Flow boiling and condensation in microscale channels 3030687031, 9783030687038

This book covers aspects of multiphase flow and heat transfer during phase change processes, focusing on boiling and con

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Flow boiling and condensation in microscale channels
 3030687031, 9783030687038

Table of contents :
Preface
Contents
Nomenclature
Chapter 1: Introduction
1.1 Problems
References
Chapter 2: Fundamentals
2.1 Basic Definitions
2.2 Flow Patterns
2.2.1 Flow Patterns During Vertical Adiabatic Flow
2.2.2 Flow Patterns for Horizontal Adiabatic Flows
2.3 Void Fraction
2.3.1 Local Void Fraction
2.3.2 Line Averaged Void Fraction
2.3.3 Area Averaged Void Fraction
2.3.4 Volume Averaged Void Fraction
2.3.5 Void Fraction Predictive Methods
2.3.5.1 Slip Ratio Method
2.3.5.2 Drift Flux Model - Zuber and Findlay Method
2.3.5.3 Minimum Entropy Generation - Zivi´s Method
2.3.5.4 Minimum Kinetic Energy Method - Kanizawa and Ribatski Method
2.4 Flow Boiling Fundamentals
2.5 In-Tube Condensation Fundamentals
2.6 Transition from Macro to Microscale Conditions
2.7 Solved Example
2.8 Problems
References
Chapter 3: Flow Patterns
3.1 Flow Pattern Identification
3.2 Flow Pattern Transition Criteria for Adiabatic Flows
3.2.1 Graphical Methods
3.2.2 Taitel and Dukler (1976)
3.2.3 Taitel, Barnea, and Dukler (1980)
3.2.4 Barnea, Shoham, and Taitel (1982a)
3.3 Predictive Methods for Convective Boiling
3.3.1 Wojtan, Ursenbacher, and Thome (2005)
3.3.2 Revellin and Thome (2007)
3.3.3 Ong and Thome (2011)
3.4 Predictive Method for Convective Condensation
3.4.1 El Hajal, Thome, and Cavallini (2003)
3.4.2 Nema, Garimella, and Fronk (2014)
3.5 Solved Examples
3.6 Problems
References
Chapter 4: Pressure Drop
4.1 Predictive Methods for Frictional Pressure Drop Parcel
4.1.1 Homogeneous Model
4.1.2 Lockhart and Martinelli (1949)
4.1.3 Chisholm (1967)
4.1.4 Müller-Steinhagen and Heck (1986)
4.1.5 Cioncolini, Thome, and Lombardi (2009)
4.2 Solved Examples
4.3 Problems
References
Chapter 5: Flow Boiling
5.1 Nucleate Boiling Concepts
5.2 Heat Transfer Coefficient for Convective Boiling
5.3 Predictive Methods for Convective Flow Boiling
5.3.1 Liu and Winterton (1991)
5.3.2 Saitoh et al. (2007)
5.3.3 Kandlikar and Co-workers
5.3.4 Wojtan et al. (2005a, b)
5.3.5 Thome and Co-workers
5.3.6 Ribatski and Co-workers (Kanizawa et al. 2016; Sempertegui-Tapia and Ribatski 2017)
5.3.7 Heat Transfer Coefficient Under Transient Heating
5.4 Solved Examples
5.5 Problems
References
Chapter 6: Critical Heat Flux and Dryout
6.1 Introduction
6.2 Hydrodynamic Model
6.3 Macrolayer Model
6.4 Critical Heat Flux During In-Tube Flow
6.5 Solved Example
6.6 Problems
References
Chapter 7: Condensation
7.1 Film Condensation on an Isothermal Surface
7.2 Predictive Methods for In-Tube Convective Condensation
7.2.1 Dobson and Chato (1998)
7.2.2 Cavallini et al. (2006)
7.2.3 Shah (2016)
7.2.4 Jige, Inoue, and Koyama (2016)
7.3 Solved Examples
7.4 Problems
References
Index

Citation preview

Mechanical Engineering Series

Fabio Toshio Kanizawa Gherhardt Ribatski

Flow boiling and condensation in microscale channels

Mechanical Engineering Series Series Editor Francis A. Kulacki, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

The Mechanical Engineering Series presents advanced level treatment of topics on the cutting edge of mechanical engineering. Designed for use by students, researchers and practicing engineers, the series presents modern developments in mechanical engineering and its innovative applications in applied mechanics, bioengineering, dynamic systems and control, energy, energy conversion and energy systems, fluid mechanics and fluid machinery, heat and mass transfer, manufacturing science and technology, mechanical design, mechanics of materials, micro- and nano-science technology, thermal physics, tribology, and vibration and acoustics. The series features graduate-level texts, professional books, and research monographs in key engineering science concentrations.

More information about this series at http://www.springer.com/series/1161

Fabio Toshio Kanizawa • Gherhardt Ribatski

Flow Boiling and Condensation in Microscale Channels

Fabio Toshio Kanizawa Laboratory of Thermal Sciences (LATERMO) Universidade Federal Fluminense, School of Engineering Niterói, Rio de Janeiro, Brazil

Gherhardt Ribatski Department of Mechanical Engineering University of São Paulo, São Carlos School of Engineering São Carlos, São Paulo, Brazil

ISSN 0941-5122 ISSN 2192-063X (electronic) Mechanical Engineering Series ISBN 978-3-030-68703-8 ISBN 978-3-030-68704-5 (eBook) https://doi.org/10.1007/978-3-030-68704-5 © The Editor(s) (if applicable) and The Author(s) 2021 All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This is an introductory textbook for multiphase flow and heat transfer during convective boiling and condensation inside channels focusing on microscale systems. A proper analysis of heat transfer with phase change inside channels requires knowledge about the flow characteristics, hence a significant parcel of the book is dedicated to the analysis and modelling of hydrodynamic aspects of liquid and gas flows and the parameters employed on their characterizations. Therefore, even though the ultimate objective of this book is the analysis of convective boiling and condensation, students focused on adiabatic flow, such as for oil and gas industry, can also adopt this book as reference because it gathers several prominent studies of multiphase flow subject into a single text with uniform nomenclature. It must be highlighted that the fundamentals of multiphase flows have been formalized and studies have begun to be published in a systematic way mainly from the 1950s. From the late 1960s, the investigations in this area have experienced a search for mathematical formality and a tendency of unifying the modelling approach. Hence, multiphase flows is a relatively young subject of research, and this book aims to contribute on this task by addressing classical and more recent studies of the area into a single text. The characteristics of multiphase flows depend on several aspects, including fluids properties, operational conditions, phases proportion, channel orientation, and geometry, and whether the flow is heated, cooled, or adiabatic, among other aspects. Therefore, the complete analysis of the flow requires a detailed investigation of the phenomena occurring along each phase and how the phases interact among themselves and with the flow boundary conditions. Such an approach is challenging; in this context, numerical studies involving Lattice Boltzmann techniques and Molecular Dynamic and Direct Numerical Simulations are being employed to investigate the microscopic aspects of two-phase flows, and based on them, complex models are being constructed. Up-to-date instrumentation such as high-speed video cameras, IR thermography, and micro PIVs, among others, are also being employed in order to investigate the microscopic aspects of heat transfer and momentum diffusion in two-phase flows. However, an analysis of such studies are out of the v

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Preface

scope of the present book that focuses mainly on analyses assuming one-dimensional flow with averaged properties along the cross section, with their variation along the channel length due to heat and/or momentum transfer. Even though modelling such a complex problem through a one-dimensional approach might seem to be over simplified, this procedure has been in use for several decades and has provided reliable results because most of the models developed based on one-dimensional approach relies on innovative modellings and adjustment of empirical constants based on broad experimental databases. Although the book’s title indicates its main focus on microscale channels, the reader shall notice that most analyses begin with descriptions of models for conventional-sized channels following a historical order and then switch to microscale channels. This sequence is adopted based on the fact that usually the models developed for microscale conditions derive from models for conventional channels. As a consequence, the basic parameters employed on the characterization of multiphase flows inside conventional channels and mini and microchannels are similar. Differences arise from distinctions between the dominant forces and the main physical mechanisms, for example, stratified flows in horizontal channels, characterized by the liquid flowing in the lower part of the channel and vapor in its upper part, are not observed in microscale channels due to the predominance of surface tension over gravitational effects on the two-phase flow distribution. The present textbook is focused on gas–liquid flow of the same substance, and most of the modelling approaches consider saturated liquid and vapor in thermal equilibrium, and when not specified differently, conditions of incompressible flow (low Mach number) are assumed. Additionally, most of the analyses presented here refer to the flow along a single channel; however, it must be highlighted that differences related to flow maldistribution effects and reverse flow, among others, might result in distinct behavior for multichannel system when compared with single channel. Based on these aspects, this book starts with an introductory and motivational chapter, which aims to explain the advantages and applications concerning phase change in microscale channels. Then, Chap. 2 presents the fundamental parameters of multiphase flow, including fundamental aspects of flow patterns, void fraction, and heat transfer processes. Chapter 3 is dedicated to flow pattern characterization and prediction, including conditions of conventional and microscale channels. Then, Chap. 4 presents the fundamental aspects of pressure drop for single- and two-phase flows and derives the dominant equations for the later. Additionally, several predictive methods for estimate of the frictional parcel of the pressure drop are addressed and compared. Chapters 5 and 6 cover pool and convective boiling, critical heat flux, and dry out, presenting an overview of the heat transfer trends, dominant mechanisms, and methods to predict the heat transfer coefficient. Finally, Chap. 7 addresses the subject of condensation, describing the film condensation model of Nusselt followed by an analysis of the condensation inside channels. Regarding the organization of this book, whenever possible, usual nomenclature of the field is adopted to facilitate the transition from and to different textbooks.

Preface

vii

Additionally, when not explicitly mentioned differently, it is assumed that the working fluid is a pure substance, hence it presents constant saturation temperature for a given pressure. Several enhancement techniques for heat transfer in microscale channels are available in the open literature, such as micro fins, re-entrant cavities, and porous coatings. Nonetheless, since these methods are not consolidated, this book is focused on heat transfer in microscale channels without any enhancement. A complete understanding of the subjects covered in this book requires a previous knowledge of thermodynamics, fluid mechanics, and convective heat transfer during single-phase flow. An experienced researcher may find the descriptions of traditional models and predictive methods a bit different from the original presentations, such as the flow pattern transition criteria presented by Taitel and Dukler (1976) and the Lockhart and Martinelli (1949) developments on pressure drop, mainly related to the basic parameters adopted to describe the mixture flow. Whenever possible and convenient, the present authors address the models based on the parameters presented in Chap. 2, which include void fraction, superficial velocities, and so on, rather than liquid height, which are not uncommon in classical literature. By completing the study of the subject addressed in this book, the reader shall be able to discuss the main aspects of multiphase flow and heat transfer processes. Additionally, it is expected that the students should be able to infer the dominant mechanisms for heat and momentum transfer, design equipment and systems operating with phase change, and continue their studies with specific literature for the subject of investigation. São Paulo, Rio de Janeiro, Brazil Niteròi, Rio de Janeiro, Brazil São Carlos, São Paulo, Brazil

Fabio Toshio Kanizawa Gherhardt Ribatski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 10 10

2

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Flow Patterns During Vertical Adiabatic Flow . . . . . . . . . 2.2.2 Flow Patterns for Horizontal Adiabatic Flows . . . . . . . . . 2.3 Void Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Local Void Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Line Averaged Void Fraction . . . . . . . . . . . . . . . . . . . . . 2.3.3 Area Averaged Void Fraction . . . . . . . . . . . . . . . . . . . . . 2.3.4 Volume Averaged Void Fraction . . . . . . . . . . . . . . . . . . 2.3.5 Void Fraction Predictive Methods . . . . . . . . . . . . . . . . . . 2.4 Flow Boiling Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 In-Tube Condensation Fundamentals . . . . . . . . . . . . . . . . . . . . . 2.6 Transition from Macro to Microscale Conditions . . . . . . . . . . . . 2.7 Solved Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 20 21 23 28 28 29 29 31 33 43 49 51 56 60 61

3

Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1 Flow Pattern Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Flow Pattern Transition Criteria for Adiabatic Flows . . . . . . . . . . . 68 3.2.1 Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.2 Taitel and Dukler (1976) . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.3 Taitel, Barnea, and Dukler (1980) . . . . . . . . . . . . . . . . . . . 85 3.2.4 Barnea, Shoham, and Taitel (1982a) . . . . . . . . . . . . . . . . . 96 3.3 Predictive Methods for Convective Boiling . . . . . . . . . . . . . . . . . 103 3.3.1 Wojtan, Ursenbacher, and Thome (2005) . . . . . . . . . . . . . . 103

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

ix

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Contents

3.3.2 Revellin and Thome (2007) . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Ong and Thome (2011) . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Predictive Method for Convective Condensation . . . . . . . . . . . . . 3.4.1 El Hajal, Thome, and Cavallini (2003) . . . . . . . . . . . . . . 3.4.2 Nema, Garimella, and Fronk (2014) . . . . . . . . . . . . . . . . 3.5 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

107 110 112 114 117 120 121 122

4

Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Predictive Methods for Frictional Pressure Drop Parcel . . . . . . . . 4.1.1 Homogeneous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Lockhart and Martinelli (1949) . . . . . . . . . . . . . . . . . . . . 4.1.3 Chisholm (1967) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Müller-Steinhagen and Heck (1986) . . . . . . . . . . . . . . . . 4.1.5 Cioncolini, Thome, and Lombardi (2009) . . . . . . . . . . . . 4.2 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

125 137 137 140 146 149 151 156 158 159

5

Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Nucleate Boiling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Heat Transfer Coefficient for Convective Boiling . . . . . . . . . . . . 5.3 Predictive Methods for Convective Flow Boiling . . . . . . . . . . . . 5.3.1 Liu and Winterton (1991) . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Saitoh et al. (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Kandlikar and Co-workers . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Wojtan et al. (2005a, b) . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Thome and Co-workers . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Ribatski and Co-workers (Kanizawa et al. 2016; Sempertegui-Tapia and Ribatski 2017) . . . . . . . . . . . . . . 5.3.7 Heat Transfer Coefficient Under Transient Heating . . . . . 5.4 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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161 163 179 189 190 191 194 196 199

. . . . .

205 208 210 213 214

Critical Heat Flux and Dryout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Hydrodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Macrolayer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Critical Heat Flux During In-Tube Flow . . . . . . . . . . . . . . . . . . . 6.5 Solved Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

217 217 219 224 228 235 238 239

6

Contents

7

Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Film Condensation on an Isothermal Surface . . . . . . . . . . . . . . . 7.2 Predictive Methods for In-Tube Convective Condensation . . . . . . 7.2.1 Dobson and Chato (1998) . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Cavallini et al. (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Shah (2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Jige, Inoue, and Koyama (2016) . . . . . . . . . . . . . . . . . . . 7.3 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

. . . . . . . . . .

241 242 250 251 254 256 257 260 262 263

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Nomenclature

A Cf d ds D E Fc f g G h H î j k K L m ṁ M p P q Q ! r r Rp Ra s

transversal area, m2 Darcy friction factor, non-dimensional diameter, m line segment, m domain energy, J enhancement factor of convective effects, non-dimensional Fanning friction factor, non-dimensional gravitational acceleration, m/s2 mass flux, kg/m2s heat transfer coefficient, W/m2K height, m specific enthalpy, J/kg superficial velocity, m/s thermal conductivity, W/mK momentum coefficient, non-dimensional length, m mass, kg mass flow rate, kg/s molar mass, kg/kmol pressure, Pa perimeter, m heat transfer rate, W volumetric flow rate, m3/s position vector, m radius, m surface peak roughness, μm surface averaged roughness, μm entropy, J/kgK xiii

xiv

ŝ S Snb T Tˆ t u V V w x X x y Ŷ z Δp Δt

Nomenclature

Laplace variable, 1/s surface area, m2 suppression factor of nucleate boiling effects, non-dimensional temperature,  C time constant, s time instant, s local velocity, m/s velocity, m/s volume, m3 in situ axial velocity, m/s axis perpendicular to z and y density function, non-dimensional vapor quality, non-dimensional horizontal axis perpendicular to the duct axis gravitational parameter of Taitel and Dukler model, non-dimensional axial direction pressure difference, Pa time interval, s

Greek Symbols α β βc ε Γ γ δ η θ λ Λ μ ρ σ τ ϕ ω ξ

void fraction, non-dimensional volumetric fraction, non-dimensional contact angle,  entrainment factor, non-dimensional propagation velocity of interfacial perturbations, m/s stratification angle, rad liquid film thickness, m generic parameter inclination relative to horizontal plane, rad mean free path, m wavelength, m viscosity, kg/m.s density, kg/m3 surface tension, N/m shear stress, Pa heat flux, W/m2 frequency, Hz liquid holdup of entrained droplets, non-dimensional

Nomenclature

Subscripts 0 1 2 3 b c cs e f fluid g h i I k l l0 lv m n nb r sat v v0 z w cap

relative to a point relative to a line segment relative to an area relative to a volume bubble relative to convective effects relative to the channel wall section equivalent frictional parcel evaluated for bulk conditions gravitational parcel hydraulic relative to phase i interface relative to kinetic energy component relative to liquid phase assuming the mixture flowing as liquid vaporization momentum parcel domain dimension relative to nucleate boiling reduced property saturation condition relative to vapor phase assuming the mixture flowing as vapor axial direction evaluated at the wall temperature relative to capillar effects

Nondimensional Parameters Bd Bo Ca Cn Co Eo Fr Ga Kn La*

Bond number Boiling number Capillary number Convection number Confinement number Eötvos number Froude number Galileo number Knudsen number Laplace number

xv

xvi

Pr Re We X^lv

Nomenclature

Prandtl number Reynolds number Weber number Lockhart and Martinelli parameter

Operators n

Space average operator Time average operator

Chapter 1

Introduction

Multiphase flow corresponds to simultaneous flow of two or more immiscible phases, which can be gas-liquid, liquid-liquid, gas-solid, liquid-solid, or a combination of these pairs, and its characteristic is strongly dependent on the interface between the phases. The simplest multiphase flow corresponds to two-phase flow that consists in simultaneous flow of two immiscible phases. For liquid-liquid flow to be treated as a multiphase flow, it is necessary that they are composed of immiscible liquids, such as oil and water. Similarly, the tendency of gases to get mixed when put in contact also precludes mixtures of different gases to be characterized as multiphase flows. Multiphase flow is present in several natural and industrial processes. The morning shower can be treated as a two-phase flow, with liquid water droplets flowing downward due to gravitational forces and interacting with air and vapor mixture. The water boiling in the kettle for the morning coffee or tea is also a condition of multiphase flow, with water vapor being formed in the kettle bottom surface, detaching and rising due to buoyance forces. The operation of internal combustion engines of the commute involves multiphase flow in several processes, such as the fuel injection that consists of a mixture of air and dispersed droplets of liquid fuel, and the lubricating oil in contact with air and engine parts in the oil pan, among others. In recreational aspects, the multiphase flow is also present, such as when pouring beer from the bottle into a pint, the condition in the bottle neck corresponds to counter current the flow of air entering the recipient and liquid exiting it. With the pressure reduction due to the opening of the bottle, the dissolved gas tends to be released in the form of bubbles forming the beer foam, and when in contact with hotter surface of the glass, additional bubbles are formed and rise due to buoyance forces. Several examples of multiphase flow can be found also in natural processes, such as rain, fog, clouds, wave formation, dust, sandstorm, etc. The cases of multiphase flow with a solid phase, such as in dust transport by the wind or slurry flow, the mixture behaves as fluid and then can be treated as multiphase flow.

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_1

1

2

1 Introduction

The subject of multiphase flow has been investigated by several industrial areas, including power generation from fossil fuels, nuclear source, or even based on renewable energy source more recently, when considering the application of organic Rankine cycles (ORC) for solar or geothermal energy harvesting. Additionally, and according to that described above, the multiphase flow is also present in refrigeration and heat management applications, oil and gas industry, and material transport, among others. The pioneer studies on multiphase problems have been addressed at the beginning of the twentieth century, by investigations of heat transfer problems with phase change. It should be highlighted that in this period, even single-phase flow was a challenging and unknown world, and Prandtl solution of the boundary layer problem was just coming up in 1904. Just a few years later, Nusselt (1916) described the modelling of condensation on a vertical isothermal plate which can be considered one of the first theoretical studies for heat transfer of phase change processes. From the 1940s, several studies focusing on multiphase flows have been published, with successive increment of complexity and agreement with empirical observations. Subsequently, from the 1970s the number of publications focused on nuclear industry has increased, and since then the overall publications related to multiphase flow have increased sharply, emphasizing the growing interest and relevance of the theme. Figure 1.1 depicts the number of publications along the decades according to different areas of applications and segregating two-phase flow multiphase flow studies. Boiling, condensation, freezing, melting, and sublimation are heat transfer processes associated to multiphase flows that provide reasonable high heat transfer

Number of publications 100000 10000 1000 100 10 1

Refrigeration

Two-phase flow

Multiphase flow

Steam generator

Nuclear multiphase Oil two-phase flow

Fig. 1.1 Number of publications along the time according to search term for abstract, title, or keyword in July of 2019, according to Science Direct database in June of 2019

1 Introduction

3

coefficients. For example, R134a single-phase liquid flow in a 1.5 mm ID channel the heat transfer coefficient is approximately 1300 W/m2K for a mass velocity of 500 kg/m2s and a fluid temperature of 30  C, while, under conditions of convective boiling, values in the order of 5000 to 10,000 W/m2K are achieved for the same mass flow rate and temperature. Recall that the heat transfer coefficient for natural convection of a surface exposed to gases is of the order of 5 W/m2K, and for forced convection of gases is of the order of 80 W/m2K. The high heat transfer coefficients typical of phase change processes allows the minimization of the equipment size compared to similar equipment working under single-phase conditions. Nonetheless, it is important to highlight that it is also possible to obtain high heat transfer coefficients under single-phase flow conditions, such as for forced convection of liquid metals and forced convection of conventional fluids at high or extremely high velocities. However, their use is restricted to special applications, such as nuclear power generation. In this context, the reader should recall from the thermodynamic course that one example of irreversibility is related to heat transfer with a finite temperature difference (ΔT ), whereas the reversible heat transfer process requires infinitesimal temperature difference (dT). Hence, recalling the Newton’s Cooling Law from the heat transfer course, the heat flux for convective heat transfer is given as follows: ϕ ¼ hΔT

ð1:1Þ

where ϕ is the heat flux in W/m2 from the surface to the fluid, ΔT is the temperature difference between the surface and the fluid, and h is the heat transfer coefficient in W/m2K in SI units. Hence, for a given heat flux, the lower the heat transfer coefficients, the higher the temperature difference and consequently the irreversibility. The condition of infinitesimal temperature difference would be achieved only for very high heat transfer coefficient, as discussed before (ϕ ¼ hΔT  hdT for very high h). Additionally, the reader shall remember that the Carnot cycle is the most efficient thermodynamic cycle, either for power generation or refrigeration, composed by two isentropic and adiabatic processes, corresponding to compression, or pumping, and expansion processes and two isothermal processes, corresponding to heat transfer processes with infinitesimal temperature difference. Hence, a single-phase liquid flowing along a channel that receives or rejects energy will present temperature variation which deviates from the isothermal process, unless conditions of very highspecific heat, density, and/or mass flow rate are imposed. The fluid temperature variation could be overcome using phase change processes for pure substance, whereas for vapor-liquid in equilibrium of pure substances the temperature variation is related only to pressure variation. Therefore, by properly designing the heat exchangers to obtain low pressure drop, the heat transfer process in the cycle can be considered as almost isothermal. Figure 1.2 schematically depicts the thermodynamic diagram of theoretical cycles for refrigeration based on vapor compression and power generation, assuming pure substance and thermal reservoirs with uniform temperature. Considering the case of refrigeration, depicted in Fig. 1.2a, the refrigerated space temperature Tref must be higher than evaporation temperature, and the

4

1 Introduction

=

cte

c

p

p=

cte

b

p=

cte

a

Tsource T

Text

T

Tref

Carnot cycle Text

Carnot cycle

Theoretical cycle s Refrigeration - Vapor compression

Theoretical cycle s Power generation - Rankine

T

Text

Carnot cycle

Tref Theoretical cycle s Refrigeration - Gas cycle

Fig. 1.2 Schematics of refrigeration and power generation cycles in comparison with Carnot cycle – temperature vs. entropy diagram (a) Refrigeration - Vapor compression cycle (b) Power generation - Rankine (c) Refrigeration - Gas cycle

condensation temperature must be higher than the temperature of the ambient to which heat is rejected Text. Similarly, in the case of power generation depicted in Fig. 1.2b, the temperature of the thermal reservoir that is the energy source to generate vapor Tsource must be higher than the fluid temperature along the heat exchanger, and the condensing temperature must be higher than the temperature of the ambient to which energy is rejected Text. In a practical case, the Tsource could be defined in two stages, with a lower temperature in the phase change region, and higher in the superheating region to reduce requirements of the hot thermal reservoir. In any case, either for refrigeration or power generation, Fig. 1.2 also depicts the Carnot cycle, which is closer to the theoretical cycle with phase change in comparison with the single-phase cycle, depicted in Fig. 1.2c for refrigeration with gas cycle. It can be added to this discussion that the high vaporization latent heat of conventional fluids, for example, for water at atmospheric pressure, the vaporization enthalpy is approximately 2.257 MJ/kg, and the specific heat at constant pressure of the saturated liquid at the same pressure is 4.217 kJ/kgK. Hence, to sustain the same heat transfer of vaporization, the same mass of liquid water would have to experience a temperature variation higher than 500 K, and would not be liquid anymore. Even considering fluid and operational conditions that support such temperature variation, the temperature of the hot source needs to be higher than the highest temperature of the working fluid, such as depicted in Fig. 1.2, which impacts the costs of the heat source and of the materials to sustain such temperatures. Conversely, when using liquid-vapor flow at almost uniform temperature, the heat source can be at a temperature slightly above the saturation temperature. Still considering refrigeration processes, the assumption of isobaric heat transfer processes in the heat exchangers is usually adopted for didactical and modelling purposes, which relies on the condition that the pressure drop in heat exchangers is null. However, in real systems the pressure reduces along the flow path due to fluid friction with the channel wall, and due to phase interactions in the case of multiphase flow. Figure 1.3 depicts schematically the pressure vs. enthalpy diagram for vapor compression cycle, similar to the one presented in Fig. 1.2a, however also with the

p

Theoretical cycle Text

s=

Fig. 1.3 Vapor compression cycle with effect of pressure drop along the heat exchangers – pressure vs. enthalpy diagram

5

cte

1 Introduction

Carnot cycle

s=c te

Tref

Δic,1

Cycle with Δic,2 pressure drop i Refrigeration - Vapor compression

effect of pressure drop along the heat exchangers. According to Fig. 1.3, the pressure drop in heat exchangers implies on variation of the fluid temperature, increasing the deviation from the Carnot cycle. Additionally, according to Fig. 1.3 the pressure in the heat exchangers implies on increment of the compressing power. In the case of theoretical cycle with no pressure drop, the compressing power is given as mass flow rate multiplied by the enthalpy variation in the compressor, denoted by Δic,1, while in the case that pressure drop is accounted for, the enthalpy difference increases to Δic,2. The reader might have noticed that even the cooling capacity has been impacted, and an increase in the mass flow rate, or superheating of the vapor, would be necessary to keep the same capacity. Based on this discussion, it can be concluded that it is impossible to treat only the heat transfer processes and not consider the momentum transfer processes, associated with pressure drop of internal flows. Therefore, this book dedicates Chap. 4 to this subject. In this context, it could be mentioned that systems with large capacity, such as air-conditioning in shopping malls or refrigeration in meat industry, usually counts with shell and tube heat exchangers as evaporator and condenser, with refrigerant operating in the shell side, hence with lower pressure drop. However, considering systems with low or intermediate capacity, such as domestic and commercial refrigerator and air-conditioning, usually the evaporator and condenser consist in coiled and finned heat exchangers with refrigerant in the tube side, which consists in a condition of relevant flow resistance. Still in the subject of refrigeration and air-conditioning, a recent report published by International Energy Agency (IEA 2018) pointed out that space cooling consumed more than 25% of USA’s total electrical energy demand during a condition of peak consumption in 2016, and the world averaged parcel is of the order of 10%. The same report indicated that the expected increment of energy demand for space cooling in the world from 2016 to 2050 is approximately 2500 GW, which is higher than the combined generation capacity of the USA, Europe, and India in 2016,

6

1 Introduction

Fig. 1.4 Share of renewable energy in global consumption in 2010. The World Bank (2018)

implying in higher demand for natural resources and impacting the environment. A possible way to mitigate this effect is the development of more efficient refrigeration systems, which rely on improvement of compressor, heat exchangers, and development of new materials and fluids. In this context, The World Bank (2018) indicated that more than 75% of the electrical energy was generated from non-renewable sources in 2010, such as depicted in Fig. 1.4, which impacts the environment and motivates additional research to develop and improve renewable energy sources to suffice the inevitable increment of demand. Hence, photovoltaic solar cells have been in development for several decades, and nowadays their use has become more common due to reduction of price. Nonetheless, the current efficiency in converting solar energy into electrical energy is between 12 to approximately 17% for single-junction and in laboratorial conditions according to the report of Fraunhofer Institute for Solar Energy Systems (Fraunhofer Institute for Solar Energy Systems, ISE 2019), and the non-converted parcel is reflected or absorbed by the material as heat, which must be dissipated to avoid superheating that would cause severe efficiency deterioration or even permanent damage of the component. Singh and Ravindra (2012) evaluated the effect of operating temperature on the open circuit tension, closed circuit current, and converting efficiency and concluded that the three parameters reduced with increment of temperature. The heat can be properly dissipated by natural convection in the case of conventional solar cells, however in the case of concentrated solar cells, the energy input flux can be of the order of 300 kW/m2 according to Sinton et al. (1986) and Green et al. (2015). In these cases, air cooled heat spreaders do not suffice the heat dissipation, and alternative approaches are needed, such as using microchannel heat sinks with phase change process, which has potential of absorbing high heat fluxes with low temperature variation. In the same context, another approach for solar energy harvesting that has gained interest is the organic Rankine cycle (ORC), which is fundamentally similar to the steam Rankine cycle (SRC), however, it operates with organic fluids rather than water. Hence, due to the characteristics of the organic fluids, the energy source for vapor generation in the ORC can be of lower temperature in comparison with SRC, which allows the operation with low grade thermal energy sources, such as from solar collectors, geothermal reservoirs, and industrial waste heat. In this context,

1 Introduction

7

Fig. 1.5 Variation of transistors count in processors between 1971 and 2018 – Moore’s Law

Tocci et al. (2017) recently presented a review about this concept and listed the main advantages and disadvantages of ORC in comparison with SRC systems. Nonetheless, even though ORC-based systems are already commercially available, several aspects of the heat and momentum transfer for the characteristic operational conditions are still unclear, because they are way distinct from those for refrigeration systems. This aspect emphasizes the need for additional studies focusing on the understanding of the main mechanisms occurring during heat transfer process in the heat exchangers, aiming to improve the design, reliability, and efficiency of systems based on ORC. Another application that heat transfer with phase change has gained attention in the recent decades is related to heat management of electrical and electronic equipment, mostly focused on semiconductor components, such as processors, high power transistors, and laser sources. In this context, in 1965 Gordon E. Moore foresaw that the number of transistors in processors would double every year, which was revised in 1975 to doubling every 2 years, and this prediction is nowadays known as Moore’s Law, which has shown to be in agreement with the evolution of computer processors as shown in Fig. 1.5. With the increment of transistor numbers and clocks, the heat dissipation per unit volume has increased accordingly, and the heat dissipation to keep the semiconductor temperature within the allowed operational range, usually below 65  C, can be accomplished by air-cooled heat sinks until recently. However, considering the trend of continuous increment of power dissipation in reduced spaces/areas, air cooled systems will not suffice and use of other working fluid is an alternative, which in turns bring the use of microchannel due to space restrictions.

8

1 Introduction

The appealing aspect of increment of efficiency of heat management of processes can also be seen on basis of the energy consumption of data centers. According to Zhang et al. (2017) and Song et al. (2015), approximately 40% of the electricity consumption of data centers is related to air-conditioning to keep the processor in the optimal temperature range. And according to Jones (2018), the share of datacenter in 2018 on the electricity consumption is approximately 1% worldwide; however, it is expected that by 2030 this parcel would increase to approximately 8%, also increasing the demand for more energy production. Again, a way of reducing this impact would be the development of more efficient processors and heat management systems, such as direct cooling of the chip instead of the entire system and building. In this context, the reduction of channel diameter itself already implies the increment of the heat transfer coefficient. For example, consider a condition of laminar flow with imposed heat flux, which is quite common since the flow rate required to reach turbulent flow regime is usually very high in microscale channel due to its reduced diameter. Therefore, the heat transfer problem in developed singlephase flow consists in a condition of constant Nusselt number of 4.364 for uniform heat flux condition, and the diameter reduction implies on the increment of the heat transfer coefficient as follows: Nu ¼ 4:364 ¼

hd k ) h ¼ 4:364 k d

ð1:2Þ

where h is the heat transfer coefficient in W/m2K, d is the channel internal diameter in meters, and k is the fluid thermal conductivity in W/m.K. Hence, the reduction of channel size implies on more efficient heat exchangers. On the other hand, the heat transfer rate for this specific condition is independent of the channel diameter, as follows: k q_ ¼ hAΔT lm ¼ 4:364 πdLΔT lm ¼ 4:364πkLΔT lm d

ð1:3Þ

where L is the length of the considered channel and ΔTlm is the log mean temperature difference between the duct wall and fluid bulk temperature. Therefore, reduction of the channel size implies on reduction of the material cost and refrigerant charge for the same heat transfer rate. It must be highlighted that, according to the analysis of this simple case, the mass flow rate presents no influence on the total amount of heat removed by the fluid, as long as the flow is kept under laminar regime. The influence of the flow rate could be perceived in the enthalpy variation, which can lead to the saturation condition and consequent two-phase flow boiling in the case of positive heat flux.

1 Introduction

9

On the other hand, the friction factor for laminar developed flow is given as follows: 16 f ¼ Gd ð1:4Þ μ

where G is the mass flux in kg/m2s and μ is the fluid viscosity in kg/ms. And the pressure drop is given as follows: Δp ¼ 2f

G2 L GL ¼ 32μ 2 ρd ρd

ð1:5Þ

Therefore, the pressure drop increases according to a square power with the reduction of the channel diameter for round duct and laminar flow. Continuing the analysis, the pumping power parcel due to the pressure drop in the considered channel can be given by the product of the pressure drop and the volumetric flow rate, as follows: Pumping power ¼ ΔpQ ¼ Δp

G πd2 GL G πd2 G2 L ¼ 32μ 2 ¼ 8πμ 2 ρ 4 ρ ρd ρ 4

ð1:6Þ

Therefore, for the same mass velocity, or mass flux, the pumping power is independent of the duct diameter. It must be emphasized that this analysis is restricted to the condition of laminar flow in round duct, and its validity depends on limitation of heat flux to avoid phase change by vaporization. Nonetheless, this analysis gives us an idea about the advantages of the use of microscale channels related to reduction of channel size and refrigerant charge for a similar heat transfer rate and pumping power. In the case of multiphase flow of pure substance, as above mentioned, the temperature variation depends basically on pressure variation, and this characteristic provides more uniform temperature of the surface to be cooled. Considering the applications for heat management of semiconductors, this aspect is also very important because the presence of hot spots along the component can deteriorate its capacity, as pointed out by Royne et al. (2005) and Baig et al. (2012) for photovoltaic cells. These studies concluded that hot spots could deteriorate the system efficiency, or even damage it permanently. Nonetheless, a similar analysis as the one presented above for phase change problems would be way more complex, as the reader will see along the book. Therefore, this textbook aims to present the fundamental concepts of multiphase flow, with successive increment of complexity in the analysis for each topic. Hence, the foregoing chapters deal with distinct aspects related to multiphase flow, and for heat transfer during boiling and condensation.

10

1 Introduction

1.1

Problems

1. List examples of multiphase flow that you faced since you woke up. 2. List examples of phase change processes that you faced since you woke up. 3. Consider a refrigeration system operating with R134a with evaporating temperature of 20  C and condensing temperature of 40  C. Evaluate the coefficient of performance COP considering Carnot cycle, theoretical cycle, and a cycle with pressure drop of 10% of the inlet pressure in each heat exchanger assuming isentropic compression. 4. Repeat Exercise 3 assuming isentropic efficiency of 75%. 5. Evaluate the required heat transfer area for a tube-in-tube counter-current heat exchanger for 1 kW operating with the following (tip: use effectiveness approach, and assume the thermal resistance related to conduction as negligible for items a, b, and c): (a) Single-phase water on both sides, with inlet temperatures of 90 and 20  C, and respective mass flow rates of 0.025 and 0.030 kg/s, and heat transfer coefficient for each side as approximately 1000 W/m2K. (b) Repeat previous item assuming isothermal fluids. (c) Single-phase water flow in the shell at 90  C and mass flow rate of 0.025 kg/s, and R134a boils inside at evaporating temperature of 20  C and averaged heat transfer coefficient of 5000 W/m2K. Estimate the vaporization rate in kg/s. (d) Condensation of water at 100  C in the shell, and convective boiling of R134a on the tube side, both with averaged heat transfer coefficient of 5000 W/m2K. 6. Consider a hot thermal reservoir with 300  C and a cold thermal reservoir with 30  C, estimate the efficiency of theoretical thermal machines operating with: (a) Rankine cycle operating with water, evaporating temperature (Tsat) of 200  C, outlet of steam generator at 300  C, condensing at 30  C. Assume isentropic pumping and expansion. (b) ORC operating with R245fa, without superheating, and fluid operating at 300  C in the vapor generator, and 30  C. Assume isentropic pumping and expansion.

References Baig, H., Heasman, K. C., Sarmah, N., & Mallick, T. (2012, October). Solar cells design for low and medium concentrating photovoltaic systems. In AIP conference proceedings (Vol. 1477, 1, pp. 98–101). AIP. Fraunhofer Institute for Solar Energy Systems, ISE. (2019). Photovoltaics report - with support of PSE GmbH. (report). November, 14th of 2019. Accessed in April, 9th of 2020, available at: https://www.ise.fraunhofer.de/content/dam/ise/de/documents/publications/studies/Photovol taics-Report.pdf.

References

11

Green, M. A., Emery, K., Hishikawa, Y., Warta, W., & Dunlop, E. D. (2015). Solar cell efficiency tables (version 45). Progress in Photovoltaics: Research and Applications, 23(1), 1–9. IEA. (2018). The future of cooling. Opportunities for energy-efficient air conditioning. Paris: International Energy Agency. Jones, N. (2018). The information factories. Nature, 561(7722), 163–166. Nusselt, W. (1916). The condensation of steam on cooled surfaces. Zeitschrift des Vereins Deutscher Ingenieure, 60, 541–546. Prandtl, L. (1904). On fluid motions with very small friction (in German). Third International Mathematical Congress (pp. 484–491). Heidelberg. Royne, A., Dey, C. J., & Mills, D. R. (2005). Cooling of photovoltaic cells under concentrated illumination: A critical review. Solar Energy Materials and Solar Cells, 86(4), 451–483. Singh, P., & Ravindra, N. M. (2012). Temperature dependence of solar cell performance—An analysis. Solar Energy Materials and Solar Cells, 101, 36–45. Sinton, R. A., Kwark, Y., Gan, J. Y., & Swanson, R. M. (1986). 27.5-percent silicon concentrator solar cells. IEEE Electron Device Letters, 7(10), 567–569. Song, Z., Zhang, X., & Eriksson, C. (2015). Data center energy and cost saving evaluation. Energy Procedia, 75, 1255–1260. The World Bank. (2018). Global tracking framework. Sustainable energy for all. 77889 v.3. Tocci, L., Pal, T., Pesmazoglou, I., & Franchetti, B. (2017). Small scale Organic Rankine Cycle (ORC): A techno-economic review. Energies, 10(4), 413. Zhang, X., Lindberg, T., Xiong, N., Vyatkin, V., & Mousavi, A. (2017). Cooling energy consumption investigation of data Center IT room with vertical placed server. Energy Procedia, 105, 2047–2052.

Chapter 2

Fundamentals

Two-phase flow conditions are characterized by the simultaneous flow of two identifiable phases, which can be of the same substance flowing as different phases and of distinct and immiscible substances, such as liquid and non-condensable gas. Single-phase flow in conventional ducts, such as straight pipes with circular crosssection and large diameters, are satisfactorily well characterized by the flow rate, fluid properties, and duct geometry. On the other hand, in the case of two-phase flow, the channel orientation, phase geometrical distributions characterized by the flow patterns, and their parcels along the duct also play an important role and preclude the unambiguous characterization of the flow conditions. Therefore, additional definitions are required to characterize the two-phase flow. These parameters ultimately influence the heat transfer coefficient and pressure drop.

2.1

Basic Definitions

Two-phase flow parameters are defined in this chapter. The characterization of the operational conditions of two-phase flows requires the definition of several parameters such as velocity, saturation temperature, phases distributions, and vapor fraction, among others; therefore, it is essential to define these parameters precisely and unambiguously. Most of the definitions presented here are similar to those adopted by Delhaye (1981a), Wallis (1969), and Collier and Thome (1994), and they are valid for two-phase flow in microscale channels. Moreover, most of the definitions are also valid for adiabatic conditions, commonly seen in oil and gas industry. Even though the gas-liquid two-phase flow rarely can be considered strictly developed due to variation of phase velocities related to phase change and/or pressure drop that causes variation of vapor density and fraction, it is possible to define its characteristics for a given cross-section, or for a “short” region of the channel. Therefore, based on the definition for an arbitrary cross-section, it is possible to extend the definitions and analysis for any cross-section along the flow © The Author(s) 2021 G. Ribatski, F. Kanizawa, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_2

13

14

2 Fundamentals

path. Moreover, even though the schematics presented here are generally for circular cross-sections, the vast majority of definitions are valid for any geometry. Due to the intrinsic fluctuation characteristic of two-phase flow, the definition of two-phase flow parameters require presentation of average operators, which can be defined for time or space domains. These operators are adopted for the definition of several parameters such as void fraction and in situ velocities. The definitions presented here are similar to those addressed by Delhaye (1981b). Phase Density Function The phase density function corresponds to a binary function that indicates the ! presence of a given phase i in a position r of the domain at instant t, and is defined as follows: 

!



Xi r , t ¼

8
n ¼

Dn η



R

 r , t dDn

!

Dn dDn

ð2:3Þ

where the subscript n denotes the domain dimension, and n denotes spatial average in the domain Dn, which can be a line segment s, an area A, or a volume V. Alternatively, the average propertie can be evaluated only in the region occupied by one of the phases as follows:     ! ! η r , t X i r , t dDn Dn   ¼ R ! Dn X i r , t dDn R

< ηðt Þ>n,i

ð2:4Þ

2.1 Basic Definitions

15

where the subscript i refers to the phase vapor or liquid, and the term Xi corresponds to the density function defined by Eq. (2.1). Time Average Operator Two-phase flow is also characterized by the temporal variation of the properties and parameters along the time. Thus, it is interesting and useful to define the time averaged operator of a parameter η that can be written as follows: Z     1 t0 þΔt ! ! η r ¼ η r , t dt Δt t0

ð2:5Þ

where over-bar denotes a time-averaged value. Additionally, it is possible to define the time average operator for a parameter η only in phase i as follows:   ! ηi r ¼

R t0 þΔt !  !  R t0 þΔt !  !  η r , t X r , t dt η r , t X i r , t dt i t0 t0   ¼ R t0 þΔt !  ! X r , t dt r Δt i i t0

ð2:6Þ

!

where the term Δti corresponds to the time interval that point r is occupied by phase i. It must be highlighted that both average operators can be applied for a given parameter. For example, the proportion of phases along the cross-section, which consists in the spatial average of the density function along the cross-section, can be averaged in time. Moreover, based on the average operator properties it is possible to infer some commutative properties of the time and space average operators as follows:  n < η>n ¼ < η>

ð2:7Þ

With the definition of the average operators, it is possible to introduce basic parameters for two-phase flows. Delhaye (1981b) is indicated as supplementary material for further analyses of average operators. Two-Phase Flow Basic Parameters Figure 2.1 schematically depicts two-phase flow in a duct with cross-sectional area A and with inclination θ relative to the horizontal plane. It is assumed that the transversal areas occupied by the vapor and liquid phases are, respectively, Av and Al, and the mass flow rates of each phase through the cross-section are ṁv and ṁl, respectively. In this figure, and along the entire text, it will be assumed that the axial direction is z, the horizontal axis is y, and x is perpendicular to these axes, all denoted by bold, italic, and lower case letters.

16

2 Fundamentals

A Av Av

x

mv

y

g

g x

m z y

ml

Al

Al

0

Fig. 2.1 Schematics of two-phase flow in a duct

The mixture instantaneous mass flow rate ṁ is given by the sum of both phases mass flow rates, as follows: m_ ¼ m_ l þ m_ v

ð2:8Þ

where ṁl and ṁv correspond to the liquid and vapor mass flow rates, respectively. The instantaneous mass flow rate can be determined as follows: Z

!

m_ i ðt Þ ¼

!

Z

X i ρV  dA ¼ A

!

!

