, in which a multicomponent mixture can be found. Let us focus here on the transfer of a determined component, i. The mass balance relative to component i is written: Ii – Oi – Gi = Ai. 4.3.1. The term Ii – Oi For the flux density vector of component i see Chapter 2.

Fick’s Second Law: Macroscopic Balances

This is the integral on S of the flux scalar product with normal (I − O) = −

f × n dS( − ) = −

89

to the surface:

×

4.3.2. The accumulation term The mass of element i contained in dV is given by: ρ dV. Hence the mass of i in V is: ∭ ρ dV. Thus, the accumulation of i in the system is given by: A =

∂ ∂t

∂ρ dV ∂t

ρ dV =

4.3.3. The generation term If the total mass of a given multicomponent system necessarily remains constant, there is no guarantee that the mass of each of its components is conserved. Indeed, if some components are generated in the system, then their mass will increase. If, on the other hand, they are consumed, then their mass will decrease. Thus, to express the term Gi, the generation velocity of component i will be designated by ri: [mass of i]/[time][vol]. Under these conditions: dGi = ri dV. Consequently: G = ∭ r dV. The balance equation is then written: ∭

dV = − ∬ f × n dS + ∭ r dV.

Yet, according to Stokes’ theorem, we have: ∬ f × n dS = ∭ ∇f dV. Thus, we have ∀V: ∭ Or ∀ :∭

dV + ∭ ∇f dV − ∭ r dV = 0.

+ ∇f − r dV = 0.

90

Mass Transfers and Physical Data Estimation

Consequently:

+ ∇f − r = 0.

NOTE.– By writing the continuity equation for the n components composing a given mixture and by adding together, the overall continuity equation is obtained. ρ

Indeed: ∑

+ ∇f − r = 0.

Yet, because the total system mass is retained: ∑ ρ

Therefore: ∑ Or:

+∑

ρ + ∇∑

∑

Yet: ∑

ρ = ρ.

And: ∑

f =∑

Consequently:

ρ

r = 0.

∇f = 0.

f = 0.

ρ v = ρ∑

w v = ρv.

+ ∇(ρv) = 0.

4.3.4. Continuity equations in molar terms Consider an arbitrary volume with a mixture with n components. Let us now focus on the number of moles of component i in the system. The balance relating to this component is written: I i – Oi + G i = A i 4.3.4.1. The Ai term With the molar concentration of species i in the system being Ci, the number of moles of species i in an element of volume dV is: dNi = CidV. The total number of moles in the system is then given by: ∭ C dV Consequently, the accumulation term of species i is given by: A =

∂ ∂t

C dV

Fick’s Second Law: Macroscopic Balances

91

4.3.4.2. The generation term To express the term Gi, the velocity at which component i appears in the system will be designated by Ri. The appearance of component i in the system can be induced by a chemical reaction or by the decomposition of an element containing component i. The dimensions of Ri are given by: = The generation of species i is thus given by: G = ∭ R dV. 4.3.4.3. The Ii – Oi term The molar flux of species i is given by (see Chapter 2): F = C v . Consequently, the term (I-O)i is given by: F × n × dS =

(E − S) =

∇F × dV

The balance equation is then written: ∀V; ∭ Subsequently:

+ ∇F − R dV = 0.

+ ∇F − R = 0.

NOTE.– The sum of the specific molar continuity equations gives the overall continuity equation in molar terms. Indeed: ∑

+ ∇F − R = 0.

We then have:

∑

C +∇ ∑

F −∑

R = 0.

Note, however, that in this case (molar velocities), the sum ∑ R is not necessarily nil. We will write: ∑ R = R. Moreover: ∑

F =∑

C v = C∑

x v = Cv ∗ .

Hence the overall molar continuity equation:

+ ∇ Cv ∗ − R = 0.

92

Mass Transfers and Physical Data Estimation

4.4. Illustration: diffusion with chemical reaction Consider a gas-liquid reactor, where a gas, noted A, is brought into contact with a liquid, noted B, with a view to make them react. The reaction mechanism can be decomposed into two phases: firstly, there is diffusion of gas A in liquid B. Next, the A molecules, penetrating into the first layers of liquid B thanks to their diffusion, react with B. Thus, as soon as A molecules penetrate into the liquid phase, they are consumed. The chemical reaction between A and B is then produced in a liquid phase saturated with B (see Figure 4.2). Gas A z=0 Liquid B z=L Figure 4.2. Chemical reaction after diffusion

Let us assume that the reaction of B on A is total and that it proceeds through a first-order mechanism. Questions 1) Determine the fading rate, RA, of component A (in moles per unit time and per unit volume of the reactor) as a function of the mole fraction of A and the reaction rate constant, k. 2) Write the continuity equation specific to component A. 3) What happens to this equation in steady state? 4) How does the above equation simplify in case of a one dimensional diffusion, in direction z? 5) Deduce therefrom the differential equation expressing the mole fraction of A, xA(z), in the reactor. Data: The rate of reaction is given by: τA = - k xA τA is the reaction rate: [moles]/cm3sec

Fick’s Second Law: Macroscopic Balances

93

k is the reaction constant: [moles]/cm3sec xA is the mole fraction of A in the liquid phase xA(0) = xA° Solutions 1) RA in terms of the mole fraction of A and the reaction constant, k The fading rate of A is given by the chemical reaction rate. As this reaction is first-order, its rate is given by: τA = - k xA We deduce therefrom: RA = τA = - k xA. 2) The continuity equation specific to component A The continuity equation is written as follows for component A: ∂C + ∇F − R = 0 ∂t 3) The continuity equation in steady state In steady state, we have:

= 0.

The continuity equation thus becomes: ∇F − R = 0. 4) A single diffusion direction: z With a one dimensional propagation, F is a function of z only; as a result, we have: ∇F

=

dF dz

The continuity equation thus becomes: Or:

= −kx .

Yet: F = J ∗ + x (F + F ).

=R .

94

Mass Transfers and Physical Data Estimation

In addition, Fick’s first law gives the molar flux, J ∗ , in the form: J∗ = −CD

dx dz

Consequently: F = −CD

+ x (F + F ).

Yet, component B is stagnant, therefore: FB = 0. .

Hence: F (1 − x ) = −CD

Moreover, with the reaction of A with B being rapid, as soon as a mole of A arrives in the liquid phase, it is consumed by the reaction, therefore the concentration of A in the liquid will always remain very low. Consequently, we will have: xA