ρV  dA

ð2:9Þ

Ai

where ρ corresponds to the phase density in kg/m3 and the integral domains correspond to the cross-sectional areas. It must be highlighted that even though the sub index i was omitted for the fluid density and velocity, the integration was restricted to the cross-sectional area occupied by the phase i. This restriction is imposed by the phase density function in the first equality and by the integration domain in the second. It is usually not feasible to determine the instantaneous mass flow rate of each phase, therefore the time-averaged values are adopted and the overbar symbol is omitted. Moreover, most of the foregoing definitions are based on time-averaged parameters, and the time-averaged operator will be omitted for simplicity and to avoid pollution of the text. Additionally, the variation of the fluid properties along the cross-section, such as density and viscosity, is usually considered negligible; therefore, whenever it is not specified differently, the properties addressed correspond to the mean value over the cross-section, which are determined based on the saturation condition for phase change processes. According to the thermodynamic definition, the vapor quality x is defined as the vapor mass fraction in a vapor-liquid mixture. In two-phase flow conditions, even considering conditions of non-equilibrium that occur during phase change processes,

2.1 Basic Definitions

17

it is assumed that the vapor quality corresponds to the relative fraction of the vapor mass flow rate, and is defined as follows: x¼

m_ v m_

ð2:10Þ

In conditions of two-phase flow with phase change, it is quite common to define the vapor quality based on the energy balance, which is recurrently referred in the literature as the equilibrium thermodynamic vapor quality. This approach is adopted because it is usually not feasible to directly measure the mass flow rate of each phase, as discussed above. Thus, the equilibrium thermodynamic vapor quality, or simply vapor quality, is determined based on the local enthalpy i assuming thermodynamic equilibrium as follows: x¼

bi  bil bilv

ð2:11Þ

where the fluid local enthalpy î is usually evaluated based on energy balance. The terms îl and îlv correspond, respectively, to the liquid and vaporization enthalpies evaluated at the local saturation pressure. Based on Eqs. (2.11) and (2.10), it is possible to evaluate the mass flow rate of the vapor phase, and from Eq. (2.8) it is possible to evaluate the liquid mass flow rate. The mass velocity G, also referred as mass flux, is defined as the mixture mass flow rate per unit of cross-sectional area, and is given as follows: G¼

m_ m_ þ m_ v m_ l m_ v ¼ l ¼ þ ¼ Gl þ Gv A A A A

ð2:12Þ

The mass velocities of each phase, Gl and Gv, are already defined in Eq. (2.12), which correspond to the mass flow rate of each phase per unit of the total crosssectional area. Based on the definitions of mass velocity and vapor quality, given by Eqs. (2.10) and (2.12), respectively, it is possible to write the mass velocities of the vapor and liquid phases as follows: m_ v m_ ¼ x¼Gx A A _ m_ m Gl ¼ l ¼ ð1  xÞ ¼ G  ð1  xÞ A A Gv ¼

ð2:13Þ ð2:14Þ

In the foregoing discussion, the overbar in the mass flow rate term will be suppressed because it is usually considered the time averaged value, hence, ṁ ¼ ṁ.

18

2 Fundamentals

The superficial velocity j, or volumetric flux, is commonly defined and adopted in the related literature to characterize the operational condition, as well as coordinated axis for some flow pattern maps. The superficial velocities correspond to the volumetric flow rate per unit of cross-sectional area, and are defined for the mixture and each phase as follows: j ¼ jl þ jv 1 jv ¼ hX vV z i2 ¼ Δt  zi ¼ 1 jl ¼ hX l V 2 Δt

Z

Z t0

t 0 þΔt t0

t 0 þΔt

1 A

Z

1 A

Z

!

!

X v V  dA dt ¼ A

!

ð2:15Þ

h i

h i !

X l V  dA dt ¼ A

m_ l ρl

A

¼

m_ v ρv

A

¼

G x ρv

G ð 1  xÞ ρl

ð2:16Þ

ð2:17Þ

where the term ρ corresponds to the fluid density and the subscripts l and v correspond to the liquid and vapor phases, respectively. The terms inside square brackets in Eqs. (2.16) and (2.17) correspond to the volumetric flow rate of vapor and liquid phases, respectively. It must be highlighted that, even though the mixture superficial velocity given by Eq. (2.15) corresponds to the mean flow velocity, this parameter is different from the actual flow velocity, and the difference is mainly related to the slip between the phases. It will be shown below that the actual flow velocity is related to the superficial velocity through the definition of the void fraction α. In an alternative definition, the superficial velocities can be considered as the mean velocity of each phase if they were flowing alone in the same duct, which is derived using Eq. (2.3). The instantaneous velocity u, which is closer to the actual flow velocity when compared to the superficial velocity, corresponds to the mean velocity evaluated along the parcel of the cross-sectional area occupied by the respective phase. It is calculated according to Eq. (2.4), and is defined as follows: R V X dA z v A V z dA uv ¼< V z >2,v ¼ R ¼ AR ¼ v Av D2 X v dD2 A X v dA R R R V z X l dD2 V z X l dA A V z dA ¼ l ¼ AR ul ¼< V z >2,l ¼ DR2 Al D2 X l dD2 A X l dA R

D2 V z X v dD2

R

ð2:18Þ ð2:19Þ

where the term Vz corresponds to the axial component of the velocity vector, and Av and Al correspond to the transversal area occupied by vapor and liquid phases, respectively. The in situ velocity is defined as the time averaged value of the instantaneous velocity adopting the definition of area averaged void fraction α2, which will be presented in item 2.3. Recognizing that the numerators of the last members of Eqs. (2.18) and (2.19) correspond to the volumetric flow rate of vapor and liquid

2.1 Basic Definitions

19

phases, respectively, and assuming steady state condition and developed flow, it is possible to write the in situ velocity as follows: uv ¼

ul ¼

R R 1 t 0 þΔt Av V z X v dAv dt Δt t 0 R R t þΔt 0 1 Av X v dAv dt Δt t 0

R R 1 t 0 þΔt Al V z X l dAl dt Δt t 0 R R t þΔt 0 1 Al X l dAl dt Δt t 0

¼

¼

j Qv Qv A Gx ¼ ¼ v¼ Av AAv α2 ρv α2

Gð1  xÞ jl Ql Ql A ¼ ¼ ¼ Al AAl 1  α2 ρl ð1  α2 Þ

ð2:20Þ

ð2:21Þ

where the term Q refers to the time averaged volumetric flow rate, and Av and Al refer to time averaged cross-sectional area occupied by vapor and liquid phases, respectively. The overbar for in situ velocity u will be kept along this textbook to ensure distinction from instantaneous velocity. Additionally, based on Eqs. (2.20) and (2.21), it is possible to conclude that the void fraction α2 is equal to Av/A, and this parameter will be more deeply described and analyzed in Sect. 2.3. Useful relationships between the in situ velocity, superficial velocity, mass flux, and void fraction can be inferred from Eqs. (2.21) and (2.22). These equalities are recurrently used along the text as well as for derivation of models and predictive methods. The drift velocities are also recurrently presented in two-phase flow textbooks, and correspond to the difference between the phases and mixture mean velocities, as follows: uvj ¼ uv  j

ð2:22Þ

ulj ¼ ul  j

ð2:23Þ

The volumetric fraction of the gas phase β, or simply volumetric fraction, is also a common parameter cited in the literature, and corresponds to the fraction of volumetric vapor flow rate, as follows: h i m_ v

ρv jv Qv ¼h i h i¼ β¼ Qv þ Ql j þ jl _mv _ml v þ ρv

ð2:24Þ

ρl

It will be shown in Sect. 2.3 that the volumetric fraction given by Eq. (2.24) corresponds to the void fraction according to the homogeneous model. The heat transfer coefficient h relates the heat flux ϕ and the temperature difference between a reference solid wall Tw and the fluid bulk temperature T. The heat transfer coefficient is given according to Newton’s cooling law as follows: h¼

ϕ Tw  T

ð2:25Þ

20

2 Fundamentals

In the following subsections, specific aspects of two-phase flow focused on phase change processes in microscale channels will be discussed.

2.2

Flow Patterns

Definition and characterization of flow patterns occurring during two-phase flow in microscale channels. The differences between two-phase flow in macro and microscale are introduced, as well as the difference between flow patterns during convective condensation and boiling. Due to the two-phase flow complexities, it is convenient to segregate the solutions of the two-phase flow problems according to intrinsic flow characteristics associated to the dominant geometrical and dynamic parameters. In two-phase flows, depending on the operational conditions, the phases are distributed according to distinct geometrical configurations. Flow topologies with similar characteristics are defined as flow patterns. The correct prediction of the flow pattern is a key aspect for the development of accurate predictive methods for heat transfer coefficient, pressure drop, flow-induced vibration and noise, etc., since they are intrinsically correlated to the flow pattern. In this context, several heat transfer coefficient and pressure drop predictive methods that account for the flow pattern are developed based on a mechanistic approach. According to Ishii and Hibiki (2011), the flow patterns can be categorized based on the continuity of the interface as follows: • Separated flows: the phases are continuum and segregated by a single interface between them. These flows can be characterized as liquid in contact with the duct walls and vapor flowing in the section core, and as a liquid film in the bottom region of the cross-section and vapor in the upper part. • Dispersed flows: one of the phases is dispersed in the other. Liquid droplets flow dispersed within the vapor, and vapor bubbles flow within the liquid phase. The interface is not continuous, but it is formed of several segments. • Mixed flows: the flow patterns pertaining this group are composed of a combination of characteristics of separated and dispersed flows, that is, large vapor bubbles (large and continuum portion of vapor) flowing intermittently separated by liquid pistons containing dispersed vapor bubbles, and liquid flowing on the duct wall along its entire perimeter with vapor flow in the test section core containing entrained liquid droplets. Detailed classifications for the flow patterns are frequently found based on the structure of the phases interface. Most of them are classified based on adiabatic flows, even though the heat transfer process itself can affect the phases distribution, such as during condensation that precludes dry regions on the duct surface, and pure stratified flow is not expected. Nonetheless, the discussion presented in this section corresponds only to description of flow patterns in conventional and microscale channels. Flow pattern prediction methods are addressed in Chap. 3. It should be

2.2 Flow Patterns

21

highlighted that different classifications are available in the open literature, adopting distinct nomenclature for the same flow pattern and the same nomenclature for different flow patterns.

2.2.1

Flow Patterns During Vertical Adiabatic Flow

Figure 2.2 depicts the flow patterns observed for vertical upward flow, which are classified based on subjective approach (visual observations) as follows: • Bubbles: characterized by the vapor phase distributed according to discrete bubbles smaller than the duct diameter within the continuum liquid flow. This flow pattern is characteristic by reduced vapor fraction and high liquid velocities. • Slug flow: as the vapor fraction increases, bubbles coalescence takes place causing the increase of bubble size. This implies on the transition to slug flow pattern, characterized by successive passage of large vapor bubbles with transversal dimensions of the same order of the duct diameter. These bubbles are denominated as Taylor bubbles with a hemispherical nose and an amorphous trailing region. During two-phase flow under certain conditions, the liquid film formed between the large bubble and the duct wall may flow downwards,

Bubbles

Slug

Churn

Annular

Fig. 2.2 Schematics of flow patterns for vertical upward flow

Wispy annular

Wispy

22









2 Fundamentals

counter-current to the rising vapor. Small vapor bubbles are observed within the liquid slug between successive Taylor bubbles. Churn flow: with additional increment of vapor fraction, the increment of inertial effects and reduction of liquid fraction disrupt liquid bridges between successive large bubbles, and the flow becomes chaotic. This flow pattern is characterized by chaotic movement of liquid and vapor phases and is observed in channels of conventional dimensions. Churn flow is frequently assumed as a transitional flow pattern. Annular flow: under conditions of high vapor velocities, the dominant inertial effects of the vapor flow moves the liquid phase to the duct walls, forming a continuous liquid film on the duct surface. The interface between the phases is perturbed and usually contains interfacial waves. Depending on the operational conditions, small bubbles are wrapped by the liquid film. Wispy annular: the high flow velocity of the vapor flow typical of annular flows might detach liquid droplets from the interfacial waves. This behavior implies on a liquid film along the tube perimeter with the vapor flowing in the center of the section containing entrained liquid droplets. Wispy flow: this flow pattern is also referred in the literature as mist flow. Under conditions of very high vapor velocities and reduced liquid fraction, the liquid film thins and becomes unstable, being ceased by the vapor flow. Then, the flow is characterized by a dry wall and a continuum vapor phase with entrained liquid droplets.

Studies focused on downward flow patterns are also found in literature, such as the one presented by Barnea et al. (1982). According to them, only annular flow is naturally occurring in downward flow because buoyancy effects preclude concurrent natural downward flow of vapor and liquid. The bubbles and slug flow patterns are reported only for significantly high liquid velocities. Figure 2.3 depicts images of downward air-water flows in square microchannel and Fig. 2.4 depicts them schematically. The annular flow corresponds to a falling film inside tubes, and it is likely to occur during condensation inside vertical tubes. In the case of conditions with higher liquid content, such as slug and bubbles flow, the liquid velocity must be high enough to overcome the slip velocity of the vapor phase, in order of resulting in co-current downward flow. Moreover, Lavin et al. (2019) observed coring effect during bubble flow, as shown in Fig. 2.3, with higher concentration of bubbles close to the center of the cross-section, and the main quoted mechanism for this phenomena are related to pressure gradient due to shear stress gradient, and transverse lift force induced by velocity gradient. In addition to concurrent upward and downward flow patterns in vertical ducts, there is also the possibility of counter-current gas-liquid flow with downward liquid flow and upward gas flow. In this case, annular or slug flow patterns are likely to occur. Depending on the superficial velocities of both phases, the counter current flow might not exist, whereas one of the phases is dragged by the other, such as discussed by Wallis (1969) for conventional sized channels.

2.2 Flow Patterns

Annular

23

Bubbles

Slug

Fig. 2.3 Images of air-water downward flow in microchannel, Lavin et al. (2019)

The condition of counter-current flow in microscale channels is unlikely to occur due to effects of bubble confinement, hence, it will not be discussed in this book, but the interested readers are encouraged to check Wallis (1969).

2.2.2

Flow Patterns for Horizontal Adiabatic Flows

During vertical flows, even for concurrent upward and downward flows, the buoyancy effects influence the slip between the phases, whereas the vapor phase presents higher vertical upward velocity component. Nonetheless, these effects do not

24

2 Fundamentals

Fig. 2.4 Schematics of vertical downward flow

Annular

Bubbles

Slug

necessarily imply on axi-asymmetric phase distribution for vertical channels. On the other hand, for horizontal flow, buoyancy effects imply on predominantly higher concentration of gas phase in the upper region of the duct, with consequent asymmetry in the phase distributions. Figure 2.5 schematically depicts the horizontal flow patterns for conventional channels that are described as follows: • Bubbles: as for vertical flows, this flow pattern is characterized by bubbles dispersed in a continuum liquid phase. Buoyancy effects are responsible for a higher concentration of bubbles in the upper region of the tube. This flow pattern is verified for reduced gas fraction and high liquid velocities. • Stratified: characterized by the liquid as a continuum film in the bottom region of the section and gas flowing in the upper region with the phases separated by a smooth interface. This flow pattern is verified for reduced flow velocities and under predominance of gravitational effects on the two-phase flow topology. The liquid-vapor interface is curved in the azimuthal direction due to inertial and capillary effects. The stratified flow is absent in microscale channels because the capillary effects tend to fill the surface with liquid film.

2.2 Flow Patterns Fig. 2.5 Schematics of horizontal flow patterns, flowing from left to the right

25

Bubbles

Stratified

Stratified wavy

Plug

Slug

Annular

Mist

• Stratified wavy: with the increment of vapor phase velocity, interfacial waves with considerable amplitude are formed, moving according to the flow main direction. This flow pattern is also verified only for channels of conventional size. • Plug: this flow pattern corresponds to a type of intermittent flow pattern, where the vapor is distributed according to discrete large bubbles, but with transversal dimensions smaller than the channel diameter. The liquid slugs between successive bubbles contain small vapor bubbles dispersed within the liquid phase. This flow pattern is verified for intermediate flow velocities. • Slug: this is another sub-classification of intermittent flows and is characterized by vapor bubbles larger than those observed for plug flow and is also referred in literature as elongated bubbles. A thin liquid film separates the upper part of the tube from the bubbles, which correspond to parcel of the liquid that remains from the liquid slug passage. Small vapor bubbles might be observed dispersed within the liquid slugs. • Annular: under conditions of high vapor velocities, the inertial effects of the vapor phase move the liquid to the duct periphery; thus, the liquid flows as a continuum liquid film along the entire tube perimeter. Distinctly than vertical flow, the liquid film thickness may present significant non-uniformity along the channel perimeter due to gravitational effects. The film is thicker in the bottom region of the channel.

26

2 Fundamentals

Fig. 2.6 Horizontal flow patterns for vapor-liquid flow in microchannel. R245fa in 1.1 mm ID tube, Tibiriçá and Ribatski (2013)

• Mist: the characteristics of this flow pattern are similar for vertical and horizontal flows. The inertial effects of the vapor phase dominate the phase distribution. It must be highlighted that these classifications and definitions are commonly found in textbooks concerning the subject of two-phase flows and are valid mostly for channels of conventional size. Moreover, the occurrence of some flow patterns is unlikely to occur during phase change phenomena; for example, the stratified and wavy-stratified flow patterns as described in this item are unlikely to occur during condensation, since the phase change process implies on the formation of a liquid film along the entire tube perimeter. In the case of two-phase flow in microchannels, the surface tension effects have significant contribution on the phase distribution, and flow patterns dominated by buoyance effects are not verified, as shown in Fig. 2.6, and schematically depicted in Fig. 2.7. According to Tibiriçá and Ribatski (2013), and similar to Ishii and Hibiki (2010), the flow patterns during horizontal flow in microchannels were classified as separated, dispersed, and mixed. The following sub-classification was proposed by Tibiriçá and Ribatski (2013): • Dispersed: – Bubble flow pattern: similar to the case of two-phase flow in conventional channels, with continuum liquid phase with dispersed vapor bubbles slightly smaller than channel diameter. It is verified only for reduced vapor content. – Mist flow: similar to the description for mist flow in conventional channels, this flow pattern is characterized by continuum vapor phase with dispersed liquid droplets, even though it was not reported by Tibiriçá and Ribatski (2013) because it is close to the condition of wall dryout. It is verified for conditions of low liquid content.

2.2 Flow Patterns

27

Dispersed (Bubbles)

Intermittent (Slug)

Intermittent (Elongated bubbles)

Intermittent (Churn)

Annular Fig. 2.7 Schematics of horizontal flow patterns in microscale channels

• Separated: – Annular flow: characterized by continuum liquid film along the channel wall, with vapor flowing in the channel core. • Mixed (intermittent): – Intermittent (slug): it is characterized by successive passage of vapor bubbles with characteristic dimensions of the same order of magnitude of the channel diameter. – Intermittent (elongated bubbles): it is similar to the slug flow, but the vapor bubble length corresponds to several diameters. – Intermittent (churn): it can be considered as transitional flow pattern between elongated bubbles and annular flow patterns, with the liquid pistons being disrupted by the vapor flow. It must be highlighted that stratified flow is not observed for microscale channels due to suppression of buoyance effects, and that dispersed bubbles usually count with aligned single bubbles.

28

2.3

2 Fundamentals

Void Fraction

Definition of area averaged void fraction, as well as volume averaged void fraction and local void fraction. A short discussion about the importance of these parameters on two-phase flow problems is presented. The void fraction α is defined as the time average fraction of vapor phase in a region of the space occupied by the two-phase mixture and can be defined according to distinct domains D including a single point (local), line segment, area and volume, denoted in this text by the subscripts 0, 1, 2, and 3, respectively. Void fraction is one of the most important parameters of two-phase flow because it is directly related to the accelerational and gravitational pressure drops. Moreover, several parameters of two-phase flow, such as local flow pattern and consequently frictional pressure drop and heat transfer coefficient, are interrelated with the void fraction. Additionally, the void fraction is a critical parameter for the evaluation of dryout occurrence, which can lead to the heat transfer surface burnout and consequent damage of the heat exchanger under conditions of imposed heat flux, which must be avoided specially in systems such as vapor generators of nuclear power plants. Even though the vapor quality is a measurement of the vapor fraction in a mixture, this parameter does not represent the actual amount of vapor in a given region of the flowing mixture. From a practical point of view, it is possible to infer this aspect when we evaluate the volume fraction that would be occupied by a mixture at a given vapor quality x. Let us assume saturated R134a at 5  C. For this condition, the liquid and vapor densities are 1311 and 12.1 kg/m3, respectively. For a motionless (non-flow) condition, a vapor quality value of 0.009 corresponds to similar volumes occupied by both phases, that is, 50% in volume for each phase. Therefore, marginal vapor quality values would correspond to the cross-section almost entirely occupied by vapor phase, which is not observed experimentally, and the slip between the phases justifies the higher liquid content. Thus, several approaches for estimation of void fraction are based on modelling of the slip between the phases.

2.3.1

Local Void Fraction !

The local void fraction corresponds to the parcel of time that a location (point r ) is occupied by the vapor phase, and it is defined based on the density function given by Eq. (2.1). The local void fraction is given as follows: α0 ¼

1 Δt

Z

t 0 þΔt

X v dt t0

ð2:26Þ

2.3 Void Fraction

29

The local void fraction can be experimentally determined using intrusive approaches, such as resistive and optical probes, as well as non-intrusive approaches such as some based on optical method. It should be highlighted that it is nonsense to evaluate the integral in the space domain defined by a point, because, theoretically, a point has no dimensions, even though the measuring probe must have a finite size.

2.3.2

Line Averaged Void Fraction

The line averaged void fraction corresponds to the time average of the space parcel occupied by the vapor phase along a line segment D1. It is defined as follows: 1 α1 ¼ Δt

t 0 þΔt Z

Z

X v dD1 =

dD1

D1

t0



Z D1

1 dt ¼ Δt

t 0 þΔt Z

Z

Z X v ds=

t0

L

 ds dt ð2:27Þ

L

Assuming a rectilinear segment, Eq. (2.27) can be rewritten as a function of parcel of the line corresponding to the vapor phase (Lv/L ) as follows: 1 α1 ¼ Δt

Z

t 0 þΔt t0

Lv dt L

ð2:28Þ

Certain experimental approaches provide the mixture density along a segment of line, such as gamma and x-ray attenuation techniques. Based on this approach and knowing the index of attenuation of each phase, it is possible also to experimentally determine the phase parcels and consequently the instantaneous vapor fraction and void fraction along the segment of line. Falcone et al. (2009) provide a careful discussion about these methods.

2.3.3

Area Averaged Void Fraction

Similar to the line averaged void fraction, the area averaged void fraction, also referred as superficial void fraction, corresponds to the time averaged parcel of area occupied by the vapor phase in a given two-dimensional domain D2, which usually refers to the cross-sectional area, and is defined as follows: 1 α2 ¼ Δt

Z   Z Z 1 t0 þΔt X v dD2 = dD2 dt ¼ X v dA= dA dt ð2:29Þ Δt t0 D2 D2 A A

t 0 þΔt Z

Z t0

Z

30

2 Fundamentals

Assuming that the instantaneous area occupied by the vapor phase along the duct cross-section is Av and the total area in analysis is A, Eq. (2.29) can be rewritten as follows: 1 α2 ¼ Δt

Z

t 0 þΔt t0

Av dt A

ð2:30Þ

In this text, when not specified differently, the term void fraction refers to the area averaged void fraction and will be denoted simply by α. There are several experimental methods for determining the area averaged void fraction, such as some optical method, capacitive sensing system, wire mesh sensor, and array of radioactive emitter-receptor pairs, such as discussed by Falcone et al. (2009). Nonetheless, the reader must be aware that experimental investigation of area averaged void fraction is challenging, and most of the studies have performed experimental determination of local, line averaged, or volume averaged void fraction. Figure 2.8 from Marchetto (2019) illustrates the evolution of the instantaneous local void fraction for intermittent flow in a small diameter channel during the passage of three consecutive elongated vapor bubbles. In this figure, the evolution of the time-averaged superficial void fraction estimated according to Eq. (2.30) is also shown. From a comparison of both curves displayed in Fig. 2.8, it can be noted that the time-averaged void fraction tends to a value of approximately 0.6 as the upper integer limit in Eq. (2.30) increases. Therefore, it can be concluded that despite of the flow intermittence, a constant time-averaged superficial void fraction can be

Fig. 2.8 Variation of the instantaneous and time-averaged void fraction over time for an intermittent flow, Marchetto (2019)

2.3 Void Fraction

31

measured under developed flow conditions if the integer upper limit in Eq. (2.30) is long enough to characterize the flow.

2.3.4

Volume Averaged Void Fraction

The volume averaged void fraction, also referred as volumetric void fraction, is frequently mentioned in literature. The term gas holdup is also found in literature mainly by engineers working in the sector of oil and gas. The volumetric void fraction is defined as the time averaged volume parcel of the gas phase, and is given as follows: α3 ¼ ¼

1 Δt 1 Δt

Z Z

t 0 þΔt t0 t 0 þΔt

Z ð Z ð

Z X v dD3 = D3

Z

X v dV= V

t0

dD3 Þdt D3

dVÞdt

ð2:31Þ

V

Supposing that the instantaneous volume occupied by the vapor phase is V v and the total volume is V, the volume averaged void fraction is given as follows: α3 ¼

1 Δt

Z

t 0 þΔt t0

Vv dt V

ð2:32Þ

The quick closing valve technique is recurrently mentioned in the literature as an experimental approach for volumetric void fraction measurements. This technique consists on trapping the two-phase mixture in a given volume of the test section by closing simultaneously valves located upstream and downstream of the measurement target region. Subsequently, the phase volumes are measured, and the fractions of liquid and gas phases are calculated. This approach is used for two-phase flow of liquid and immiscible gas, as well as for two-phase flow of saturated substance. The volumes of each phase can be measured directly in the first case because the phases are separable; on the other hand, in the case of two-phase flow of a single substance (or solution), the heat transfer between the fluid and its neighborhood might imply on variation of phase proportion between the trapping and measuring instants. Thus, researchers usually measure the trapped mass, and based on the phase densities for the experimental condition, they evaluate the phase volumetric fraction. The distance between the upstream and downstream valves must be long enough to avoid effects of intermittency, characteristic of intermittent flow patterns. Moreover, when applied to two-phase developing flow, the results obtained through the quick closing valve method become typical of the measurement length and the distance between the valves and the test section inlet, therefore, they cannot be considered as general.

32

2 Fundamentals

Ac

La

Pa

Lb

Aa

Ab

Flow direction Lc

Ld

Pc Pb

Fig. 2.9 Schematics of a vapor bubble passage in a duct

Falcone et al. (2009) present several experimental techniques for evaluation of two-phase flow parameters, including void fraction, and is here indicated as supplementary material. It covers details about installation, application, and range of validity mostly focused on applications in the oil and gas industry. It can be recognized at this point that, similarly to the density function defined by Eq. (2.1), the time averaged fraction of the liquid phase in a given domain is equal to the unity minus the void fraction (1 – αn). In the same context, specialists of the oil and gas industry refer to the volumetric liquid content as liquid holdup, or simply holdup. It must be mentioned that the point, line, area, and volume averaged void fractions are not necessarily similar due to variation of flow parameters in space and time. These four parameters would correspond to the same value only in conditions of homogenous flow, corresponding to uniform distribution of the phases along the entire domain region. Figure 2.9 schematically depicts these differences in the case of elongated bubble flow patterns. According to this figure, it can be noticed that the density function for points Pa, Pb, and Pc are distinct from each other, since the upper and central points are in contact with the vapor phase while point Pc is in contact with the liquid phase. Therefore, the local void fraction, which consists in the time averaged density function of the vapor phase given by Eq. (2.26), is distinct for each point. Similarly, the instantaneous vapor fraction along the lines La, Lb, Lc, and Ld are distinct from each other, and the differences get even higher due to buoyancy effects that make the flow non-axisymmetric. Therefore, the void fraction for distinct line segments would be different as well. Moreover, the determination of the area averaged void fraction based on the line averaged void fraction requires a previous knowledge about the flow pattern and phases distributions, and this information is as difficult to obtain as the area averaged void fraction. Some authors present experimental studies, such as described by Falcone et al. (2009), in which they determine the area averaged void fraction based on measurement of three lines’ averaged void fractions. Based on empirical relationships between line and area average void fractions for a given phase distribution, the area averaged void fraction is estimated.

2.3 Void Fraction

33

Therefore, based on this discussion, it can be concluded that the point, line, and area averaged void fraction are distinct from each other, and some of these parameters are strongly dependent on the measurement location. The volume averaged void fraction is also a function of the region of analysis. According to Fig. 2.9, distinct instantaneous vapor volume fractions would be obtained if we consider the volumes formed between section pairs Ac - Aa, Ac - Ab, and Aa - Ac. Thus, even considering that the flow is completely developed, a high number of samples of these volumes would be required to obtain a reasonably low deviation from the area averaged void fraction, because the instant that the sample is trapped affects the result. Alternatively, a larger volume would reduce the influence of the instant of time of the sample for a completely developed flow. However, for two-phase flow in microscale channels the high pressure drop might result in significant variation of vapor-specific volume, with consequent variation of phase fractions and velocities along the channel length. In conclusion, the integral in time of the vapor area parcel is similar to the volume averaged void fraction only if the flow is fully developed, under steady state condition, non-intermittent, for a uniform cross-section, adiabatic, and under conditions of negligible compressibility effects. In conclusion, the reader might have noticed that experimental determination of void fraction is challenging, and even today several research groups are engaged in developing instrumentation and methods for precise determination of void fraction. Additionally, several of the measurement techniques are intrusive, that is, they correspond to instrumentation and sensor inserted into the flow stream, disturbing the velocity and temperature profiles, and consequently altering the parameter that was intended to be measured. In this context, it should be mentioned that when analyzing a real thermal cycle, such as refrigeration or power generation, the experimental determination of void fraction would help to obtain the thermodynamic state along the cycle. For instance, consider the outlet of expansion device in refrigeration cycle, whereas the conventional instrumentation would correspond to pressure and/or temperature transducers. But considering that it is two-phase flow in this point, the pressure and temperature data does not allow determination of thermodynamic state. Alternatively, the expansion device could be considered as adiabatic, and by evaluating the inlet condition, which is probably subcooled liquid, it is possible to infer the outlet condition. However, this approach requires the hypothesis of adiabatic flow that is not completely attained and knowing the phase fraction can help in determining the local vapor quality. Moreover, in some industries, such as in oil and gas, the correct determination of each phase fraction is needed to calculate amount, taxes, prices, etc.

2.3.5

Void Fraction Predictive Methods

Several approaches for void fraction prediction are available in the open literature. Woldesemayat and Ghajar (2007) presented an extensive review of literature

34

2 Fundamentals

concerning void fraction predictive methods, and classified them as slip correlation, drift flux models, general correlations, and as correction factors for the void fraction estimated according to homogeneous model. In the present textbook, an alternative classification is adopted as follows: • Kinematic approaches: these methods consist in correlations that assume some characteristic of the velocity profiles and based on average operators and continuity it is possible to estimate the void fraction. This category includes, for example, the slip ratio methods and drift flux models. • General methods: these methods consist in purely empirical correlations and methods based on physical principles, other than the assumption of velocity characteristics. These approaches include those developed based on Lockhart and Martinelli (1949) parameter, minimum entropy generation, minimum kinetic energy, and purely empirical methods.

2.3.5.1

Slip Ratio Method

The simplest modeling approach for two-phase flow consists in the homogeneous model that is based on the assumption of the liquid and gas phases flowing as a single pseudo-fluid with averaged properties. This is a kinematic model and considers absence of slip between the phases and uniform velocity profiles, thus the in situ velocities of the liquid and gas phases are similar. Hence, from Eqs. (2.20) and (2.21), eliminating the mass flux term, we obtain the following expression: ð 1  xÞ x ¼ ρl ð1  α2 Þul ρv α2 uv

ð2:33Þ

Solving Eq. (2.33) for α2, we obtain the void fraction given as a function of the slip ratio as follows: α2 ¼

1



  uv ρv 1x ul ρl x

ð2:34Þ

where the term uv / ul corresponds to the slip ratio, and according to the homogeneous model assumption is equal to the unity (uv / ul ¼ 1). It is common sense in literature that the void fraction is a function of vapor quality x and fluids properties, as well as of the flow velocity and channel geometry, which effects on the void fraction are not totally captured by the homogenous model. Moreover, in the case of vertical upward flow, buoyancy effects might imply on higher velocity of vapor phase, contributing to deviation from the homogenous assumption. Even during horizontal flow, the vapor phase usually possesses higher velocity than the liquid phase, which is related to the friction between the

2.3 Void Fraction

35

phase and duct wall and/or the bubble nose shape. Therefore, the assumption of no-slip is not representative of the real phenomena for these conditions, and the homogenous model provides the highest void fraction value possible, otherwise the results would be physically incorrect because it would correspond to liquid flowing faster than vapor. An in situ liquid velocity higher than the vapor velocity would be possible in the case of liquid flowing downward along the duct wall, due to gravitational effects, and the vapor flowing concurrently dragged by the liquid phase. Some researchers adjust correlations for the slip ratio in order of developing a void fraction predictive method that is dependent on flow conditions and duct geometry, such as proposed by Tibiriçá et al. (2017) for microchannels, according to which the slip ratio is given as follows:  0:31   uv 1  x 0:267 0:1082 ρv ¼ 1:2364 Fr x ul ρl

ð2:35Þ

where the Froude number is evaluated for the mixture as follows: Fr ¼

2.3.5.2

G2 ðρl  ρv Þ2 gd

ð2:36Þ

Drift Flux Model – Zuber and Findlay Method

Several approaches have been proposed to account for the effects of slip between the phases and non-uniform distribution and velocity profile of the phases along the cross-section, and among them, the drift flux model is one of the most used methods for prediction of void fraction and was firstly presented by Zuber and Findlay (1965). The derivation of this method assumes non-uniform velocity profiles and phase distributions and takes into account the slip between the phases. The method is derived based on spatial average operator, as given by Eq. (2.4), for steady state condition. Based on the definition of the drift velocity of vapor phase, given by Eq. (2.22), the instantaneous and local vapor drift velocity are written as follows: V z,vj ¼ V z,v  j

ð2:37Þ

which can vary along the cross-section. Thus, rearranging this equation as follows and multiplying it by the vapor density function, we get: V z,v X v ¼ V z,vj X v þ jX v

ð2:38Þ

Therefore, by evaluating the mean value of this function along the entire crosssection, we obtain:

36

2 Fundamentals

R

A VRz,v X v dA A dA

R ¼

A VRz,vj X v dA A dA

R

jX v dA þ AR A dA

ð2:39Þ

We can recognize that the numerator of the first member of Eq. (2.39) corresponds to the vapor volumetric flow rate, which divided by the cross-sectional area corresponds to the vapor superficial velocity jv. Multiplying this equation by the inverse of the area averaged void fraction, and adopting the relationship between the void fraction, vapor superficial velocity and void fraction, we can re-write Eq. (2.39) as follows1: R

A V z,v X v dA=A

α2

j ¼ v¼ α2

R

A V z,vj X v dA=A

α2

R þ

A jX v dA=A

α2 j

j

ð2:40Þ

where a term j / j was multiplied by the last term of the second member. Zuber and Findlay (1965) rewrote Eq. (2.40) as follows: jv e ¼ V vj þ C 0 j α2

ð2:41Þ

where the term C0 corresponds to the distribution parameter that takes into account e vj correthe non-uniform distribution of the phases along the cross-section, and V sponds to the drift parameter that takes into account the slip between the phases. These parameters are defined empirically, and, according to the original authors, are dependent on the local flow pattern. The distribution and drift parameters are written explicitly as follows: R C0 ¼ R e vj ¼ V

A jX v dA=A

α2 j

A V z,vj X v dA=A

α2

ð2:42Þ ð2:43Þ

Recall that the numerator in Eq. (2.43) does not correspond to the superficial velocity, because it consists in the integral of the slip velocity. By multiplying Eq. (2.41) by α2/j, the following relationship is obtained: jv ¼ j

! e vj V þ C 0 α2 j

ð2:44Þ

1 This presentation is slightly different than the original description, which is based on weighted averages of the two-phase parameters, but the final relationship for void fraction is exactly the same.

2.3 Void Fraction

37

where the first member corresponds to the volumetric ratio β, defined by Eq. (2.24). Thus, rearranging the terms, it is possible to obtain a relationship for void fraction given as follows: α2 ¼

β C0 þ

e V vj

ð2:45Þ

j

In order to check the validity of this method for the case of homogeneous flow, it is expected that the phase distribution is uniform, thus C0 is equal to the unity. e vj is null. Therefore, Additionally, the drift between the phases is negligible, thus V the void fraction is equal to the volumetric fraction, which is similar to the void fraction according to the homogenous model. According to the original authors (Zuber and Findlay, 1965), the distribution and drift parameters depend on the flow pattern, and consequently, regressions for e vj are required for each flow pattern. Collier determination of values for C0 and V and Thome (1994) present some values and relationships from literature for these parameters depending on the local flow pattern. The rising velocity of a bubble in stagnant media is recurrently adopted as the drift parameter, even for horizontal flow. The drift term is generally related to gravitational effects on the phases, and in the case of flow in horizontal channels the bubble nose geometry is the main parameter defining the drift between the phases. As the channel size decreases, the importance of gravitational effects on two-phase flow topology reduces, the flow tends to be axisymmetric, and the drift between the phases tends to zero, as shown by Sempertegui-Tapia et al. (2013). Subsequently to the Zuber and Findlay (1965) study, several authors proposed adjustments for the distribution and drift parameters. Constants were proposed for narrow ranges of operational conditions and correlations for wider ranges. In this context, Rouhani (1969) and Rouhani and Axelsson (1970) proposed correlations for the distribution and drift parameters for horizontal and vertical upward flows, respectively. Considering conditions of phase change, j and β vary along the channel length as function of vapor quality, therefore it is more convenient to present Eq. (2.45) as a function of mass velocity and vapor quality, as follows: ( α2 ¼

"  #)1  e vj V ρv x 1x þ C0 þ ρv ρl x G

ð2:46Þ

The mass velocity and vapor quality were introduced because in applications involving phase change processes, the first is constant in case of fixed cross-section and the second can be obtained directly from an energy balance. Rouhani (1969) and Rouhani and Axelsson (1970) proposed the rising velocity of a bubble in a stagnant media as the drift parameter, as proposed by Zuber and Findlay (1965), given as follows:

38

2 Fundamentals

 14 e vj ¼ 1:18 gσ ðρl  ρv Þ V ρ2l

ð2:47Þ

For horizontal flow, Rouhani (1969) proposed the following relationship for the distribution parameter:  1 gdρ2l 4 C 0 ¼ 1 þ 0:2ð1  xÞ G2

ð2:48Þ

where the term inside the brackets can be considered as the inverse of the Froude number. For vertical upward flow, Rouhani and Axelsson (1970) proposed the following values for the distribution parameter:

C0 ¼

1:10

for G  200 kg=m2s

1:54

for G < 200 kg=m2s

ð2:49Þ

The methods proposed by Rouhani (1969) and Rouhani and Axelsson (1970) were developed originally focused on nuclear applications, thus based on experimental results for liquid water and steam. Nonetheless, several studies, such as Wojtan et al. (2005a, b), indicate their validity for refrigerant flow during phase change processes.

2.3.5.3

Minimum Entropy Generation – Zivi’s Method

Differently than the kinematic approaches, general approaches are also available in the open literature, which are not necessarily based on the characteristics of velocity profile. Zivi (1964) proposed a method based on the minimization of the entropy generation of the two-phase flow. Zivi assumed that the minimum entropy generation condition is attained for minimum kinetic energy flux integrated along the crosssection, assuming uniform velocity profiles. The kinetic energy of a fluid particle is equal to u2/2, thus the kinetic energy flux along the cross-section is given as follows: Z Ek ¼

u2 ! ! ρV  dA ¼ 2A

A

2

ul 2A

Z

ul 2 ! ! ρV  dA þ 2A

Al

!

!

ρV  dA þ Al

Z

2

uv 2A

Z

Av

uv 2 ! ! ρV  dA ¼ 2A

Av !

!

ρV  dA ¼

G 2 ul ð1  xÞ þ uv 2 x 2

Z

2

ul m_ l uv 2 m_ v þ ¼ 2 A 2 A

ð2:50Þ

2.3 Void Fraction

39

Writing the in situ velocities as a function of the mass flux, vapor quality, and fluid densities, given by Eqs. (2.20) and (2.21), and then minimizing the kinetic energy flux through derivation in relation to α2, the following is obtained: ( " #) 3 ∂Ek ∂ G 3 ð 1  xÞ 1 x3 1 ¼ þ ¼ ∂α2 ∂α2 2 ρ2l ð1  α2 Þ2 ρ2v α22 " # 3 3 ð 1  x Þ x þ ¼0 G3  ð1  α2 Þ3 ρ2l α32 ρ2v

ð2:51Þ

Solving for α2, a relationship for void fraction is obtained, given as follows: α¼ 1þ

1  23   ρv 1x ρl

ð2:52Þ

x

Similar to the homogenous model, the Zivi (1964) method does not take into account mass flow rate effects. Nonetheless, this method has shown reasonably good agreement with experimental results for distinct operational conditions.

2.3.5.4

Minimum Kinetic Energy Method – Kanizawa and Ribatski Method

Considering a similar approach of Zivi (1964), Kanizawa and Ribatski (2016) proposed a void fraction predictive method based on the minimization of the kinetic energy along the cross-section assuming non-uniform velocity profile of the phases along the cross-section. They assumed that the void fraction is associated to a condition of minimum momentum flux along the cross-section of the pipe (instead of kinetic energy flux as adopted by Zivi, 1964). In their method, the momentum flux of each phase is averaged in relation to its respective area, given by (2.4), as follows: R 2 ! ! V z,l dA  X v ÞρV  dA K l u2l Al ¼ ρl ARl ¼ ρl ¼ K l ρl u2l A ð 1  X ÞdA dA l v A Al

R < E k >2,l ¼

ð1 A V z,l R R

< E k >2,v ¼

!

X ρV A V z,v R v

A X v dA

!

 dA

ð2:53Þ

R

2 K v u2v Av A V z,v dA ¼ ρv Rv ¼ ρv ¼ K v ρv u2v Av Av dA

ð2:54Þ

where the term K corresponds to the momentum coefficient (form factor of velocity profile) and corresponds to the contribution R 2 of the non-uniformities of the velocity V z dAÞ profile and phases distribution (K ¼ A u2 A ). In conditions of uniform velocity

40

2 Fundamentals

profiles, even considering slip between the phases, the form factor of each phase becomes equal to the unity. The kinetic energy of the two-phase mixture is given as the sum of the parcels of both phases. Similar to the Zivi (1964) method, writing the in situ velocities as a function of the mass flux, vapor quality, and fluid density, and minimizing the function in α2, the following relationship is obtained:  ∂ < E k >2 ∂ ¼ K l ρl ul 2 þ K v ρv uv 2 ¼ ∂α2 ∂α2 (  2 )  2 G ð 1  xÞ ∂ Gx K l ρl þ K v ρv ¼ ρv α 2 ρl ð1  α2 Þ ∂α2 ( " #  2 ) 2 1 K v x2 ∂ 1 2 K l ð 1  xÞ ∂ þ ¼ G ρl ρv ∂α2 α2 ∂α2 ð1  α2 Þ2 ( ) 2 K v x2 2 K l ð1  xÞ  ¼0 2G ð1  α2 Þ3 ρl α32 ρv

ð2:55Þ

Assuming G different from zero (flow condition) and solving for α2, the following relationship is obtained: α2 ¼ 1þ

1  13  13  2 ρv Kl 1x 3 Kv

ρl

ð2:56Þ

x

As above discussed, the momentum factor of each phase is related to the non-uniformities of the velocity profiles, therefore it is expected that their values depend on phase distribution and, consequently, on the flow pattern. Thus, the momentum coefficient ratio was correlated by Kanizawa and Ribatski (2016) as a function of non-dimensional parameters for horizontal and vertical upward flows. It should be highlighted that this method is valid only for conditions under which the estimated void fraction is smaller than the estimative according to the homogenous model, and this condition is given as follows: 

Kl Kv

13

 23  1 ρv 1x 3  x ρl

ð2:57Þ

hence, in the case that Eq. (2.57) is not satisfied, the homogeneous model must be considered. For horizontal flows, the two-phase flow topology is dominated mainly by the balance between buoyancy and inertial effects. Thus, the momentum coefficient ratio was correlated as a function of the Froude number and the viscosity ratio as follows:

2.3 Void Fraction

41



Kl Kv

13

¼ 1:021  Fr 0:092 m

 0:368 μl μv

ð2:58Þ

where the Froude number of the mixture Frm corresponds to the balance between gravitational and inertial effects, and is given as follows: Fr m ¼

G2 gðρl  ρv Þ2 d

ð2:59Þ

The correlation given by (2.58) was adjusted for a database with wide experimental conditions comprising results for R12, R22, R134a, and R410A, for flow in ducts with diameters ranging from 0.5 to 13.8 mm, saturation temperatures between 5 and 50  C, and mass flux between 70 and 800 kg/m2s. For vertical flows, the two-phase flow is dominated by a balance between inertial and interfacial effects, thus the momentum coefficient ratio is correlated as a function of the liquid-vapor viscosity ratio and the Weber number of the mixture, as follows: 

Kl Kv

13

¼ 14:549  We0:222 m

 1:334 μl μv

ð2:60Þ

where the Weber number of the mixture corresponds to the balance between inertial and surface tension effects, and is given as follows: Wem ¼

G2 d ðρl  ρv Þσ

ð2:61Þ

Equation (2.60) was adjusted for a database comprising basically in-tube flows of water and heavy water, hydraulic diameters ranging from 6 to 13.4 mm, saturation temperatures between 164 and 275  C, and mass velocities between 52 and 2030 kg/m2s. Unfortunately, the number of studies for refrigerant flow in vertical channels is quite limited, and no data was capable of validating this model for other fluids. In the case that the estimative of the momentum coefficient ratio according to Eqs. (2.58) and (2.60) does not satisfy the restriction given by Eq. (2.57), the homogenous model should be considered for void fraction estimative. Figures 2.10 and 2.11 depict the void fraction predicted values according to the described methods in this section. It can be noticed according to these figures that the homogeneous void fraction model predicts the highest values among all the methods, which is coherent because this method is based on the assumption of equal flow velocity for both phases. Void fraction values higher than the homogeneous model correspond to liquid in situ velocities higher than the vapor velocity for horizontal and vertical upward flows. In most applications, the liquid velocity should be lower due to gravitational and frictional forces.

2 Fundamentals 1.0

1.0

0.8

0.8

0.6

0.6

Homogeneous model Kanizawa and Ribatski (2016) Rouhani (1969) Zivi (1964)

0.4 0.2

a [-]

a [-]

42

0.0 0.0

0.0 0

0.2

0.4

x [-]

0.6

0.8

1

1.0

1.0

0.8

0.8

0.0 0.0

a [-]

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4 0.2

Horizontal flow R134a, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.2

0.4

0.6

0.6

a [-]

0.4 0.2

Horizontal flow R134a, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

Homogeneous model Kanizawa and Ribatski (2016) Rouhani (1969) Zivi (1964)

0.4

x [-]

0.6

0.8

0.6

0.8

1.0

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4 0.2

Vertical flow R134a, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

0.2

x [-]

Vertical flow R134a, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.0 0.0

1.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 2.10 Predicted void fraction values for R134a in 1.0 mm ID tube 1.0

1.0

0.8

0.8 Homogeneous model Homogeneous model Kanizawa andRibatski Ribatski (2016) Kanizawa and (2016) Rouhani (1969) Rouhani (1969) Zivi (1964) (1964)

0.4 0.2 0.0 0.0

Horizontal flow Isobutane, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

0.2

0.4

x [-]

0.6

0.8

0.2 0.0 0.0

1.0

1.0

0.8

0.8

α [-]

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4

0.0 0.0

Horizontal flow Isobutane, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.2

0.4

x [-]

0.6

0.8

1.0

0.6

0.6

α [-]

0.4

1.0

0.2

Homogeneous model Homogeneous model Kanizawa andRibatski Ribatski (2016) Kanizawa and (2016) Rouhani (1969) Rouhani (1969) Zivi (1964) Zivi (1964)

0.6

α [-]

α [-]

0.6

0.2

Vertical flow Isobutane, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

0.2

0.4

x [-]

0.6

0.8

1.0

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4

0.0 0.0

Vertical flow Isobutane, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.2

0.4

Fig. 2.11 Predicted void fraction values for isobutane in 1.0 mm ID tube

x [-]

0.6

0.8

1.0

2.4 Flow Boiling Fundamentals

43

Up to date, there is no reliable predictive method for downward flows, even though the condition of condensation inside channels during downward flow is not uncommon (Fig. 2.11).

2.4

Flow Boiling Fundamentals

The fundamentals of convective boiling (also named as flow boiling) are presented in this section. Nucleate boiling and convective boiling mechanisms are addressed. Heat transfer during convective boiling is dominated by a combination of convective and nucleate boiling effects with both schematically depicted in Fig. 2.12. The convective effects are strictly associated to the fluid flow effects, while the nucleate boiling effects are related to the heat transfer due phase change process, characterized by bubble formations and detachment. Conditions dominated by convective effects are characterized by significant influence of flow velocity associated to mass velocity and vapor fraction, with phase change occurring mainly on the vapor–liquid interface. On the other hand, under conditions dominated by nucleate boiling mechanism, the influence of flow velocity on heat transfer is minimized and the phase change process is associated with the bubble nucleation, growth, and departure from the channel wall. Figure 2.13 schematically depicts the evolution of the two-phase flow characteristics, wall and bulk temperatures, and main heat transfer mechanisms for convective boiling in vertical upward flow with imposed heat flux, assuming subcooled condition (mean cross-sectional fluid enthalpy lower than the saturated liquid enthalpy at the local pressure) at the inlet section. Under single-phase flow conditions, the fluid temperature rises along the duct length due to the heat transferred from the wall through forced convection. Moreover, the temperature difference between the bulk and the wall is almost constant due to a heat transfer coefficient that presents only Convective effects

Nucleate boiling effects

Hydrodynamic boundary layer

Thermal boundary layer

Velocity profile

+

Temperature profile

Fig. 2.12 Schematics of convective and nucleate boiling effects for convective flow boiling inside ducts

44

2 Fundamentals

Averaged fluid temperature

Vapor single-phase flow

Forced convection to the vapor flow

Mist flow

Drywall + droplets deposition

x=1 Fluid temperature in the center of the flow

Wall temperature Wall dryout

Annular wispy flow

Fluid temperature

Annular flow

Wall temperature

Fluid temperature in the center of the flow

x=0

Saturation temperature T

Forced convection heat transfer through liquid film

Plug flow

Nucleate saturated boiling

Bubbles

Averaged fluid temperature

Subcooled boiling Liquid single-phase flow

z

Forced convection to the liquid flow

Fig. 2.13 Schematics of convective flow boiling during vertical upward flow, modified from Collier and Thome (1994)

minor variations associated to the changes of the fluid transport properties with increasing temperature. The bubble nucleation process is triggered for a wall temperature higher than the saturation temperature at the local pressure. This characterizes a phenomenon known in literature as the onset of nucleate boiling (ONB) and may occur even for bulk fluid temperature inferior than the saturation temperature because the fluid closer to the wall is hotter than the temperature at the channel core. Bergles and Rohsenow (1964), based on the approach of Hsu and Graham (1961), proposed a model for the prediction of the ONB, which assumes vapor contained into cavities along the surface, such as schematically depicted in Fig. 2.14, and estimates the superheating or heat flux necessary to promote the bubble growth. According to this approach, the pressure difference between the vapor trapped into cavities with radius r and the external liquid is evaluated based on the force balance as follows: pv  p l ¼

2σ r

ð2:62Þ

2.4 Flow Boiling Fundamentals

45

Tv (r) - Eq. (2.65)

Liquid

y=r Tl (y) = Tv (r) dTl (y) = dTv (r) dy dr Eq. (2.67)

↑ϕ

y Tl (y) - Eq. (2.66)

r

r

r

↑ Tw Temperature Surface with cavities Fig. 2.14 Schematics of vapor bubble into cavity for fixed Tw, adapted from Bergles and Rohsenow (1964)

where pv and pl correspond to the pressure inside and just outside the bubble, respectively, σ corresponds to the surface tension and r to the cavity radius. Therefore, the pressure inside the bubble is higher than the liquid pressure, and consequently the vapor is superheated in relation to local liquid pressure (Tv > Tsat ( pl)). It is possible to relate the corresponding superheating temperature based on the pressure difference using the Clausius–Clapeyron equation for ideal gas, given as follows for vaporization: bilv bi dpv p bi ¼  lv ¼ v lv2 dT T v T v ð vv  vl Þ Rv T v v v

ð2:63Þ

where îlv corresponds to the latent heat of vaporization, Rv to the gas constant of the substance, and Tv to the vapor temperature. Hence, by integrating Eq. (2.63) from the liquid to the vapor condition, the following relationship is obtained:     bi ðT  T sat Þ pv pv  pl ln þ 1 ¼ lv v ¼ ln pl pl Rv T v T sat

ð2:64Þ

which combined with Eq. (2.62) results in the following relationship: T v  T sat ¼

  Rv T v T sat 2σ ln þ1 rpl bilv

ð2:65Þ

which represents the superheating temperature of the vapor in relation to the saturation condition at pl to maintain the bubble stable.

46

2 Fundamentals

Bergles and Rohsenow (1964) assumed pure heat conduction along the liquid to estimate the temperature profile, which results in the following relationship: Tl ¼ Tw 

ϕy kl

ð2:66Þ

where ϕ corresponds to the heat flux, kl to the liquid thermal conductivity, and Tw to the surface temperature. Therefore, considering that at the bubble tip the liquid and vapor must be in equilibrium, whereas: T l ¼T v dT l dT v ¼ dy dr

ð2:67Þ

for y ¼ r, Bergles and Rohsenow (1964) assumed Eq. (2.67) as the condition for the onset of nucleation boiling. It must be mentioned that Eqs. (2.65), (2.66) and (2.67) cannot be solved analytically and an interactive method is required for calculating Tv (or Tl), ϕ (or Tw), and r (or y). Figure 2.15 depicts the predictions of parameters for ONB according to the model proposed by Bergles and Rohsenow (1964). According to Fig. 2.15a, the wall superheating increases asymptotically with heat flux for ONB, and according to Fig. 2.15b, the cavity radius that would be activated as a vapor formation nuclei decreases with increasing wall superheating and/or heat flux. It must be mentioned that by solving Eqs. (2.65), (2.66) and (2.67), a single cavity radius value is obtained, which corresponds to the critical value. However, according to the analysis of Bergles and Rosehnow (1964) the bubble would form and grow as long as the liquid temperature, given by Eq. (2.66), is higher than the vapor temperature of equilibrium, given by Eq. (2.65). Hence, a range of active cavities can be estimated by imposing Tl ¼ Tv, as shown schematically by the dotted line on b) 104

100

R134a R1234ze(Z) R600a Water

103

r = y [m m]

T w - T sat [K]

a) 101

10-1

R134a R1234ze(Z) R600a Water

Tsat = 20 °C

102

101 T sat = 20 °C

10-2 10

100

1000 2 f [W/m ]

10000

100 10-2

10-1

100

101

Tw - Tsat [K]

Fig. 2.15 Onset of nucleate boiling according to the approach proposed by Bergles and Rohsenow (1964), depicting variation of (a) wall superheating with heat flux, and (b) cavity sizes with wall superheating

2.4 Flow Boiling Fundamentals

47

the right in Fig. 2.14. Nonetheless, keep in mind that increasing the heat flux results in an increment of wall temperature and vice versa. Based on this analysis, it can be concluded that high wall superheating and heat flux are needed to activate small cavities, as shown in Fig. 2.15b. Therefore, by combining an energy balance equation with a correlation for forced convection, such as Gnielinski (1976) valid for Reynolds higher than 2500, and the above described method, the ONB position along the flow path can be estimated. Recently, Kandlikar (2006) described a predictive method developed by his research group, which consists in a modification of the model proposed by Hsu and Graham (1961) and Bergles and Rohsenow (1964) for microchannels during water flow. According to his proposal, the wall superheating is given as follows: T w  T sat ¼ 1:1

2σ sin ðβr Þ T sat rϕ þ r k l sin ðβr Þ ρvbilv

ð2:68Þ

where r stands for the cavity radius and βr to the receding contact angle, which depends on the pair fluid-surface, as well as on the thermodynamic state of the fluid. In Fig. 2.13, the bubble nucleation is established upstream the state of saturated liquid, characterizing subcooled flow boiling. Downstream the ONB, as the fluid enthalpy increases, the number of nucleation sites rises, and nucleate boiling becomes the main heat transfer mechanism. Eventually, the high vapor fraction causes bubbles coalescence with consequent formation of large bubbles, characterizing plug flow. With additional increments of vapor fraction, the liquid plugs between consecutive vapor bubbles vanish and the flow becomes characterized by a continuum liquid film on the duct wall and vapor flowing in the core of the channel, characterizing annular flow. Due to the higher specific volume of vapor phase compared to liquid, the vapor velocity and, consequently, the two-phase velocity are increased as the liquid is evaporated. Under this circumstance, the shear of the vapor at high velocity on the liquid film might cause detachment of liquid droplets. Thus, the flow transitions from pure annular to wispy flow characterized by a liquid film along the tube perimeter with vapor flow in the center of the section and entrained liquid droplets. For annular and wispy flows, the heat transfer rate is given as a result of convective effects and the contribution of nucleate boiling effects is suppressed. As shown in Fig. 2.13 for the region comprising bubbles, plug, annular, and annular wispy flow patterns, assuming negligible pressure drop, the fluid temperature is almost constant and corresponds to the saturation temperature. On the other hand, the heat transfer coefficient is high and as a consequence the difference between the fluid and wall temperature is low. Eventually the liquid film will vanish due to a combination of droplet detachment and their vaporization, and the liquid phase becomes distributed as small dispersed droplets within the vapor phase, characterizing mist flow. As result of the surface dryout, the heat transfer coefficient decreases significantly, and the wall temperature presents a sudden increase. The heat transfer from the wall to fluid under mist flow conditions occurs firstly from the wall to the vapor contacting the duct, and then,

48

2 Fundamentals

Intermittently dry

Singlephase liquid

Bubbles

Forced Subcooled convection to boiling the liquid flow T

Plug

Slug

Nucleate saturated boiling

Annular

Forced heat transfer though liquid film

Singlephase vapor

Stratified

Forced convection to the vapor flow

Dry wall + droplets deposition

Wall temperature

x=0

Average fluid temperature

Fluid temperature in the center of the section

Saturation temperature x=1

Fig. 2.16 Schematics of convective flow boiling during horizontal flow

from the vapor to the liquid. Droplet deposition on the tube wall also contributes to the overall heat transfer process, promoting a slight increase in the heat transfer coefficient, with a proportional reduction on the wall temperature. Downstream the surface dryout, the droplets will eventually vanish, and single-phase forced convection to the vapor flow becomes the heat transfer mechanism. Based on the above discussion, it should be emphasized that the main heat transfer mechanisms during flow boiling depend on the phases’ distribution, characterized as flow patterns. The flow patterns, in turns, depend on the vapor fraction. Similarly, Fig. 2.16 depicts the evolution of the flow characteristics along the duct length for horizontal flow in channels of conventional size under condition of imposed heat flux. This figure also depicts the draft of wall and fluid temperatures, as well as main heat transfer mechanisms in each region. Differently than vertical flow, buoyancy effects exert a crucial effect on the phase distribution along the cross-section making it non-axisymmetric. Additionally, even though the pressure variation along the cross-section is small, the onset of bubble nucleation is more likely to occur in the upper region of the channel due to local slightly smaller pressure and predominance of hotter liquid in this region due to buoyance effects. The non-equilibrium and higher temperature of the duct wall cause bubble nucleation even under conditions of subcooled liquid flow, which is depicted on the left side in Fig. 2.16 as subcooled flow boiling. With additional increment of the fluid enthalpy, the flow will eventually become saturated, which is denominated as nucleate saturated boiling. For reduced vapor quality, the flow is characterized by the flow of vapor as small dispersed bubbles, hence corresponding to the bubble flow pattern. With subsequent increment of vapor fraction, the bubble size and number increase, implying on their coalescence and flow pattern transition from plug to slug flow. Under this condition, the contribution of convective effects becomes more significant, even though the heat transfer process is dominated by nucleate boiling effects.

2.5 In-Tube Condensation Fundamentals

49

As the liquid slugs between successive vapor bubbles are disrupted, the flow shifts either to stratified flow pattern under conditions of reduced flow velocity or annular flow pattern under conditions of intermediate and high flow velocities. For stratified flow, the upper portion of the duct wall is in contact with the vapor phase, implying on reduced heat transfer coefficient and higher wall temperature along this region. The occurrence of stratified flow pattern is restricted to conventional channels and low mass velocity flows; therefore, it is expected that the contribution of the convective effects to heat transfer plays a small role while the nucleate boiling is the major heat transfer mechanism. This aspect is observed empirically, because in general, the heat transfer coefficient during stratified flow presents negligible variation with the increment of vapor quality; even a slight reduction of heat transfer coefficient with vapor quality is reported in some studies for stratified flow, which is attributed to reduction of the liquid portion in contact with the channel wall, and consequently increment of the surface exposed to the vapor phase. Annular flow pattern is characterized by high heat transfer coefficients along the entire duct perimeter and, consequently, a perimeter-averaged heat transfer coefficient higher than the value for stratified flow. Similar to the case of vertical flows, the heat transfer process during annular flow pattern in horizontal channel presents a significant contribution of convective effects. Nonetheless, for annular flow in horizontal ducts, the liquid film in the upper region is thinner due to gravitational effects; therefore, additional increment of vapor fraction progressively dries the liquid film from the upper region to the tube bottom. Some authors, such as Wojtan et al. (2005a), define the condition of gradual wall drying as a dryout flow pattern, assumed as a transitional condition between annular and mist flow (observed under high mass velocity conditions).

2.5

In-Tube Condensation Fundamentals

The fundamentals of in-tube condensation are presented in this section. The main heat transfer mechanisms during condensation are presented and discussed. Condensation during forced convection inside ducts is dominated by convective and gravitational effects, as schematically depicted in Fig. 2.17. Convective effects are dominant for high flow velocities. Under this condition, the heat transfer coefficient is mainly a function of the two-phase flow velocity and its value increases with raising vapor quality and mass velocity, and effects on the heat transfer coefficient of wall subcooling are negligible. On the other hand, condensation inside horizontal tubes dominated by gravitational effects are typical of low flow velocities. The heat transfer process is basically controlled by the vapor condensation on the upper part of the duct and the resulting liquid is driven by gravity to the bottom region of the tube and is simultaneously propelled according to the main flow direction. The heat transfer process in the upper part of the duct is somewhat similar to the condensation in vertical walls as addressed by Nusselt (1916). Under this condition, the heat transfer coefficient is mainly a function of the wall subcooling.

50

2 Fundamentals

Convective effects

Gravitational effects

Hydrodynamic boundary layer

+

Thermal boundary layer

Velocity profile

Temperature profile

Fig. 2.17 Schematics of convective and gravitational effects for convective flow condensation inside ducts

For small diameter tubes, surface tension effects are also indicated as relevant to the heat transfer process, as pointed out by Rossato et al. (2017). Figure 2.18 schematically depicts the two-phase flow evolution, fluid temperature, and heat transfer coefficient variations for condensation along horizontal channels. This figure also depicts the dominant mechanisms affecting the heat transfer coefficient along the condensation process. This description is similar to the one addressed by Ribatski and Da Silva (2016). The onset of liquid condensation requires a duct wall temperature lower than the saturation temperature at the local pressure, hence, liquid is condensed even for conditions corresponding to thermodynamic vapor qualities higher than the unity. Initially, the condensate forms a liquid film along the duct internal perimeter, and with additional vapor quality reduction and high flow velocities waves are present on the liquid–vapor interface. This flow structure corresponds to a wavy-annular flow pattern. Additional vapor quality reductions imply lower vapor shear effects on the liquid film with the interface becoming smoother and the flow transitioning to annular flow. The heat transfer coefficient for wavy-annular flow is higher than for annular flow due to both a thinner perimeter-averaged liquid film and high two-phase flow velocity. Under conditions of high flow velocities, the vapor flow may detach liquid droplets from the wavy film, and the droplets flow entrained within the vapor phase. As the vapor quality decreases, a significant parcel of the liquid accumulates in the bottom region of the tube due to buoyancy effects, and the flow becomes stratified with liquid segregated in the lower part of the tube and as a film on the tube wall and vapor flowing in the upper part of the tube. Depending on the vapor velocity, the vapor–liquid interface may be either smooth or wavy, corresponding to stratified and wavy-stratified flow patterns, respectively. The heat transfer for these flow patterns is dominated by gravitational effects. Convective effects are dominant for annular and wavy-annular flow patterns while gravitational effects dominate under stratified flow conditions.

2.6 Transition from Macro to Microscale Conditions

Superheated vapor

Wavyannular

Annular

Wavy stratified

51

Stratified

Slug

x = 1.0

Subcooled liquid x = 0.0

Convective effects

Gravitational effects

Convective effects

T Fluid temperature in the center of the flow

Fluid temperature

Wall temperature

Heat transfer coefficient

Fig. 2.18 Schematics of convective flow condensation inside horizontal channel, adapted from Ribatski and da Silva (2016)

Successive reductions of vapor quality will eventually result in liquid slugs between elongated bubbles, as depicted in Fig. 2.18. According to Ribatski and Da Silva (2016), the heat transfer process for intermittent flow patterns is dominated by convective effects, since the heat transfer coefficient is mainly associated to the flow characteristics close to the tube wall. Finally, further vapor quality reductions promote bubbles to shrink and, ultimately, their collapse when liquid single-phase flow is established. However, as shown in Fig. 2.12, some vapor bubbles may remain within the flow even for subcooled flow, characterizing a non-equilibrium condition. For slug flow, as the vapor quality decreases the two-phase flow velocity and, consequently, the heat transfer coefficient are reduced.

2.6

Transition from Macro to Microscale Conditions

Several approaches and criteria that define the transition between micro and macroscale available in the open literature are presented and discussed. Approaches based on manufacturing techniques, channel geometry, bubble confinement, and two-phase flow characteristics are addressed. In small diameter channels, the duct dimension becomes of the same order of magnitude of the bubble departure diameter, hence it is expected that surface tension forces present significant influence on the flow behavior and heat transfer process, which justifies the need for different modelling and analysis approaches for micro and conventional channels. Moreover, it is expected that the flow restriction caused

52

2 Fundamentals

by the bubble growth itself induces an earlier bubble detachment than observed for conventional channels and pool boiling conditions in a stagnant media as pointed out by Jacobi and Thome (2002). Therefore, the characterization of the channel size scales might be important based on the fact that the predominant forces and mechanisms change as the channel dimension reduces. In this context, Thome (2004) pointed out several aspects that justify the classification of channels according to the dimensions. In his review, Thome gathered experimental results from the literature and indicated axisymmetric phase distribution in the case of microscale channels, as well as differences of trends and values for heat transfer coefficient, critical heat flux, void fraction, and pressure drop between micro and macroscale conditions. Considering the hydrodynamic aspect, it is common sense that two-phase flow in conventional, also named as macroscale channels, is dominated by inertial and gravitational forces with transport properties such as phase densities and viscosity playing the dominant role. Nonetheless, although surface tension also influences the phase distribution and interface geometry for two-phase flow in conventional channels, such as pointed out by Rodriguez and Castro (2014) setting the curvature of interface for stratified flow and affecting two-phase flow transitions, its effects are not dominant. On the other hand, for two-phase flows in microscale channels, the reduced curvature radius of the interface associated to the two-phase flow confinement implies on the predominance of surface tension effects. In the state-of-the-art review of Mehendal et al. (2000), the authors pointed out distinguished behaviors between micro and mesoscale channels for Nusselt number and friction factor, even for single-phase flow, compared to conventional channels. Based on the literature review, these authors also speculated that the transition from laminar to turbulent flow regimes occurs for Reynolds number within the range of 200–900 for microscale channels, well below the usual values observed for conventional channels quoted around 2300. In fact, currently it is well established for single-phase flow that the hydrodynamics and heat transfer behaviors are similar for channels with dimensions larger than the condition for which rarefaction effects become relevant. As pointed out by Morini (2006), these differences observed in the earlier studies for single-phase flows were linked mainly to experimental uncertainties associated to surface roughness and channel dimension evaluation. Based on this discussion and on contrary to single-phase flow, it can be concluded that the classification of micro and conventional scale channels for two-phase flows is not just a matter of channel size, but it is related to the dominant forces that govern the flow and dominant heat transfer mechanisms. Researchers have proposed criteria for the classification of channels according to size scales, and Mehendal et al. (2000) proposed their classification according to an order of increasing their size as: nano, micro, mini (or meso), and macrochannels (or conventional). Classifications based on manufacturing processes with the transitional diameters defined arbitrarily were proposed by Mehendal et al. (2000) and Kandlikar and Grande (2003). Mehendal et al. (2000) proposed the following

2.6 Transition from Macro to Microscale Conditions

53

classification based on the hydraulic diameter: micro 1 ~ 100μm; meso 100 ~ 1000μm; compact 1000 ~ 6000μm; conventional >6000μm. Kandlikar and Grande (2003) introduced in this discussion aspects related to the range of dimensions for which the random motion of the particles affects the flow conditions, and classified the channel scales as follows: molecular nanochannel 3000μm. According to Thome (2008), Bretherton (1961) was the first author to propose a channel scale classification based on bubble confinement, even though not intentionally focused on microscale flows, who indicated the pipe diameter below which a Taylor bubble would not rise due to buoyancy effects. Bretherton (1961) concluded that a Taylor bubble would not rise for Eötvos numbers smaller than 6.736. Kew and Cornwell (1997) proposed also a classification based on mechanistic aspects of two-phase flow and convective boiling. According to them, prediction methods for the heat transfer coefficient developed for conventional channels are not valid for confinement numbers higher than 0.5; hence, the flow boiling process is considered to occur under microscale conditions for Co higher than 0.5, with the confinement number defined as follows: Co ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ gðρl  ρv Þd2

ð2:69Þ

which is associated to the ratio between surface tension and buoyance forces. It must be emphasized that even though this method takes into account surface tension and buoyance effects, it does not include inertial effects associated to two-phase flow velocity that may favor the detachment of smaller bubbles compared to quiescent conditions. Moreover, the evaluation of the degree of flow confinement for diabatic conditions based on two-phase topology may depend on heat flux intensity and if the fluid is being evaporated or condensed. Based on the same approach of Kew and Cornwell (1997), Ullman and Brauner (2007) proposed a transition criterion based on the Eötvos number, given as follows: Eo ¼

gðρl  ρv Þd 2 8σ

ð2:70Þ

which corresponds to the inverse of squared confinement number divided by eight. According to Ullman and Brauner (2007), a channel can be considered a microscale channel for conditions with Eötvos number smaller than 0.2. It must be emphasized that this approach is focused on two-phase flow inside channels, taking into account characteristics of both phases, and assumed that the condition to characterize a microchannel corresponds to the impossibility of occurring stratified flows. Kawaji and Chung (2003) proposed a transition criterion based on six dimensionless numbers that takes into account the influence of flow velocity. The transition criteria proposed by them is given as follows:

54

2 Fundamentals

ðρl  ρv Þgd 2 1 4σ j2 dρ G 2 ð 1  xÞ 2 d ¼ l l¼ 1 ρl σ σ j2 dρ G2 x2 d ¼ v v¼ 1 ρv σ σ ρ j d Gð1  xÞd ¼ l l ¼ < 2000 μl μl ρ j d Gd ¼ v v ¼ < 2000 μv μv μ G ð 1  xÞ μj 1 ¼ l l¼ l ρl σ σ

Bd ¼ Wel Wev Re l Re v Cal

ð2:71Þ

where Cal is the capillary number based on the liquid velocity and viscosity, and Bd is the Bond number, which consists of two times the Eötvos number. In their method, all the criteria must be satisfied, and the smallest diameter should be considered as the threshold value between micro and macroscale conditions. Even though the capillary number does not take into account the tube hydraulic diameter, it is used to evaluate the balance between viscous dissipation and surface tension, and must be satisfied to characterize the two-phase flow in a pipe as under microscale conditions. Similarly, Ong and Thome (2011) proposed a transition criterion to classify micro, meso, and macroscale channels based on their investigation of channel effect on flow pattern transitions. The criteria to classify a certain channel as micro or macro was based on whether flow patterns characteristic of microscale occurred. The mesoscale channels consisted of dimensions with thermohydraulic characteristics of micro and macroscales. Hence, according to Ong and Thome (2011), a channel can be considered microchannel for confinement number higher than 1.0, macrochannels correspond to confinement number smaller than 0.34, and mesoscale for the intermediate range, where the confinement number Co is given by Eq. (2.69). Macro-to-microscale segregation methods based on the bubble departure diameter are suitable only for flow boiling, whereas the transition for adiabatic flows and convective condensation cannot be predicted based on these approaches. Therefore, Li and Wang (2003) proposed a method to characterize micro-to-macroscale transitions based on the Young–Laplace equation accounting for the pressure difference between both sides of the interface imposed by the surface tension. In their method, the transition is given as a function of capillary length, defined as follows: Lcap ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ gð ρl  ρv Þ

ð2:72Þ

Li and Wang (2003) evaluated the symmetry of the phase distribution during condensation inside round channels, and proposed criteria for identification of channel dimensions for which the occurrence of stratified diameter is possible, defined as dcritical, and the channel dimension for which the flow would be minimally axisymmetric, defined as dtransition. Their transitional dimeters are given as follows:

2.6 Transition from Macro to Microscale Conditions

d [m]

5 x10

R134a, G = 100 kg/m²s, ∆ T = 10 °C, x = 0.10

-3

10

10

Ong and T Brethert home (2011), c on (196 1) - c , macro Kew an d Cornw ell (199 Li and W 7) - c ang (20 03), criti cal Ullman and Bra uner (20 Ong and 07) - c Thome (2011), c, micro

-3

-4

Kutate Nishikawa et al. (1976) Li and W ladze and Gog o ang (20 03), tran nin (1979) sition Kandlikar and Grande (2003) 003) - c (2 ng hu C d Fritz (1 Kawaji an 935) Mehendal et al. (2000)

0,0

4 x10

0,2

-3

-4

p r [-]

0,6

0,8

1,0

Ong an d Thom Brethe e (2011), c, macro rton (1 961) Kew a nd Li and Cornwell ( 19 Wang (2003) 97) Ullma , critic n and B al rauner Ong an (2007) d Thom e (201 1), c, m icro

d [m] 10

0,4

Isobutane, G = 100 kg/m²s, ∆ T = 10 °C, x = 0.10

-3

10

55

Kuta Nishikawa et al. (1976) Li and teladze and G Wang (2003) ogonin (197 9) , transit ion Kandlikar and Grande (2003) Fritz ( Kawaji and Chung (2003) 1935) Mehendal et al. (2000)

0,0

0,2

0,4

p r [-]

0,6

0,8

1,0

Fig. 2.19 Micro and macroscale transition diameters for R134a and isobutene at wall superheat of 10  C, x of 0.10, and mass velocity of 100 kg/m2s

dtransition ¼ 0:255  Lcap

ð2:73Þ

d critical ¼ 1:749  Lcap

ð2:74Þ

Figure 2.19 depicts the variation of the transition diameter according to distinct criteria discussed in the present section, for R134a and isobutane for reduced

56

2 Fundamentals

pressure pr varying from values close to zero to values close to unity. According to this figure, as the working pressure increases, the transition diameter that characterizes the microscale channels is reduced according to most methods proposed based on fluid properties, on the contrary to the criteria of Mehendal et al. (2000) and Kandlikar and Grande (2003), which consist on fixed transitional values. In conclusion, a criterion to characterize macro to microscale conditions is suitable only if different behaviors for heat transfer, pressure drop, flow induced vibration, and noise are observed, since those are the parameters that a designer of a heat transfer device should care. Moreover, if this is the case, different criteria should be considered according to the two-phase flow conditions such as adiabatic flows, flow boiling, and convective condensation. Aspects such as flow patterns should also be considered since a bubble flowing under narrowed conditions does not represent the physics of confined annular flow.

2.7

Solved Example

1. Consider two-phase flow of R134a inside a 1 mm ID channel, with saturation temperature of 15  C, mass flux of 250 kg/m2s, and vapor quality of 0.15. Present the mass flow rate. Superficial velocities of each phase and the mixture velocity. Volumetric fraction. Void fraction according to homogeneous model, Zivi approach, minimum kinetic energy, and Rouhani approaches for horizontal and vertical upward flow. (e) The in situ velocities assuming the four void fraction predictions. (f) Does this condition correspond to macro or microscale channel according to Kew and Cornwell criteria?

(a) (b) (c) (d)

Firstly, let us list the input data: Fluid ¼ R134a; d ¼ 1 mm ¼ 0.001 m; Tsat ¼ 15  C; G ¼ 250 kg/m2s; x ¼ 0.15. Considering that this problem comprises simultaneous flow of liquid and vapor phases, it is interesting to list the thermophysical properties of both phases for Tsat ¼ 15  C, as follows: ρl ¼ 1243 kg/m3; ρv ¼ 27.78 kg/m3; σ ¼ 0.009359 N/m; μl ¼ 0.00022 kg/ms; μv ¼ 1.153105 kg/ms. (a) The mass flux consists to the ratio between the mass flow rate and the crosssectional area, as follows: G¼

m_ m_ ¼ A πd2 4

Then, solving for ṁ, the mass flow rate is equal to 7.854107 kg/s.

2.7 Solved Example

57

(b) The superficial velocity, or volumetric flux, consists in the ratio between the volumetric flow rate and the cross-sectional area. Therefore, for each phase, the volumetric flux can be given as follows: jv ¼

Qv Q and jl ¼ l A A

Considering the available input parameters, it is easier to represent the volumetric flow rate as a function of the mass flux, as follows: jv ¼

Qv _ Gx m m_ mx ¼ v ¼ ¼ ¼ 1:577 s A ρv A ρv A ρv

Similarly, for the liquid phase: jl ¼

Ql Gð1  xÞ m ¼ 0:171 ¼ ρl s A

And the mixture superficial velocity is given as the sum of the volumetric flux of each phase, as follows: j ¼ jv þ jl ¼ 1:748

m s

Notice that even for a relatively low vapor quality value, the liquid velocity corresponds to only approximately 10% of the mixture velocity. (c) The volumetric fraction corresponds to the ratio between the volumetric flow rate of the vapor phase and the mixture volumetric flow rate, as follows: β¼

Qv Qv þ Ql

Dividing the numerator and denominator by the cross-sectional area A, it is possible to use directly the results for superficial velocities obtained in the previous item, as follows: β¼Q

v

A

Qv A

þ QAl

¼

jv j ¼ v ¼ 0:902 jv þ jl j

(d) According to the homogenous model, it is assumed that the slip velocity ratio is equal to the unity, hence, the void fraction is given as follows: αH ¼

1 1 þ uuvl ρρv l

1x x

58

2 Fundamentals

where uv/ul ¼ 1.0. Hence, substituting the values, the void fraction according to the homogeneous model is 0.902, which is identical to the volumetric fraction, as discussed along the chapter. According to the Zivi method, the void fraction is given as follows: αZivi ¼ 1þ

1  23 ρv ρl

1x x

and substituting the corresponding values, the resulting value is 0.712, which is considerably lower than αH. According to the minimum kinetic energy approach, the void fraction depends on the momentum coefficient ratio, and is given as follows: α min ,kinetic ¼ 1þ

1  13  13  2 ρv Kl 1x 3 Kv

ρl

x

whereas for horizontal flow, the momentum coefficient ratio is given as: 

Kl Kv

13

¼ 1:021  Fr 0:092 m

 0:368 μl μv

with Froude number given as: Fr m ¼

G2 ¼ 4:284 gðρl  ρv Þ2 d

resulting in momentum coefficient ratio of 0.3018. And for vertical upward flow, the momentum coefficient ratio is given as follows: 

Kl Kv

13

¼ 14:549  We0:222 m

 1:334 μl μv

with Weber number given as: Wem ¼

G2 d ¼ 5:476 ðρl  ρv Þσ

which results in momentum coefficient ratio of 0.1954. Both values must be evaluated to verify whether they satisfy the condition that the maximum void fraction possible corresponds to the homogenous model. Hence, considering the inequality:

2.7 Solved Example

59



Kl Kv

13



 23  1 ρv 1x 3 x ρl

the right-hand side is equal to 0.1275. Hence, this restriction is satisfied for horizontal and vertical flows. Therefore, by substituting both values for void fraction, we obtain void fraction values of 0.796 for horizontal flow and 0.858 for vertical flow. Finally, the predictions according to the drift flux model adjusted by the Rouhani research group are given as follows: ( α2 ¼

"  #)1  e vj V ρv x 1x C0 þ þ ρv ρl x G

where the slip parameter is estimated for gas bubbles rising in stagnant liquid, given as follows:  1 gσ ðρl  ρv Þ 4 m e V vj ¼ 1:18 ¼ 0:1088 s ρ2l For horizontal flow, the distribution parameter is given as follows: C 0 ¼ 1 þ 0:2ð1  xÞ

 1 gdρ2l 4 ¼ 1:119 G2

and for vertical flow, it is equal to 1.10 because the mass flux is higher than 200 kg/m2s, otherwise it would be 1.54. Therefore, substituting in the relationship for void fraction, the obtained values are, respectively, 0.764 and 0.776 for horizontal and vertical flows. (e) The in situ velocity depends on the local void fraction, as follows: uv ¼

jv jl u ¼ α l 1α

Hence, considering the results from the previous item, the following in situ velocity values are obtained: Void fraction Homogenous model Zivi (1964) model Kanizawa and Ribatski (2016) horizontal Kanizawa and Ribatski (2016) vertical Rouhani (1969) horizontal Rouhani and Axelsson (1970) vertical

α [] 0.902 0.712 0.796 0.858 0.764 0.776

jv [m/s] 1.577

jl [m/s] 0.171

uv [m/s] 1.748 2.032 1.982 1.839 2.065 2.032

ul [m/s] 1.748 0.593 0.837 1.200 0.723 0.764

60

2 Fundamentals

Notice the difference between superficial and in situ velocities. Additionally, the void fraction value estimated according to the homogenous model is considerably higher than according to other methods. (f) The Kew and Cornwell criterion is based on the confinement number, whereas a channel is considered as micro for Co higher than 0.5. Hence, the confinement number is given as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ Co ¼ ¼ 0:885 > 0:5 gðρl  ρv Þd2 Hence, it can be considered as a microchannel according to this criterion.

2.8

Problems

1. Derive the relationships given by Eqs. (2.20) and (2.21). 2. Derive the relationships for superficial velocities based on average operators. 3. Show that the volumetric fraction β is similar to the void fraction estimated according to the homogenous model. 4. Assume an axisymmetric annular flow absent of liquid droplets in the vapor stream. What is the area averaged void fraction if the linear void fraction, along a rectilinear segment passing by the center of the cross-section, is (a) 85, (b) 90, and (c) 95%? Are they different from the volume averaged void fraction in this case? 5. Redo exercise 4 by assuming stratified flow, with linear void fraction measurement along vertical direction in the central plane of the channel. 6. Redo exercise 5 by assuming the same conditions, but consider that the surface tension is equal to 0,01073 N/m and that the contact angle is equal to 45 . For simplicity, consider vertical wall for the estimation of liquid portion due to surface tension. 7. Consider R134a flowing along a horizontal microscale channel with 0.5 mm of internal diameter. Assuming that a uniform heat flux of 10 kW/m2 is imposed in the internal wall, the fluid in the duct inlet is at 5  C subcooling temperature for a pressure of 293 kPa at mass flux of 300 kg/m2s, determine: (a) the vapor quality 0.1 m away from the inlet? (b) the void fraction according to the homogenous model, Kanizawa and Ribatski method, and drift flux model. Assume that the pressure drop parcel is negligible. 8. Repeat exercise 7 assuming vertical upward flow. 9. Repeat exercise 7, and calculate the phases and mixture superficial velocities for vapor qualities of 0.1, 0.5, and 0.8. 10. Compare the void fraction predicted values for vapor quality ranging from 0 to 1 (increment of 0.1) according to the predictive methods described in this study

References

11. 12.

13.

14.

61

for vertical and horizontal flows. Assume internal diameter of 1 mm, and two-phase flow of R22, R134a, CO2, and R1234ze at 0 and 40  C. Discuss the differences between the estimated values. Based on the definitions presented in this chapter, derive Eq. (2.46) starting from Eq. (2.45). Discuss if the condition of α higher than the homogenous model for horizontal or vertical upward flow is feasible. Why, or why not? What about the case of downward flow? The line averaged void fraction determined with a radioactive densitometer indicated a value of α1 for annular flow in a round channel, measured transversal to the tube axis and passing by its center line. What is the corresponding area averaged void fraction? Estimate for α1 ¼ 95%. Now, assume a stratified flow whereas the line averaged void fraction measured vertically and transversally to the channel axis, passing by its centerline, is equal to α1. Determine the corresponding area averaged void fraction. Estimate for α1 of 95%, 80%, 50%, and 25%. Compare with previous problem.

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63

Ullmann, A., & Brauner, N. (2007). The prediction of flow pattern maps in minichannels. Multiphase Science and Technology, 19(1), 49–73. https://doi.org/10.1615/MultScienTechn. v19.i1.20 Wallis, G. B. (1969). One-dimensional two-phase flow. New York: McGraw Hill Book Company. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005a). Investigation of flow boiling in horizontal tubes: Part I—A new diabatic two-phase flow pattern map. International Journal of Heat and Mass Transfer, 48(14), 2955–2969. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005b). Measurement of dynamic void fractions in stratified types of flow. Experimental Thermal and Fluid Science, 29(3), 383–392. Woldesemayat, M. A., & Ghajar, A. J. (2007). Comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes. International Journal of Multiphase Flow, 33(4), 347–370. Zivi, S. M. (1964, May). Estimation of steady state steam void fraction by means of the principle of minimum entropy production. Journal of Heat Transfer, 86, 247–252. Zuber, N., & Findlay, J. (1965). Average volumetric concentration in two-phase flow systems. Journal of Heat Transfer, 87(4), 453–468.

Chapter 3

Flow Patterns

This chapter is focused on the analysis of flow patterns during gas-liquid flows inside channels. The flow patterns commonly found in literature for adiabatic two-phase flow in conventional channels are described in Sect. 2.2. These flow patterns are schematically illustrated in Figs. 2.2, 2.3, and 2.4 for vertical upward, vertical downward, and horizontal flows, respectively. According to the discussion presented in Sect. 2.6, the definition of microscale channel itself is usually based on the phase distribution within the flow and the dominant forces, as well as the dominant heat transfer mechanisms. Therefore, it is expected that the flow patterns during two-phase flows in microscale channels present different characteristics from those for conventional channels, or at least, the transition between flow patterns are not the same. Despite these aspects, this chapter also comprises descriptions of the predictive methods for flow pattern transitions in conventional channels, because they are used as the basis for the development of predictive methods for microscale channels.

3.1

Flow Pattern Identification

The flow patterns can be identified subjectively, based on visual observation and the subjective judgement by an observer, and objectively, based on an analysis of time response of some flow parameters, such as void fraction and pressure drop. Subjective Approach Generally speaking, flow patterns are classified based on the geometrical characteristics of the gas–liquid interface, as discussed in Chap. 2, and named through intuitive denominations such as bubbles, annular, and so on. This procedure allows to identify them based on the judgement of an observer considering visual observations of the flow pattern at naked eye as well as helped by flow images and videos captured along a translucid section. Considering that different flow pattern © The Author(s) 2021 G. Ribatski, F. Kanizawa, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_3

65

66

3

Flow Patterns

judgments may be provided by distinct observers, this approach is nominated as subjective. Flow pattern segregations based only on visual observations are appropriate for conditions of low mass velocities and separated flows, while under conditions of high flow velocities and/or non-continuous flow patterns, flow pictures and videos are necessary. The subjective approach provides reasonable results and, generally, the flow patterns segregated according to this method are associated with different heat transfer and pressure drop behaviors. In this context, predictive methods for heat transfer coefficient and frictional pressure drop are frequently developed based on the local flow pattern characterized through the subjective approach. Objective Approach Under certain conditions, the visual access to the flow is not possible, such as for test sections with metallic walls, opaque fluids, and applications involving extremely high pressure and/or temperature. Consequently, under these circumstances, flow patterns are inferred from time varying properties of the flow. Jones and Zuber (1975) were among the pioneers to implement an objective approach to segregate flow patterns. They identified flow patterns for vertical gas-liquid flow in a rectangular channel based on the analysis of chordal void fraction measurements performed with X-ray densitometry with the presence of liquid phase corresponding to higher attenuation of the X-ray. The probability density function (PDF) of the void fraction signal was used to identify the following flow patterns: bubbles, intermittent, and annular. Figure 3.1 schematically depicts the void fraction signature with time of the normalized X-ray signal attenuated by the flow and the corresponding power spectral density of the chordal void fraction α for bubbles, intermittent, and annular flow patterns. Two-phase flow under bubble flow patterns corresponds predominantly to a continuum liquid with dispersed bubbles, therefore, the PDF peak corresponds to reduced void fraction values, as shown in Fig. 3.1. On the other hand, the annular flow pattern corresponds to a thin liquid film flowing on the duct wall, and gas a)

b) Bubbles Intermittent Annular

Probability Density Function

Vapor 1.0

α [-]

0.8 0.6 0.4 0.2 Liquid 0.0

Time

0.0 Liquid

0.2

0.4

0.6 α [-]

Fig. 3.1 Schematics of power spectral density analysis for different flow patterns

0.8

1.0 Vapor

3.1 Flow Pattern Identification

67

flowing in the core, therefore, corresponds to conditions of predominantly high void fraction values, as also shown in Fig. 3.1. The intermittent flow pattern corresponds to the intermittent passage of elongated bubbles separated by liquid slugs containing small dispersed bubbles (refer to the slug flow schematically depicted in Fig. 2.2). Hence, during the elongated bubble passage across the sensor signal, the void fraction value is similar to that of the annular flow, and during the liquid passage, the void fraction signal is similar to the condition of bubble flow pattern. As shown in Fig. 3.1, this condition corresponds to two peaks, at low and high void fraction values. Even though the approach based on the power spectral density is less dependent on the judgement of an observer, the classification still relies on the definition of threshold values based on arbitrary judgements. In this context, methods to detect and segregate the flow patterns based on artificial intelligence such as clustering and neural network algorithms have become a promising approach, such as performed by Mi et al. (1998, 2001). The clustering methods are independent of subjective judgments and their principle is grouping the data points according to similarities. One example of clustering method is the k-means that groups the data points to form clusters of data based on the Euclidean distance among them and the centroid of each cluster. The k-means method consists in an iterative method, and as a result, each experimental data is attributed to a data cluster that can be considered as a flow pattern. The correspondence between the obtained groups and the flow patterns visually identified is not necessarily attained. Moreover, according to the literature, the final groups might depend on the initial guess for the centroids. Routines to implement such methods are not a major problem since they are already available in most engineering software, such as Matlab, Scilab, and Canopy, which provide libraries with these functions. The implementation of the clustering methods to flow pattern identification initially requires the definition of the typical flow parameters to be used as data input in the algorithm. The mean value, standard deviation, and peak to valley value of the instantaneous void fraction along a period, among others, can be used as input parameters for the clustering method. Other parameters, such as local pressure, pressure drop along a channel length, and temperature, could also be used as input parameters for the clustering methods, as well as combinations of them. However, it is desirable to use non-dimensional parameters; therefore, the standard deviation, peak to valley values, etc., can be normalized by the mean value of each experimental/operational condition. It must be mentioned that flow pattern identification should mainly consider the effects of two-phase flow topology on heat exchanger design parameters, such as heat transfer coefficient, pressure drop, flow-induced vibration, flow-induced noise, and critical heat flux, since these are the aspects considered by the designer. The next sections are dedicated to the presentation of flow pattern predictive methods.

68

3.2

3

Flow Patterns

Flow Pattern Transition Criteria for Adiabatic Flows

This section addresses flow pattern predictive methods that were developed focused on conventional channels. Usually, these methods are based on experimental results for channels with dimensions typical of applications in nuclear and oil and gas industries, and quite commonly air and water mixtures close to atmospheric pressure and ambient temperature are used as working fluids. The methods nominated as flow pattern maps are simply graphical representations characterizing regions associated to the flow patterns, that is, based on the operational conditions, two coordinates are defined that characterize a data point contained within a region of the graph that corresponds to a specific flow pattern. Prediction methods based on phenomenological approaches are also available. They consider the physical mechanisms responsible for the transitions. These methods are detailed in the present text, since they are used as the basis for the development of flow pattern predictive methods for two-phase flow in microscale channels. The graphical methods, even though are relevant from a historical point of view, are usually developed based on restrict experimental databases, and therefore, because they do not include a mechanistic approach, are recommended only to the range of experimental conditions considered in their development. On the other hand, predictive methods based on physical mechanisms can be considered as general, as long as the dominant mechanisms are correctly chosen. In this context, this chapter describes the methods developed by Taitel and coworkers, which states the main flow pattern transitional mechanism for adiabatic two-phase flows adopted in most studies after the classical paper published by Taitel and Ducker (1976). Due to the differences that the heat transfer process may impose on flow pattern transitions, this chapter also addresses predictive methods for flow patterns during evaporation and condensation.

3.2.1

Graphical Methods

Baker (1954) was pioneer to propose one of the first flow pattern maps for oil and gas horizontal flows focused on applications in the oil and gas industry. His map shown in Fig. 3.2 comprises in its coordinated axes flow conditions and dimensionless transport properties referred to the corresponding properties of air and water, aiming to give a universal characteristic to the map. This method does not account for the effect of tube diameter on the flow pattern transitions, even though it is expected that this parameter influences significantly the transitions. In the subsequent years, several flow pattern maps were proposed based on different experimental databases. Even though the maps are useful and the flow pattern information is easily obtained from them, these methods are usually valid for a narrow range of operational conditions and working fluids as mentioned above.

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

69

105

Dispersed flow

Annular

Wavy

4

Bubbles

G x ρair,STP ρwater,STP ρv ρl

1/2

10

Slug 10³

Stratified Plug 10²

10-1

100

101

1 - x ρv x ρair,STP

1/2

ρl ρwater,STP

1/6

102

σwater,STP μ σ μwater,STP

1/3

103

104

Fig. 3.2 Flow pattern map for horizontal gas and oil flows, Baker (1954)

3.2.2

Taitel and Dukler (1976)

In 1976, Taitel and Dukler proposed an analytical model to predict flow pattern transitions for gas and liquid flows in horizontal and near horizontal round channels, which has been used as the basis for several foregoing predictive methods. This method accounts for the effects of fluids properties, channel dimension, and duct inclination on the flow pattern transitions. The original development presented by Taitel and Dukler (1976) is based on the liquid height Hl, as schematically shown in Fig. 3.3, associated to stratified flows. Additionally, the original authors pointed out that their method is valid for adiabatic flows, implying on neglecting aspects such as possible wall dryout due to vaporization, or film formation due to condensation. The focus of the method is on the oil and gas industries, under conditions that heat transfer processes do take place, but slightly affect the phase distribution. Nonetheless, the transition criteria proposed by Taitel and Dukler (1976) are extended to applications involving two-phase vapor-liquid flow with phase change by several authors, for example, Kind et al. (2010) and Wojtan et al. (2005). The basic assumptions of their model are the following: steady state flow, hence, constant time averaged properties and parameters, completely developed flow, and small duct inclinations. Taitel and Dukler (1976) proposed transition criteria considering the following flow patterns: smooth stratified, wavy stratified, intermittent (slug and plug), annular with dispersed liquid, and dispersed bubble. In their method, the flow parameters are non-dimensionalized adopting as the scaling parameters the pipe diameter d and the

70

3

d

uv τwv

Sv

g

τi



τi ul

τwl

Flow Patterns

Si θ

Hl

A(1 - α)

Sl

z Fig. 3.3 Schematics of stratified flow in inclined duct

superficial flow velocities, jl and jv. The prediction of stratified wave emphasizes the fact that this method might not be valid for microscale channels, in which the stratification effects are suppressed. Hence, subsequent to the analysis of momentum balance for both phases, the transition criteria are described according to physical aspects. Based on the schematics presented in Fig. 3.3 and by neglecting pressure gradients along the channel cross-section, the force balance in the axial direction for an element dz long for the liquid and vapor phases are given as follows, respectively: dp  τwl Sl þ τi Si  ρl Að1  αÞgsinðθÞ ¼ 0 dz

ð3:1Þ

dp  τwv Sv  τi Si  ρv AαgsinðθÞ ¼ 0 dz

ð3:2Þ

Að1  αÞ Aα

where the term τwl and τwv refer to the shear stress between the liquid and vapor phases and the duct wall, respectively, Sl and Sv correspond to the contact area between the phases and the wall, τi corresponds to the interfacial shear stress, and Si corresponds to the interfacial area between the phases. Eliminating the pressure gradient term of both equations, we obtain the following relationship: τwl Sl  τi Si τ S þ τ i Si þ ρl Agsinθ ¼ wv v þ ρv Agsinθ 1α α

ð3:3Þ

and rearranging the terms, we obtain: 

τwl Sl τwv Sv τ i Si þ þ  ðρl  ρv ÞAgsinθ ¼ 0 1α α α ð1  α Þ

ð3:4Þ

The shear stress among the phases and the duct wall can be estimated based on the Fanning friction factor f as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

71

τwl ¼ f l

ρl u2l 2

ð3:5Þ

τwv ¼ f v

ρv u2v 2

ð3:6Þ

τi ¼ f i

ρv ð uv  ui Þ 2 2

ð3:7Þ

The friction factor can be evaluated via Blasius correlation for turbulent flow, which in turn is a function of the Reynolds number given as a function of the in situ velocity. Taitel and Dukler (1976) considered that the friction factor for the interfacial shear stress is similar to the one of the vapor phase ( fi ¼ fv). Moreover, it is assumed that the vapor velocity is considerably higher than the interface velocity, u̅v  u̅i, and the interfacial tension can be evaluated as a function of the velocity only of vapor phase. Hence, it is necessary to define characteristic lengths for both phases and Taitel and Dukler adopted the hydraulic diameters for both phases defined as follows: d hl ¼ 4

Að1  αÞ Sl

ð3:8Þ

Aα Sv þ Si

ð3:9Þ

d hv ¼ 4

where it is considered that the wall imposes the main friction to the liquid phase; conversely, the wall and gas–liquid interface imposes restriction to the vapor phase flow. Hence, the Reynolds number and friction factor are given as follows: Re hl ¼

ρl ul d hl μl

ð3:10Þ

Re hv ¼

ρv uv dhv μv

ð3:11Þ

hl f l ¼ C hl Re n hl

ð3:12Þ

fv ¼

hv C hv Re n hv

ð3:13Þ

where the constant Ch and exponent nh are, respectively, 16 and 1 for laminar flow (Reh < 2300), and 0.046 and 0.2 for turbulent flow. Hence, Eq. (3.4) can be rewritten as follows:

72

3

Flow Patterns

  ρl u2l Sl ρv u2v Sv Chl C hv Si n n þ ðρl  ρv ÞAgsinθ ¼ 0  þ þ ρl ul d hl hl 2 1  α ρv uv d hv hv 2 α αð1  αÞ μl

μv

ð3:14Þ Developing the above equation, we find the following relationships: 

Chl Þ 4 ρl ul AμðS1α l

nhl

l

2

ρl u2l Sl þ 2 1α 4

C hv

nhv

ρv uv Aα μv ðSv þSi Þ

3

  ρv u2v Sv Si þ  ðρl  ρv ÞAgsinθ ¼ 0 2 α α ð1  α Þ

2

3

ð3:15Þ

ρ j2 7 S nhl ρ j2 7S þ Si nhv 1 Sl 6 C 6 C 4 hlnhl l l 5 l þ 4 hvnhv v v 5 v 3 ρv jv d ρl jl d 2 πd 2 πd α2 ð1  α Þ μl

  S Si  vþ  ðρl  ρv ÞAgsinθ α αð1  αÞ

μv

¼0

ð3:16Þ

We can recognize that the terms inside the square brackets of the first and second terms of Eq. (3.16) correspond, respectively, to the frictional pressure drop for only the liquid and vapor phases flowing in the channel. Therefore, dividing all the terms by the vapor single-phase pressure drop, we obtain the following non-dimensional relationship: b X

2



Sl πd

nhl

    Sl Sv þ Si nhv 1 Sv Si b ¼0 þ þ Y d α2 αd αdð1  αÞ d ð1  α Þ3

ð3:17Þ

where the term X^ is the Lockhart and Martinelli parameter, named after the authors for their study presented in 1949, which corresponds to the ratio between the pressure gradients for only liquid and gas flowing in the channel, given as follows: nhl

ρl j2l 2d

ρ Cj dhv nhv

ρv j2v 2d

 2

b ¼ X

C hl

ρl j l d μl

vv μv

 

¼

dp dz  l dp dz v

ð3:18Þ

and in terms of mass flux and vapor quality, the Lockhart and Martinelli parameter is given as follows: n

2nh,l h,l b 2 ¼ C hl ðGd Þnh,v nh,l μnl ρv ð1  xÞ X h,v 2n Chv x h,v μv ρl

ð3:19Þ

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

73

Fig. 3.4 Schematics of the geometric distribution of phases during stratified flow

d Sv Aα γ Hl

Si Sl

A(1 - α) Stratified flow

This parameter is presented again in Chap. 4, and is recurrently mentioned in two-phase flow studies, as well as along this book. And the term Ŷ in Eq. (3.17) takes into account the channel inclination and is given as follows: b ¼ ðρl  ρv ÞAgsinθ Y ¼ ρ j2 d ρv Cjv dhv nhv v2 v μv

ðρl ρv ÞAgsinθ  d dp dz v

ð3:20Þ

Despite the fact that transitions from stratified to annular or intermittent flow patterns, between intermittent and annular, and between annular and dispersed bubbles are expected, the method proposed by Taitel and Dukler considers only the force balance for stratified flow, hence using the liquid height Hl, which is schematically depicted in Fig. 3.4. Assuming the phases are distributed according to stratified configuration, all parameters can be written as a function of the liquid height Hl as follows1:  2H l 1 d   Sv 2H l ¼ cos 1 1 d d   Sl 2H l ¼ π  cos 1 1 d d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 Si 2H l ¼ 1 1 d d γ ¼ cos 1

1



ð3:21Þ ð3:22Þ ð3:23Þ ð3:24Þ

The cross-sectional area occupied by the liquid phase Al can be easily evaluated based on the circle section (0!π/2-γ and π/2 + γ!0) plus the isosceles triangle area (π/2-γ!π/2 + γ). Similarly, the interfacial perimeter Si can be evaluated based on the Pythagoras theorem for the same isosceles triangle.

74

3

1α¼

Flow Patterns

Al πd2 =4

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2H l 1 1 d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      2 A cos 1 2H l 1 2H l 2H l 1 α ¼ 2v ¼ 1  1 1 π d π d d πd =4     cos 1 2H l 1 2H l ¼1 1 þ 1 π d π d

ð3:25Þ ð3:26Þ

Therefore, all the terms of Eq. (3.17) are a function of the liquid height to diameter ratio Hl / d, or 2Hl/d-1, Lockhart and Martinelli parameter X^, and other known operational parameters, such as flow velocities and fluids properties. Hence, the variation of non-dimensional liquid height with Lockhart and Martinelli parameter can be represented graphically, as shown in Fig. 3.5. The reader is encouraged to implement computationally Eqs. (3.21) to (3.26) and Eq. (3.17), varying Hl / d from unity to zero, to construct a graph similar to the one shown below. In the case that the reader would like to study the original paper of Taitel and Dukler, please be advised that the channel inclination angle is defined differently than in this study, which implies in negated values of Ŷ, hence, Fig. 3.5 is not identical to the one available in the original paper. We might be interested in representing Eq. (3.17) as a function of the area averaged void fraction, rather than the liquid height. Hence, the geometrical terms, more specifically Sl, Sv, and Si, can be written as a function of the area averaged void

1.0

0.8

100

0.6

Y = 10.0

0.0 10 -3

10 -2

10 -1

0

Turbulent - Turbulent

-10

0.2

-10

0.4 0.0

(Hl / d ) [-]

1000

10 0

Laminar - Turbulent

10 1

10 2

10 3

10 4

X [-] Fig. 3.5 Variation of non-dimensional liquid height with Lockhart and Martinelli parameter for stratified flow

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

75

1.0 Turbulent - Turbulent Laminar - Turbulent

-10

0.0

-100

0.8

Stratified flow

α [-] a

0.6 Y = 10

0.4

0.2

100 1000

0.0 10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

X [-] Fig. 3.6 Variation of void fraction with Lockhart and Martinelli parameter for stratified flow pattern

fraction α, and Fig. 3.4 depicts the schematics for stratified flow. The stratification angle γ can be related to the void fraction α in a transcendental formulation as follows: α¼

γ sin 2γ  π 2π

ð3:27Þ

And the perimeters Sl, Sv, and Si can be written, respectively, as a function of γ as follows: Sl ¼πγ d Sv ¼γ d Si ¼ sin γ d

ð3:28Þ ð3:29Þ ð3:30Þ

Hence, by knowing the Lockhart and Martinelli parameter X^, it is possible to infer the void fraction α based on Eq. (3.17), and analogous to Fig. 3.5, Fig. 3.6 depicts the variation of void fraction with Lockhart and Matinelli parameter for stratified flow pattern, which again is based on Eq. (3.17) obtained from the momentum balance of liquid and vapor phases.

76

3

Flow Patterns

Hence, by knowing the operational conditions, it is possible to estimate the Lockhart and Martinelli parameter, and based on Eq. (3.17), it is possible to estimate the liquid height or void fraction, which could also be performed through Figs. 3.5 and 3.6, respectively. Based on these parameters, it is possible to determine the in situ velocities of liquid and vapor phases, which are used for determination of flow pattern transitions described in the following paragraphs. The stratified flow pattern is characterized by the dominance of gravitational forces over inertial and/or surface tension forces, hence for very low flow velocities and for channels with conventional dimensions. In this context, Taitel and Dukler (1976) considered smooth and wavy stratified flows as different flow patterns. The first one corresponds to the flow of liquid in the bottom region and vapor in the upper region of the cross-section separated by a smooth interface. In this condition, the gas/vapor velocity is not high enough to disturb the interface and to generate waves. Taitel and Dukler adopted the method addressed by Stewart (1967), which in turn is based on studies by Jeffreys (1925, 1926). According to this approach, interfacial waves for gas and liquid flow are formed when: ð uv  uw Þ 2 u w >

4μl gðρl  ρv Þ sρl ρv

ð3:31Þ

where uw is the wave velocity, which is considerably lower than the vapor velocity (u̅v  uw), and s is a sheltering coefficient, equal to 0.3. The wave velocity is approximated by the liquid film velocity, u̅l; therefore, based on these simplifications the transition between smooth and stratified flow patterns is given as follows:  1=2 4μl gðρl  ρv Þ cos θ uv > ul sρl ρv

ð3:32Þ

where the phase in situ velocities are determined based on the void fraction determined through Eqs. (3.17), (2.18), and (2.19). The wavy stratified is similar to the smooth stratified flow pattern, however, the interface is rough with interfacial waves. Equation (3.32) can be made non-dimensional for a modified Froude number and liquid Reynolds number as follows: 1=2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2αð1  αÞ1=2 ρv ρl j l d jv > d ðρl  ρv Þgcosθ μl s1=2 Fr v Re l

1=2

>

2αð1  αÞ1=2 s1=2

ð3:33Þ ð3:34Þ

where α is determined based on Eq. (3.17) and Eqs. (3.27) to (3.30) as a function of X^ and Ŷ. The modified Froude number and liquid Reynolds number are given, respectively, as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows Fig. 3.7 Schematics of a solitary wave flowing in a closed channel

77

p' uv'

uv p ul

Fr v ¼ jv

Hv Hl

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρv d ðρl  ρv Þgcosθ

Re l ¼

ρl jl d μl

Hv' Hl'

ð3:35Þ ð3:36Þ

With the increment of vapor velocity, the formed waves tend to grow and can occur transitional from stratified wavy to annular or intermittent flow patterns, which is given based on the balance between liquid wave weight and pressure reduction in the wave tip, caused by vapor flow acceleration imposed by the restriction generated by the wave, as schematically depicted in Fig. 3.7. Assuming that interfacial waves already exist, whose occurrence is given by the condition of Eq. (3.32), the transition from wavy stratified to annular or intermittent flow patterns occurs for the condition that the wave will grow and block the channel section. Therefore, a force balance is performed in the channel cross-section comprising the wave tip, assuming that the wave is subjected to gravitational forces, depicted by the darker region in Fig. 3.7, and the pressure reduction in the vapor region, which is determined based on Bernoulli’s equation. Hence, the pressure in the wave tip, denoted by p’ is estimated based on the gas velocity variation from a flat region to a region with wave, neglecting variation of potential energy, resulting in the following relationship: p  p0 ¼ ρ v

u0 2v  u2v 2

ð3:37Þ

Assuming that the pressure along the liquid film is uniform and equal to p, and that the wave and liquid film velocities are similar, it is possible to perform a force balance for the dark region of Fig. 3.7 in the “vertical” direction, and the criterion that dictates when the wave will grow is related to the condition that the force due to pressure reduction is higher than the buoyance forces as follows:  ðp  p0 ÞdA  ðρl  ρv Þ H 0l  H l gcosθdA

ð3:38Þ

Hence, combining Eq. (3.37) in Eq. (3.38) we obtain:  ρv

u0 2v  u2v 2

  dA  ðρl  ρv Þ H 0l  H l gcosθdA

ð3:39Þ

78

3

Flow Patterns

or:   ρ  ρv  0 2 u0 v  u2v  2 l H l  H l gcosθ ρv

ð3:40Þ

where the in situ velocity u̅v and liquid height Hl are directly related to the void fraction α, hence, determined based on Eq. (3.17); in the same token, the restriction vapor velocity u̅v’ and liquid height Hl’ can be given as a function of a local void fraction α’. Moreover, according to the schematics depicted in Fig. 3.7, it is possible to evaluate the velocity u̅v’ based on the continuity equation assuming incompressible flow, and a local void fraction α’; therefore, Eq. (3.40) can be rewritten as follows: ρ  ρv u2v  2 l ρv

H 0l  H l ðα=α0 Þ2  1

! gcosθ

ð3:41Þ

For small disturbance, the term inside the brackets can be approximated by a Taylor series, namely, the term H’l, resulting in the following relationships: u2v

 02  ρl  ρv α ∂H l   gcosθ ρv α ∂α

ð3:42Þ

where, again for small disturbance α’  α, and the liquid height can be given as a function of stratification angle γ as follows: Hl ¼ 1 þ cos γ d

ð3:43Þ

which is ultimately a function of void fraction given by Eq. (3.20). Therefore: u2v 

  ρl  ρv sin γ dgcosθ α 1  cos 2γ ρv

j2v ρl  ρv α π dgcosθ  sin γ 2 ρv α2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi ρv πα3 jv  2 sin γ ðρl  ρv Þdgcosθ

ð3:44Þ ð3:45Þ ð3:46Þ

where the term on the left side is recognized as a modified vapor Froude number Frv*, and Eq. (3.46) can be written as follows: Fr v 

rffiffiffiffiffiffiffiffiffiffiffiffi πα3 2 sin γ

ð3:47Þ

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

79

a) Hl/d > 0.5

Hl/d > 0.5

b) Hl/d < 0.5

Hl/d < 0.5 Fig. 3.8 Schematics of sinusoidal wave growth

The term on the right side of (3.47) is basically related to the void fraction α, which is determined based on Eqs. (3.17) and (3.27) as a function of X^ and Ŷ. Therefore, for values of modified Froude number smaller than that defined by Eq. (3.47), the wave growth would imply in transition to annular or intermittent flow patterns, depending on the amount of liquid. In conditions of high liquid level, the amount of liquid is enough to form a stable liquid slug, blocking the channel section, and then the vapor flows as discrete bubbles. On the other hand, for reduced liquid heights, the grown wave tends to block the vapor path, but this slug is unstable and the liquid is pushed to the channel periphery, which corresponds to an annularlike flow. Therefore, the transition between intermittent and annular-like flows depends basically on the liquid height, or void fraction, and Taitel and Dukler proposed and justified a value of 50% for Hl / d and α as transition criteria. To reinforce the assumption of Hl / d ¼ 0.5 (α ¼ 0.5) for the transition between intermittent and annular-like flow patterns, the original authors claim the reader to imagine a sinusoidal wave, such as depicted schematically in Fig. 3.8. The growth of the wave requires liquid from the vicinity, which results in valleys in the interface; therefore, if the interface mean line is below the channel center (Hl / d < 0.5), in conditions of growing wave amplitude, the interface would reach the bottom channel wall before the wave tip reaches the top channel wall, with consequent unstable slug. On the other hand, for liquid height above the channel center for smooth interface, when increasing the wave amplitude, the wave tip would reach the top channel wall before the interface valley reaches the channel bottom. Therefore, by defining Hl / d ¼ 0.5 (or α ¼ 0.5), it is possible to determine the corresponding X^ from Eq. (3.17), or from Figs. 3.5 or 3.6.

80

3

Flow Patterns

Nonetheless, it is intuitive that for Hl / d ¼ 0.5, the stratification angle γ is equal to π/2, Si / d ¼ 1.0, and Sl / d ¼ Sv / d ¼ π/2. Therefore, by substituting these values in Eq. (3.17), the Lockhart and Martinelli parameter is given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2nkl 1 1 nkv b b X¼ Y þ 5 þ 2 π π

ð3:48Þ

which, for turbulent flow regimes for both phases (nkl ¼ nkv ¼ 0.2) and horizontal flow (Ŷ ¼ 0.0), results in a constant value of 1.325. Therefore, according to the definition of the parameter X^, given by Eq. (3.19) for turbulent flow regimes, we find:    0:1 ρ 0:5 1  x 0:9 μl v b X tt ¼ x μv ρl

ð3:49Þ

equal to 1.325, where the subscripts tt refers to turbulent flow regimes for liquid and vapor phases. Therefore, for a fluid at a given saturation temperature the transition between intermittent and annular-like flow patterns occurs for constant vapor quality values. Subsequently, Barnea et al. (1982b) proposed that his transition occurs for Hl/d of 0.35, and the difference is attributed to the vapor fraction within the liquid slugs during the transition, which would require less liquid to close the cross-section. Nonetheless, in this text we will keep the original proposal. It will be seen later, in Sect. 3.3.1, that assuming constant α of 0.5 for the transition between intermittent and annular flow patterns is not confirmed experimentally, and alternative proposals for this transition are available in the open literature. In conditions of intermittent flow pattern with high liquid content, the vapor bubbles can be broken due to high flow turbulence. Hence, the intermittent flow pattern transits to dispersed bubbles in conditions that the liquid flow turbulence overcomes the buoyance forces of the vapor phase. In this context, Taitel and Dukler addresses relationships for the buoyancy force as follows: F b ¼ gcosθðρl  ρv ÞαA

ð3:50Þ

The turbulence forces are given according to the method proposed by Levich (1962) as follows: Ft ¼

 ρl u0l 2 Si 2

ð3:51Þ

where ul’ is the radial velocity fluctuation of the liquid phase due to turbulence, whose root mean square is evaluated as a function of the friction factor and liquid velocity as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows



u0l 2

1=2

 ¼ ul

fl 2

81

1=2 ð3:52Þ

Therefore, by defining the condition of Ft > Fb, the following relationship is obtained: ρl u2l f l Si > gcosθðρl  ρv ÞαA 4

ð3:53Þ

The equation above can be rewritten as a function of non-dimensional parameters as follows: fl

ρl j2l Si πd > gcosθðρl  ρv Þα 4 2d 4ð1  αÞ2

ð3:54Þ

ρl j2l gcosθðρl  ρv Þαð1  αÞ2 π > 2d Si =d

ð3:55Þ

ρl j2l 4πgcosθðρl  ρv Þαð1  αÞ2 > 2d sin γ

ð3:56Þ

fl 4fl 

dp dz



¼ Cl l

ρl j2l 4πgcosθðρl  ρv Þαð1  αÞ2 > 2d sin γ

ðdp=dzÞl 4παð1  αÞ2 > sin γ gcosθðρl  ρv Þ T>

4παð1  αÞ2 sin γ

ð3:57Þ ð3:58Þ ð3:59Þ

where the term T on the left side can be considered as the ratio between turbulence and buoyance forces acting on the vapor phase and is evaluated according to the flow conditions and fluid properties, and the term on the right side is a function of α. Therefore, the non-dimensional transitions given by Eqs. (3.34), (3.47), (3.48), and (3.59) can be presented graphically, as shown in Fig. 3.9. It must be emphasized that all the listed transitions can be given as a function of the non-dimensional liquid height or void fraction, which are a function of the Lockhart and Martinelli parameter, as given by Eq. (3.17) and shown by Figs. 3.5 or 3.6. Hence, by knowing the operational conditions, namely, fluid properties, mass velocity, vapor quality, and channel diameter, it is possible to infer the Lockhart and Martinelli parameter, given by Eq. (3.19), and from Eq. (3.17) or from Figs. 3.5 and 3.6, it is possible to infer the non-dimensional liquid height or void fraction. From these parameters, it is possible to estimate all the transitions, depicted in Fig. 3.9, which is convenient due to the validity for several fluids and channel sizes.

82

3 3

1 x10

Frv* . Rel0.5 [-]

Annular

10

2

10

1

Dispersed bubbles

1

Wavy stratified

1 x10

-1

1 x10

-2

1 x10 10 4

-3

Intermittent

10

0

Frv* or T [-]

10

Flow Patterns

Smooth stratified

10

-1

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

X [-] Fig. 3.9 Non-dimensional flow pattern transitions for horizontal flow (θ ¼ 0 ) according to Taitel and Dukler (1976)

Alternatively, the flow pattern transitions can be given as a function of mass velocity and vapor quality for a given fluid, saturation temperature, and channel diameter as shown in Fig. 3.10. This form of presentation is more appropriate for heat transfer problems because usually the fluid flows along a channel of uniform cross-section, and due to the heat transfer process, there is variation of its enthalpy, and consequently of its vapor quality. In this presentation form, it is possible to infer the flow pattern evolution along the channel as the fluid enthalpy increases or decreases. The reader can observe in Fig. 3.10 that smooth stratified flow occurs only for very low flow velocities, which is coherent with the intuition and experimental observations. With the increment of flow velocity, the flow pattern can suffer a transition to wavy flow, and additional increments of flow velocity can result in a transition to annular, or intermittent/dispersed bubble flow patterns. Conversely, assuming a tube of an evaporator and saturated liquid at its inlet for G fixed at 600 kg/m2s, at very low x values, the local flow pattern corresponds to wavy stratified, and with sensible increment of vapor quality a transition to intermittent flow patterns occurs. With additional increments of x, the flow pattern will transit to annular flow, and this transition occurs at a fixed vapor quality of approximately 0.17 for this fluid and saturation temperature, with occurrence of annular flow pattern until total liquid dryout at a vapor quality equal to 1. On the other hand, for a mass velocity of 10 kg/m2s, the same fluid, channel, and saturation temperature, at the beginning of the channel, corresponding to a vapor quality close to 0, the flow is smooth stratified due to buoyancy forces and negligible interfacial tension and

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

10

83

5

Dispersed bubbles

10

4

10

3

10

2

R134a, d = 10 mm, Tsat = 50 °C

G [kg/m²s]

Intermittent Annular

Wavy stratified

10

1

Smooth Stratified 0

10 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.10 Flow pattern transitions for horizontal flow of R134a at 50  C in 10 mm ID channel

turbulence. This condition is kept for successive increments of vapor quality until achieving a value of approximately 0.5, which corresponds to a transition to wavystratified flow maintaining this flow pattern until the total liquid dryout. The above mind exercise is not entirely correct, because practical applications of evaporation at 50  C in 10 mm ID tube is uncommon in real applications, and during the heat transfer process the local flow pattern can be affected by the heat transfer rate and direction (evaporation or condensation), as will be seen in Sects. 3.3 and 3.4. Additionally, even though this method is valid for adiabatic flow, subsequent developments in this area, such as the study of Wojtan et al. (2005) presented in Sect. 3.3.1, used it as a basis for the derivation of transition criteria for two-phase flow during evaporation. Additionally, with the reduction of the channel diameter, the surface tension effects present more significant contribution to the phases distribution, and in this context, Barnea et al. (1983) proposed in their pioneer study a flow pattern predictive method for microscale channels. The experimental database used by the authors to develop their predictive method comprises results for air-water flow, close to atmospheric pressure, for channels with internal diameter ranging between 4.0 and 12.3 mm. Based on the evaluated experimental results, the main difference was verified by the authors for the transition between stratified and non-stratified flows, whereas for conventional channels it is given by the conditions of wave growth due to vapor acceleration. In the case of microscale channel, the surface tension starts to play an important contribution pushing the liquid upward and promoting an earlier transition from stratified flow. Figure 3.11 schematically depicts the liquid slug

84

3

Fig. 3.11 Schematics of liquid slug formation due to surface tension forces

R

Hv

Flow Patterns

σ g

V2

V1

formation according to the model proposed by Barnea et al. (1983), who considered a two-dimensional wave, whereas the wave formation requires liquid from the vicinity and results in a valley. Moreover, by assuming a wave with circular format and radius R, centered in the upper wall, it is possible to infer the radius R by equating the volumes V 1 and V 2 , which results in the following relationship: 4 R ¼ Hv π

ð3:60Þ

Therefore, by equating the weight of the wave (darker region in Fig. 3.11) with the surface tension force σ, we find: π Hv ¼ 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ gð ρl  ρv Þ ð 4  π Þ

ð3:61Þ

And, based on this relationship it is possible to determine the non-dimensional liquid height for the transition: Hl H π ¼1 v ¼1 2d d d

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ gð ρl  ρv Þ ð 4  π Þ

ð3:62Þ

which, substituting in Eq. (3.17), provides a constant value for the Lockhart and Martinelli parameter X^, and consequently a constant vapor quality for a given fluid and saturation pressure. Additional to the transition from stratified to non-stratified flows, Barnea et al. (1983) also adopted the modified value for the transition between annular and intermittent-like flows, as suggested by Barnea et al. (1982b), according to which Hl / d ¼ 0.35 and also results in a constant X^ and vapor quality. Figure 3.12 depicts the flow pattern transitions for R134a in a 4 mm ID channel, and it can be noticed according to this figure that the transitions are considerably different from those for conventional channels, depicted in Fig. 3.10, and the main differences are due to the surface tension effects, neglected by Taitel and Dukler (1976). Based on the analysis of the flow pattern transitions for small channels depicted in Fig. 3.12b, it can be speculated that the absence of transition from wavy stratified to dispersed bubbles for high mass velocities is incoherent, since the stratified flow patterns for very high mass velocities is unexpected. The reader should be aware that the dominant mechanism for flow pattern transitions during horizontal flow are distinct from those for vertical flow. Therefore, below we present the classical predictive method also proposed by the research group of Taitel and Dukler, concerning vertical flows.

3.2 Flow Pattern Transition Criteria for Adiabatic Flows 1

1 x10

5

10

4

10

3

10

2

10

1

Dispersed bubbles

Dispersed bubbles

R134a, d = 4 mm, Tsat = 35 °C

1

0

T [-]

10

b) 10

Intermittent

1 x10

-1

Smooth stratified

G [kg/m²s]

Frv* . Rel0.5 [-]

Annular

1

R134a, d = 4 mm, Tsat = 35 °C

Wavy stratified

a) 10

85

Wavy stratified Intermittent

Annular

Smooth Stratified

10

-1

10

-3

10

-2

10

-1

10

0

10

X [-]

1

10

2

10

3

1 x10 10 4

-2

0

10 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.12 Flow patterns transition for microscale channels according to Barnea et al. (1983)

3.2.3

Taitel, Barnea, and Dukler (1980)

Subsequent to the proposal of the method for flow pattern transitions in horizontal and near horizontal flows, Dukler and coworkers proposed criteria for flow pattern transitions during vertical upward adiabatic flow that considers transitions among slug, churn, bubble, finely dispersed bubbles, and annular flow patterns. Analogous to horizontal flow, the authors defined the dominant mechanisms responsible for the transitions, and then described the governing equations associated with these mechanisms. On the other hand, different from the case of horizontal flow, where the liquid height was considered as the main characteristic length of the flow, in the case of vertical flow these authors took the void fraction α as the main geometrical parameter. Additionally, in this method the transition criteria do not result in non-dimensional relationships. According to Taitel et al. (1980), the bubble and dispersed bubble flow patterns are expected to occur in conditions of reduced void fraction, and the limiting conditions for their occurrence are related to the turbulence intensity and void fraction. The turbulence intensity should be high enough for the occurrence of bubble flow, in order of breaking large bubbles into small ones, or to inhibit their coalescence into larger bubbles. Conversely, the void fraction should be small enough to inhibit the collisions among the bubbles, favoring their coalescence. Therefore, assuming a cubic lattice distribution for the bubbles, such as depicted in Fig. 3.13a, the corresponding maximum void fraction is 0.52. Nonetheless, any random movement of the bubbles for this packaging results in collision among them, favoring their coalescence and the transition to slug flow. Therefore, at least some space between the bubbles is required to avoid collision and coalescence, and in this context the authors proposed to adopt a separation of half of the bubble radius, such as depicted in Fig. 3.13b, for which the void fraction is approximately 0.25, as the transitional criterion. Taitel, Barnea, and Dukler justify this consideration based on experimental results, which show that slug flow is not observed for void fraction lower than 0.20 and that bubble flow rarely occurs for void fraction higher than 0.30. Additionally, this hypothesis is coherent with previous studies available in the literature at that time. Therefore, assuming a constant void fraction α of 0.25 it is

86

3

Flow Patterns

a)

d=a

α = 0.52

a a

b)

d=(4/5)a

d/4

α = 0.268 ≈ 0.25

a a Fig. 3.13 Schematics of bubbles distribution according to cubic lattice

possible to determine a relationship between the liquid and vapor superficial velocities. The liquid in situ velocity is given according to Eq. (2.21) as follows: ul ¼

jl 1α

ð3:63Þ

In the case of vapor flow, the in situ velocity is given as the sum of u̅l and the slip velocity between the bubble and liquid phase U0, which is mainly related to buoyancy forces as follows: uv ¼

jl jv þ U0 ¼ ul þ U 0 ¼ 1α α

ð3:64Þ

The slip velocity is given according to the correlation proposed by Harmathy (1960) for relatively large bubble rising velocity in stagnant liquid media, which is insensitive to the bubble characteristic diameter, and is given as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

 U 0 ¼ 1:53

gðρl  ρv Þσ ρ2l

87

1=4 ð3:65Þ

Based on the above equations, it is possible to determine a relationship between the liquid and vapor superficial velocities as follows: jl ¼

 1=4 gðρl  ρv Þσ 1α jv  1:53 ð1  αÞ α ρ2l

ð3:66Þ

From which for α ¼ 0.25, we obtain the following relationship:  1=4 gðρl  ρv Þσ jl ¼ 3 jv  1:15 ρ2l

ð3:67Þ

The above equation provides the transition criterion between bubble and slug flow for conditions under which dispersion forces are not dominant. The proposers have reinforced the validity of this criterion by comparing its predictions with flow pattern maps available in the open literature. The bubble flow pattern delimited by this restriction corresponds to the vapor phase flowing as bubbles of intermediate diameter in a zig-zag manner due to the bubble non-spherical form with higher velocity than the liquid phase due to buoyance forces. In conditions of high liquid velocity, the flow turbulence can break large bubbles into small ones even for conditions of intermediate void fractions, including α higher than 0.25, and the bubbles tend to be more spherical due to their reduced size. Based on the studies presented by Hinze (1955), who theoretically evaluated the characteristic dimension of a dispersion caused by turbulence, and by Brodkey (1967), who presented a criterion to define the critical diameter of a bubble to be spherically stable, Taitel, Barnea, and Dukler proposed the transition criterion between bubble and dispersed bubble flow patterns. According to their criterion, the maximum diameter of a bubble generated due to flow turbulence is given as the balance between surface tension and turbulence forces as follows: dmax

 3=5 σ ¼k E 2=5 ρl

ð3:68Þ

where the term k is an empirical constant equal to 1.14 for gas dispersed in liquid, and E is the rate of energy dissipation per unit mass, which is evaluated based on the pressure drop, and is given as follows:   dp j Ε ¼ dz m ρH

ð3:69Þ

88

3

  ρ j2 dp ¼ 2f H dz m d  0:2 ρ jd f ¼ 0:046 l μl

Flow Patterns

ð3:70Þ ð3:71Þ

Therefore, rewriting Eqs. (3.68) to (3.71), we obtain the following relationship:

dmax

 3=5  0:2 3 !2=5 ρl jd σ j ¼k 0:092 ρl d μl

ð3:72Þ

In contrast, the critical diameter of the bubbles to maintain their spherical shape is given as follows:  dcrit ¼

0:4σ ðρl  ρv Þg

1=2 ð3:73Þ

For dmax > dcrit the bubble rising velocity in stagnant liquid is almost independent of the bubble size and is given by Eq. (3.65), which corresponds to the bubble flow pattern. Conversely, for dmax < dcrit the rising velocity sharply decreases with bubble diameter. Under conditions that the turbulence intensity is high enough, the coalescence is suppressed and there is no transition to slug flow even for void fraction values higher than 0.25. Therefore, by equating dmax and dcrit in Eqs. (3.72) and (3.73), it is possible to infer the transition between bubble and dispersed bubble flow patterns, which is given as follows:  3=5  0:2 3 !2=5  1=2 ρl jd σ j 0:4σ k 0:092 ¼ ρl d μl ðρl  ρv Þg 1=1:12 σ 0:1 d0:48 1=2 j ¼ 4:0 0:52 0:08 ððρl  ρv ÞgÞ ρl μ l  0:1 0:48 1=1:12 σ d 1=2 jl ¼  jv þ 4:0 0:52 0:08 ððρl  ρv ÞgÞ ρl μ l

ð3:74Þ



ð3:75Þ ð3:76Þ

We should recall the geometric analysis previously performed, such as depicted in Fig. 3.13, according to which it is impossible for a bubble flow pattern to exist for void fraction higher than 0.52. Therefore, the transition given by Eq. (3.76) is limited by this condition. Additionally, for high flow velocity and turbulence intensity and reduced bubble diameter, the slip between the phases is almost negligible, hence u̅l  u̅v. Then, by considering α ¼ 0.52 it is possible to determine the complementary transition condition as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

89

jl ¼ 0:92 jv

ð3:77Þ

Therefore, the conditions of liquid superficial velocity higher than those given by Eqs. (3.76) and (3.77) the vapor-liquid flow corresponds to dispersed flow of the vapor phase as small bubbles, moving within the liquid phase with negligible slip. In the case of two-phase vertical flow in small dimension channels, the possibility of existing a sporadic vapor slug existing must be evaluated. The rising velocity of a Taylor bubble inside a round channel is proportional to the square root of the channel diameter, and can be evaluated according to the Nicklin (1962) method as follows: U T ¼ 0:35

pffiffiffiffiffi dg

ð3:78Þ

where d is the channel internal diameter and g is the gravitational acceleration. Therefore, assuming simultaneous flow of bubbles with intermediate dimensions and Taylor bubbles, whenever UT > U0, the smaller bubbles would be swept around the Taylor bubble nose to the channel periphery, such as schematically depicted in Fig. 3.14a, and then directed to the wake region without coalescence. Therefore, there is no transition to slug flow. On the other hand, assuming that the smaller bubbles present higher velocity than the Taylor bubble, UT < U0, verified for reduced diameter channel and schematically depicted in Fig. 3.14b, the migration of the bubbles toward the Taylor bubble tail favors coalescence, and consequent transition to slug flow. Therefore, depending on the channel diameter and two-phase mixture velocity, the bubble flow pattern will not occur. For instance, for air-water flow at 20  C and atmospheric pressure, the limiting channel diameter is 52 mm; while for R134a at 20 and 50  C, the limiting channel diameters are, respectively, 19.8 and 12.5 mm, whereas no bubble flow occurs in channels with smaller dimensions than these.

Fig. 3.14 Schematics of simultaneous flows of Taylor bubbles and bubbles with intermediate dimensions relative to liquid flow

a)

b)

U0

UT

UT

g

U0

UT > U0

UT < U 0

90

3

Flow Patterns

Before analyzing the transition between slug and churn flow patterns, Taitel et al. (1980) expatiated about the definition of the churn flow itself. In fact, since several classifications are based on visual observations and subjective method, therefore sensible to personal judgement of the researcher, distinct experimental results can be obtained for a certain set of results depending on who classifies the local pattern. Therefore, according to Taitel et al. (1980) the churn flow pattern is characterized by the oscillatory flow of liquid slugs and vapor Taylor bubbles, with successive destruction and construction of these entities along the channel. Complimentary to this classification, Taitel et al. (1980) stated that churn flow corresponds to a transitionary flow pattern, occurring only in the inlet region of the section. In the entrance region, short lengthened liquid slugs and Taylor bubbles are formed and flow in the main flow direction, and since these liquid slugs are unstable due to their thickness, they break up, fall, and combine with the liquid slug in the lower region increasing the size of the liquid slugs. Then the liquid slug is propelled again by the stream and a net flow is verified. Simultaneously, the successive Taylor bubbles combine creating longer bubbles. This process occurs along the channel length until a region where the liquid slugs are stable enough to sustain the slug flow pattern. Therefore, the transition criterion is actually based on the channel length required to develop the slug flow, whereas churn flow occurs upstream to this point, and the derivation is performed based on the required conditions to obtain stable liquid slugs and Taylor bubbles. The Taylor bubble velocity in flowing liquid is given according to the proposal of Nicklin (1962) as follows: uv,T ¼ 1:2ul þ 0:35

pffiffiffiffiffi gd

ð3:79Þ

where the second term on the right side corresponds to the rising velocity of a Taylor bubble inside a circular duct with stagnant liquid, and the first term corresponds approximately to the liquid velocity at the center of the tube, which is  20% higher than the average velocity for turbulent flow. In this development, it is also assumed that the liquid slugs separating successive Taylor bubbles contain dispersed bubbles with intermediate dimensions, such as schematically depicted in Fig. 3.15a. In these regions, it is assumed that the void fraction is equal to 0.25, in a similar way to the characterization of the bubble flow pattern (refer to Fig. 3.13). Moreover, it is considered that the smaller bubbles move with the same velocity of the Taylor bubbles; therefore, the vapor in situ velocity u̅v is given according to Eq. (3.79). The mixture superficial velocity is a known parameter, and can be related to the velocities of each phase as follows: j ¼ jv þ jl ¼ uv α þ ul ð1  αÞ

ð3:80Þ

By solving Eqs. (3.79) and (3.80) for the in situ velocities, we obtain the following expressions:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows Fig. 3.15 Schematics of the flow of stable slugs

91

a)

b)

ul r

c)

ul u̅v

r z*

ls

d) u l

1.2 u̅l

r α = 0.25 lT g r

pffiffiffiffiffi j 1:2 1α þ 0:35 gd α 1:2 1α þ1 pffiffiffiffi 0:35α gd j  ul ¼ 1α α 1α 1:2 1α þ 1

uv ¼

ð3:81Þ

ð3:82Þ

During the passage of a large bubble, the liquid film in the channel periphery flows downstream in order to obey the mass conservation, such as depicted schematically in Fig. 3.15b. Therefore, a jet is formed in the liquid slug just upstream the bubble tail, Fig. 3.15c, which can make the liquid slug unstable due to reverse flow. Hence, the occurrence of slug flow requires stable and consequently quite thick

92

3

Flow Patterns

liquid slugs, to allow the reestablishment of the liquid velocity profile, such as schematically depicted in Fig. 3.15d. In the case of short liquid slugs, the slug length would not be enough to allow reestablishment of the flow, therefore the liquid bridge would collapse and fall to the lower liquid portion, implying on the transition to churn flow. Based on experimental results obtained in their laboratory for air-water flows, Taitel et al. (1980) state that the liquid slug length observed for stable slug flow is approximately 16 times the channel diameter, ls / d ¼ 16 (Fig. 3.15a), independently of the channel diameter. Hence, just downstream the inlet section of the vapor-liquid mixture, liquid slugs, as gas bubbles, of short length are formed and collapse forming larger slugs and bubbles. This process continues and evolves downstream until a point that stable liquid slug is formed; recalling that the stable liquid slug length is approximately 16 times the diameter, two slugs of 8 diameters in length are required to form a stable slug. Therefore, referring to the z* coordinate depicted in Fig. 3.15a, the liquid velocity in the channel center (r ¼ 0) must vary from u̅v in z* ¼ 0 to 1.2 u̅l in z* ¼ ls, and this velocity variation can be modelled as an exponential function as follows:   ul,r¼0 ¼ uv eϑz=ls þ 1:2ul 1  eϑz=ls

ð3:83Þ

where the term ϑ corresponds to the decay constant, and it is assumed as equal to ln (100) ¼ 4.6. Hence, by considering the in situ velocities of both phases given by Eqs. (3.81) and (3.82), with the void fraction of the slug region equal to 0.25 (recall the modelling approach for the bubble flow pattern), the following relationship is obtained: ul,r¼0 ¼

pffiffiffiffiffi  pffiffiffiffiffi  1:6j þ 0:35 gd ϑz=ls 1:6j  0:35 gd=3 e þ 1  eϑz=ls 1:4 1:4 pffiffiffiffiffi   0:35 gd 4 ϑz=ls 1 1:6 jþ e  ul,r¼0 ¼ 1:4 3 3 1:4

ð3:84Þ ð3:85Þ

Therefore, by assuming the entrance region, the approaching velocity between two consecutive bubbles is given as follows: ul,r¼0 jz¼0 z¼ls ¼

pffiffiffiffiffi 4 0:35 gd  1  eϑ 3 1:4

ð3:86Þ

By integrating Eq. (3.86), it is possible to infer the time interval required for collision between successive bubbles in the developing region as follows: 48d t i ¼ pffiffiffiffiffi gd ð1  eϑ Þ

ð3:87Þ

The product of the period required for the collision of two consecutive bubbles and the vapor phase in situ velocity, estimated through Eq. (3.81), gives the length

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

93

along which slug collapses occur, which corresponds to the entrance length le, where the churn flow pattern is present2: pffiffiffiffiffi 1:6j þ 0:35 gd 48d t i uv ¼ le ¼ pffiffiffiffiffi 1:4 gd ð1  eϑ Þ   le j ¼ 55:41 pffiffiffiffiffi þ 0:22 d gd

ð3:88Þ ð3:89Þ

Hence, in a set of flow conditions that would correspond to slug flow, there can be either slug or churn flow pattern depending on the position along the channel; whereas, in the entrance region, z < le, churn flow is present and for developed region, z > le, slug flow is present. Subsequently in this section, the flow pattern transitions will be plotted in charts according to the flow conditions; therefore, it is interesting to present Eq. (3.89) in the form of superficial velocities in a similar way to the previously described transitions as follows: 

 pffiffiffiffiffi le gd jl ¼  jv þ  0:22 55:41d

ð3:90Þ

Hence, upstream a given position defined by le / d the flow corresponds to churn, otherwise it corresponds to slug flow, as long as the conditions do not correspond to bubble or dispersed bubble flow. During annular flow in vertical channels, the vapor flows in the central region of the channel, and the liquid flows predominantly as a film adjoined to the channel wall. The higher density of the liquid phase, summed to the friction imposed by the wall, makes the liquid film velocity considerably inferior to the vapor flow velocity, with consequent high interfacial shear stress between the phases. Therefore, interfacial waves tend to be formed, which favor detachment of portions of liquid from the interfacial disturbances and consequently part of the liquid phase flows as dispersed droplets within the vapor phase. Hence, the transition to annular flow is given as a function of the balance of forces in a liquid droplet flowing within the vapor stream, whereas the annular flow pattern occurs when the drag forces overcome the gravity forces as follows: Cd

πd2d ρv u2v πd3d ¼ gðρl  ρv Þ 4 2 6

ð3:91Þ

where Cd is the drag coefficient that can be assumed as constant and equal to 0.44, and dd is the droplet diameter, which is estimated according to the method proposed by Hinze (1955) for the maximum stable droplet size as follows:

2 The coefficient of this equation is 36% higher than the original proposal due to few simplifications adopted in this book to be more didactical.

94

3

dd ¼

Wecrit σ ρv u2v

Flow Patterns

ð3:92Þ

where the critical Weber number Wecrit is assumed to be 30 as suggested by Hinze (1955). Hence, by substituting the droplet diameter into Eq. (3.91) we obtain the corresponding vapor velocity for transition to annular flow. Moreover, under conditions of annular flow the liquid film is considerably thin, and the assumption of void fraction of approximately unity leads to negligible difference in the superficial velocities. Therefore, the transition to annular flow is given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4Wecrit σgðρl  ρv Þ 4 σgðρl  ρv Þ ¼ 3:09 jv ¼ 3C d ρ2v ρ2v

ð3:93Þ

Therefore, the transition to annular flow is independent of the liquid flow velocity. Figure 3.16 depicts the flow pattern transitions for air-water flow in 26 mm ID tube, where the coordinate axis is given as a function of the superficial velocities of each phase. Again, under phase-change conditions, it is common to face a situation with constant mass flow rate, and variation of vapor quality along the evaporator or condenser, which are more properly represented by pairs of G and x for the transition 10

1

Dispersed bubble

10

0

500 200

jl [m/s]

Bubble

10

Churn

100

-1

Air-water T = 20 °C, p = p atm ,

le / d = 50

10

-2

10

-3

d = 26 mm, d crit = 52 mm

Slug

10 -2

Annular

10 -1

10 0

10 1

10 2

jv [m/s] Fig. 3.16 Flow pattern transitions for vertical upward flow of air-water in 26 mm ID channel at 20  C and patm

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

95

instead of jl and jv. Therefore, Eqs. (3.67), (3.76), (3.77), (3.90), and (3.93) can be rewritten, respectively, as follows:  1=4 1:15 gðρl  ρv Þσ G ¼ 1x x ρ2l ρ  3ρ

ð3:94Þ

 0:1 0:48 1=1:12 4:0 σ d G ¼ 1x x 0:52 0:08 ððρl  ρv ÞgÞ1=2 μl ρ þ ρ ρl

ð3:95Þ

l

l

v

v



1 ρl x ¼ 1 þ 0:92 ρv   pffiffiffiffiffi gd le G¼  0:22 1x x 55:41d ρl þ ρv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3:09 4 G¼ σgðρl  ρv Þρ2v x

ð3:96Þ ð3:97Þ ð3:98Þ

Figure 3.17 depicts the transitions among flow patterns for vertical flow of R134a in a 20 mm ID tube. This diameter was selected based on the critical diameter, which is considerably large for this fluid, otherwise the bubble flow pattern would not occur. Independently, the dispersed bubble flow is not observed in the present analysis, which is mainly related to the restriction of α 0.52 and no-slip between 100 R134a, Tsat = -20 °C, d = 20 mm, d crit = 19.8 mm

Annular

G [kg/m²s]

75

Slug

50 500 Churn

25

200 Bubble

0 0.0

le / d = 100

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.17 Flow pattern transitions for R134a vertical upward flow in 20 mm ID tube at 20  C

96

3

Flow Patterns

the phases. In fact, the prediction of bubble flow pattern is only possible due to the slip velocity between the phases, otherwise the void fraction would be higher than 0.52. From Fig. 3.17 and considering a mass velocity of 75 kg/m2s and vertical evaporation of R134a at 20  C as saturated liquid at the channel inlet, it can be figured out that as the fluid enthalpy increases, bubbles are formed on the channel wall and, then, they flow dispersed within the liquid. With the increment of the vapor quality due to heat addition along the channel length, the coalescence of the bubbles takes place, implying on a transition to churn or slug flow, depending on the flow length, at a vapor quality of approximately 5%. Further addition of heat to the fluid promotes the transition to annular flow at a vapor quality of approximately 40%. It is important to highlight that this analysis neglects the effect of heat flux intensity and its direction on the flow pattern transitions. The flow pattern transition method proposed by Mishima and Ishii in 1984 for vertical upward flow of vapor-liquid mixtures can also be cited. Analogous to the method of Taitel et al. (1980), the method proposed by Mishima and Ishii (1984) is based on the physical mechanisms mostly related to the geometric distribution of the phases. This method provides values of void fraction associated to the transitions between the flow patterns. The method is mainly based on the drift-flux model proposed by Zuber and Findlay (1965) and described in Sect. 2.3, to determine the superficial velocities associated to the flow pattern transitions.

3.2.4

Barnea, Shoham, and Taitel (1982a)

This subsection addresses the predictive method for vertical downward flow proposed by Taitel and his coworkers. Even though vertical downward flow is not commonly verified in practice for convective flow boiling, this flow direction is commonly observed for condensation applications. As described in Chap. 2, basically three flow patterns are expected to occur during vertical downward co-current vapor-liquid flow, namely, annular, bubble, and slug (intermittent) flows. Different from the condition of upward flow, where the vapor in situ velocity is higher or equal to the liquid phase velocity, due to the gravity and density differences, the same does not occur for downward flow, when a liquid velocity higher than the vapor velocity is possible. Analogous to the case of horizontal flow method proposed by Taitel and Dukler (1976), the modelling and derivation of the flow pattern transitions for vertical downward flow begin by the analyzes of the momentum balance of vapor and liquid phases, considering the schematics depicted in Fig. 3.18. The most natural flow pattern during vertical downward flow inside a duct of co-current vapor and liquid phases is the annular flow, which would occur even without a propelling system for the flow.

3.2 Flow Pattern Transition Criteria for Adiabatic Flows Fig. 3.18 Schematics of downward flow inside a channel

97

z ul τw,l

g

τi

uv

τi

δ

τw,l

τi τi

τw,v = 0

Assuming a coordinate axis in the main flow direction, such as depicted in Fig. 3.18, the force balance for liquid and vapor phases in the z direction are given, respectively, as follows: τw,l Sl  τi Si  dp Al þ ρl Al g dz ¼ 0

ð3:99Þ

þτi Si  dp Av þ ρv Av g dz ¼ 0

ð3:100Þ

where, analogous to horizontal flow, the term S corresponds to the contact areas and τ to the shear stresses, and the subscripts w and i correspond to the wall and interface, respectively. By eliminating the pressure difference in both equations, the following relationship is obtained: ðρl  ρv ÞgAdz ¼

τ S τ i Si þ w,l l α ð1  α Þ 1  α

ð3:101Þ

Whereas, the terms involving dimensions such as the liquid perimeter can be rewritten as functions of the channel diameter and void fraction. Recalling the definition of the interfacial perimeter between liquid and channel wall, and vapor– liquid interface, are given as follows: Sl ¼ πdz d pffiffiffi Si ¼ π αdz d

ð3:102Þ ð3:103Þ

Substituting these relationships in Eq. (3.101), the following equation is derived: ðρl  ρv Þgd τ τ ¼ pffiffiffi i þ w,l 4 α ð1  α Þ 1  α

ð3:104Þ

where the shear stress terms are given as functions of the friction factor as follows:

98

3

τw,l ¼ f w,l τi ¼ f i

ρl u2l 2

Flow Patterns

ð3:105Þ

ρv ð uv  ul Þ 2 2

ð3:106Þ

The friction factors are estimated based on their respective Reynolds numbers based on the hydraulic diameters as follows:  nl ρl ul dh,l f w,l ¼ C l μl  nv ρv uv dh,v f i ffi f v ¼ Cv μv

ð3:107Þ ð3:108Þ

where the constants C and exponent n are, respectively, equal to 0.046 and 0.2 for turbulent flow, and 16 and 1.0 for laminar flow. The hydraulic diameters are given as follows: dh,l ¼

4Al 4πd2 ð1  αÞ ¼ dð1  αÞ ¼ 4πd Sl

ð3:109Þ

pffiffiffi 4Av 4πd2 α pffiffiffi ¼ d α ¼ Si 4πd α

ð3:110Þ

dh,v ¼

where it is considered that for momentum transfer issues, only the channel wall imposes resistance to the liquid flow. And the in situ velocities are given as functions of the superficial velocities and void fraction, as described in Chap. 2. Therefore, substituting Eqs. (3.102) and (3.103), and (3.105) to (3.110) into Eq. (3.104), the relationship for the momentum balance of both phases is given as follows: ðρl  ρv Þgd 1 ¼ 4 ð1  αÞ3   nv  nl 2  ρv j2v ðð1  αÞ=α  jl = jv Þ2 ρl j l ρ jd ρl j l d  Cv v v þ C l μv μl 2 2 αð1nv Þ=2 ð3:111Þ Then, dividing the equation above by the frictional pressure drop gradient of vapor single-phase flow, the following relationship is derived: 2

 nl 2 3 ρl jl ρl jl d C l μl 2d ðρl  ρv Þg=4 ðð1  αÞ=α  jl = jv Þ 1 6  nv 2 ¼ nv 2 7 þ  5 ð3:112Þ 34 ð1nv Þ=2 ρv jv ρ ρv jv d ρ j d α v jv ð1  αÞ C v vμ v Cv μ 2d 2d 2

v

v

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

99

where the second term inside the square brackets on the right side can be recognized as the square of Lockhart and Martinelli parameter X^. The term on the left side can be considered as the ratio between buoyancy forces and the vapor friction forces, and is represented by Ŷ. Hence, the previous equation is re-written as follows: b¼ Y

  ðð1  αÞ=α  jl = jv Þ2 b 2 1 þX αð1nv Þ=2 ð1  αÞ3

ð3:113Þ

from which it is possible to determine the void fraction, or liquid film thickness, as a function of the superficial velocities and fluids properties. Different from that performed for horizontal and vertical upward flow, a mental exercise is performed assuming initially reduced liquid flow rate with successive increment, rather than increment of the vapor phase fraction. This analysis is more coherent with the process occurring inside a vertical condenser. Analogous to the case of horizontal flow, the transition between annular and slug flow for vertical downward flow depends basically on the liquid holdup. Therefore, Barnea et al. (1982a) considered that the transition from annular to slug flow occurs when the liquid holdup (1 – α) during the slug flow is approximately twice the liquid holdup of the annular flow immediately before the transition. Hence, by assuming the same void fraction value taken for transition between bubbles and slug flow for vertical upward flow, which was of approximately 25% ((1 – α)slugflow ¼ 0.75  0.7), it is possible to determine the annular to slug transition condition as follows: 2 ð1  αÞtransition ¼ ð1  αÞslugflow

ð3:114Þ

αtransition ¼ 0:65

ð3:115Þ

The reason for defining factor 2 as a transition criteria is not clear, but this assumption results in a transition condition that agrees with the experimental results. By substituting this value in Eq. (3.113), we can obtain the transition condition as follows: " jl ¼ jv 0:538  0:65

ð1nv Þ=4



b Y b2 X 23:32

0:5 # ð3:116Þ

With successive increment of the liquid flow rate, eventually the slug flow pattern will transition to dispersed bubbles, and analogous to the cases of vertical upward flow and horizontal flow, this transition is characterized by the condition that turbulence forces overcome the interfacial tension forces. Barnea, Shoham, and Taitel stated the transition given by Eq. (3.76) between slug and dispersed bubbles for vertical upward flow is also reasonable for downward flows, but the prediction can be improved by considering the effect of void fraction on the maximum possible diameter of a bubble generated by turbulence, which is given by Calderbank (1958) as follows:

100

3

dmax ¼ 4:15α1=2 þ 0:725 3=5 2=3 ðσ=ρl Þ E

Flow Patterns

ð3:117Þ

where again σ is the surface tension, and E is the rate of energy dissipation per mass unit, given by Eqs. (3.69) to (3.71), that results in the following relationship:  0:2 3 ρl jd j Ε ¼ 0:092 d μl

ð3:118Þ

and the homogeneous model is assumed for the estimative of void fraction α as follows: α¼

jv j

ð3:119Þ

Therefore, the transition criterion can be determined by equating the maximum bubble diameter generated by turbulence during downward flow with the critical diameter for a bubble to keep its spherical shape. Different from the upward flow, for downward flows Barnea, Shoham, and Taitel indicated that for bubbles with diameter twice the one given by Hinze (1955) and Eq. (3.73) the shape is still spherical. Therefore, by considering these conditions, the transition criterion is given as follows: 0:487

μl 0:08 ρl 0:52 j1:62 ððρl  ρv ÞgÞ1=2 σ 0:1 d 0:48

 0:725 j0:5 ¼ 4:15 jv 0:5

ð3:120Þ

which is transcendental on j, hence cannot be explicitly solved on j and a numerical method is needed for its solution. In a similar way to the case of upward flow, the downward flow cannot sustain dispersed bubble flow pattern for void fraction higher than 0.52, and this restriction is related to the maximum packaging of a cubic lattice distribution, schematically depicted in Fig. 3.13. Analogous to vertical upward flow, the transition is determined based on the definition of the in situ velocities of the vapor phase as follows: uv ¼ ul  U 0

ð3:121Þ

where the buoyance forces tend to decrease the vapor in situ velocity in relation to the liquid velocity, and the difference is given by Eq. (3.65), which corresponds to the rising velocity of bubbles in quiescent liquid. Therefore, by rewriting the in situ velocities as a function of the superficial velocities and void fraction, the following relationship can be obtained:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

jl ¼ 0:923 jv þ 0:734

101

 1=4 gðρl  ρv Þσ ρ2l

ð3:122Þ

In a similar way to the case of vertical upward flow, the existence of the bubble flow pattern during downward flow is also restricted to large channels, whereas the transition value is given by comparing the velocities of a Taylor bubble, given by Eq. (3.78), and of regular amorphous bubbles, given by Eq. (3.65). Anyway, it should be remembered that during downward flow the vapor phase tends to be slower than the liquid due to buoyance forces, and under conditions that the amorphous bubbles have higher velocity in absolute value than the Taylor bubble, the bubble flow pattern can occur. Otherwise, it is impossible for such a flow pattern to exist. This restriction is given as follows:  d ¼ 4:372

ðρl  ρv Þσ gρ2l

1=2 ð3:123Þ

which for air-water mixture at patm and 20  C corresponds to 52 mm, and R134a corresponds to 16 mm. Finally, under conditions of very low vapor velocities and high liquid holdup, the amount of liquid might be insufficient to form a Taylor bubble or the continuum core vapor stream of annular flow. Hence, bubble flow occurs for these conditions, and analogous to the analysis performed for upward flow, the transition to slug flow would occur for void fraction of approximately 25%, which is restricted by the maximum packaging of bubbles in a cubic lattice distribution with half radius of spacing between the bubbles. Therefore, by assuming the slip given by Eq. (3.65), this transition is given as follows: jl ¼ 3 jv þ 1:148

 1=4 gðρl  ρv Þσ ρ2l

ð3:124Þ

Figures 3.19 and 3.20 depict the flow pattern transitions for air-water and R134a, respectively, for two diameters. In Fig. 3.19 for air-water flow, the transition between slug and dispersed bubbles is not represented for the channel with 25 mm of internal diameter because it is smaller than the critical diameter, given by Eq. (3.123). The flow pattern transitions for R134a are presented as a function of the mass velocity and vapor quality, since for heat transfer problems it is common to face constant mass flow rate and fluid enthalpy variation, rather than fixed superficial flow velocity of one of the phases. In any case, it can be speculated that the transition from annular to slug flow for high jv or x is incoherent because both are associated to the increment of the vapor fraction. These conditions would be favorable for annular flow condition; therefore, it is reasonable to assume an extrapolation of the transition curve with positive or null derivative rather than attaining to the transition given by Eq. (3.116) for the entire range of vapor quality and superficial vapor velocity.

102

3

10

Flow Patterns

1

Dispersed bubbles

jl [m/s]

Slug

10

0

Annular Air - water flow at p atm and T = 20 °C, d crit = 52 mm d = 60 mm d = 25 mm

10

-1

10 -2

10 -1

10 1

10 0

10 2

jv [m/s] Fig. 3.19 Flow pattern transitions for vertical downward flow of air water flows at patm and 20  C

3 x10

4

10

R134a, Tsat = 35 °C, dcrit = 13.7 mm

4

G [kg/m²s]

Dispersed bubbles

Slug

10

3

d = 50 mm

Annular

d = 25 mm 2

3 x10 10 -4

10 -3

10 -2

10 -1

x [-] Fig. 3.20 Flow pattern transitions for vertical downward flow of R134a at 35  C

10 0

3.3 Predictive Methods for Convective Boiling

3.3

103

Predictive Methods for Convective Boiling

The transitions for flow patterns during adiabatic flows were addressed in the previous subsection for conventional and minichannels. During the heat transfer process, the flow pattern transitions, and the flow patterns themselves can be different. For example, the pure stratified flow is not expected during the condensation process because the cooler surface implies on a liquid film being formed along the wall. Conversely, during convective flow boiling in horizontal channel, the gravitational effects during annular flow act in order to reduce the film thickness in the upper region of the tube; therefore, the film in the upper region tends to dry first, creating a transitional flow pattern, here defined as dryout. It is recurrently mentioned in the open literature that the gravitational effects are less pronounced during two-phase flow in microchannels, hence the flow patterns might be different from those for conventional channels. Therefore, this subsection addresses some predictive methods for flow patterns during convective boiling, and the next for convective condensation. Several flow pattern predictive methods for convective boiling are available in the open literature, such as Kattan et al. (1998) and subsequently Wojtan et al. (2005), who included the effects of heat transfer process on the flow pattern transitions. Most of them take the physical mechanisms of the Taitel and Dukler (1976) method to predict the transitions, and adjusted new empirical constants and exponents according to their experimental database. Moreover, since these methods are focused on heat transfer problems, the transitions are presented as a function of the mass velocity and vapor quality, which is more elucidating for phase change processes than superficial velocities.

3.3.1

Wojtan, Ursenbacher, and Thome (2005)

The method of Wojtan et al. (2005) proposed to predict flow pattern transitions during convective boiling in conventional channels is an updated version of the method proposed by Kattan et al. (1998). The later method, in turn, is a modification of the method proposed by Steiner (Kind et al. 2010, which was available in 1993 in the first edition of VDI Heat Atlas) and ultimately is based on adjustments of the Taitel and Dukler (1976) predictive method. Even though the experimental results used to validate the method were obtained for channels with internal diameter of 8.0 and 13.8 mm, this condition still corresponds to conventional scale channel according to the threshold values indicated by Kew and Cornwell (1995), described in Chap. 2, for refrigerants R410A and R22. The main difference between the Wojtan et al. (2005) and the Taitel and Dukler (1976) methods is related to the prediction of a dryout flow pattern, which corresponds to the annular flow being gradually drought from the top to the tube bottom region. Additionally, the method also predicts some transitional flow patterns,

104

3

Flow Patterns

such as Slug + Stratified wavy flows, which possesses characteristics of both patterns. Moreover, the authors expatiated about the flow pattern identification method that based on void fraction fluctuations during time, in a similar way to the description presented by Jones and Zuber (1975); the void fraction is determined based on an optical method with the use of fluorescent dye dispersed in the liquid. In this textbook, basically the implementation of the Wojtan et al. (2005) is presented, and those interested are encouraged to check the original paper to get the steps of its development. In a similar way to Taitel and Dukler, it is considered the geometrical distribution of stratified flow as a basis for the remaining parameters, however the authors considered the stratification angle γ, shown in Fig. 3.4. Moreover, instead of determining the void fraction, or liquid height, based on the momentum balance of both phases, α is determined based on the Rouhani (1969) method, such as described in Sect. 2.3. This approach eliminates the iterative method required for the liquid height evaluation, that is, a function of the Lockhart and Martinelli parameter X^, as performed by Taitel and Dukler (1976). The transition from smooth stratified and wavy stratified is given according to the proposal of Kattan et al. (1998) as follows:  1=3 800:18ð1  αÞα2 ρv ðρl  ρv Þμl g G¼ x2 ð1  xÞ

ð3:125Þ

The transition from stratified wavy to non-stratified flows is given by the following relationship, which comes from successive adjustments of the Taitel and Dukler (1976) proposal: 2

!

31=2

πα3 gdρv ρl π3 Fr l0 6 7 ffi G ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ1 5 2 2 25ðH l =d Þ Wel0 2 4x 1  ð2H l =d  1Þ

þ 50 kg=m2 s

ð3:126Þ

where the Froude and Weber numbers for the mixture flowing as liquid phase are given, respectively, as follows: Fr l0 ¼ G2 =ρ2l gd

ð3:127Þ

Wel0 ¼ G2 d=ρl σ

ð3:128Þ

hence, the ratio Frl0 / Wel0 is independent of the mass velocity. The vapor quality corresponding to the transition between intermittent and annular-like flow patterns is given as follows, which comes from the assumption of constant Lockhart and Martinelli parameter:

3.3 Predictive Methods for Convective Boiling

" x ¼ 0:34

1=0:875

105

#1  1=1:75  1=7 ρl μv þ1 ρv μl

ð3:129Þ

The condition corresponding to the dryout inception is given by the following relationship, which comes from the definition of the vapor quality for dryout inception, proposed by Mori et al. (2000) and adjusted by Wojtan et al. (2005): ( G¼

      0:25  0:7 )1=1:08 ½gdρv ðρl  ρv Þ 0:37 ρv σ 0:17 ρl ϕcrit 0:58 þ 0:52 ln x 0:235 ρv ϕ d ð3:130Þ

where the critical heat flux ϕcrit is given according to Kutateladze (1948) as follows: 0:25 ϕcrit ¼ 0:131ρ0:5 v ilv ½gσ ðρl  ρv Þ

ð3:131Þ

Similarly, the end of the dryout region is given as a function of the vapor quality for dryout completion, as proposed by Mori et al. (2000) and adjusted by Wojtan et al. (2005), as follows: ( G¼

      0:09  0:27 )1=1:06 ½gdρv ðρl  ρv Þ 0:15 ρv σ 0:38 ρv ϕcrit 0:61 þ 0:57 ln x 0:0058 ρl ϕ d ð3:132Þ

Figure 3.21 depicts the transitions according to the aforementioned conditions for R134a flow at 10  C in an 8 mm ID tube with heat flux of 10 kW/m2. This figure also depicts the limits for the transitional flow patterns proposed by Wojtan et al. (2005). It must be highlighted that, analogous to Taitel and Dukler’s (1976) original method, the transition between intermittent and annular flow patterns is predicted to occur at constant vapor quality. However, according to Barbieri et al. (2008) and Felcar et al. (2007), the vapor quality corresponding to this transition reduces with the increment of mass velocity, hence the curve presents a negative derivative on x. According to Barbieri et al. (2008), this transition is given as a function of the liquid Froude number and Lockhart and Martinelli parameters as follows: 2:40

b tt Fr l ¼ 3:75X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "     #1:2 3:75ρ2l gd 1  x0:9 ρv 0:5 μl 0:1 G¼ x ρl μv ð 1  xÞ

ð3:133Þ ð3:134Þ

Figure 3.22 depicts the flow pattern transitions depicted in Fig. 3.21, as well as the transition between intermittent and annular flow patterns described above,

106

3

Flow Patterns

500 Mist

R134a, Tsat = 10 °C, d = 8 mm, φ = 10 kW/m²

Intermittent

300

ut Dryo

G [kg/m²s]

400

200

Annular

Slug

100

Slug + Stratified wavy

Stratified wavy x ia

0 0.0

0.2

Smooth stratified

0.4

x [-]

0.6

0.8

1.0

Fig. 3.21 Flow pattern transition for R134a flow at 10  C in an 8 mm ID tube according to Wojtan et al. (2005)

500 Barbieri, Jabardo and Bandarra Filho (2008)

Mist

400

R134a, Tsat = 10 °C, d = 8 mm, φ = 10 kW/m²

200

Annular

Slug

100

Slug + Stratified wavy

Stratified wavy x ia

0 0.0

ut Dryo

G [kg/m²s]

Intermittent

300

0.2

Smooth stratified

0.4

x [-]

0.6

0.8

1.0

Fig. 3.22 Flow pattern transition according to Wojtan et al. (2005) and the intermittent to annular flow patterns according to Barbieri et al. (2008)

3.3 Predictive Methods for Convective Boiling

107

highlighting the differences in this transition. Therefore, for flow with heat transfer process, the consideration of constant void fraction or liquid height might not be coherent with reality. It must be advised that the predictive method proposed by Wojtan et al. (2005) was developed based on experimental results for conventional channels (d  8 mm); nonetheless, it is presented in this book for didactical purposes. Moreover, this aspect can be recognized by the fact that even this method and the Taitel and Dukler (1976) method do not consider surface tension forces on flow pattern transitions. Conversely, Barnea et al. (1983) consider this mechanism for the transition between stratified wavy and non-stratified flow patterns, as discussed in Sect. 3.2.2. Most recently, Cheng et al. (2008) modified the method of Wojtan et al. (2005) to predict the flow pattern transitions of CO2 based on results from literature for flow patterns and heat transfer coefficient of carbon dioxide. Their updated prediction method is capable of providing accurate flow pattern predictions for tube diameters from 0.6 to 10 mm, therefore being suitable to micro and macroscale conditions. The database used for its formulation covers mass velocities from 50 to 1500 kg/m2s, heat fluxes from 1.8 to 46 kW/m2, and saturation temperatures from 28 to +25  C (reduced pressures from 0.21 to 0.87).

3.3.2

Revellin and Thome (2007)

Revellin and Thome (2007) proposed a flow pattern predictive method for convective flow boiling in small-scale channels. As previously discussed, during two-phase flow in reduced diameter channels, the surface tension forces play a significant role on the occurring phenomena and can overcome the buoyance forces under some conditions. Therefore, the gravity-dominated flow patterns, namely, the stratifiedlike flow patterns, are unlikely to occur during two-phase flow in microchannels, and basically bubbles, slug, and annular flow patterns are expected. Revellin and Thome (2007) also characterized the transitional flow patterns that correspond to flow patterns combining the characteristic of two flow patterns, but did not address the transition conditions among them. The flow patterns identified by them are schematically depicted in Fig. 3.23. The transitions predicted by their method were compared with experimental results for R134a and R245fa in 0.509 and 0.790 mm ID tubes, and reasonable agreement was found. These authors performed an objective method for flow pattern identification based on bubble length and their frequency. The transition between bubbles (nominated by them as isolated bubbles) and slug flow patterns was characterized by the vapor quality corresponding to the maximum bubbles’ frequency. For vapor qualities higher than the one corresponding to the vapor quality associaty to the bubbles frequency peak, further increase in vapor quality implies on the bubble coalescence and reduction of their frequency. It was found by them that this transition depends on the mass velocity, heat flux, fluids properties including the surface tension, and geometrical parameters. Therefore,

108

3

Flow Patterns

Bubbles

Bubbles to Slug transition

Slug

Slug to annular transition

Wavy annular

Annular

Flow direction Fig. 3.23 Schematics of flow patterns in mini and microchannels

Revellin and Thome proposed that the vapor quality corresponding to the transition can be correlated as a function of the liquid only Reynolds number, Boiling number, and vapor only Weber number, given as follows:  x ¼ 0:763

Re l0 Bo Wev0

Re l0 ¼ Bo ¼

Gd μl

ϕ Gilv

Wev0 ¼

G2 d ρv σ

0:41 ð3:135Þ ð3:164Þ ð3:137Þ ð3:138Þ

Equation 3.135 can be rewritten as a function of the mass velocity, given as follows:

3.3 Predictive Methods for Convective Boiling



109

 1=2 0:719 ϕρv σ x1=0:82 μl ilv

ð3:139Þ

Analogous to the transition between bubbles and intermittent flow patterns, the transition from slug to annular flow is given as a function of the bubble frequency when its value tends to zero, annular flow is achieved, corresponding to the absence of bubbles. Hence, based on their experimental results, Revellin and Thome identified that this transition depends mainly on the liquid Reynolds and Weber number, and is given as follows: 1:23 x ¼ 1:4 104 Re 1:47 l0 Wel0

ð3:140Þ

where the liquid only Weber number Wel0 is given as follows: Wel0 ¼

G2 d ρl σ

ð3:141Þ

The transition from annular to dryout conditions is estimated based on the critical heat flux estimated according to the method of Katto and Ohno (1984) modified by Wojtan et al. (2006), based on data for flow boiling of halocarbon refrigerants in microscale channels and given as follows: ϕcrit

 0:073  0:72 ρ d Gilv ¼ 0:437 v L ρl We0:24 l0

ð3:142Þ

Then, based on energy balance by knowing the inlet enthalpy iin, it is possible to determine the vapor quality for the transition as follows: x¼

ϕ L iin  il L þ 4 crit ¼ xin þ 4Bocrit d ilv Gilv d

ð3:143Þ

Figure 3.24 depicts the flow pattern transitions according to the Revellin and Thome (2007) predictive method for R134a. According to this figure, the dryout condition would only occur for very high mass velocities. Conversely, the bubble flow pattern is observed for the entire mass velocity range, however, only for very low vapor quality values. Again, the stratified flow pattern is absent during two-phase flow in minichannels because the surface tension forces overcome the buoyance forces. Nonetheless, the experimental database for flow patterns during convective boiling of refrigerants in vertical microchannels is limited, and consequently, as far as the present authors know, a flow pattern prediction method for flow boiling in vertical small diameter channels is still not available in the open literature. However, due to the predominance of surface tension over gravitational effects for small diameter channels, it can be speculated that prediction methods developed for horizontal microchannels are also suitable for flows boiling in vertical small diameter channels.

110

3

2000

Dryout R134a, T = 20 °C, φ = 10 kW/m², φcrit = 3.4 MW/m²

1600

G [kg/m²s]

Flow Patterns

d = 0.509 mm

1200

d = 0.790 mm

800 Annular

400 Slug Bubbles

0 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.24 Flow pattern transitions for microchannels during convective flow boiling of R134a according to Revellin and Thome (2007)

3.3.3

Ong and Thome (2011)

Ong and Thome (2011) proposed flow pattern transitions for two-phase flow in microchannels. The experimental database obtained by the authors to adjust the empirical coefficients and exponents in their method includes results for R134a, R236fa, and R245fa flowing inside channels with 1.03, 2.20, and 3.04 mm, for mass fluxes ranging between 100 and 1500 kg/m2s, and saturation temperature between 25 and 35  C. They also add to this database the data previously obtained by Revellin and Thome (2007). The flow patterns were identified by an objective method using two sets of laser and photodiode installed along the visualization section of their experimental facility, and the signal from this instrumentation was evaluated through a PDF analysis. Based on the experimental results for flow patterns, the authors proposed transitions criteria between flow patterns by data fitting with governing non-dimensional parameters. The authors adopted non-dimensional parameters, such as Reynolds and Weber numbers, but also included confinement number Co, to account for the effect of a confined bubble, and the Froude number Fr to account for the relative effect of inertial and gravitational effects. Based on all parameters, it is claimed that the predictive methods capture the transition from conditions governed by confinement effects, such as for bubble flow, and conditions governed by shear effects, such as for annular flow. In addition to bubble (isolated bubbles), slug (coalescing bubbles), and annular flow patterns, Ong and Thome (2011) also defined the slug-plug flow that is constituted of a long

3.3 Predictive Methods for Convective Boiling

111

vapor bubble separated by liquid plugs that exhibit strong buoyancy effects and a thick stratified layer of liquid at the bottom of the elongated bubbles. The vapor quality for the transition between bubbles (isolated bubbles) and slug (coalescing bubbles) (from the top of Fig. 3.23, transition from second to fourth flow patterns) is given as follows: x ¼ 0:36Co

0:20

 0:65  0:90 μv ρv 0:25 Re 0:75 We0:91 v0 Bo l0 μl ρv

ð3:144Þ

where the Reynolds number is evaluated assuming the mixture flowing as vapor, the Boiling number Bo is evaluated according to Eq. (3.137), the confinement number Co is given as follows: Co ¼

1 d

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ gðρl  ρv Þ

ð3:145Þ

and the Weber number for the mixture flowing as liquid is given as follows: Wel0 ¼

G2 d σρl

ð3:146Þ

The transition from slug (coalescing bubbles)-to-annular and annular flow patterns is given as follows: x ¼ 0:047Co

0:05

 0:70  0:6 μv ρv 0:91 Re 0:8 v0 Wel0 μl ρl

ð3:147Þ

The transition between slug to slug-plug flow patterns is given as follows: x ¼ 9Co0:20

 0:9 ρv Fr 1:2 Re 0:1 l0 l0 ρl

ð3:148Þ

where the Reynolds number is evaluated assuming the mixture flowing as liquid, and the Froude number for the mixture flowing as liquid is given as follows: Frl0 ¼

G2 ρ2l gd

ð3:149Þ

According to the proposal, the value estimated according to Eq. (3.148) should be considered only when it is lower than the vapor quality estimated according to Eq. (3.147). Figure 3.25 depicts the flow pattern transitions according to the proposal of Ong and Thome (2011) for R134a and R245fa in 1 and 2 mm ID channels.

112

3

a) 1200

b) 1200

1000

1000

R134a, T sat = 20 °C, φ = 10 kW/m²

Annular

G [kg/m²s]

100

Slug

R245fa, T sat = 20 °C, φ = 10 kW/m²

ar ul nn -a -to ug Sl

Slu g-t o-a nnu lar

G [kg/m²s]

Flow Patterns

d = 1 mm d = 2 mm

Annular

100

Slug Bubbles

12 0.0

0.2

Bubbles

d = 1 mm d = 2 mm

0.4

x [-]

0.6

0.8

1.0

12 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.25 Flow patterns transition during convective boiling inside microchannels according to the Ong and Thome (2011) proposal

3.4

Predictive Method for Convective Condensation

At the present, the number of studies focused on the fluid mechanics and heat transfer problems during convective condensation are considerably inferior to those for convective boiling. Nonetheless, in this textbook we address the predictive method proposed by El Hajal et al. (2003) for conventional sized channels. For practical applications, sometimes it is satisfactory defining the flow as gravity dominated or convection dominated, such as performed by Cavallini et al. (2006), who concluded that this transition can be predicted based on the non-dimensional vapor velocity jv*, such as defined by Wallis (1969), as follows: jv ¼

h

1=3  i3 3 b 1:111 7:5= 4:3X þ 1 þ C tt t

ð3:150Þ

where the constant Ct depends on the fluid type and is equal to 1.6 for hydrocarbons and 2.6 for other refrigerants. The non-dimensional vapor velocity is given as follows: xG jv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gdρv ðρl  ρv Þ

ð3:151Þ

which is somewhat similar to the Froude number, corresponding to the ratio between inertial and gravitational forces. This transition is depicted in Fig. 3.26 for hydrocarbons and other refrigerants, and according to this figure the difference between types of fluids vanishes for high values of the Lockhart and Martinelli parameter, which corresponds to low vapor quality conditions. It should be mentioned that Cavallini et al. (2006) indicated conditions dominated by wall subcooling, which will be seen in Chap. 7 to correspond to gravity-dominated conditions; however, the interpretation is similar to the one addressed in this book.

3.4 Predictive Method for Convective Condensation

10

113

1

jv* [-]

Predominance of inertial effects

10

0

Predominance of gravitational effects

Refrigerants in general Hydrocarbons

10

-1

0,01

0,1

Xtt [-]

1

10

Fig. 3.26 Transition of predominant effects during condensation, Cavallini et al. (2006)

Hence, in conditions that the actual non-dimensional vapor velocity, given by Eq. (3.151), is higher than the transitional value, given by Eq. (3.150), the flow condition is dominated by inertial effects and the heat transfer is dominated by convective effects, with a corresponding specific predictive method for the heat transfer coefficient, such as depicted in Fig. 3.27. Conversely, if the actual jv* is smaller than the transitional value, the heat transfer process is governed by gravitational effects, and the heat transfer coefficient predictive method is estimated as a function of the temperature difference, and terms that represent the buoyancy forces, such as thermal expansion coefficient, schematically depicted in Fig. 2.17. These aspects are well addressed in Chap. 7. Despite the fact that the correlation given by Eq. (3.150) had been developed for conventional sized channels, Matkovic et al. (2009), Del Col et al. (2015), and Lopez-Belchi et al. (2016) concluded that this method is also appropriate for microscale channels. In a similar analysis, Jige et al. (2016) proposed a predictive method for heat transfer coefficient during condensation inside rectangular microchannels segregating into intermittent and annular flow patterns. According to these authors, the annular flow pattern in turn can be dominated by vapor shear stress or surface tension effects, while the intermittent flow pattern is given by combination of annular and liquid single-phase flow conditions. Nonetheless, the prediction of heat transfer coefficient is given by an asymptotic combination of all parcels, as described in Chap. 7, and no transition of predominant effects can be inferred.

114

3

3.4.1

Flow Patterns

El Hajal, Thome, and Cavallini (2003)

El Hajal, Thome, and Cavallini (2003) proposed a flow pattern predictive method for condensation inside horizontal channels of conventional dimensions. This method is similar to the Wojtan et al. (2005) method for flow patterns during convective boiling, in the sense that it is based on the method of Kattan et al. (1998), which in turn is based on the method of Steiner, described in Kind et al. (2010), and ultimately is based on the method of Taitel and Dukler (1976) for horizontal flows. As expected, some flow patterns related to the heat transfer process predicted by the Wojtan et al. (2005) method are absent, such as dryout. This is reasonable because this flow pattern depends on the heat transfer process. The physical mechanisms that govern the transitions are analogous to the method of Taitel and Dukler (1976), and the differences are related to the transition values, which are adjusted by means of the empirical constants to experimental results for condensation. Therefore, the complete description and discussion about the transitions are not presented in this chapter, and the reader is encouraged to turn to the corresponding section to remember the physical aspects of the transitions. It must be mentioned that a genuine stratified flow is not expected during the condensation process, because the contact between the saturated vapor with the cooler wall would result in the formation of a liquid film or droplets along all the channel perimeter. Nonetheless, the authors keep the prediction of this transition 600

G [kg/m²s]

Predominance of inertial effects

d = 10 mm d = 5 mm d = 2 mm

400

200 Predominance of gravitational effects R134a, Tsat = 20 °C

0 0,0

0,2

0,4

x [-]

0,6

0,8

1,0

Fig. 3.27 Transition between conditions dominated by inertial or gravitational effects, Cavallini et al. (2006)

3.4 Predictive Method for Convective Condensation

115

considering that the liquid in the upper region does not significantly disturb the vapor phase flow, and, therefore, stratified flow becomes a reasonable definition. Analogous to the Wojtan, Ursenbacher, and Thome (2005) method for prediction of flow pattern during convective boiling, the El Hajal et al. (2003) method also takes into account a predictive method for void fraction α to avoid the iterative method required by the Taitel and Dukler (1976) for determination of the liquid height. In the present method, the authors adopted the log mean value of the void fractions estimated according to the homogeneous model, given by Eq. (2.34), and the Rouhani (1969) method, based on the Zuber and Findlay (1965) predictive method, given by Eqs. (2.46), (2.47), and (2.48). Therefore, the log mean void fraction is given as follows: α¼

αH  αR ln ðαH =αR Þ

ð3:152Þ

where the subscripts H and R attain to the homogeneous model and Rouhani (1969) methods, respectively. Based on the void fraction given by Eq. (3.152), it is possible to infer the non-dimensional liquid height and other geometrical parameters according to Eq. (3.21). The transition from smooth to wavy stratified flow is determined for mass velocities given as follows:  G¼

800:18ð1  αÞα2 ρv ðρl  ρv Þμl g x2 ð 1  xÞ

1=3



þ 20 kg=m2 s x

ð3:153Þ

And the transition between stratified wavy and non-stratified flows is given for the following mass velocity:



8 >
:4x2

3

"

dπ 5H l

2 

Fr l0 Wel0

1:023 þ1

9 #>0:5 = > ; ð3:154Þ

where the influence of the heat flux was neglected. The transition between intermittent and annular flows is given in a similar way to Taitel and Dukler’s (1976) original method, defined for a constant Lockhart and Martinelli value of 0.34. Therefore, the vapor quality for this transition is given according to Eq. (3.129).

116

3

5000

Bubbly

Flow Patterns

Mist flow

1000

G [kg/m²s]

Intermittent Annular

100

Stratified wavy

T = 40 °C

Smooth stratified

10 0.0

T = 60 °C

0.4

0.2

x [-]

0.8

0.6

1.0

Fig. 3.28 Flow pattern transitions according to El Hajal et al. (2003) for condensation of R134a in 8 mm ID tub, G ¼ 300 kg/m2s

The transition to mist flow occurs for conditions of high mass velocities, and intermediate and high vapor qualities, and is given as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"  2 # 480α2 gdρl ρv Fr l0 8 1:138 þ ln G¼ Wel0 3ð1  αÞ x2

ð3:155Þ

where the Froude and Weber for the mixture flowing as liquid are given, respectively, by Eqs. (3.127) and (3.128). The transition to bubbly flow is given as follows: " G¼

4παð1  αÞ2 d 5=4 ρl ðρl  ρv Þg 0:3164ð1  xÞ7=4 Pid μ0:25 l

#4=7 ð3:156Þ

For all the calculations in this method, it is considered that the void fraction is evaluated for a given mass flux, that is, it is not evaluated for the mass velocity of the corresponding transition. Figure 3.28 depicts the transitions according to this method for R134a in a channel with an internal diameter of 8 mm.

3.4 Predictive Method for Convective Condensation

3.4.2

117

Nema, Garimella, and Fronk (2014)

Nema et al. (2014) proposed predictive methods for flow pattern transitions during condensation inside mini and microchannels. In their map, the flow patterns were classified as wavy, intermittent, annular, and dispersed bubbles. The database used by the authors to adjust and compare the proposed transition criteria includes results for condensation of R134a for mass fluxes between 150 and 750 kg/m2s, circular, square, and rectangular horizontal channels with hydraulic diameters ranging between 1.00 and 4.91 mm. Given the range of the channel size, part of the results comprise a stratified-like flow pattern, hence justifying the inclusion of transition between discrete and dispersed wave flow patterns. Additionally, the authors pointed out that the dispersed bubble flow pattern is unlikely to occur in refrigeration applications because it requires very high mass fluxes. The intermittent flow pattern is expected to occur for reduced vapor content, and Nema et al. (2014) adopted a transition criterion similar to Barnea et al. (1983), schematically represented in Fig. 3.11. By solving Eq. (3.61) for σ, and using the result of Eq. (3.60), the following relationship was obtained:   π σ ¼ R2 g 1  ðρl  ρv Þ 4

ð3:157Þ

where R is the radius of the bubble represented in Fig. 3.11. According to Nema et al. (2014), the criterion for transition from wavy or annular to intermittent flow pattern depends basically on the liquid content. For mini and microchannels, this condition corresponds to the minimum liquid content required to block the cross-section, which is obtained for R ¼ dh. Hence, by equating R to the channel hydraulic diameter (R ¼ dh), it is possible to obtain a critical Bond number based on the surface tension evaluated according to Eq. (3.157) as follows: Bd crit =

ðρl 2 ρv Þgd 2h 4 ¼ 4:66 ¼ σ crit ð 4  πÞ

ð3:158Þ

The critical Bond number given by Eq. (3.158) corresponds to the transition criterion between micro and conventional scale, whereas for Bond numbers smaller than Bdcrit it corresponds to microscale. The two-phase mixture Bond number is given as follows: Bd ¼

ðρl  ρv Þgd 2h σ

ð3:159Þ

Hence, for mini and microchannels (Bd Bdcrit) the transition to intermittent flow pattern is similar to the criterion proposed by Barnea et al. (1983) for conventional channels, given by a constant Lockhart and Martinelli parameter (X^tt ¼ 0.3521) assuming turbulent regime for both phases. The Lockhart and

118

3

Flow Patterns

Martinelli parameter for assuming both phases as turbulent X^tt is given by Eq. (3.49). In the case that the Bond number is higher than the critical value (Bd > Bdcrit), Nema et al. (2014) performed a regression with their data and proposed the following transition criteria: b tt ¼ 0:3521 þ X

1:2479 1 þ 5:5=ðBd  Bd crit Þ

ð3:160Þ

which tends asymptotically to 1.6 as Bd increases, and according to Nema et al. (2014) this corresponds to the condition of intermittent flow pattern transition proposed by Taitel and Dukler (1976) for conventional size channels. However, as we have seen in Eqs. (3.48) and (3.49) of Sect. 3.2.2, the condition of α equal to 0.5 corresponds to X^tt ¼ 1.325. Nonetheless, the original proposal presented by Nema et al. (2014) will be kept in this textbook. For sufficiently large Bond numbers (Bd  Bdcrit), corresponding to large channels, Nema et al. (2014) proposed a transition between discrete and dispersed wave, both corresponding to a stratified-like flow pattern; however, the discrete wave pattern corresponds to a condition of identifiable waves with a long amplitude and wavelength, which is more common for low velocities. On the other hand, a dispersed wave flow pattern corresponds to a condition that the waves have small amplitude and wavelength, and the liquid–vapor interface is almost indistinguishable, being more common for intermediate flow velocities. The transition between discrete and dispersed wave flow patterns is given by a constant value of modified vapor Froude number of 2.75, and the modified vapor Froude number is defined as follows: Gx Fr v ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh gρv ðρl  ρv Þ

ð3:161Þ

Recall that the transition between or to/from stratified flow pattern is not observed for microscale channels (Bd Bdcrit) according to the literature. Nema et al. (2014) also proposed distinct transitions to annular flow pattern depending on the channel dimensions. According to the authors, the annular flow pattern is characterized by the dominance of vapor inertial effects over surface tension effects. In the case of microscale channels (Bd Bdcrit), the occurrence of annular flow is given by: Wev  35

ð3:162Þ

Wev < 35 and bX tt 0:3521

ð3:163Þ

or

where the Weber number of the vapor phase is given as follows:

3.4 Predictive Method for Convective Condensation

Wev ¼

119

dh G2 x2 ρv σ

ð3:164Þ

and the Lockhart and Martinelli parameter for both phases as turbulent X^tt is given by Eq. (3.49). In the case of flow in channels with macroscale characteristics (Bd > Bdcrit), the transition to annular flow is also given as a function of the vapor Weber number as follows: Wev ¼ 6 þ 7ðBd  Bdcrit Þ1:5

ð3:165Þ

The authors identified the occurrence of the mist flow pattern for Wev higher than 700 and X^tt lower than 0.175, independent of the channel diameter. Additionally, Nema et al. (2014) indicated that the number of experimental results for dispersed bubbles in their database was limited, and consequently they suggested to adopt for this transition the criterion proposed by Taitel and Dukler (1976), described in Sect. 3.2.2. Figure 3.29 depicts the flow pattern transitions according to the predictive method proposed by Nema et al. (2014) for condensation inside 1 mm ID channel. In this figure, the transition to dispersed bubbles was not included because it occurs for G much higher than the values expected in most applications, and the transitions to and between stratified-like flow patterns were not included because the displayed conditions correspond to a microscale case (Bd < Bdcrit). 1500 dh = 1 mm,Tsat = 40 °C

1200

G [kg/m²s]

Mist

900

600

300

Annular

Intermittent

R134a (Bd = 1.76) R410A (Bd = 2.58)

0 0.0

0.2

0.4

x [-]

0.6

Fig. 3.29 Flow pattern transitions according to Nema et al. (2014)

0.8

1.0

120

3.5

3

Flow Patterns

Solved Examples

Suppose that, similar to the study presented by Felcar et al. (2007), you have performed a series of experiments to infer the transition between intermittent and annular flow patterns for R134a flowing inside a 10 mm ID tube at 5  C. By performing a regression analysis by minimum square, you have found that the relationship between the mass flux and vapor quality for the transitional condition can be given according to the following dimensional relationship: G ¼ 57:531 x1 where G is given in kg/m2s and x as non-dimensional (0–1). Assuming that this transition can be given by the vapor inertial forces overcoming buoyance forces of the mixture, it is reasonable to consider the following modified Froude number as a first attempt to correlate: Fr  ¼

ðGxÞ2 ðρl  ρv Þgd

Find the corresponding non-dimensional correlation for this transition, and evaluate the root mean square deviation between it and the assumption of constant vapor quality for mass velocities between 100 and 400 kg/m2s. Solution: For saturation temperature of 5  C, the following properties can be found for R134a: ρl ¼ 1311 kg/m3 ρv ¼ 12.09 kg/m3 μl ¼ 2.832 104 kg/m s μv ¼ 1.071 105 kg/m s Based on the given information, d ¼ 0.010 m and g ¼ 9.806 m/s2. Hence, by substituting the obtained dimensional correlation into the definition of the modified Froude number, we can notice that the numerator term results in a constant. Additionally, since there is mention about variation of operation conditions, the denominator is also a constant. Therefore, the transition can be estimated based on a constant value of Froude number as follows: Fr trans ¼ 0:02 where it is an annular flow pattern for Fr* > Fr*trans, and intermittent flow pattern for lower values. We can recall Eq. (3.129) to evaluate the transition based on the assumption of constant liquid height as follows:

3.6 Problems

121

" xTaitel ¼ 0:34

1=0:875

#1  1=1:75  1=7 ρl μv þ1 ρv μl

which is a function of the fluid properties only and is equal to 0.2734 for the informed values. Therefore, it is possible to infer the difference in the root mean square deviation between both approaches as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u G¼400kg=m2 s u Z u ðxtrans  xTaitel Þ2 Dev ¼ u dG t G G¼100kg=m2 s

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u h i2 2s u G¼400kg=m Z ð0:02ðρl  ρv ÞgdÞ0:5 =G  xTaitel u ¼u dG t G G¼100kg=m2 s

which results in a deviation of 0.15.

3.6

Problems

1. Analyze the momentum balance equation obtained based on the Taitel and Dukler (1976) approach for annular flow, given by Eq. (3.17). (a) Is it valid for homogeneous flow condition, specifically for non-slip condition? (b) What can you conclude about annular flow? (c) Do the same analysis for stratified flow, Eq. (3.17). 2. Verify whether Eq. (3.76) is dimensionally correct. 3. Derive Eq. (3.77). 4. Assume that the transition between bubbles and intermittent flow patterns proposed by Taitel et al. (1980) is given by the bubbles distant one diameter from each other, rather than half diameter. Derive the corresponding relationship between jl and jv. 5. Demonstrate the derivation of Eqs. (3.94) to (3.98). 6. Present the transitions for horizontal flow according to Taitel and Dukler (1976) as a function of G and x. 7. Rewrite all the flow pattern transitions according to the Revellin and Thome (2007) method in terms of mass velocity.

122

3

Flow Patterns

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123

Kind M. et al. (2010) H3 Flow Boiling. In: VDI e. V. (eds) VDI Heat Atlas. VDI-Buch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77877-6_124 Kutateladze, S. S. (1948). On the transition to film boiling under natural convection. Kotloturbostroenie, 3, 10–12. Levich, V. G. (1962). Physicochemical hydrodynamics. Englewood Cliffs: Prentice Hall. López-Belchí, A., Illán-Gómez, F., Cascales, J. R. G., & García, F. V. (2016). R32 and R410A condensation heat transfer coefficient and pressure drop within minichannel multiport tube. Experimental technique and measurements. Applied Thermal Engineering, 105, 118–131. Matkovic, M., Cavallini, A., Del Col, D., & Rossetto, L. (2009). Experimental study on condensation heat transfer inside a single circular minichannel. International Journal of Heat and Mass Transfer, 52(9–10), 2311–2323. Mi, Y., Ishii, M., & Tsoukalas, L. H. (1998). Vertical two-phase flow identification using advanced instrumentation and neural networks. Nuclear Engineering and Design, 184(2–3), 409–420. Mi, Y., Ishii, M., & Tsoukalas, L. H. (2001). Flow regime identification methodology with neural networks and two-phase flow models. Nuclear Engineering and Design, 204(1–3), 87–100. Mishima, K., & Ishii, M. (1984). Two-fluid model and hydrodynamic constitutive relations. Nuclear Engineering and Design, 82(2–3), 107–126. Mori, H., Yoshida, S., Ohishi, K., & Kakimoto, Y. (2000). Dryout quality and post-dryout heat transfer coefficient in horizontal evaporator tubes. In European thermal sciences conference, pp. 839–844. Nema, G., Garimella, S., & Fronk, B. M. (2014). Flow regime transitions during condensation in microchannels. International Journal of Refrigeration, 40, 227–240. Nicklin, D. J. (1962). Two-phase bubble flow. Chemical Engineering Science, 17(9), 693–702. Ong, C. L., & Thome, J. R. (2011). Macro-to-microchannel transition in two-phase flow: Part 1– Twophase flow patterns and film thickness measurements. Experimental Thermal and Fluid Science, 35(1), 37–47. Revellin, R., & Thome, J. R. (2007). A new type of diabatic flow pattern map for boiling heat transfer in microchannels. Journal of Micromechanics and Microengineering, 17(4), 788. https://doi.org/10.1088/0960-1317/17/4/016 Rouhani, S. Z. (1969). Modified Correlations for Void-Fraction and Pressure Drop. AB Atomenergi Sweden, AE-RTV-841, 1–10. Stewart, R. W. (1967). Mechanics of the Air—Sea Interface. The Physics of Fluids, 10(9), S47–S55. Taitel, Y., & Dukler, A. E. (1976). A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AICHE Journal, 22(1), 47–55. Taitel, Y., Barnea, D., & Dukler, A. E. (1980). Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AICHE Journal, 26(3), 345–354. Wallis, G. B. (1969). One-dimensional two-phase flow. McGraw Hill Book Company. Wojtan, L., Revellin, R., & Thome, J. R. (2006). Investigation of saturated critical heat flux in a single, uniformly heated microchannel. Experimental Thermal and Fluid Science, 30(8), 765–774. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005). Investigation of flow boiling in horizontal tubes: Part I – A new diabatic two-phase flow pattern map. International Journal of Heat and Mass Transfer, 48(14), 2955–2969. Zuber, N., & Findlay, J. (1965). Average volumetric concentration in two-phase flow systems. Journal of Heat Transfer, 87(4), 453–468. https://doi.org/10.1115/1.3689137

Chapter 4

Pressure Drop

The pressure drop corresponds to the pressure reduction during the fluid passage through a channel segment. It is an important parameter in the design of heat exchanger since it is directly related to the pumping power, which impacts the overall efficiency of the thermal system. Additionally, and as expected especially for two-phase flows, the variation of fluid pressure along the flow path affects fluid properties, such as vapor-specific volume and vapor quality, which might cause increment of flow velocity. The pressure reduction during gas-liquid flow of a substance under saturation conditions cause reduction of fluid temperature, which may impact the heat transfer performance and cause non-uniformity of the solid surface temperature. The pressure drop imposed to the flow can be related to the friction between the phases, friction between the fluid and the wall, variation of the flow height, and of flow velocity. These contributions are classified as frictional, gravitational, and accelerational parcels, respectively. The first parcel is dissipative and hence cannot be recovered, while the gravitational and accelerational parcels are conservative and can be recovered. In this textbook, a mathematical relationship for these parcels is derived based on the Reynolds transport theorem for the conservation of mass and momentum quantity for a short duct segment for each phase. The Reynolds transport theorem for conservation of mass and momentum quantity, which can be found in undergraduate textbooks, for one of the phases is given, respectively, as follows: 0¼

∂ ∂t

Z

Z

!

ρdV þ CV

!

ρV  dA

ð4:1Þ

CS

Z Z X! X! ! ! ! ! ∂ F surface ¼ V ρdV þ V ρV  dA F body þ ∂t CV CS

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_4

ð4:2Þ

125

126

4 Pressure Drop

dz τwv

AI

Awv τI

τI τwl

m Awl

θ z

A x α p ρl ρv ul uv

g

A + dA x + dx α + dα p + dp ρl + dρl ρv + dρv ul + duv uv + duv

Fig. 4.1 Schematics of two-phase flow in a duct

Figure 4.1 schematically depicts the parallel and concurrent flow of vapor and liquid in a duct dz long with a not necessarily uniform cross-section. In the point z, the duct has area A and the flow is characterized by a vapor quality x, void fraction α, pressure p, liquid ρl and vapor ρv densities, and liquid ul and vapor uv axial velocities. Assuming that all the parameters might present variations along a duct length dz, these variables are represented by their original values plus an infinitesimal variation. Figure 4.1 also depicts the shear stress between the liquid and vapor phases with the duct wall, denoted, respectively, by τwv and τwv, and between the phases along the interface, denoted by τI. Since under conditions of positive inclination θ, it is expected that the vapor phase presents higher velocity than the liquid phase, the interfacial shear stress tends to reduce the vapor velocity and increase the liquid velocity. On the other hand, the shear stress with duct wall always tends to reduce the velocity of both phases. Based on the schematics presented in Fig. 4.1, it is convenient defining the mean values of the parameters for the length dz. In the

4 Pressure Drop

127

present development, the domain dimension for the evaluation of the space average operator will be suppressed: A þ A þ dA dA ¼Aþ 2 2 α þ α þ dα dα ¼αþ < α >¼ 2 2 p þ p þ dp dp < p >¼ ¼pþ 2 2 ρ þ ρl þ dρl dρ < ρl >¼ l ¼ ρl þ l 2 2 ρv þ ρv þ dρv dρv < ρv >¼ ¼ ρv þ 2 2 < A >¼

ð4:3Þ ð4:4Þ ð4:5Þ ð4:6Þ ð4:7Þ

Moreover, the definition of the volume occupied by each phase is useful in the foregoing development and is given as follows:    dA dα Aþ αþ dz 2 2    dA dα 1α dz V l ¼< A > ð1 < α >Þdz ¼ A þ 2 2    d ð1  αÞ dA ¼ Aþ 1αþ dz 2 2 V v ¼< A >< α > dz ¼

ð4:8Þ

ð4:9Þ

In this development, it is assumed that the system is under steady state condition (∂/∂t ¼ 0) and that the velocity profiles are uniform along the cross-sections; therefore, even though the over bar sign was not introduced in this development to avoid excessive pollution of the equations that are already long, the flow velocities depicted in Fig. 4.1 consist on the mean values evaluated for each phase along the cross-section. Additionally, the following simplifying hypotheses are assumed along a cross-section: uniform transport properties for liquid and gas, and the pressure is uniform. Mass Conservation The mass conservation given by Eq. (4.1) assuming the above-mentioned hypothesis for the gas phase is given as follows:

Z

!

!

ρV  dA ¼ ρv uv Aα þ ðρv þ dρv Þðuv þ duv ÞðA þ dAÞðα þ dαÞ  m_ I

0¼ CS,v

ð4:10Þ where ṁI corresponds to the mass flow rate from the liquid to the vapor phase across the interface, and its occurrence is associated to the phase change process. It is

128

4 Pressure Drop

assumed that the sign of ṁI is negative for condensation; therefore, the present development is valid for both cases. This term can be rewritten as ṁdx, where dx is the variation of vapor quality along the length dz. Simplifying Eq. (4.10), neglecting multiplication of infinitesimal terms, the following relationship is obtained: _ 0 ¼ dðρv uv AαÞ  mdx

ð4:11Þ

The first term of the right side in Eq. (4.11) corresponds to the variation of vapor mass flow rate along the infinitesimal length dz, and the last one corresponds to the mass transfer due to phase change, which must be equal for non-penetrating walls. Similarly, the same approach can be applied for the liquid phase as follows: Z 0¼

!

!

ρV  dA CS,l

¼ ρl ul Að1  αÞ þ ðρv þ dρv Þðuv þ duv ÞðA þ dAÞð1  α  dαÞ þ m_ I

ð4:12Þ

and after the simplifications we obtain the following relationship: _ 0 ¼ dðρl ul Að1  αÞÞ þ mdx

ð4:13Þ

where the first term of the right side corresponds to the variation of liquid mass flow rate along the infinitesimal length dz, and the last term corresponds to the mass transferred due to phase change, and can be considered as the linking term between the phases. Momentum Quantity Conservation In a similar way to the development performed for mass conservation, the momentum conservation is evaluated for each phase separately, using an interfacial term to link the phases. The momentum balance for the control volume defined by the vapor phase in the axial direction is given by Eq. (4.2) for the z direction as follows: τwv Awv  τi Ai þ pAα  ðp þ dpÞðA þ dAÞðα þ dαÞþ < p > dA < α > ρv V v gsinθ ¼ 0  uv ρv uv Aα þ ðuv þ duv Þ½ðρv þ dρv Þðuv þ duv ÞðA þ dAÞðα þ dαÞ _  mdxV ð4:14Þ I where VI corresponds to the axial component of the velocity of mass flowing across the interface. Notice that the fifth term on the left-hand side refers to the force that pressure imposes to the vapor due to variation of the cross-sectional area, and the mean value of pressure and void fraction were considered for its estimative. The term in square brackets of Eq. (4.14) is equal to the second term of the last member in Eq. (4.10). Additionally, the term of mean values can be rewritten based

4 Pressure Drop

129

on Eqs. (4.3), (4.4), (4.5), (4.6), and (4.7). Therefore, the above equation can be simplified as follows, where the terms corresponding to product of infinitesimal terms are neglected: τwv Awv  τi Ai  Ad ðpαÞ  Aαρv gsinθdz ¼ ρv Aα

du2v _  V I mdx _ ð4:15Þ þ uv mdx 2

Similarly, for the liquid phase the momentum conservation equation can be derived for the liquid phase, and the resulting relationship is given as follows: τwl Awl þ τi Ai  Adp þ Ad ðpαÞ  Að1  αÞρl gsinθdz ¼ ρl Að1  αÞ

du2l _ þ V I mdx _  ul mdx 2

ð4:16Þ

We can evaluate the momentum conservation for the mixture in the element based on the sum of Eqs. (4.15) and (4.16). In order to simplify the relationship: ðτwv Awv þ τwl Awl Þ  Adp  A½αρv þ ð1  αÞρl gsinθdz ¼ ρv Aα

du2v du2 _  ul mdx _ þ ρl Að1  αÞ l þ uv mdx 2 2

ð4:17Þ

Dividing Eq. (4.17) by A.dz and simplifying, we obtain the following relationship: 

ðτwv Awv þ τwl Awl Þ dp   ½αρv þ ð1  αÞρl gsinθ Adz dz ¼

ρv α du2v ρl ð1  αÞ du2l m_ dx þ þ ð uv  ul Þ A dz dz 2 dz 2

ð4:18Þ

Hence, rearranging:   ðτwv Awv þ τwl Awl Þ dp  f½αρv þ ð1  αÞρl gsinθg ¼ Adz dz   ρv α du2v ρl ð1  αÞ du2l m_ dx  þ þ ð uv  ul Þ 2 A dz dz 2 dz

ð4:19Þ

Therefore, according to Eq. (4.19) the pressure drop gradient can be considered as a combination of three main parcels. The first term on the right-hand side of this equation corresponds to the friction between both phases and the duct wall, and can therefore be considered as a frictional parcel. The second term corresponds to the pressure variation due to gravity, where the fluid density is averaged based on the void fraction. The last term corresponds to pressure variation due to the change of fluid momentum. Thus, Eq. (4.19) can be rewritten as follows:

130

4 Pressure Drop

      dp dp dp dp ¼   dz dz frictional dz gravitational dz accelerational

ð4:20Þ

where each parcel is given as follows:     ðτwv Awv þ τwl Awl Þ dp ¼ Adz dz frictional   dp ¼ f½αρv þ ð1  αÞρl gsinθg dz gravitational     ρv α du2v ρl ð1  αÞ du2l dp m_ dx þ þ ðuv  ul Þ ¼ 2 dz accelerational A dz dz 2 dz

ð4:21Þ ð4:22Þ ð4:23Þ

The shear stress between each phase with the duct wall, as well as the contact area between each phase and the duct wall, are difficult to estimate during two-phase flows by analytical and numerical methods, because these parameters depend on the phase distribution and velocity profile. Therefore, the frictional parcel of the pressure drop is usually evaluated based on prediction methods developed considering the adjustment of empirical constants. This parcel depends on the flow velocity, fluids properties, and duct geometry, among other parameters. The gravitational parcel of the pressure drop given by Eq. (4.22) depends on the area averaged void fraction. When integrating along a length of interest, the area averaged void fraction can be similar to the volume averaged void fraction. The correct estimation of this parameter is critical under conditions of reduced flow velocity, or for conditions of reduced friction, because this parcel can correspond to more than 90% of the total pressure drop. In fact, under conditions of vertical and inclined flows, the estimative of this pressure drop parcel is strongly dependent of the prediction method adopted for the estimative of the area averaged void fraction. On the other hand, under conditions of horizontal flow the duct inclination in relation to the horizontal plane is null, θ ¼ 0, therefore this parcel is null as well. The accelerational parcel of the pressure drop, given by Eq. (4.23), can be simplified adopting the relationship between the in situ velocity, mass flux, vapor quality, and fluid densities, given by Eqs. (2.20) and (2.21). Therefore, Eq. (4.23) can be rewritten as follows: 

( )   2  2   ρl ð1  αÞ d Gð1  xÞ ð1  xÞ ρv α d Gx dp x 2 dx  G ¼ þ þ dz accelerational dz ρl ð1  αÞ ρ v α ρ l ð 1  αÞ dz 2 dz ρv α 2

  2    ð 1  xÞ 2 ð1  xÞ2 dG2 d x 1 x2 2 ¼ þ þ G  dz 2 αρv ð1  αÞρl dz αρv ð1  αÞρl

ð4:24Þ

4 Pressure Drop

131

Notice that Eq. (4.24) can be derived from Eq. (4.23) only under conditions of two-phase flow, otherwise either α or (1-α) would be null, and the relationships between in situ and mass velocities would be invalid. Hence, imposing the limiting values for x (or α) tending to unity or zero directly in Eq. (4.24) is incoherent. For steady-state two-phase flow along a uniform cross-section duct and with null mass transfer across the walls, the mass flow rate and mass velocity are constant and hence the second term of the last member of Eq. (4.24) is null. In this case, the term G2 of the first term can be taken out of the derivative and the integration of pressure gradient can be evaluated based on the Fundamental Calculus Theorem, which is basically related to variation of phase fraction and densities. The contribution of this parcel can be considered negligible under conditions of adiabatic horizontal two-phase flow with low or moderate flow velocities in conventional channels; however, under conditions that the frictional or gravitational pressure drop parcels are significant, and during phase change along the channel, the variation of flow momentum cannot be neglected and this parcel must be taken into account. In this context, under conditions of high-pressure gradient, even during adiabatic flow, the variation of saturation pressure may impact the heat transfer process. Figure 4.2 depicts the pressure-enthalpy diagrams for R134a and R600a, and assuming a condition of isenthalpic two-phase flow, it can be noted in these plots that the reduction of pressure causes increment of vapor quality, which in turn results in increment of flow velocity, enhancing the frictional pressure drop parcel. According to Fig. 4.2, this effect is more prominent for R600a than for R134a for high specific enthalpy values, while for reduced enthalpy, such as exemplified in this figure, the variation of vapor quality for isenthalpic processes are of the same order for both fluids. Thus, the contribution of the vapor quality increment due to pressure reduction must be evaluated during the analysis of heat exchangers. The process of increasing the vapor content due to pressure drop is known as flashing effect and is important for conditions of high pressure drop such as occurring in reduced diameter channels, for example, in capillary tubes of refrigeration systems. Based on these derivations, it is possible to infer the importance of a precise estimation of the phases fraction, namely, void fraction and vapor quality. Chapter 2 introduced approaches for evaluation of vapor quality x based on energy balance and inlet properties. Moreover, Sect. 2.3 introduced predictive methods for estimating void fraction α. Some predictive methods for the frictional parcel of the pressure drop are addressed below. Nonetheless, some general aspects relating frictional pressure drop during two-phase flow can be discussed. Due to the fact that this textbook is focused on phase change processes, corresponding to evaporation and condensation, it is reasonable to analyze the parameters for a given mass flow rate. The reader might find some studies that deal mainly with superficial velocities, which is reasonable for adiabatic (or almost adiabatic) flows, such as verified for oil and gas industry.

132

4 Pressure Drop

Fig. 4.2 Pressure and enthalpy diagrams for R134a and R600a

Therefore, assuming a fixed mass velocity and saturation pressure, it is possible to estimate the frictional pressure drop parcels for liquid and vapor single-phase flows as follows: 

dp dz

 ¼ 2fi f ,i

G2i ρi d

ð4:25Þ

4 Pressure Drop

133

20000 R134a, d = 5 mm, G = 300 kg/m²s

Linear interpolation (dp/dz)f,v

dp/dz [Pa/m]

15000

Tsat = -10 °C

10000

Tsat = -5 °C

5000

0 0.0

(dp/dz)f,l

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 4.3 Pressure drop for single and two-phase flows (linear interpolation)

where the friction factor for round channel can be determined based on the Hagen– Poiseuille velocity profile for laminar flow, or Blasius correlation for turbulent flow regime as follows: f i ¼ c1 = Re m i

ð4:26Þ

with c1 and m, respectively, equal to 16 and 1 for laminar flow regime, usually assumed for Reynolds numbers below 2300 for round channels, and c1 ¼ 0.079 and m ¼ 0.25 for turbulent flow regime, assumed to occur for Reynolds numbers higher than 10,000, and the region between these limits is considered as a transitional condition. Again, c1 ¼ 0.046 with m ¼ 0.2 are also commonly seen in the literature for turbulent flow regime. The frictional pressure drop gradient for liquid and vapor single-phase flows are estimated according to Eq. (4.25), and are depicted in Fig. 4.3 for R134a flowing in a 5 mm ID channel at 10  C, and they must be the limits for two-phase flow conditions, corresponding to x!0 and x!1, respectively. Even though the viscosity of the liquid phase is higher than that of the vapor phase, the pressure drop of the vapor phase is significantly higher due to the respective phases’ velocities, which are related to the specific volumes. For example, for R134a at 10  C the liquid and vapor densities are, respectively, 1327 and 10.05 kg/m3; therefore, for the same mass velocity, the vapor single-phase flow velocity for a given mass flow rate would be more than 130 times higher than the liquid velocity. In comparison, for the same saturation temperature the liquid and vapor viscosities are approximately,

134

4 Pressure Drop

respectively, 3.02104 and 1.05105 kg/m.s, a ratio of approximately 30 times. Similar velocity ratios are also found for different fluids and saturation pressures. Another important point is related to the effect of saturation temperature and pressure. The property that has higher impact on the pressure drop is the specific volume of the vapor phase. For instance, considering the conditions depicted in Fig. 4.3 for R134a, namely, saturation temperatures of 10 and  5  C, the corresponding liquid viscosities are 3.02104 and 2.83104 kg/m.s (variation of 6.3%), and the vapor viscosities are, respectively, 1.05105 and 1.07105 kg/m.s (variation of 1.9%); while the vapor-specific volume varied from 0.09952 m3/kg at 10  C to 0.08274 m3/kg (variation of 16.9%). The frictional pressure drop parcel for vapor single-phase flow reduced by 16.5% for the same temperature variation (10 to 5  C), which indicates that the variation of flow velocity is the dominant effect on the pressure drop variation. Again, similar results are verified for other fluids and saturation pressures/temperatures. Returning to the discussion about the frictional pressure drop parcel during liquid and vapor simultaneous flow, the reader might be wondering whether considering a linear interpolation between the pressure drop for liquid and vapor flows to estimate the frictional pressure drop during liquid and vapor simultaneous flow as a function of the vapor quality is reasonable, such as depicted in Fig. 4.3. In fact, this hypothesis makes sense, however due to the flow complexity and interaction between the phases, the mechanical energy dissipation is higher than a combination of the pressure drop gradient for both phases. Therefore, the frictional pressure drop parcel for two-phase flow should be somewhat like the curve depicted for homogeneous flow in Fig. 4.4. 20000 R134a, d = 5 mm, G = 300 kg/m²s

dp/dz [Pa/m]

15000

(dp/dz)f,v

Homogeneous model

Tsat = -5 °C Tsat = -10 °C

10000

5000

0 0.0

(dp/dz)f,l

0.2

0.4

x [-]

0.6

0.8

Fig. 4.4 Pressure drop for single and two-phase flows (homogeneous model)

1.0

4 Pressure Drop

135

Alternatively, the increment of pressure drop beyond a linear interpolation can be interpreted as partial restriction of the channel cross-section by each phase, that is, the liquid portion constrains the vapor flow and vice versa. Hence, the pressure gradient increases for intermediate vapor qualities, and for reduced or high vapor qualities the flow tends to present characteristics more similar to single-phase flow. As shown in Fig. 4.4 and observed in most of the experimental studies from literature, the frictional pressure gradient increases with quality until it reaches a peak. Further increments in the vapor quality promote the reduction of the pressure drop gradient. Based on experimental results for the liquid film behavior during annular flow of R245fa, Moreira et al. (2020) pointed out that at high vapor quality, when the liquid film is very thin, the liquid–vapor interface becomes smoother and disturbance waves slow down and vanish. They linked this shift in the trend of the pressure gradient with vapor quality to the disappearance of disturbance waves. It was mentioned in Chap. 3 that the local flow pattern should impact several twophase flow parameters, including the pressure drop, which is verified in practice. Therefore, few predictive methods for the frictional pressure drop parcel account for the local flow pattern. Another important aspect when selecting a predictive method for the frictional pressure drop parcel is related to the attainment of the limiting conditions of saturated vapor and liquid. Some predictive methods are developed based on adjustment of empirical parameters with experimental results for a limited range, for example, comprising only low vapor quality values. Therefore, it is not uncommon to find predictive methods whose estimative for x equal unity is different than that for vapor single-phase flow. Hence, the reader must check the range of validity of the proposed methods. It has been verified experimentally by several researchers that the vapor quality value corresponding to the maximum pressure drop reduces with the increment of mass velocity, as shown in Fig. 4.5. Therefore, it is desirable that the predictive method satisfactorily captures this trend. In the case of flow in vertical or inclined channels, the gravitational pressure drop parcel might correspond to a significant parcel of the total pressure drop, specifically for reduced or intermediate flow velocities and high liquid content, which emphasizes the need for reliable and precise void fraction predictive methods, as indicated by Eq. (4.22). In this context, Lavin, Kanizawa, and Ribatski (2019) presented an analysis of the effect of channel inclination on flow patterns and pressure drop, and based on their analyses, the authors pointed out that the experimental determination of the frictional pressure drop parcel for non-horizontal channels relies on the selection of appropriate void fraction predictive method, and depending on the chosen method, even negative values for Δpf might be obtained, such as illustrated in Fig. 4.6 from their study, which is incoherent with the physical problem. Hence, when evaluating frictional pressure drop parcel for vertical or inclined channels by using a method developed based on experimental results, it is necessary to check which void fraction predictive method was adopted for data regression, and adopt it to estimate the total pressure drop. Otherwise, the error on the estimation of the total pressure drop might be significant.

136

4 Pressure Drop

G1

dp/dz [Pa/m]

G2 G3 G1 > G 2 > G 3

0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 4.5 Schematics of the effect of mass velocity on the pressure drop

1.0

5 4 Downward flow θ =-90°

0.8

2 1

0.6

0

α [-]

∆ pf / L [kPa/m]

3

-1 Channel rotation =0°

-2 -3

j l = 0.195 m/s G È 195 - 200 kg/m²s

-4 -5 0.1

0.4

Channel rotation = 30° Channel rotation = 60° Channel rotation = 90°

1

jv [m/s]

10

0.2 20

Fig. 4.6 Variation of pressure drop with gas superficial velocity for air-water flow in rectangular channel with hydraulic diameter of 6.24 mm (Lavin et al. 2019)

Finally, we try to attain to the original author when describing the predictive methods, hence, depending on the school and habit, some researchers prefer to use the Darcy friction factor Cf while others prefer the Fanning friction factor f. The reader should remember their definition from the fluid mechanics course.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

137

Nonetheless, their equivalence is simply given by a product Cf ¼ 4f (the Darcy type friction factor is equivalent to 4 times the Fanning type friction factor), and the frictional pressure drop relationship corresponds to each accordingly.

4.1

Predictive Methods for Frictional Pressure Drop Parcel

This section addresses predictive methods for the frictional pressure drop parcel, given by Eq. (4.21). Due to the complexity of the two-phase flow, it is not feasible to determine the velocity profiles and precise geometrical distribution of each phase, therefore it is a difficult task to determine the frictional pressure drop parcel analytically. Thus, the pressure drop predictive methods are usually based on simplifying assumptions followed by adjustment of empirical parameters. One of the simplest approaches, denominated as homogeneous model, is based on the assumption that the mixture behaves as a pseudo fluid with averaged properties, and for this approach the main difficulty is the determination of the mixture equivalent viscosity for the estimative of the Reynolds number and friction factor. One of the most common approaches for frictional pressure drop estimative is based on the two-phase multipliers, which consists in multiplying the single-phase pressure drop, evaluated based on the friction factor for the corresponding flow regime, by an estimated factor. For laminar flow, the friction factor is determined based on the Hagen–Poiseuille velocity profile, and for turbulent flow regime it can be estimated based on Blasius, Colebrook (1939), Churchill (1977), or any other method. The first proposal for the two-phase flow multiplier was developed by Lockhart and Martinelli (1949), and is described below. Subsequent to these authors, several adjustments and methods were proposed based on this method, such as Friedel (1979) and Grönnerud (1979). Finally, there are also predictive methods that are purely empirical, and one of the most known and reliable among them is the method proposed by Müller-Steinhagen and Heck (1986). Additionally, there are methods based on the pre-definition of the local flow pattern, which adopts distinct predictive methods for each flow pattern, such as the methods proposed by Moreno-Quibén and Thome (2007). Cheng et al. (2008) for CO2, and Cioncolini et al. (2009) for annular flow in micro and conventional channels. In general, reasonable pressure drop predictions are obtained through methods that do not take into account the flow patterns, and, therefore, are easier to implement.

4.1.1

Homogeneous Model

The homogeneous model is the simplest frictional pressure drop predictive method, and assumes that the mixture is a pseudo-fluid with averaged properties, which is

138

4 Pressure Drop

A

A

Av mv x

x

z y

m

Al

g

z y

m

g

ml

Fig. 4.7 Assumptions of the homogeneous model, from two-phase flow to a pseudo fluid

schematically depicted in Fig. 4.7, and the pressure drop is evaluated assuming single-phase correlations. Therefore, this approach is more coherent with conditions of dispersed flow such as dispersed bubbles or mist flow. In this model, the frictional pressure drop parcel is evaluated as follows:   dp G2 ¼ 2fm dz tp ρm d

ð4:27Þ

where the mixture density ρm is evaluated based on the specific volume of each phase assuming equilibrium condition as follows:  ρm ¼

x 1x þ ρv ρl

1 ð4:28Þ

In Eq. (4.27), the sub index tp refers to the frictional pressure drop parcel during two-phase flow, for cleanliness of the text. Since we are considering that the mixture behaves as a pseudo-fluid, the friction factor can be evaluated based on the mixture Reynolds number as follows: fm ¼

C Re nm

ð4:29Þ

where the constant C and exponent n are, respectively, 16 and 1 for laminar flow and, respectively, 0.046 and 0.2 for turbulent flow (0.079 and 0.25 are also commonly used). The mixture Reynolds number is given as follows: Re m ¼

Gd μm

ð4:30Þ

The main challenge here is determining the mixture viscosity, which cannot even be determined for a mixture in equilibrium as performed for other parameters, such

4.1 Predictive Methods for Frictional Pressure Drop Parcel

139

as mixture-specific volume and enthalpy that can be estimated based on the vapor quality. In this token, Wallis (1969) addresses several models to estimate the mixture viscosity. For dispersed flow with low concentration of the dispersed phase, especially for a condition of solid aspherical particles, the Einstein model provides reasonable predictions and is given by the following relationship: μm ¼ μc ð1 þ 2:5αd Þ

ð4:31Þ

where μc stands for the continuous phase viscosity and αd refers to the volumetric fraction of the dispersed phase. In the case of dispersed gas bubbles in continuum liquid, where the gas has low viscosity, the mixture viscosity is given as follows: μm ¼ μc ð1 þ αd Þ

ð4:32Þ

For conditions distinct than dispersed flow, rheological models are usually not capable of correctly capturing the mixture viscosity behavior. In these cases, several methods have been proposed, accounting for the constraint of single-phase flows. The method proposed by McAdams et al. (1942) follows the same approach adopted for the estimative of mixture density, and is given as follows:  μm ¼

x 1x þ μv μl

1 ð4:33Þ

On the other hand, the method proposed by Cicchitti et al. (1959) is similar to the evaluation of the mixture-specific volume, and is given as follows: μm ¼ xμv þ ð1  xÞμl

ð4:34Þ

Finally, the method proposed by Dukler et al. (1964) considers the volumetric fraction of each phase as pondering parameter, and the mixture viscosity is estimated as follows: μm ¼ βμv þ ð1  βÞμl

ð4:35Þ

where the volumetric fraction β is given by Eq. (2.24). Several studies available in the open literature compared the experimental results for two-phase flow of liquid and condensable vapor with the homogeneous model, and the adoption of Cicchitti et al. (1959) approximation of the mixture viscosity provided reasonable agreement with experimental results for microchannels. This result seems to be associated with the fact that the phase slip is deteriorated in microchannels in comparison with conventional channels. Recall the homogeneous model for void fraction, which explicitly assumes that the phases flow at the same velocity.

140

4 Pressure Drop

(dp dz )

Av A

x

mv

z y

Av

+

mv x

z y

v

m

Al

dp dp (dp dz ) = (dz ) = (dz ) v

l

tp

g

ml

(dp dz )

l

x

Al

z

g

ml y

Fig. 4.8 Assumptions of the Lockhart and Martinelli model, from two-phase flow in a channel to single-phase flows in two channels

4.1.2

Lockhart and Martinelli (1949)

As above described, and according to the present authors’ knowledge, Lockhart and Martinelli (1949) were the first to propose the prediction of the frictional pressure drop parcel during two-phase flow based on two-phase multipliers. Even though the method does not provide satisfactory predictions of experimental results for mini and microchannels, it is presented here for historical reasons and because it is didactical for two-phase flow students. Nonetheless, subsequent improvements and adjustments, as performed by Chisholm (1967) among others, have turned this method appropriate for a variety of applications, including phase-change flows in minichannel, and across tube bundles. This method can be considered as a two-fluid model because it considers the existence of two parallel flow streams, one corresponding to the liquid phase and another to the vapor phase, each one flowing along an imaginary channel with a characteristic hydraulic diameter, as schematically depicted in Fig. 4.8. Additionally, the model is based on the following simplifying assumptions: • The static pressure in each section of the liquid flow is equal to the static pressure in the vapor flow. Therefore, the pressure gradient is similar for both phases. • The sum of the volume (area) occupied by liquid and vapor phase must equal to the channel volume (area).

4.1 Predictive Methods for Frictional Pressure Drop Parcel

141

Table 4.1 Possible combinations of flow regimes ll lt tl tt

Liquid phase Laminar Laminar Turbulent Turbulent

Vapor phase Laminar Turbulent Laminar Turbulent

Condition Rehl < 1000 Rehl < 1000 Rehl > 2000 Rehl > 2000

Rehv < 1000 Rehv > 2000 Rehv < 1000 Rehv > 2000

c1,l 16 16 0.046 0.046

c1,v 16 0.046 16 0.046

ml 1.0 1.0 0.2 0.2

mv 1.0 0.2 1.0 0.2

Based on the above-mentioned hypothesis, the following relationships can be addressed: 

 ρ u2 dp ¼ 2 f hl l l dz tp dhl   ρ u2 dp ¼ 2 f hv v v dz tp dhv

ð4:36Þ ð4:37Þ

where dhl and dhv are the hydraulic diameters of the liquid and vapor phases, respectively, and fh corresponds to the friction factor which is a function of the Reynolds number based on the hydraulic diameter of each phase, as schematically depicted in Fig. 4.8. At this point the reader might be wondering which flow regime, laminar or turbulent, should be considered to estimate the friction factor, and in fact, it is possible to have four combinations of flow regimes, as shown in Table 4.1, which also addresses the flow regime transition criteria that will be discussed subsequently. The pressure drop during two-phase flow is higher than that for the case of only one of the phases flowing in the channel, which is a consequence of the cross-section reduction of each phase and the interfacial forces that results in non-reversible work. Therefore, it is reasonable to expect that the hydraulic diameters dhl and dhv are smaller than the channel diameter. Moreover, Lockhart and Martinelli (1949) defined the hydraulic diameters based on classical definition as follows: dhl ¼ 4Að1  αÞ=dπψ l ¼ dð1  αÞ=ψ l

ð4:38Þ

dhv ¼ 4Aα=dπψ v ¼ dα=ψ v

ð4:39Þ

where the terms ψ l and ψ v correspond to the correction factor for the non-circularity of the phases’ geometries. These parameters are in principle determined from experiments; therefore, they consider the slip between the phases. The friction factor is given as a function of the Reynolds number, assuming the Blasius form as follows: f hi ¼

c1i Re mi h,i

ð4:40Þ

142

4 Pressure Drop

where the coefficient and exponent are presented in Table 4.1, for each phase i. The Reynolds number in this equation is given as a function of the in situ velocity and of the hydraulic diameter, given by Eqs. (4.38) or (4.39), as follows: Re h,i ¼

ρi ui d h,i μi

ð4:41Þ

By equating Eqs. (4.36) and (4.37), which is one of the main assumptions of the model proposed by Lockhart and Martinelli, the following equation is obtained:   ρ u2 ρ u2 dp ¼ 2 f hl l l ¼ 2 f hv v v dz tp dhl dhv

ð4:42Þ

2v c1,l ρl u2l c1,v ρv u ¼ m,v dhl Re h,v dhv Re m,l h,l

ð4:43Þ

2v ρl u2l ρv u c1,l c1,v m,l d ¼ ρ u d m,v d ρl ul d h,l hl v v h,v hv

ð4:44Þ

ρl j2l ρv j2v c1,l c1,v m,l dð1αÞ3 ¼ ρ j d m,v dα3

ð4:45Þ



μv

μl



ρl jl d μl ψ l



ψl

v v

ψv

μv ψ v



m,l

c1,l

1  α 3 ψ 1þm,v v ¼ l α ψ 1þm

ρ l

ρl j l d μl

c1,v

v jv d μv

m,v

ρl j2l d ρv j2v d

ð4:46Þ

We can recognize that the term on the right side of Eq. (4.46) corresponds to the pressure drop gradient ratio, considering only the liquid and vapor phases flowing in the channel, even though the friction factor coefficients and exponents are estimated based on the actual velocities and flow area. This ratio is defined as the square of Lockhart and Martinelli parameter, given as follows: #  )1=2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdp=dzÞl ρl j2l ρv j2v c1l c1v ¼ = ðdp=dzÞv ðρv jv d=μv Þmv d ðρl jl d=μl Þml d

(" b¼ X

ð4:47Þ

where X^ is the Lockhart and Martinelli parameter, also used by Taitel and Dukler for the development of their flow pattern transition criteria as previously shown in Chap. 3. Conversely, the left-hand side of Eq. (4.46) represents the unknown parameters that we want to estimate.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

143

Returning to the pressure drop relationship, given by Eq. (4.36):   ρ u2 ρl j2l dp c1l ¼ 2 f hl l l ¼ 2 dz tp d hl ðρl jl d hl =μl ð1  αÞÞml ð1  αÞ2 dhl   ρl j2l dp c1l ¼2 ml dz tp ðρl jl d=μl ψ l Þ ð1  αÞ3 d=ψ l   ψ 1þml ρl j2l dp c1l l ¼ 2 dz tp ð1  αÞ3 ðρl jl d=μl Þml d

ð4:48Þ ð4:49Þ ð4:50Þ

that can be rewritten as follows: 

dp dz

 ¼ tp

Φ2l

  dp dz l

ð4:51Þ

where the term Φ2l is the liquid two-phase multiplied, and (dp/dz)l is the pressure drop evaluated assuming only the liquid phase flowing in the channel with diameter d. A similar result is obtained for the vapor phase, from Eq. (4.37), which is given by: 

dp dz

 tp

    ψ 1þmv dp 2 dp v ¼ ¼ Φv dz v dz v α3

ð4:52Þ

Both two-phase multipliers, Φ2l and Φ2v , are experimentally determined, and according to Eqs. (4.46), (4.47), (4.51), and (4.52), they seem to be related to the Lockhart and Martinelli parameter X^. Hence, using two-phase experimental results for air-water, air-benzene, and air-kerosene for diameters ranging from 1.5 to 25.8 mm, the original authors verified whether these parameters are in fact a function of X^, which was confirmed. The authors plotted a variation of two-phase multipliers as a function of the Lockhart and Martinelli parameter for distinct combinations of flow regimes and confirmed that X^ is the main variable for representation of the two-phase multipliers. It should be mentioned that this confirmation was based on analysis of di-log graphs, which hide significant data dispersion, therefore it might not be so convincing and concluding. Nonetheless, deviations higher than 30% for studies in two-phase area, even among experimental results for similar conditions and between experimental and predicted results, are reasonable, common, and marginally acceptable; hence, the conclusions presented by Lockhart and Martinelli are reliable enough for applications. Moreover, this study was the first to address the concept of two-phase multiplier, which is adopted until nowadays. Lockhart and Martinelli (1949) addressed tabular data for the relationship between two-phase multiplier and the parameter X^, which are depicted graphically in Fig. 4.9.

4 Pressure Drop 2

1.0 x10

2

Φ l [-]

2.0 x10

1.0 x10

Turbulent - Turbulent Laminar - Turbulent Turbulent - Laminar Laminar - Laminar

1

0

1.0 x10 10 -2

10 -1

2

1.0 x10

2

1.0 x10

1

1.0 10 2

10 1

10 0

2.0 x10

Φ v [-]

144

X [-] Fig. 4.9 Two-phase multiplier according to Lockhart and Martinelli (1949)

10

-1

a [-]

5 x10

0

2 x10

-1

10

-1

10 -2

10 -1

10 0

10 1

10 2

X [-] Fig. 4.10 Void fraction value according to Lockhart and Martinelli (1949)

According to Eq. (4.46), it can be speculated that the void fraction should also be a function of the Lockhart and Martinelli parameter, which was also proofed from analysis of experimental results by the original authors, and the relationship is depicted graphically in Fig. 4.10.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

145

The term X^ can be presented in a more rapidly useful form, as a function of phase properties, and in the case of phase change problems, as a function of the vapor quality as follows: b ¼ X 2

ρl j2l c1l ðρl jl d=μl Þml d ρv j2v c1v ðρv jv d=μv Þmv d

b¼ X



c1l c1v

¼

1=2

1ml ðGð1  xÞ=ρl Þ2ml c1l mvml μml l ρl d mv 1mv c1v μ v ρv ðGx=ρv Þ2mv

ðdGÞðmvmlÞ=2



μml l ρv μmv v ρl

1=2

ð1  xÞ1ml=2 x1mv=2

ð4:53Þ

ð4:54Þ

from which, the most seen is the Lockhart and Martinelli parameter for turbulent flow regimes for both phases (c1l ¼ c1v, and ml ¼ mv ¼ 0.2):  0:9  μ 0:1 ρ 0:5 l v b tt ¼ 1  x X x μv ρl

ð4:55Þ

which takes the sub index tt, which was suppressed until this point. Then, along this textbook, whenever the Lockhart and Martinelli parameter is mentioned with distinct combinations of sub-indexes (ll, lt, tl or tt), the first one is relative to the liquid phase and the second to the gas phase. Another point is related to the flow regime transition criteria presented in Table 4.1. Lockhart and Martinelli performed a qualitative analysis of the conditions that would contribute to the transition from laminar to turbulent flow regime, and concluded that, different from that verified for single-phase flows in round channels, the laminar flow regime occurs for Reynolds numbers lower than 1000, and turbulent flow regime occurs for Reh > 2000. These limits are distinct from the ones commonly assumed for single-phase flow in round channels, which correspond to laminar flow for Re  2300 and turbulent for Re > 10,000, and the difference is caused by the interaction between the phases. Nonetheless, when determining which group of parameters c1 and m would be selected, the reader does not necessarily have the in situ velocity; therefore, the present authors suggests to evaluate the Reynolds number based on the actual diameter and superficial velocities, and then correct the estimations based on void fraction values if the obtained values are close to the regime transition. The Lockhart and Martinelli method is graphically based, therefore it is not promptly implemented computationally; hence, subsequent improvements were proposed by other researchers, such as the Chisholm (1967) approach, which is more widely used than Lockhart and Martinelli’s original method. In this context, it is appropriate to introduce alternate forms of the two-phase multipliers, given by Φ2l0 and Φ2v0 that are related to the factor to be multiplied by the pressure drop of the mixture flowing as liquid and vapor, respectively.

146

4 Pressure Drop

Table 4.2 Coefficients C for Chisholm (1967) correlation of two-phase multiplier

4.1.3

Liquid Turbulent Laminar Turbulent Laminar

Vapor Turbulent Turbulent Laminar Laminar

C 20 12 10 5

Chisholm (1967)

Subsequently to the proposal presented by Lockhart and Martinelli (1949), several authors have contributed to the development and improvement of predictive methods for frictional pressure drop parcel during vapor-liquid two-phase flow, and several of them are based on the concept of two-phase multiplier, such as Chisholm (1967). In his study, Chisholm presented an extended theoretical analysis of the Lockhart and Martinelli model including several aspects in the analysis, such as shear force. However, in conclusion the author proposed a simplified correlation to estimate the two-phase multiplier, which is given as follows: Φ2l ¼ 1 þ

C 1 þ 2 b b X X

ð4:56Þ

where the term X^ is the Lockhart and Martinelli parameter, defined by Eq. (4.47) or (4.54). The parameter C depends on the flow regimes, presented in Table 4.2, which are valid for two-phase flow mixtures with density ratio similar to that of air-water close to atmospheric pressure. Different from Lockhart and Martinelli (1949), Chisholm adopted a value of 2000 for Reynolds number as transition between laminar and turbulent flow regimes for each phase. Alternatively, Eq. (4.56) can be rewritten as a two-phase multiplier for the vapor phase as follows: bþX b2 Φ2v ¼ 1 þ C X

ð4:57Þ

with the C parameter also given by Table 4.2. Subsequently to these studies, several investigators have tried to correlate the parameter C of Eq. (4.56) for distinct flow condition and duct geometry. This approach is adopted even for external flow across tube bundles, such as in the study presented by Ishihara et al. (1980), who assumed the condition of turbulent– turbulent regimes for both streams and assumed the friction factor for internal flow in round smooth pipes. Specifically for the case of internal flow inside channels with reduced diameter, the correlation for the parameter C proposed by Mishima and Hibiki (1996) was developed for reduced scale channels, which can be considered as an adjustment of the constant value 20 of Table 4.2 as a function of the channel diameter. The

4.1 Predictive Methods for Frictional Pressure Drop Parcel

147

experimental database used for its development comprises results for air-water flows in channels with internal diameter ranging from 1.05 to 4.08 mm, made of glass and aluminum, and the resulting correlation to be used with Eq. (4.56) is given as follows1:  C ¼ 21 1  ed=d0

ð4:58Þ

where d0 is a reference diameter equal to π mm. Equation (4.58) is indicated for conditions comprising density ratios similar to that of air-water flows (ρl / ρv  800) for horizontal and vertical channels, in round and rectangular channels assuming the hydraulic diameter as the characteristic dimension. This relationship seems to be adequate for turbulent–turbulent flow regimes. Adopting the same approach, Zhang et al. (2010) proposed alternate correlations for the parameter C of Chisholm’s method, which were adjusted based on experimental results for air-water, water-N2, R113, R134a, R22 among others fluids, in horizontal and vertical channels with internal diameter ranging from 0.014 to 6.25 mm for diabatic and adiabatic conditions, and for liquid and vapor Reynolds numbers below 2000. The parameter C is given as follows:  C ¼ 21 1  e0:358=La

ð4:59Þ

where La* is the modified Laplace number, which is given as follows: La ¼

1 d

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ gðρl  ρv Þ

ð4:60Þ

where the internal diameter d is substituted by the hydraulic diameter for non-circular channels. The reader shall find several alternate forms for the parameter C and/or for the two-phase multipliers, Φ2l and Φ2v, as a function of Froude, Weber, and Reynolds numbers, among others non-dimensional parameters. Alternatively, the reader might also find predictive methods for two-phase multipliers that assume the pressure drop of the mixture flowing as liquid or vapor, which are given, respectively, as Φ2l0 and Φ2v0. In this context, the predictive method proposed by Friedel (1979) can be addressed, according to which the frictional pressure drop parcel during two-phase flow is given as follows:

The original correlation is given as C ¼ 21(1  e0.319d), but to be coherent with the units, the present authors presented the exponential as a diameter ratio.

1

148

C

f ,i0

4 Pressure Drop

¼

    dp 2 dp ¼ Φl0 ð4:61Þ dz tp dz l0   ρ C f ,v0 Φ2l0 ¼ ð1  xÞ2 þ x2 l ρv C f ,l0  0:91  0:19  0:7 0:78 3:21x ð1  xÞ0:224 ρl μv μv 1 ð4:62Þ þ ρv μl μl Fr 0:0454 We0:035 tp tp  64= Re i0 for Re i0  1055 f0:86859 ln ½ Re i0 =ð1:964 ln ð Re i0 Þ  3:8215Þg2

ð4:64Þ

G2 gdρ2H

ð4:65Þ

G2 d ρH σ

ð4:66Þ

Wetp ¼  ρH ¼ 

dp dz



x 1x þ ρl ρv ¼C

l0

Re i0 > 1055 ð4:63Þ

Gd μi

Re i0 ¼ Fr tp ¼

for

f ,l0

1

G2 2dρl

ð4:67Þ ð4:68Þ

Notice that the Reynolds number and the single-phase pressure drop are evaluated for the two-phase mixture flow rate (G) rather than the phase flow rate (G(1-x) for liquid, or Gx for vapor), which characterizes the difference from the subscripts l0 and l. This method was developed based on an experimental database comprising more than 25,000 experimental results for frictional pressure drop during single and two-component mixtures flowing in channels with internal diameter ranging from 1 to 260 mm. At this point, the significance of the non-dimensional parameters adopted in this method are discussed. The Froude number is associated to the balance between inertial and buoyancy effects, whereas high values indicate predominance of inertial forces. It is commonly observed in literature that gravitational effects on the two-phase topology become negligible during two-phase flows in minichannels; therefore, this method might not be appropriate for two-phase flow in miniscale channels. For instance, the Weber number corresponds to the balance between inertial and surface tension effects, where low values represent predominance of surface tension effects, and in the case of two-phase flow inside minichannels the contribution of surface tension effects is significant. Considering the fact that the two-phase multiplier reduces with the increment of Froude and Weber numbers, which possess exponents of similar magnitude, the effect of surface

4.1 Predictive Methods for Frictional Pressure Drop Parcel

149

tension effects compensates the negligible effect of buoyance forces during two-phase flow in microchannels.

4.1.4

Müller-Steinhagen and Heck (1986)

Müller-Steinhagen and Heck proposed an empirical method for prediction of frictional pressure drop parcel during two-phase flow, which is given as a combination of the pressure drop during single-phase flows of vapor and liquid podered as a function of the vapor quality. This method has been elaborated based on the qualitative variation of pressure drop with vapor quality, as discussed before and depicted in Figs. 4.3 and 4.4, that is, the frictional pressure drop parcel should present positive first-order derivative, for low and intermediate vapor quality values, and negative second-order derivative, attaining to the single-phase pressure drops. The coefficients and exponents of this method have been adjusted based on an experimental database comprising more than 9300 experimental data points for wide ranges of channel diameter (down to 5 mm), reduced pressure, and working fluids. This method is given as follows: 

dp dz

 ¼ tp

          dp dp dp dp þ2  x3 x ð1  xÞ1=3 þ dz l0 dz v0 dz l0 dz v0

ð4:69Þ

where: 

dp dz

 ¼C i0

f ,i0

G2 2dρi

ð4:70Þ

and:  C

f ,i0

¼

64= Re i0 0:3164= Re 0:25 i0

for for

Re i0  1187 Re i0 > 1187

ð4:71Þ

By assuming the limits of x!0 and x!1, it can be observed that this method obeys the limits of single-phase flows. Moreover, as mentioned above, it has been reported in several studies available in the open literature that the frictional pressure drop estimated according to Müller-Steinhagen and Heck (1986) agrees reasonably well with experimental results for two-phase flow inside conventional and miniscale channels. Despite the reasonably good predictions, some researchers have proposed adjustments for the Müller-Steinhagen and Heck (1986) predictive method in order of turning it more adequate for miniscale channels, such as performed by SemperteguiTapia and Ribatski (2017). These authors have proposed modifications of constants

150

4 Pressure Drop

and exponents of the Müller-Steinhagen and Heck method, which were adjusted based on experimental database obtained with low global warming potential (GWP) refrigerants, in circular, triangular, and square channels. The resulting method is given as follows: 

dp dz

 ¼ tp

   

dp dp þ 3:01 exp 4:64  106 Re v0,e dz l0 dz v0     dp dp  Þxð1  xÞ1=2:31 þ x2:31 dz l0 dz v0

ð4:72Þ

where the Reynolds number for the mixture flowing as vapor is evaluated based on the equivalent diameter de,2 hence, the subscript e, which is given as follows: Re v0,e ¼

Gd e μv

ð4:73Þ

The frictional pressure drop parcel for the mixture flowing as liquid and vapor are evaluated according to the flow regime, hence differentiating between laminar and turbulent flow regimes with transition defined for Reynolds number of 2300. In the case of circular channels, the friction factor is given by the already familiar Eq. (4.40), with constant of 16 and exponent of 1.0 for laminar, and constant of 0.0791 and exponent of 0.25 for turbulent flow regime. Due to the fact that the database of Sempertegui-Tapia and Ribatski (2017) included results for non-circular channels, with equivalent diameter ranging from 0.835 to 1.1 mm (hydraulic diameter between 0.634 and 1.1 mm), the friction factor also depends on the geometry for laminar flow regimes. On the other hand, for turbulent flow regimes, the friction factor can be considered as independent of the channel geometry. For laminar flow inside rectangular channel, the friction factor is given according to Shah and London’s (1978) proposal as follows:

24 1  1:3553ζ þ 1:9467ζ 2  1:7012ζ 3 þ 0:9564ζ 4  0:2537ζ 5 de f ¼ ð4:74Þ Re i0,e dh where ζ is the aspect ratio ( 1.0), and the Reynolds number Rei0,e is evaluated based on the hydraulic diameter and total mass flow rate. The terms de and dh refer, respectively, for equivalent and hydraulic diameters. For flows inside triangular channels with sharp and rounded corner, the singlephase friction factor is given, respectively, by:

Equivalent diameter is defined as de ¼ (4A/π)1/2 and corresponds to the diameter of a circular channel with same cross-sectional area A.

2

4.1 Predictive Methods for Frictional Pressure Drop Parcel

151

f ¼

13:333 d e Re i0,e dh

ð4:75Þ

f ¼

15:993 d e Re i0,e dh

ð4:76Þ

For non-circular channels in turbulent flow regime (Rei0,e > 2300), the friction factor is evaluated according to the relationship that is used for circular channels, with the Reynolds number evaluated with the actual velocity and equivalent diameter.

4.1.5

Cioncolini, Thome, and Lombardi (2009)

Cioncolini et al. (2009) proposed a predictive method for pressure drop during annular flow in micro and macrochannels accounting for the effect of liquid entrainment in the vapor core flow. According to the proposal presented by the authors, the frictional pressure drop parcel is similar to the homogenous model, however, predictive methods for the friction factor were proposed by the authors mainly as a function of a modified Weber number, rather than only Reynolds number conventionally used for estimation of f for single-phase flow. According to the proposal, the total pressure gradient for two-phase developed flow along channels with uniform cross-section is given as follows:  

dp dz

 ¼ 2 f tp

  G2c εð 1  xÞ 2 ð1  εÞ2 ð1  xÞ2 d x2 þ G2 þ þ dz ρv α ρl εξð1  αÞ ρl ð1  ξÞð1  αÞ ρc d

þ ½ρl ð1  αÞ þ ρv αg sin θ

ð4:77Þ

where the terms on the right-hand side correspond, respectively, to frictional, accelerational, and gravitational parcels. In this textbook, Eq. (4.77) will be kept for this development rather than Eqs. (4.20), (4.21), (4.22), and (4.23) to emphasize the fact that Cioncolini, Thome, and Lombardi (2009) accounted for the entrainment fraction ε, which corresponds to the fraction of liquid that flows as dispersed droplets in the vapor stream, and can be derived by a similar approach adopted for the derivation of Eq. (4.20). In Eq. (4.77), the term ξ corresponds to the liquid holdup of the entrained droplets and is given as follows: ξ¼ε

α 1  x ρv 1  α x ρl

ð4:78Þ

The entrainment fraction ε is evaluated according to Oliemans, Pots, and Trompé (1986) as follows:

152

4 Pressure Drop

Table 4.3 Exponents for entrainment factor according to Oliemans, Pots, and Trompé (1986)

Exp. C0 C1 C2 C3 C4 C5 C6 C7 C8 C9

Par. 10 ρl [kg/m3] ρv [kg/m3] μl [kg/ms] μv [kg/ms] σ [N/m] d [m] jl [m/s] jv [m/s] g [m/s2]

All data points Standard Value error 2.52 0.40 1.08 0.05

Film Reynolds number, Relf 103– 3 100–300 300–10 3103 0.69 1.73 3.31 0.63 0.94 1.15

3103– 104 8.27 0.77

104– 3104 6.38 0.89

3104– 105 0.12 0.45

0.18

0.06

0.96

0.62

0.40

0.71

0.70

0.25

0.27

0.04

0.80

0.63

1.02

0.13

0.17

0.86

0.28

0.11

0.09

0.50

0.46

1.18

0.55

0.05

1.80 1.72 0.70 1.44 0.46

0.08 0.05 0.03 0.05 0.03

0.88 2.45 0.91 0.16 0.86

1.42 2.04 1.05 0.96 0.48

1.00 1.97 0.95 0.78 0.41

0.17 1.16 0.83 1.45 0.32

0.87 1.67 1.04 1.27 0.07

1.51 0.91 1.08 0.71 0.21

ε¼





1

1 10C0 ρCl 1 ρCv 2 μCl 3 μCv 4 σ C5 d C6 jCl 7 jCv 8 gC9

ð4:79Þ

which is a dimensional expression, with all parameters in SI units. The exponents of Eq. (4.79) are presented in Table 4.3, where the reference Reynolds number Relf is estimated for the liquid film, given as follows: Relf ¼ ð1  εÞð1  xÞ

Gd μl

ð4:80Þ

Notice that the determination of entrainment factor depends on the liquid film Reynolds number, which in turn depends on ε, hence, an iterative method is needed. Cioncolini et al. (2009) adopted the void fraction predictive method proposed by Woldesemayat and Ghajar (2007), which is an adjustment of the Dix (1971) proposal, that is based on the drift flux model proposed by Zuber and Findlay (1965), given as follows3:

3 The original proposal of Woldesemayat and Ghajar (2007) includes a diameter d inside the second square brackets of the denominator, which makes the equation dimensionally incorrect. Additionally, the original proposal of Dix (1971) does not count with d in the drift parameter. Hence, in this textbook we will keep the dimensionally correct representation. Additionally, beware that for some programming languages, such as F-Chart EES, setting the exponent of an exponent requires additional brackets, such as x^(y^z) instead of simply x^y^z.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

α¼

153

jv ð4:81Þ  0:1 3 ρ  ρv h i0:25 patm l 5 þ 2:9 gσ ð1þ cos 2θÞðρl ρv Þ jv 41 þ jjl ð1:22 þ 1:22 sin θÞ p ρ 2

v

l

where θ is the channel inclination in relation to horizontal plane, and patm corresponds to 1 atm (101.325 kPa). The two-phase friction factor ftp for macrochannels is given as follows: f tp ¼ 0:172We0:372 c

ð4:82Þ

where the modified Weber number Wec, corresponding to core Weber number, is given as follows: Wec ¼

G2c d c ρc σ

ð4:83Þ

and the core mass flux Gc is given as follows: Gc ¼

4m_ ½ x þ εð 1  xÞ  πd 2c

ð4:84Þ

and the core diameter is given as follows: dc  d

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α þ ξ  αξ

ð4:85Þ

The core density ρc is given as follows: ρc ¼ ð1  αc Þρl þ αv ρv

ð4:86Þ

where αc is the droplet laden core void fraction, given as follows: αc ¼

α α þ ξð1  αÞ

ð4:87Þ

In the case of microchannels, the two-phase friction factor is given as follows: Re 0:318 f tp ¼ 0:0196We0:372 c lf

ð4:88Þ

154

4 Pressure Drop

Cioncolini et al. (2009) used the criterion proposed by Kew and Cornwell (1997) to characterize whether a channel is micro or macro, which is based on the Bond number, given as follows: Bd ¼

gðρl  ρv Þd2 σ

ð4:89Þ

whereas if Bd is lower than 4, it corresponds to microchannel, and if Bd > 4, it corresponds to macrochannel. The reader now counts with a myriad of predictive methods. Despite the fact that all the methods should predict similar values, since they intend to represent the real phenomena, this aspect is not necessarily true. Figure 4.11 depicts predicted values for frictional pressure drop parcel during two-phase flow according to the methods described in this chapter, where the homogeneous model assumes the mixture viscosity given by the Cichitti model, and the method of Lockhart and Martinelli (1949) was implemented based on cubic interpolation of the tabular data. Assuming two-phase flow of R134a in channel with 3 mm of internal diameter at saturation temperature of 20  C and mass velocity of 100 kg/m2s as baseline for the analysis (Fig. 4.11.a), we can evaluate the effects of changing some operational conditions on the pressure drop gradient. The increment of mass velocity, depicted by Fig. 4.11.b, causes increment of the pressure drop, which is properly captured by all methods; similarly, reduction of channel diameter implies on the increment of pressure drop, which can be concluded by comparing Fig. 4.10a, c, and both responses are already expected and common sense. Finally, the increment of saturation temperature (and pressure), results in reduction of vapor-specific volume and consequently of the pressure drop, which can be inferred based on the comparison between Fig. 4.10a, d. As discussed previously in this chapter, the reduction of pressure drop in this case is mainly related to the reduction of the specific volume of vapor phase, rather than on the reduction of liquid viscosity, which also presents some contribution but it is not dominant. Now we shall discuss the differences between different predictive methods, which are significant as can be observed for all conditions depicted in Fig. 4.11. One important aspect that must be mentioned is related to discontinuities of the predicted values, which are present for some of the predictive methods. In the case of homogeneous model, the discontinuities are related to the transition between laminar and turbulent flow regimes. These discontinuities also occur for the Lockhart and Martinelli (1949), Chisholm (1967), Friedel (1979), and Cioncolini, Thome, and Lombardi (2009) methods, which are mostly related to transition between laminar and turbulent flow regimes. One important aspect to be mentioned is related to the quite weird behavior of the predictions according to Lockhart and Martinelli, which include also the effects of flow regime transition and a numerical issue related to the interpolation of tabular data. Regarding the differences among them, we can notice that in any operational conditions the differences can be as high as 100%, therefore, the reader must check the experimental database used in the development of the method to select a method

Fig. 4.11 Predicted values for frictional pressure drop parcel during two-phase flow inside channels for R134a and a) d ¼ 3 mm, G ¼ 100 kg/m2s, Tsat ¼ 20  C; b) d ¼ 3 mm, G ¼ 250 kg/m2s, Tsat ¼ 20  C; c) d ¼ 1 mm, G ¼ 100 kg/m2s, Tsat ¼ 20  C; d) d ¼ 3 mm, G ¼ 100 kg/m2s, Tsat ¼ 0  C.

4.1 Predictive Methods for Frictional Pressure Drop Parcel 155

156

4 Pressure Drop

developed for similar experimental conditions, which includes working fluids and pressure/temperature, channel geometry, and flow velocities. The maximum pressure drop predicted by several methods is independent of the flow velocity, therefore, do not properly capture this experimental behavior.

4.2

Solved Examples

Consider R600a flowing inside a 2 mm ID tube according to mass fluxes of G1 ¼ 100 kg/m2s and G2 ¼ 400 kg/m2s, and vapor quality of 0.5 for saturation temperature of 50  C. Assuming horizontal and vertical upward flows, evaluate the frictional and gravitational pressure drop parcels and the respective fraction of the total pressure drop. Neglect the accelerational pressure drop parcel and assume the Müller-Steinhagen and Heck (1986) predictive method for frictional pressure drop parcel, and minimum kinetic energy model for void fraction. Solution For Tsat ¼ 50  C, the following properties are obtained: ρv ¼ 17.59 kg/m3 ρl ¼ 516 kg/m3 μv ¼ 8.642106 kg/ms μl ¼ 1.144104 kg/ms σ ¼ 7.112 N/m Based on the Müller-Steinhagen and Heck (1986) approach, given by Eqs. (4.69), (4.70), and (4.71), it is possible to estimate the frictional pressure drop parcel, which, respectively for G ¼ 100 and 400 kg/m2s, results in:   dp ¼ 3349 Pa dz f and   dp ¼ 337893 Pa dz f In the case of horizontal flow, the void fraction values according to the Kanizawa and Ribatski (2016) model is estimated as 0.893 and 0.915 for G of 100 and 400 kg/m2s, respectively. Nonetheless, for horizontal flow the gravitational pressure drop parcel is null, and the frictional parcel corresponds to 100% of the total pressure drop. In the case of vertical flow, the void fraction values according to the same approach, accounting for the difference in the formulation, is 0.907 and 0.948, which results in the gravitational pressure drop parcel of 626.8 and 428.9 Pa/m

4.2 Solved Examples

157

(Eq. (4.22)), for G of 100 and 400 kg/m2s, respectively. Hence, for G1 ¼ 100 kg/m2s, the frictional pressure drop parcel corresponds to 15.8% of the total pressure drop, and for G2 ¼ 400 kg/m2s, this contribution reduces to 1.1%. Further Discussion This example illustrates the importance of the correct estimative of void fraction by selecting reliable predictive methods, since the gravitational pressure is directly a function of α, as shown in Eq. (4.22). This aspect is more pronounced for two-phase flow in macroscale channels, where the frictional parcel is naturally smaller due to larger diameter, and gravitational parcel corresponding to more than 90% of the total pressure drop is not uncommon. Additionally, the reader might be wondering why the accelerational pressure drop parcel was not evaluated, and this aspect is related to the fact that the channel length was not informed. Recall that this parcel corresponds to the variation of flow kinetic energy between two points, and in this case only a point was informed, which corresponds to the location with vapor quality of 50%. If at least a channel length was provided, for example, 1 m, even for adiabatic flow it is possible to estimate iteratively the accelerational pressure drop parcel. Assuming uniform frictional pressure drop parcel for horizontal flow, by performing an iterative method, it is possible to estimate the accelerational pressure drop parcel according to the following simple algorithm for adiabatic flow: Evaluate (dp/dz)f according to Müller-Steinhagen and Heck (1986). Outlet-specific enthalpy ¼ inlet-specific enthalpy (îout ¼ îin) Initialize Δpa ¼ Δpa,aux ¼ 0 Pa Error ¼ high value Tolerance ¼ low value While (Error > Tolerance) pout ¼ pin – (dp/dz)f L – Δpa Evaluate Δpa as function of îout, pout, îin, pin according to Eq. (4.24) for constant G Error ¼ |Δpa – Δpa,aux| Δpa,aux ¼ Δpa End while Based on this approach, the accelerational pressure drop parcel for 1 m in channel length assuming adiabatic and uniform frictional pressure drop parcel for horizontal flow is 1.5 and 286.3 Pa for G of 100 and 400 kg/m2s, respectively. The corresponding vapor quality variation is 0.13% and 1.5% for both mass fluxes. In the case of diabatic flow, with uniform heat flux, for example, it is important to account for vapor quality variation along the length, because the frictional and gravitational pressure drop parcels are significantly affected by vapor quality variation, as discussed in this chapter, as well as accelerational pressure drop parcel. Moreover, these parameters are not linearly dependent on the vapor quality, hence getting the mean value between inlet and outlet is incorrect. Therefore, it is interesting to discretize the domain in short segments and sum all the contributions.

158

4.3

4 Pressure Drop

Problems

1. Derive Eq. (4.11) starting from Eq. (4.10), adopting the necessary hypothesis. Do the same for Eq. (4.13). 2. Derive step by step Eq. (4.19). 3. Check whether Eq. (4.54) is non-dimensional. 4. Derive Eq. (4.57). 5. Assume homogenous model for void fraction and frictional pressure drop estimative during water downward flow inside vertical pipe of 6 mm of internal diameter, at 40  C. Estimate the pressure drop gradient due to frictional and gravity for mass flux of 250 kg/m2s and vapor quality of 50%. 6. Now, assume vertical upward flow for the conditions described in exercise 5. 7. Now, assume horizontal flow for the conditions described in exercise 5. 8. Check which predictive methods described in this chapter satisfy the conditions of single-phase flow. 9. Consider adiabatic flow of R134a inside a 1 mm ID tube, with inlet condition corresponding to saturated liquid at 40  C, and mass flux of 600 kg/m2s. Evaluate the required tube length to reduce the pressure until saturation temperature of 5  C. What is the outlet vapor quality? (In this exercise, practice the flashing effect, even though in real cases it is possible to obtain blocked flow due to the high flow velocity). Assume horizontal flow. 10. Assume an annular flow in horizontal round channel with liquid film along the channel surface, and vapor flow in the core region, both in laminar regime. Present the velocity profiles and wall shear stress, as well as pressure gradient. Consider α varying from 0.80 to 0.99 and present the variation of pressure drop for R134a flowing at 5  C in a channel of d ¼ 3 mm, at mass flux of 50 kg/m2s, and plot the variation of pressure drop with void fraction. 11. Consider R134a flow in a 3 mm ID tube at 0  C, G of 100 kg/m2s, and vapor quality of 25%. Based on the Lockhart and Martinelli (1949) approach, and assuming homogeneous void fraction model with both phases distributed according to circular geometry (which is not feasible, but it is a good exercise), evaluate: (a) (b) (c) (d)

The hydraulic and equivalent diameter of both phases. The parameters Ψ l and Ψ v based on Eqs. (4.38) and (4.39). Confirm the validity of Eq. (4.46) with the obtained values. Evaluate the two-phase multiplier values, and the two-phase pressure drop gradient.

12. Repeat exercise 11 assuming the minimum kinetic energy model for horizontal flow.

References

159

References Cheng, L., Ribatski, G., Moreno-Quibén, J., & Thome, J. R. (2008). New prediction methods for CO2 evaporation inside tubes: Part I – A two-phase flow pattern map and a flow pattern based phenomenological model for two-phase flow frictional pressure drops. International Journal of Heat and Mass Transfer, 51, 111–124. Chisholm, D. (1967). A theoretical basis for the Lockhart-Martinelli correlation for two-phase flow. International Journal of Heat and Mass Transfer, 10(12), 1767–1778. Churchill, S. W. (1977). Friction-factor equation spans all fluid-flow regimes. Chemical Engineering, 84(24), 91–92. Cicchitti, A., Lombardi, C., Silvestri, M., Soldaini, G., & Zavattarelli, R. (1959). Two-phase cooling experiments: pressure drop, heat transfer and burnout measurements (No. CISE-71). Milan: Centro Informazioni Studi Esperienze. Cioncolini, A., Thome, J. R., & Lombardi, C. (2009). Unified macro-to-microscale method to predict two-phase frictional pressure drops of annular flows. International Journal of Multiphase Flow, 35(12), 21138–21148. Colebrook, C. F. (1939). Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers, 11(4), 133–156. Dix, G. E. (1971). Vapor void fractions for forced convection with subcooled boiling at low flow rates. GE Report, Berkeley: University of California. Dukler, A. E., Wicks, M., & Cleveland, R. G. (1964). Frictional pressure drop in two-phase flow: A. A comparison of existing correlations for pressure loss and holdup. AICHE Journal, 10(1), 38–43. Friedel, L. (1979). Improved friction pressure drop correlations for horizontal and vertical two phase pipe flow. 3R International, 485–491, July 1979. Grönnerud, R. (1979) Investigation of liquid hold-up, flow resistance and heat transfer in circulation type evaporators. Part IV. Two-phase flow resistance in boiling refrigerants. Bulletin de L’Institut International Du Froid, 1972-1. Ishihara, K., Palen, J. W., & Taborek, J. (1980). Critical review of correlations for predicting two-phase flow pressure drop across tube banks. Heat Transfer Engineering, 1(3), 23–32. Kanizawa, F. T., & Ribatski, G. (2016). Void fraction predictive method based on the minimum kinetic energy. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(1), 209–225. https://doi.org/10.1007/s40430-015-0446-x Kew, P. A., & Cornwell, K. (1997). Correlations for the prediction of boiling heat transfer in smalldiameter channels. Applied thermal engineering, 17(8–10), 705–715. https://doi.org/10.1016/ S1359-4311(96)00071-3 Lavin, F. L., Kanizawa, F. T., & Ribatski, G. (2019). Analyses of the effects of channel inclination and rotation on two-phase flow characteristics and pressure drop in a rectangular channel. Experimental Thermal and Fluid Science, 109, 109850. Lockhart, R. W., & Martinelli, R. C. (1949). Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chemical Engineering Progress, 45(1), 39–48. McAdams, W. H., Woods, W. K., & Heroman, L. C. (1942). Vaporization inside horizontal tubesII-benzene-oil mixtures. Transactions of the ASME, 64(3), 193–200. Mishima, K., & Hibiki, T. (1996). Some characteristics of air-water two-phase flow in small diameter vertical tubes. International Journal of Multiphase Flow, 22(4), 703–712. Moreira, T. A., Morse, R. W., Dressler, K. M., Ribatski, G., & Berson, A. (2020). Liquid-film thickness and disturbance-wave characterization in a vertical, upward, two-phase annular flow of saturated R245fa inside a rectangular channel. International Journal of Multiphase Flow, 132, 103412. Moreno-Quibén, J., & Thome, J. R. (2007). Flow pattern based two-phase frictional pressure drop model for horizontal tubes, Part II: New phenomenological model. International Journal of Heat and Fluid Flow, 28(5), 1060–1072.

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Müller-Steinhagen, H., & Heck, K. (1986). A simple friction pressure drop correlation for two-phase flow in pipes. Chemical Engineering and Processing: Process Intensification, 20 (6), 297–308. Oliemans, R. V. A., Pots, B. F. M., & Trompe, N. (1986). Modelling of annular dispersed two-phase flow in vertical pipes. International Journal of Multiphase Flow, 12(5), 711–732. Sempértegui-Tapia, D. F., & Ribatski, G. (2017). Two-phase frictional pressure drop in horizontal micro-scale channels: Experimental data analysis and prediction method development. International Journal of Refrigeration, 79, 143–163. Wallis, G. B. (1969). One-dimensional two-phase flow. McGraw Hill. Woldesemayat, M. A., & Ghajar, A. J. (2007). Comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes. International Journal of Multiphase Flow, 33(4), 347–370. Zhang, W., Hibiki, T., & Mishima, K. (2010). Correlations of two-phase frictional pressure drop and void fraction in mini-channel. International Journal of Heat and Mass Transfer, 53(1), 453–465. Zuber, N., & Findlay, J. (1965, November). Average volumetric concentration in two-phase flow systems. Journal of Heat Transfer, 87(4), 453–468.

Chapter 5

Flow Boiling

This chapter concerns an analysis of heat transfer during in-tube convective boiling (also named in literature as convective boiling), focusing on small-scale channels. The heat transfer process during flow boiling is composed of two main mechanisms, namely, nucleate boiling and forced convection, such as schematically depicted in Fig. 2.12. Based on this fact, the majority of the predictive methods for heat transfer coefficient during flow boiling are based on the superposition of the contribution of these effects. As proposed by Kutateladze (1961), this approach is given as follows: h¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ðSnb  hnb Þn þ ðF c  hc Þn

ð5:1Þ

where, usually, hnb and hc are the heat transfer coefficients estimated based on methods for pool boiling and single-phase forced convection, respectively; Snb and Fc are the factors associated to the suppression of nucleate boiling effects and enhancement of convective effects, respectively, as the vapor quality increases; and n corresponds to the asymptotic exponent that is usually 1 or 2, depending on the predictive method. The approach given by Eq. (5.1) for heat transfer coefficient estimation is recurrently referred in the open literature as the Chen (1966) approach, who adopted unitary asymptotic exponent (n ¼ 1). In general, the value of the nucleate boiling suppression factor varies from zero to the unity (0  Snb  1) and reduces with the increment of flow inertial effects. Conversely, the convective enhancement factor is usually higher than one (Fc  1), and its value increases with increasing flow inertial effects, as follows: Inertial effects "!Snb # Inertial effects "!Fc "

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_5

161

162

5 Flow Boiling

Additionally, under typical operational conditions corresponding to conventional channels, the heat transfer coefficient associated to nucleate boiling effects is usually much higher than the value associated to the forced convection parcel (hnb  hc); on the contrary, for convective boiling in microchannels both parcels tend to present a similar order of magnitude. Under conditions such as reduced flow velocities and/or high heat fluxes, the nucleate boiling mechanism is dominant, and the convective effects play a reduced role on the heat transfer process, implying on negligible variations of the heat transfer coefficient with mass velocity and vapor quality prior to the surface dryout. Conversely, for conditions of high mass velocities and/or low heat fluxes, convective effects prevail over nucleate boiling, hence the heat transfer coefficient increases with increasing two-phase flow velocity associated with the increment of mass flux and/or vapor quality. The required wall superheating for the onset of nucleate boiling (ONB) was already introduced in Chap. 2 and is deeply discussed in this chapter. Nucleate boiling consists of the vapor formation as discrete bubbles in the liquid phase and can be characterized as homogeneous when it occurs away from a surface and as heterogeneous when it occurs on a surface, usually solid. Under both conditions, beginning from a state of subcooled liquid, heating the liquid may eventually lead the fluid to a saturated state, with no phase change because a temperature higher than Tsat is necessary to trigger the vaporization process. This additional energy parcel is related to several aspects, including the interfacial energy, the surface tension effects, the inertia due to displacement of the liquid in the neighborhood, and the thermodynamic non-equilibrium between inside and outside of the bubble, among other effects. In this context, consider the force balance of a stable vapor bubble with diameter db in a liquid media, such as schematically depicted in Fig. 5.1. Based on Newton’s

pe

σ db

pi

Fig. 5.1 Schematics of vapor bubble in mechanical equilibrium with liquid media.

5.1 Nucleate Boiling Concepts

163

Second Law, and assuming only pressure and surface tension forces, we can infer the pressure difference between inside pi and outside pe of the bubble as follows: Forcesurface tension  Forcepressure ¼ πdb σ  ðpi  pe Þ ð pi  pe Þ ¼

4σ db

πd2b ¼0 4

ð5:2Þ ð5:3Þ

Hence, if we consider that the vapor would be in thermal equilibrium with a saturated liquid, the pressure inside the bubble would be higher than the saturation pressure, then the bubble would collapse. Therefore, additional energy is needed to overcome the listed effects related to bubble formation, where the force balance described above is only one of them. In the subsequent sections, these effects are discussed in more detail.

5.1

Nucleate Boiling Concepts

Homogeneous nucleation is characterized by vapor formation away from a wall within the liquid media under the addition of uniform energy to the fluid along its volume. The heating of a cup of water in a microwave until the establishment of phase change condition can be presented as an approximate experiment, even though this does not really correspond to homogeneous nucleation. You are advised in NOT doing this experiment because under such a condition the phase change from liquid to vapor may start suddenly, spilling the hot water over you. Different from the condition of a conventional pan, where the energy is supplied from a hot surface, the heating in the microwave is directly into the water molecules, with higher heat transfer intensity in the periphery. Hence, the liquid will receive energy and the temperature in the peripheric region will increase to be even higher than the saturation temperature without previous phase change, and a perturbation will eventually promote phase change at some point. Remembering that the specific volume of the vapor phase at atmospheric condition is considerably higher than that of the liquid, the initial vapor formation will disturb the entire volume in the cup due to fluid displacement, which can promote phase change from liquid to vapor phase at several points along the fluid in an explosive way. Hence, even though the liquid was superheated, it did not vaporize at saturation temperature, remaining in a metastable condition. Another day-to-day example of metastable state is the subcooled liquid, such as beer in a refrigerator, that is below the freezing point but turns into a solid-liquid mixture only when it is disturbed, which can be by abrupt opening or by holding it with bare hands. Even though the fluid was at a temperature below the freezing point, it was in liquid state.

164

5 Flow Boiling

Heating

Liquid

Liquid

T < Tsat

T ≥ Tsat

Fluid molecules with Brownian motion

Motion intensifies and energy of parcel of the molecules increases

Heating

Nucleation Vapor

Liquid

Liquid

T ≥ Tsat

T ≥ Tsat

That promotes transition to vapor phase, displacing the fluid in the vicinity and disturbing the media

Eventually, molecules with high energy form a cluster (region) of high energy

Fig. 5.2 Schematic representation of homogeneous nucleate boiling

The vapor formation requires liquid superheating that under a metastable condition changes to the vapor phase. There is no consensus about the mechanisms that favor the bubble nucleation, but one of the proposals argues that the following steps should occur to promote bubble formation, as schematically represented in Fig. 5.2, assuming the absence of a surface: the fluid molecule energy increases due to the fluid heating; the random movement of the molecules promotes diffusion of energy by net displacement of molecules with high energy, and/or by energy transfer to other molecules that get in contact with them; hence, portions of molecules with higher energy will eventually form a cluster (therefore some statistical parameter to

5.1 Nucleate Boiling Concepts

165

Liquid βc = 0 Perfect Hydrophilic

0° < βc < 90° Hydrophilic

Vapor

Vapor

90° < βc < 180° Hydrophobic

βc

βc

Vapor

βc

Solid Vapor 0° < βc < 90° Hydrophilic

90° < βc < 180° Hydrophobic Liquid βc

βc Liquid

βc = 180° Perfect Hydrophobic Liquid βc

Solid Fig. 5.3 Schematics of contact angle

consider this aspect is required), which corresponds to a region of higher energy and with capacity to promote vapor formation. It must be emphasized that the vapor bubble formation requires displacement of the neighboring liquid, which corresponds to additional work of displacement against the pressure and to increase the fluid kinetic energy of the high energetic cluster. Additionally, an energy parcel related to surface tension forces is needed for bubble formation due to the liquid–vapor interface. Methods available in literature for homogenous nucleation such as Blander and Katz (1975) predict water superheating higher than 200  C for the onset of vapor formation under atmospheric conditions. This value is considerably higher than those verified when water is boiled to prepare the morning coffee. Different from the homogeneous vapor nucleation that requires a uniform heating of a liquid, the most common approach to increase the liquid energy/temperature in practical usage is through a hot surface. Under such a condition, the liquid in contact with the surface tends to be at the same temperature as the surface, with a temperature gradient along the fluid. In addition to the defined region with higher temperature, the presence of a surface favors the bubble formation whose behavior is affected by the surface wettability and roughness. Under conditions that the surface-liquid pair presents low affinity defined as hydrophobicity, characterized by a contact angle βc higher than 90 , as schematically depicted in Fig. 5.3, the required superheating for the formation of bubble is inferior than for the condition of high wettability (hydrophilic surface), characterized by βc smaller than 90 . The reader can experience qualitatively the wettability contribution on the bubble formation in the kitchen by the conditions of water boiling in a nonstick (Teflon® covered) and

166

5 Flow Boiling

Liquid Vapor

βc=βc’

βc

βc’

Cavity angle

Solid

βc

βc’

Cavity angle

βc

βc’

Cavity angle

Fig. 5.4 Schematics of the effect of surface roughness on the effective contact angle

conventional pan. In this experiment, the reader shall notice that for similar conditions, the number of bubbles in the nonstick surface is considerably higher than that in the conventional pan. Usually, metallic surfaces and water present a contact angle close to 80 , and other fluids, such as oils and halocarbon refrigerants, and metallic surfaces usually present angles ranging from 7 to 30 , being hydrophilic. Conversely, non-stick surfaces present a contact angle of the order of 110 , thus with hydrophobic characteristics. Even though contact angles higher than 90 are desirable when regarding the bubble formation, industrially it is difficult to maintain a non-stick surface for long-term service, due to surface degradation and fouling. Additionally, as will be discussed in Chap. 6, surfaces with low wettability (high contact angle), usually result in low critical heat flux (CHF), which can also limit the operational range of the equipment. In fact, it can be shown that the energy necessary for the formation of a vapor nucleus on a surface is lower than the energy necessary for the formation within a liquid medium; moreover, this amount of energy is inversely proportional to the contact angle. Nonetheless, even considering the wettability properties of the surfaces and assuming smooth surfaces, the resulting superheating would be considerably higher than the value experimentally observed, and this difference can be partially justified on the surface roughness, which acts to increase the effective contact angle and to reduce the required superheating. Figure 5.4, adapted from Collier and Thome (1994), illustrates the influence of the cavity angle on an effective contact angle βc0 , whereas for a fixed βc the reduction of cavity angle implies on the increment of the angle between the surface and the bubble surface, βc0 , and hence reduces the required superheating of the surface for bubble formation. In this context, several research groups have been investigating surface modifications, such as roughness modification, covering with porous material, deposition of nanoparticles, among others, to enhance the heat transfer coefficient during nucleate boiling. In order to reinforce this discussion, the example of glass recipients for laboratory applications can be addressed, such as Becker recipients. These accessory surfaces present very low surface roughness due to the material characteristics and the manufacturing processes, which makes bubble formation more difficult and require higher superheating to initiate bubble formation along the surface. Hence, lab users

5.1 Nucleate Boiling Concepts

Liquid

Noncondensable gas

Solid

Cavity t1

167

Liquid

Liquid

Liquid

Solid

Solid t2

t3

Trapped non-condensable gas t4

t1 < t2 < t3 < t4

Fig. 5.5 Schematics of liquid front trapping non-condensable gas into surface cavity along time

usually intentionally introduce surface roughness mechanically or by chemical attack to facilitate bubble formation. In the case of a non-smooth surface initially dry and exposed to non-condensable gas, such as air, that is wetted with the working fluid in liquid phase, such as schematically depicted in Figure 5.5, the non-condensable gas can be trapped in the cavity, which reduces the required surface superheating. Recall Eq. (5.3), which would also be valid for a hemispherical bubble in a cavity, such as schematically depicted in Fig. 5.4, where db can be read as cavity diameter; however, the internal pressure corresponds to the sum of the vapor and non-condensable gas partial pressures, and the external pressure corresponds to the liquid pressure, as follows: pv þ pa  pl ¼

4σ db

ð5:4Þ

where pv and pa correspond to the vapor and non-condensable-gas partial pressures, respectively. Again, similar to the approach proposed by Bergles and Rohsenow (1964) and described in Sect. 2.4, the Clapeyron equation for ideal gas can be used, which is given as follows: dpv ^ilv ρv p ^i  ¼ v lv2 dT v Tv Rv T v

ð5:5Þ

where Rv stands for the specific gas constant. Equation. (5.5) is a differential equation that can be solved for pv and Tv by separation of variables, as follows: ^i 1 dp  lv dT pv v Rv T 2v v   ^i ðT  T l Þ ^i 1 1 lnðpv Þ  lnðpl Þ  lv  ¼ lv v Rv T l T v Rv T l T v

ð5:6Þ ð5:7Þ

168

5 Flow Boiling

And by using the result of Eq. (5.4), assuming that the liquid is at saturation temperature, the following relationship can be obtained for the required vapor superheating for equilibrium: T v  T sat ¼

p Rv T sat T v 4σ lnð þ 1  aÞ ^ilv d b pl pl

ð5:8Þ

Then, recalling the vapor superheating for equilibrium without non-condensable gas given in Sect. 2.4 by Eq. (2.65), the vapor superheating is given as follows: T v  T sat ¼

Rv T v T sat 2σ lnð þ 1Þ ^ilv rpl

ð5:9Þ

and by comparing it with Eq. (5.8), it can be concluded that the presence of non-condensable gas trapped into the cavities reduces the required superheating for bubble equilibrium, and consequently to bubble nucleation and growth. Recall that the determination of the condition that the bubble will grow requires to estimate the liquid temperature as well, as given by Eqs. (2.66) and (2.67). Hence, bubbles shall be formed preferentially in cavities along the heated surface, either for the cases with or without non-condensable gases, and successive energy transfer from the surface to the fluid result in the bubble growth due to vaporization. The bubble size will grow until eventually buoyance forces and drag forces (relevant in case of forced flow) overcome surface tension and inertial forces, implying on the bubble detachment from the surface. In the active cavities, defined as the cavities that propitiated the bubble formation, a small portion of vapor tends to remain; hence, the formation of the next bubble in the same cavity does not require the same superheating, and successive bubbles will be formed in the same region until other cavities also become active due to the increase of the surface heating. In the case that the bubble nucleation was facilitated by the presence of non-condensable gases, fractions of the gas will be displaced with the vapor when the bubble departs, until the obtainment of only vapor into the cavity. The reader can notice this aspect by boiling water in a kettle at your kitchen, when in the beginning of the boiling process, the bubbles are successively formed in specific locations. Recall that in the beginning of the boiling process, the non-equilibrium temperature non-equilibrium implies on higher temperature close to the pan solid surface, hence away from the surface the water might be below the saturation temperature. Consequently, the reader will observe that the bubble collapses just after its detachment, because it transfers energy to the subcooled liquid and is not self-sustained anymore, despite the reduction of pressure due to the liquid column. Hence, up to this point, the requirements for bubble formation in a heated surface have been briefly discussed, and the reader is encouraged to review the discussion presented in Sect. 2.4 regarding ONB. Focused on historical and didactical aspects, it is interesting to present the boiling curve as shown in Fig. 5.6, which can be obtained either by controlling the heat flux

5.1 Nucleate Boiling Concepts

169

7

a

b 6-7 6' - 7

6'

5

5-6

4 5 CHF 3'

3-4

2-3 ONB 3'' 1

2

3

6

1-2

1'

ΔT = Tw - Tsat Fig. 5.6 Schematic boiling curve for imposed heat flux and dominant heat transfer mechanisms. (a) Representation of boiling curve, (b) schematics of heat transfer processes in each condition

or the temperature difference between the surface and the fluid, whereas the fluid away from the heater is stagnant, which is referred in the literature as pool boiling condition. Let us assume a cylindrical heater with small diameter, disposed horizontally in the subcooled liquid media. The heating can be performed by Joule effect via electrical current applied directly to the heater surface, which provides approximately uniform and imposed heat flux condition along the external surface, or by considering vapor condensing internally to a microtube that can be approximated as a condition of imposed surface temperature. The heat transfer process from the heater to the neighbor liquid results in temperature gradient along the liquid, whereas the hotter regions are closer to the heater wall. This experiment was firstly reported by Nukiyama (1934) and even though it seems to be simple, it gives important insights about the heat transfer processes associted to the nucleate boiling phenomenon. The condition of imposed heat flux as displayed in Fig. 5.6 may correspond to the thermal management of electronic components, concentrated solar collectors, and nuclear reactors. For reduced heat flux, region 1–2 in Fig. 5.6a, the heat transfer process corresponds to natural convection from a horizontal cylinder, and well established predictive methods for the heat transfer coefficient are available in the

170

5 Flow Boiling

open literature for distinct geometries, such as in Lienhard and Lienhard (2020). The heat transfer coefficient increases with the heat flux due to the increment of density difference and induced flow around the heater. Under this condition, the liquid adjoined to the heater can be hotter than the saturation temperature without bubble formation, as previously pointed out in the discussion about the conditions necessary for bubble formation and its growth. When the heater surface achieves the temperature corresponding to the required superheating, bubble formation will eventually occur and propagate over all the surface, corresponding to point 2 in Fig. 5.6, denominated as ONB. In the imminence of bubble formation, the heat transfer coefficient presents a sharp increment, with consequent reduction of the heater wall temperature, such as depicted by the gap from point 2 to 3 in Fig. 5.6a. The reader should keep in mind the Newton’s Cooling Law, which states that the heat flux during convective heat transfer is proportional to the temperature difference between the surface and the fluid, whereas the heat transfer coefficient is the proportionality parameter and is given as follows: ϕ ¼ hðT w  T fluid Þ

ð5:10Þ

Hence, a subtle increment of h for a fixed ϕ implies on a sharp reduction of ΔT ¼ Tw – Tfluid. The reduction of the wall temperature does not necessarily eliminate the bubble generation, and this aspect is related to the presence of active cavities, such as previously discussed, which corresponds to regions with vapor nuclei that eliminate the need of high superheating, and vapor bubbles just grow and detach. The difference between the pool boiling curves obtained for increasing and decreasing heat flux is nominated as hysteresis. This phenomenon is illustrated in Fig. 5.7 that 120 Increasing f

100

Reducing f

f [kW/m2]

80

Nucleate boiling

60 40 Natural convection

20 0 0

20

Tw - Tsat [K]

40

60

Fig. 5.7 Variation of wall superheating with heat flux for pool boiling of R11 at Tsat ¼ 5.5  C in horizontal tube with 19 mm of external diameter. (Ribatski 2002)

5.1 Nucleate Boiling Concepts

171

depicts experimental results for R11 pool boiling on a horizontal tube with 19 mm of external diameter, under conditions of increasing and decreasing heat flux. When increasing the heat flux, depicted by empty symbols, the wall temperature tends to increase with heat flux until a point where the nucleate boiling begins, and from this point on the temperature difference is inferior to the condition of single-phase flow. Successive increment of ϕ results in sensible increment of the wall temperature, indicating high heat transfer coefficient. Successive increments of the heat flux beyond the ONB implies on the achievement of the fully developed boiling regime, as nominated by Jabardo et al. (2004), with active cavities propitiating successive generation, growth, and detachment of bubbles. This condition corresponds to considerably high heat transfer coefficients, which usually are higher than 1 kW/m2K and can reach values of the order of 10 kW/ m2K (or even 100 kW/m2K for microchannels depending on the fluid and operational conditions), depicted by a variation of ϕ with ΔT according to a power law with a constant exponent usually between 2 and 4 for the wall superheating as schematically depicted in Figs. 5.6 and 5.7. There is still no consensus about the dominant mechanisms that result the high heat transfer coefficients observed during the nucleate boiling process. In fact, it is expected that more than one mechanism contributes to the overall heat transfer with their relative influence varying according to the reduced pressure, gravitational acceleration, heat flux, surface roughness, wettability, fluid type, and degree of fluid subcooling. Earlier models such as Rosenhow (1952) and Forster and Zuber (1955) consider bubble agitation and microconvection effects as the main mechanisms responsible for the heat transfer in pool boiling and, then, based on an analogy between nucleate boiling and forced convection a correlation for the heat transfer coefficient is proposed. Kim (2009) pointed out the following three main mechanisms as the most important to explain the high heat transfer coefficients observed under nucleate boiling conditions: • Liquid replacement: the bubble detachment causes disruption of the thermal boundary layer close to the hot surface, and part of the hot liquid is dragged by the bubble movement. Hence, colder liquid away from the surface flows toward the surface, as described by Rohsenow (1971), and schematically represented in Fig. 5.8a. In general, the models based on such mechanism neglect wall heat transfer during the bubble growth process and assume transient conduction during the bubble waiting time as proposed by Han and Griffith (1965) and Mikic and Rohsenow (1969). Most recently, Gerardi et al. (2010), based on high-speed video of the boiling hydrodynamics and infrared thermometry of the heated wall, pointed out transient conduction following bubble departure as the dominant contribution to nucleate-boiling heat transfer. • Microlayer evaporation: Cooper (1969) and Cooper and Lloyd (1969) based on their experimental results have proposed a model considering that a thin layer of liquid (the microlayer) is trapped between the liquid–vapor interface and the solid wall as the bubble grows, as schematically depicted in Fig. 5.8b. This liquid film

172

5 Flow Boiling

Condensation (evaporation) into bulk of fluid

Bubble Microlayer evaporation Evaporation from thermal boundary layer

Hot surface

Vapor Li qu id

Tra ns T flowverse i

Micro region Adsorbed film

Hot surface

Warmer liquid

Cold liquid

Tw,mic δ

Triple point

Local heat flux

Detail¹

Detail²

Microconvection

Hot surface

Hot surface

Fig. 5.8 Schematics of heat transfer enhancement mechanisms for nucleate boiling. (a) liquid replacement, (b) microlayer evaporation, adapted from Cooper and Lloyd (1969), (c) evaporation at the triple contact line, adapted from Stephan and Hammer (1994), (d) microconvection due to Marangoni effect

is denominated as a microlayer that rapidly evaporates with subsequent heating implying on high heat transfer rates. As the liquid film evaporates, the bubble grows and eventually detaches, promoting rewetting of the surface. Near the dry patch, where the microlayer is thinnest, the heat transfer rate is maximum. In the models based on this mechanism, the higher heat transfer rates are observed during the bubble growth period. Judd and Hwang (1976) proposed a model considering the following mechanisms during the bubble life span: (i) transient conduction during the bubble waiting time, as proposed by Mikic and Rohsenow (1969); (ii) microlayer evaporation during the bubble growing time; and (iii) natural convection during the bubble life span in the area not influenced by the bubble. According to their model, microlayer evaporation is responsible for approximately one-third of the total heat transfer. This mechanism seems to prevail for hemispherical bubbles of non-metallic fluids boiling under conditions of low reduced pressures and microgravity. For bubbles growing in a subcooled

5.1 Nucleate Boiling Concepts

173

liquid, a significant amount of condensation over the curved surface of the bubble, as shown in Fig. 5.8b, may happen, leading to a slower bubble growth. • Evaporation at the triple contact line: In this mechanism, as proposed by Stephan and Hammer (1994), the main mode of heat transfer is the evaporation close to the triple (or three-phase) contact line (schematically depicted in Fig. 5.8c). According to their model, in this region, there is a liquid film adsorbed between the wall and the bubble, which consists only of a few molecular layers and cannot be evaporated due to adhesion pressure, also known as disjoining pressure. This adhesion force is inversely proportional to the fourth power of the distance from the wall and causes the liquid to spread out over the wall. In the thicker part of this region, capillary forces due to an increase of interfacial curvature generates a driving pressure gradient that drives liquid against the triple contact line direction. Assuming one-dimensional conduction through the liquid film due to the difference between the wall and the saturation temperatures, the maximum heat transfer occurs when the liquid film is sufficiently thick, such that the disjoining forces become small, and thin enough so that its thermal resistance is low. Stephan and Hammer (1994) suggested heat fluxes in the region close to the triple contact line 100 times larger than the CHF. Although thermocapillary convection, also known as Marangoni Convection, was not proved to occur for saturated pool boiling because most of bubble interface is at saturation temperature, this mechanism can be pointed as relevant for subcooled boiling mainly under microgravity conditions. Surface tension gradients are induced in the bubble interface due to temperature gradients along its curvature (whereas usually the higher the temperature the lower the surface tension), therefore, at the region of the bubble surface away from the heating wall, the temperature is inferior than the temperature close to the hot surface, hence the surface tension increases from the triple contact line to the bubble top, such as schematically depicted in Fig. 5.8d. Therefore, the tangential stress caused by the gradients in surface tension are balanced by viscous stresses associated with liquid motion since shear force exerts by vapor phase can be neglected due to the low vapor viscosity. Therefore, the Marangoni Convection promotes the circulation of warmer liquid from the wall to the colder region of the pool. Jets of warmer liquid looking as mushroom-like clouds originating from the top of the bubble are also pointed in the literature as promoted by Marangoni effects (Hetsroni et al. 2015). It is important to highlight that chemical composition variations along the bubble interface can also generate surface tension gradients and, consequently, Marangoni Convection. The region between points 3 and 4 depicted in Fig. 5.6 corresponds to the condition desired in most of engineering applications, characterized by high heat transfer coefficient, with vapor formation under stable and safe condition. The increment of heat flux will eventually lead the system to the CHF, defined by point 5 in Fig. 5.6, which corresponds to a condition that the vapor generation in the surface is high enough to inhibit the rewetting of the surface, hence the heat transfer process to the liquid is deteriorated and occurs partially from the surface to a vapor layer adjoined to the surface, and the vaporization occurs at the liquid–vapor

174

5 Flow Boiling

interface. Additionally, the heat transfer by radiation starts to have a significant contribution to the total heat transfer process due to subtle increment of the surface temperature. Assuming that the heat flux is the controlled parameter, subsequent increment of heat flux beyond the CHF condition implies on the formation of a vapor cushion adjoined to the surface, and the heat transfer coefficient presents a sharp reduction with consequent sharp increment of the surface temperature, as indicated by the gap from the point 5 to 60 in Fig. 5.6a. It must be mentioned that this condition is usually undesired for practical applications because it can result in failure of the heat transfer surface by excessive temperature, referred in literature as burnout. Starting the reduction of heat flux from point 7, the curve would follow the same path from 7 to 60 , and then to 6, which corresponds to the minimum heat flux that is able to maintain stable film boiling. Further reduction of the heat flux implies on the surface rewetting and the establishment of nucleate boiling with the boiling curve jumping to point 300 . From point 300 to 3 the expected curve is similar to the one for the condition of increasing ϕ, but since the surface cavities have already been activated, the wall temperature does not change abruptly to point 2 that corresponds to a sudden reduction of the heat transfer coefficient, but keeps a relatively high heat transfer coefficient and curve reach point 10 , where the bubble nucleation is not sustained anymore and the main heat transfer mechanism is natural convection. Subsequent heating processes of the surface do require lower superheating, since parcel of cavities are already active with bubble nuclei. Conversely, under conditions that the experiment is performed through the control of the surface temperature, the heat transfer behavior for single-phase natural convection is similar to that previously discussed for controlled heat flux. When the ONB condition is achieved, the heat transfer coefficient h presents a sharp increment, and since the surface temperature is the controlled parameter, there is a sharp increment of the heat flux (recall Eq. (5.10)), and the operational condition jumps from point 2 to 30 of Fig. 5.6, corresponding to a higher heat flux for the same wall superheating. Subsequent increments of the surface temperature provide the same behavior of the relationship between heat flux and ΔT as observed for conditions of controlled heat flux until the condition of CHF. For wall superheating higher than the one corresponding to the CHF condition, the portion of the surface that is occupied by vapor increases and the heat transfer coefficient sharply decreases. Hence, successive increments of surface temperature result in a reduction of heat flux values, depicted by the line region 5–6 in Fig. 5.6, and commonly referred in literature as transition boiling (Collier and Thome 1994), which is characterized by a combination of conditions of nucleate boiling, and regions partially occupied by vapor cushions. Continuing the increment of surface temperature, it will eventually become dry corresponding to point 6 in Fig. 5.6a, and the heat transfer from the surface to the liquid will be performed partially by convection through the vapor cushion formed around the heater, and a significant parcel will be transferred by thermal radiation. Hence, subsequent increments of the surface temperature will increase the heat flux due to the increment of radiative parcel, even though the convective parcel itself has been compromised, up to point 7.

5.1 Nucleate Boiling Concepts

175

Analogous to the case of controlled heat flux, by reducing ΔT from point 7 to point 1, the curve would present the same shape in the regions 7–6 and 6–5 as for increasing ΔT. However, the path of the curves of increasing and decreasing ΔT may differ in the region 60 –5 corresponding to the transition from film to nucleate boiling due to the wetting characteristics of the liquid on the solid surface. After achieving the condition of CHF, point 5 in Figure 5.6, subsequent reductions of the wall temperature result in a condition of stable nucleate boiling condition. In a similar way to the condition of controlled heat flux, the active cavities for nucleation keep the vapor generation, and there is no sudden reduction of heat transfer coefficient from point 30 to 2, but the heat transfer process occurs according to curve 5-4-3. Additionally, and similar to the condition of controlled heat flux, when reaching point 3 the cavities should still be active, and the heat transfer process should follow steps 3-10 -1. Again, when operating a system with a surface cooled by convective and/or nucleate boiling, it is desirable to achieve the conditions in the region of stable bubble formation, represented by the region 3–4 in Fig. 5.6, which provides very high heat transfer coefficients and relatively low difference between the surface and fluid temperatures. The achievement of the CHF is usually catastrophic for conditions of imposed heat flux, due to sudden increment of the surface temperature, and leads to system failure due to burnout, fluid leakage, and other related problems. This text does not aim to provide a broad discussion concerning nucleate boiling mechanisms, but it is intended to discuss the general aspects without further details and to provide the fundamental aspects needed for introducing the heat transfer mechanisms concerning convective boiling and how they are affected by the operational conditions. The reader is encouraged to check dedicated publications concerning this subject for a deep knowledge on nucleate boiling. Several predictive methods for nucleate boiling in stagnant fluids (named in literature as pool boiling) are available in the open literature, such as Cooper (1984), Gorenflo and Kenning (2010), and Ribatski and Saiz-Jabardo (2003), for the region 3–4 of Fig. 5.6. Most of the predictive methods for prediction of the heat transfer coefficient under pool boiling conditions accounts for heat flux and fluid properties, including thermal conductivity, molar mass, and reduced pressure. It has been verified experimentally that the heat transfer coefficient for pool boiling is approximately proportional to the heat flux to approximately 0.7, as follows: h / ϕ0:7

ð5:11Þ

Additionally, some of the methods such as Rohsenow (1952) takes into account the surface roughness and surface-fluid characteristics, such as contact angle βc and wettability, through an empirical parameter. However, it is important to highlight the difficulties associated with the quantitative characterization of the effects related to the interactions between surface and fluid and the possibility of surface contamination and boiling aging effects. Hence, contact angle values available in the open literature might not be reliable for design purposes due to oxidation of the heating surface and fouling effects.

176

5 Flow Boiling

The predictive method proposed by Cooper (1984) is one of the most employed to evaluate the heat transfer coefficient during pool boiling due to its simplicity and reasonable predictions. This method was developed based on an extensive analysis of influencing parameters during pool boiling, including fluids and surface properties, considering a database gathered from several independent laboratories containing more than 5800 data points, comprising experiments with water, synthetic refrigerants, and cryogens, among other fluids, on surfaces made of copper and stainless steel, among other materials. In his analysis, Cooper evaluated the effects of several fluid properties on the heat transfer coefficient for pool boiling, and concluded that the most important parameters defining the heat transfer coefficient are the heat flux, reduced pressure, molar mass, and surface roughness. The resulting dimensional correlation proposed by Cooper (1984) is given as follows: h ¼ h0



ϕ ϕ0

 

0:67 pr

0:120:2log10

Rp Rp0

ðlog10 pr Þ0:55



M M0

0:5

ð5:12Þ

where the heat flux ϕ is given in W/m (Bergles and Rohsenow 1964), the surface peak roughness Rp in μm, the molar mass M in kg/kmol, and the resulting heat transfer coefficient is given in W/m2K. The reference values are h0 ¼ 55 W/m2K, ϕ0 ¼ 1 W/m (Bergles & Rohsenow, 1964), Rp0 ¼ 1 μm, and M0 ¼ 1 kg/kmol. Similarly, Gorenflo and Kenning (2010) presented a correlation for prediction of heat transfer coefficient under pool boiling conditions given as a function of the fluid and surface parameters, as well as heat flux. The correlation is given as follows1: h ¼ 3580

h

i W F F F F 2 m K ϕ pr w f

ð5:13Þ

where the terms Fϕ, Fpr Fw, and Ff stand, respectively, for the effects of heat flux, reduced pressure, surface, and fluid property effects. These parameters are given as follows:  Fϕ ¼

ϕ 20, 000ϕ0

0:950:3p0:3 r

1:4pr 1  pr  152   ðkρcp Þw 0:25 Ra Fw ¼ 0:4Rp0 ðkρcp ÞCu F pr ¼ 0:7p0:2 r þ 4pr þ

1

Check the original reference for water as working fluid.

ð5:14Þ ð5:15Þ ð5:16Þ

5.1 Nucleate Boiling Concepts

177

" Ff ¼

dp ðdT Þsat

#0:6 ð5:17Þ

σ  1½μmK 1

where pr is the reduced pressure (p/pcrit), ϕ is the heat flux in W/m2, Ra is the surface average roughness in μm, and (kρcp)0.5 is the thermal effusivity of the material, whereas w stands for the wall material and Cu stands for copper material, (dp/dT)sat is the derivative of pressure with temperature for saturation condition, σ is the surface tension in N/μm, and ϕ0 and Rp0 stand for unitary heat flux in W/m2 and roughness in μm, respectively. Ribatski and Saiz-Jabardo (2013) also proposed a predictive method for heat transfer coefficient during pool boiling of synthetic refrigerants, including R11, R123, R12, R134a, and R22, for surfaces made of copper, brass, and stainless steel. The resulting predictive method consists in a correlation given as follows:  h ¼ Fw

ϕ ϕ0

0:90:3p0:2 r

0:8 p0:45 r ðlogpr Þ



Ra Rp0

0:2 

M M0

0:5

ð5:18Þ

where the term Fw depends on the surface material and is equal to 100, 110, and 85 W/m2K, respectively, for copper, brass, and stainless steel. The heat flux is given in W/m2, molar mass M in kg/kmol, and surface averaged roughness Ra in μm, and the resulting heat transfer coefficient in W/m2K. The terms ϕ0, Rp0, and M0 stand for unitary heat flux in W/m2, roughness in μm, and molar mass in kg/kmol, respectively. Finally, it worth describing the heat transfer coefficient predictive method for pool boiling conditions proposed by Stephan and Abdelsalam (1980), who also gathered more than 5000 data points from independent laboratories. These authors presented correlations for each specific group of fluids. Their correlation for water is given as follows (recommended for 104  pr  0.886, βc  45 ): !1:58 !1:26   0:673 5:22 ^ilv d 2b cpl T sat d2b ρl  ρv h db ϕ db 6 ¼ 2:46  10 kl kl T sat ρl ðk l = ρl cpl Þ2 ðk l = ρl cpl Þ2 ð5:19Þ For hydrocarbons, Stephan and Abdelsalam (1980) presented the following relationship (5103  pr  0.9, βc  35 ): !0:248   0:670  0:335 4:33 ^ilv d2b ρv ρl  ρv h db ϕ db ¼ 0:0546 ð5:20Þ kl k l T sat ρl ρl ðkl = ρl cpl Þ2 For cryogenic fluids, the correlation is given as follows (4103  pr  0.97, βc  1 ): 

178

5 Flow Boiling

!0,374 !0:329  0:624  0:257   bilv d2b ρs cps k s 0:117 cpl T sat d 2b ρv hd b ϕdb ¼ 4:82  2  2 kl k l T sat ρl ρl cpl k l k l =ρl cpl k l =ρl cpl

ð5:21Þ where subscript s attains for the surface properties. And for halocarbon refrigerants, the correlation is given as follows (3103  pr  0.78, βc  35 ):  0:745  0:581 ρv h db ϕ db ¼ 207 Pr0:533 l kl kl T sat ρl

ð5:22Þ

where the characteristic length db corresponds to the equilibrium break-off diameter of a bubble detaching from a heated surface, given as follows: d b ¼ 0:146 βc

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2σ gðρl  ρv Þ

ð5:23Þ

where βc corresponds to the contact angle, and in the absence of reliable data, it can be adopted 45 for water, 35 for refrigerants and hydrocarbons, and 1 for cryogenic fluids. It should be noticed that the heat transfer coefficient predictive methods proposed by Cooper (1984) (Eq. (5.12)), Gorenflo and Kenning (2010) (Eqs. (5.13) to (5.17)), Ribatski and Saiz-Jabardo (2013) (Eq. (5.18)), and Stephan and Abdelsalam (1980) (Eqs. (5.19) to (5.23)) present several similarities among them, namely, the exponent of the heat flux is close to 0.7 for all methods, and the heat transfer coefficient is proportional to a function of the reduced pressure, and in most of them is correlated as a function of the surface roughness and molar mass. Figure 5.9 depicts the variation of heat transfer coefficient with reduced pressure and heat flux for pool boiling of R134a on a copper surface. According to this figure, the predictive method proposed by Gorenflo and Kenning (2010) presents steeper 5 a ) 10

b ) 2000

10

4

1000

h [W/m²K]

h [W/m²K]

Cooper (1984) Gorenflo and Kenning (2010) Ribatski and Saiz-Jabardo (2013) Stephan and Abdelsalam (1980) R134a, copper, f = 50,000 W/m², Ra = 0.4 µm, Rp = 1.0 µm

Cooper (1984) Gorenflo and Kenning (2010) Ribatski and Saiz-Jabardo (2013) Stephan and Abdelsalam (1980)

500 R134a, copper, pr = 0.08, Ra = 0.4 µ m, Rp = 1.0 µm

3

10 0.01

0.1

p r [-]

0.5

200 1000

2000

5000

10000

2 f [W/m ]

Fig. 5.9 Variation of heat transfer coefficient for pool boiling with (a) reduced pressure pr and (b) heat flux

5.2 Heat Transfer Coefficient for Convective Boiling

179

variation of heat transfer coefficient with reduced pressure than the other methods, while all of them present similar trends with variation of heat flux for pr of 0.08.

5.2

Heat Transfer Coefficient for Convective Boiling

Firstly, it is important to highlight that a predictive method should not be only statistically accurate, but should also capture the trends observed experimentally. Figure 5.10 illustrates the expected behavior of the heat transfer coefficient with vapor qualities under conditions of low heat fluxes and mass velocities. For subcooled liquid, the heat transfer coefficient depends basically on the channel geometry and Reynolds and Prandtl numbers, which in turn are functions of the fluid properties and present reduced variation with fluid enthalpy, which is mainly related with variations of liquid viscosity and thermal conductivity. It has been pointed out in this chapter that the ONB for stagnant fluid is characterized by a sudden increment of the heat transfer coefficient, and a similar behavior is observed under conditions of forced convection. For vapor qualities just higher than the one corresponding to the ONB, the heat transfer coefficient presents a sudden increment, as depicted in Fig. 5.10, which is related to phase change due to nucleate boiling. In this context, Chen (1966) pointed out that the heat transfer during convective boiling is given as the superposition of macroconvection, associated to the fluid flow, and microconvection related to bubble nucleation, growth, and detachment. Based on this, he proposed the heat transfer coefficient given as the sum of both parcels, with proper weighting factors.

rve

ted cu

expec

M

you t

Onset of Nucleate Boiling (ONB)

Dr

h [W/m²K]

less ore or

hONB Linear interp olation

hl Liquid single-phase f low

hv Vapor single-phase f low 0.0

0.2

0.4

Fig. 5.10 Expected heat transfer coefficient curve

x [-]

0.6

0.8

1.0

180

5 Flow Boiling

Under pool boiling conditions, characterized by nucleate boiling in stagnant liquid, a temperature gradient is formed along the fluid, whereas the liquid is hotter close to the heater surface and cooler away from it, and the fluid movement is mainly associated with buoyancy effects associated with the bubbles’ growth and their detachment. Therefore, as discussed before, the higher liquid temperature close to the heater surface favors bubble formation due to liquid superheating. Conversely, in the case of forced convection, the temperature profile is dependent of the velocity profile and the temperature gradient close to the hot surface is affected by the boundary layer characteristics, what implies on higher temperature gradients close to the wall compared to nucleate boiling under quiescent conditions. Therefore, the achievement of conditions for bubble nucleation and growth is mitigated by the flow, and, therefore, the contribution of nucleate boiling effects to the overall heat transfer under convective boiling conditions is reduced when compared to pool boiling. Such an effect is taken into account in predictive methods based on the superposition of effects for the heat transfer coefficient under flow boiling conditions through a nucleate boiling suppression factor, Snb, that usually, in these methods, multiplies a heat transfer coefficient given by a pool boiling correlation. Therefore, the contribution of nucleate boiling to the heat transfer coefficient for conditions close to the ONB during forced convection tends to be lower than during pool boiling. In this context, the suppression factor of nucleate boiling effects should tend to unite when the flow velocity tends to zero (G ! 0 ) Snb ! 1). Therefore, it is possible to conclude that, different from the two-phase pressure drop curve, the heat transfer coefficient for convective boiling with x tending to 0.0 (x ! 0+) does not necessarily tend to the heat transfer coefficient of liquid singlephase flow. Moreover, since subcooled liquid boiling is possible depending on heat flux, fluid properties, and surface characteristics, the condition of heat transfer coefficient of single-phase liquid flow is unattainable even for slightly negative thermodynamic vapor qualities (Eq. (2.11)). It must be emphasized that the occurrence of subcooled nucleate boiling is common, and is characterized by the formation of vapor bubbles with fluid averaged enthalpy lower than that of the saturated liquid (î < îl), which is a consequence of the temperature profile with hotter fluid close to the wall and cooler fluid in the core region of the channel; therefore, the point of ONB occurs for negative vapor quality values (x < 0.0), such as represented in Fig. 5.10. Hence, bubbles are formed and grow in the channel surface, and the drag force facilitates their detachment. Just after detachment, the bubbles tend to collapse under conditions of subcooled boiling because the liquid away from the solid surface is colder. With successive increment of fluid enthalpy along the channel, the liquid temperature away from the surface eventually will be high enough to allow the migration of the bubble without collapsing, and the simultaneous flow of vapor and liquid is possible. In this context, Tibiriçá and Ribatski (2014) presented an experimental investigation based on flow visualizations of bubble formation and departure during R134a and R245fa boiling inside microchannels, inferring the size of the bubbles and departure frequency. Based on their results, as well as based on the literature review addressed by them, it was concluded that the bubble diameter reduces with the increment of flow velocity,

5.2 Heat Transfer Coefficient for Convective Boiling

181

and Tibiriçá and Ribatski (2014) proposed a correlation for bubble departure diameter during convective boiling in microchannels, given as follows: db 130 ¼ Rel0 d

ð5:24Þ

where d is the channel diameter, and Rel0 corresponds to the Reynolds number estimated for the mixture flowing as liquid. Tibiriçá and Ribatski (2014) also analyzed the bubble growth and detachment frequency, and in a similar way to Mikic and Rohsenow (1969), and is given as follows: f reqb ¼

1:195085 ½m2 =s d2b

ð5:25Þ

where db is given by Eq. (5.24), and the resulting frequency is in Hz. Consequently, after nucleation, successive increment of the fluid enthalpy, implies on the increment of the flow velocity. In this context, the mixture superficial velocity that corresponds to the mean velocity can be recalled:   Gð1  xÞ G x 1x x Gx j ¼ jl þ jv ¼ þ ¼G þ  ρl ρv ρl ρv ρv

ð5:26Þ

Considering that the liquid density is usually in the order of 103 kg/m3 while the vapor density is usually in the order of 10 to 101 kg/m3, the parcel of Eq. (5.26) related to liquid phase can be neglected, as shown in the last term. Hence, the flow velocity increases along the channel for convective boiling conditions (increasing x) for a fixed mass velocity, and the convective parcel tends to present higher contribution with the increment of vapor quality, denoted by the positive slope of the curve for conditions with enthalpy higher than the ONB in Fig. 5.10. The velocity of vapor phase tends to be higher than that of the liquid phase because the liquid phase tends to adhere to the channel wall due to wettability properties. Additionally, as described in Chaps. 2 and 3, the two-phase flow tends to be annular for intermediary and high vapor content or for conditions dominated by vapor inertial effects. Hence, with successive evaporation of the liquid phase, the wall will eventually become dry and partially exposed to the vapor phase, which is characteristic of smaller heat transfer coefficient. Therefore, the averaged heat transfer coefficient in a given cross-section tends to reduce after the dryout initiation, which is schematically depicted in Fig. 5.10 by the region with negative slope. Additionally, since the transition from two-phase flow to vapor single-phase flow is smooth with successive reduction of the heat transfer coefficient, rather than abrupt variation such as verified for the ONB, it is expected that the heat transfer coefficient must tend to the values estimated for vapor single-phase flow.

182

5 Flow Boiling

i.a

G1

i.b

i.c

G3

h

i.d

G1,G2,G3

G2 h

G1 G2

h

G1,G2,G3

h G3

x G1>G2>G3

x G1>G2>G3 ii.b

ii.a ϕ2 ϕ3

ϕ2 ϕ3

h

x ϕ1 > ϕ2 > ϕ3

x ϕ1 > ϕ2 > ϕ3

ii.d ϕ1

h

ϕ2

ϕ1

Tsat,1 Tsat,2 Tsat,3 x Tsat,1 >Tsat,2 >Tsat,3

ϕ2

h

ϕ3

ϕ3

x ϕ1 > ϕ2 > ϕ3

x ϕ1 > ϕ2 > ϕ3

iii.b

iii.a h

x G1 >G2 >G3

ii.c ϕ1

ϕ1 h

x G1>G2>G3

iv.a

d1 d2 d 3

Tsat,1 h Tsat,2

h Tsat,3

x Tsat,1 >Tsat,2 >Tsat,3

x d1 G 2

ϕ1, G1 ϕ2, G2 xdi,1

xdi,2 x

Table 5.1 Predictive method for dryout inception (xdi) and completion (xdc) according to Mori et al. (2000) Dryout inception

 0:86 xdi,1 ¼ 0:94  1:75  106  ðRev0 BoÞ1:75 ρρv l   0:08  ρv 5 0:96 xdi,2 ¼ 0:58  exp 0:52  2:1  10 Wev0 Fr 0:02 v0 ρl 0  0:16 1 0:05 ρv ;C 0:98Fr 0:05 B v0 Bo ρl C B C B xdi,3 ¼ min B  0:21 C A @ 0:04 0:40 0:09 ρv 0:172Fr v0 Bo Wev0 ρl 8 xdi,2  xdi,3 > < xdi,3 if xdi ¼ xdi,2 if xdi,2 < xdi,3 , and xdi,2  xdi,1 > : xdi,1 if xdi,2 < xdi,3 , and xdi,2 < xdi,1 Dryout completion  0:86 xdc,1 ¼ 1:02  1:75  106  ðRev0 BoÞ1:75 ρρv l   0:08  ρv 5 0:94 xdc,2 ¼ 0:61  exp 0:57  2:65  10 Wev0 Fr 0:02 v0 ρ 0 B xdc,3 ¼ min @ 8 > < xdc,3 xdc ¼ xdc,2 > : xdc,1

1

1:01;  0:16 C A 0:22 0:09 ρv Wev0 0:690Fr 0:02 v0 Bo ρl if if if

xdc,2  xdc,3 xdc,2 < xdc,3 , and xdc,2  xdc,1 xdc,2 < xdc,3 , and xdc,2 < xdc,1

l

186

5 Flow Boiling

The non-dimensional parameters are the Boiling number Bo, Reynolds, Froude, and Weber number for the mixture flowing as vapor, and are given, respectively, as follows: ϕ G ^ilv

ð5:27Þ

Gd μv

ð5:28Þ

G2 g d ρv ðρl  ρv Þ

ð5:29Þ

G2 d ρv σ

ð5:30Þ

Bo ¼

Rev0 ¼ Fr v0 ¼

Wev0 ¼

Subsequent to the study of Mori et al. (2000), Wojtan et al. (2005a) adapted the second conditions (xdi,2 and xdc,2), which provided the best agreement their results, according to their experimental results. The correlations proposed by Wojtan et al. (2005a) are given as follows: " xdi ¼ 0:58  exp 0:52 

0:37 0:235We0:17 v0 Fr v0

 0:25  0:70 # ρv ϕ ϕcrit ρl

" xdc ¼ 0:61  exp 0:57  5:8  10

3

0:15 We0:38 v0 Fr v0

 0:09  0:27 # ρv ϕ ϕcrit ρl

ð5:31Þ ð5:32Þ

where the Froude number of the mixture flowing as vapor is given as Frv0 ¼ G2/ (ρv2gd), and ϕcrit is given according to Kutateladze’s (1948) proposal, as follows: 0:25 ^ ϕcrit ¼ 0:131 ρ0:5 v ilv ½g σðρl  ρv Þ

ð5:33Þ

An alternative approach is based on the fact that subsequent to the dryout inception, additional increments of the fluid enthalpy (vapor quality) imply on the reduction of the heat transfer coefficient, which is similar to the occurrence of CHF, even though the first phenomenon is not as critical for the heat transfer problem as the CHF. Additionally, the precise experimental identification of dryout inception is complicated, while the identification of CHF is more unambiguous, and several reliable predictive methods for CHF are available in the open literature, such as Zhang et al. (2006) and Kutateladze (1948) above presented, in this last case for pool boiling conditions. Therefore, Kanizawa et al. (2016) proposed the adoption of the CHF predictive method equaling the CHF to the actual heat flux (ϕcrit ¼ ϕ) for conditions with known and/or imposed heat flux, and by an inverse problem it is possible to infer the conditions for dryout inception.

5.2 Heat Transfer Coefficient for Convective Boiling

187

This subject will be discussed with more detail in the next chapter, but for instance consider the predictive method for CHF proposed by Zhang et al. (2006), which is originally given as follows: "

Bocrit

#   0:361 0:295 Lcrit 2:31 ρv ¼ 0:0352  Wel0 þ 0:0119 d ρl " #  0:170   ρ Lcrit 0:311  2:05 v  xin d ρl 

ð5:34Þ

where L stands for the heat length of the section, and xin corresponds to the inlet vapor quality. The Weber number for the mixture flowing as liquid is given by Wel0 ¼ G2d/ (σ ρl) and the boiling number is a function of the actual heat flux Bocrit ¼ ϕcrit/Gilv. Therefore, by assuming that the CHF is equal to the actual heat flux (ϕcrit ¼ ϕ), the corresponding heated length (Lcrit) is determined, and based on energy balance, it is possible to determine the vapor quality for dryout inception.2 Therefore, the following steps are taken for determination of vapor quality for dryout inception adopting a predictive method for CHF: • Evaluate Bocrit, Wel0, and other properties for the saturation temperature assuming ϕcrit ¼ ϕ. • Evaluate the inlet vapor quality or assume as null (xin ¼ 0.0). • Determine Lcrit/d. • Evaluate the vapor quality for dryout inception based on energy balance along a test section Lcrit long. In this context, Tibiriçá et al. (2015) experimentally investigated CHF during convective boiling in microchannels and found that the CHF occurrence depends on the system stability, whereas fluid instabilities lead to lower CHF values, and consequently lower corresponding vapor quality values. It has been shown that imposing flow restrictions upstream the evaporator results in higher flow stability, and consequently higher CHF and corresponding vapor quality. Additionally, these authors pointed out that the vapor quality corresponding to CHF reduces with increment of mass velocity, therefore, the higher the mass flux, the higher the instability of the system. Even though this study was focused on conditions for CHF, which is distinct from dryout, it provides insight about dryout inception in microchannels. It is worth mentioning that the condition for CHF occurrence in the evaporator depends on the other components of the system, such as pumping system and flow restrictions.

2

For this case, it is impossible to directly determine the Lcrit, hence, an iterative method is required for the solution. In the case of Zhang et al.’s (2006) method, it is possible to isolate the term (Lcrit/ d )0.311, and then iteratively find new values for the critical length.

188

5 Flow Boiling

a ) 1.2

R134a, d = 1.5 mm, Tsat = 30 °C, f = 10,000 W/m²

1.0

x [-]

0.8 0.6 0.4 0.2 0.0 10

Dryout inception: Mori et al. (2000) Wojtan, Ursenbacher and Thome (2005a) Saitoh, Daiguji, Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Dryout completion: Mori et al. (2000) Wojtan, Ursenbacher and Thome (2005a)

100

G [kg/m²s]

1000

10000

b ) 1.2 R134a, d = 1.5 mm, Tsat = 30 °C, G = 300 kg/m²s 1.0

x [-]

0.8 0.6 0.4 0.2 0.0 100

Dryout inception: Mori etal. (2000) Wojtan, Ursenbacher and Thome (2005a) Saitoh, Daiguji, Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Dryout completion: Mori et al. (2000) Wojtan, Ursenbacher and Thome (2005a)

1000

f [W/m²]

10000

100000

Fig. 5.13 Variation of vapor quality for dryout inception and completion with (a) mass flux and (b) heat flux for R134a in a 1.5 mm channel

Figure 5.13 depicts the variation of the dryout inception and completion with mass and heat flux for R134a at 30  C in a 1.5 mm channel. This figure also includes the prediction according to the approach presented by Saitoh et al. (2007) that is described in Sect. 5.3.2. According to Fig. 5.13a, it can be observed that in general

5.3 Predictive Methods for Convective Flow Boiling

189

the vapor quality for dryout inception and completion tends to reduce with the increment of mass flux, and a similar relationship is verified for the effect of heat flux in Fig. 5.13b. Only the predictions for dryout inception xdi according to Mori et al. (2000) predicted an increment of vapor quality for dryout inception with mass velocity, which is not expected because the higher the flow velocity, the higher is the probability of occurrence of mist flow, characterized by continuum vapor flow with entrained liquid droplets. It should also be mentioned that some of the methods predict vapor qualities higher than unity, which is related to thermodynamic non-equilibrium between the liquid and vapor phases, whereas the vapor is superheated and the liquid flows as entrained droplets within the vapor stream.

5.3

Predictive Methods for Convective Flow Boiling

Up to date predictive methods for heat transfer coefficient during convective flow boiling available in the open literature are presented and discussed. Several predictive methods for heat transfer coefficient during convective boiling in microchannels have been recently proposed, and most of them are based on the approach firstly proposed by Chen (1966), which combines the contributions of nucleate boiling and convective effects. Conversely, other more complicated methods were also proposed, such as the three zones model proposed by Thome et al. (2004) for convective boiling in microchannels. Considering convective boiling in conventional channels, other approaches have also been proposed, such as the phenomenological method proposed by Wojtan et al. (2005a, b), according to which the heat transfer coefficient is estimated according to the corresponding flow pattern, which in turn is predicted assuming the method for flow pattern prediction described in Chap. 3. Additionally, regarding historical aspects, methods based on the approach of twophase multipliers, as usually adopted for the prediction of pressure drop (see item 4.1), were also proposed for the heat transfer coefficient, where a factor given by a function multiplies the corresponding value of the heat transfer coefficient for singlephase flow, such as presented by Bergles and Rohsenow (1964) and Gungor and Winterton (1987). However, these methods capture only behaviors similar to the curves presented in Fig. 5.11i.a, corresponding to conditions dominated by convective effects. It must be remembered that it is expected that the heat transfer coefficient trends should be similar to a combination of curves, Fig. 5.11i.a and ii.c, hence with more pronounced contribution of the nucleate boiling mechanisms at least for specific ranges of operational conditions, characterized by low vapor qualities, high heat fluxes, and high saturation temperatures. Therefore, the approach presented by Chen (1966) is more appropriate to capture the contribution of nucleate boiling mechanisms; nonetheless, the original method is

190

5 Flow Boiling

graphic based, since it requires plots to infer some parameters such as enhancement factor and will not be presented in this book since it is complicated to implement it computationally. Nonetheless, several methods have been subsequently proposed based on the combination of both parcels and some of them are presented in the following items.

5.3.1

Liu and Winterton (1991)

Liu and Winterton (1991) proposed a predictive method for the heat transfer coefficient that proved to be slightly better than the previous Gungor and Winterton (1986, 1987) methods regarding the agreement with experimental results. The method proposed by Liu and Winterton (1991) is based on the Chen (1966) approach, considering the superposition of convective and nucleate boiling effects, and is given as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ¼ ðec F c hl0 Þ2 þ ðenb Snb hnb Þ2

ð5:35Þ

where Fc and Snb are, respectively, the factor of enhancement of convective effects and the factor of suppression of nucleate boiling effects, hl0 and hnb are the heat transfer coefficients estimated for the convective and nucleate boiling parcels, respectively. The terms ec and enb were included by Liu and Winterton (1991) into their method to account for stratification effects for horizontal flow that tends to reduce the perimeter-averaged heat transfer coefficient; then, for vertical flow or for horizontal flow with Froude number higher than 0.05 (Frl0 ¼ G2/(ρl2gd) > 0.05) these parameters are equal to unity, and horizontal flows with Froude number lower than 0.05 (Frl0  0.05) are given as follows: l0 ec ¼ Fr 0:12Fr l0 pffiffiffiffiffiffiffiffi enb ¼ Fr l0

ð5:36Þ ð5:37Þ

The heat transfer coefficient for convective parcel hl0 is given according to the Dittus and Boelter (1985) correlation for turbulent flow assuming the mixture flowing as liquid, as follows: hl0 ¼ 0:023 

kl 0:8 0:4 Re Pr d l0 l

ð5:38Þ

with the Reynolds number for the mixture flowing as liquid given by Rel0 ¼ Gd/ μl. The heat transfer coefficient for nucleate boiling parcel is given according to the Cooper (1984) predictive method given by Eq. (5.12) assuming the peak roughness equal to 1μm.

5.3 Predictive Methods for Convective Flow Boiling

191

The intensification factor of convective effects is given as a function of the vapor quality (the higher x, the higher F), density ratio, and liquid Prandtl number that accounts for the effect of convective heat transfer along the liquid in contact with the wall and is given as follows:   0:35 ρl F c ¼ 1 þ xPr l 1 ρv

ð5:39Þ

The suppression factor of nucleate boiling effects must be smaller than unity, because it is intended to suppress and tends to reduce its value with the increment of convective effects. Therefore, it is given as follows:   0:16 1 Snb ¼ 1 þ 0:055F 0:1 c Rel0

ð5:40Þ

The condition of dryout occurrence is not predicted by this method, therefore, the heat transfer coefficient tends to always increase with vapor quality. This method was compared by the original authors with an extensive database of experimental results for saturated and subcooled convective boiling, comprising results for mass flux between 12 and 8150 kg/m2s, heat flux between 348 and 2.62106 W/m2, and channels with internal diameter ranging from 2.95 to 32 mm. It could be mentioned that the adoption of an asymptotic exponent equal 2 in Eq. (5.35) emphasizes the dominant heat transfer mechanism, whereas the resulting value tends to the highest value of the sum, and the smallest one present reduced contribution, which is even negligible depending on the orders of magnitude and exponents (1002 + 12  100 100 + 1 ¼ 101). For additional details on asymptotic correlations, it is suggested to check the study presented by Churchill (2000).

5.3.2

Saitoh et al. (2007)

The predictive method for heat transfer coefficient during convective boiling proposed by Saitoh et al. (2007), focused on microchannels, is based on the Chen (1966) approach by combining according to an additive manner the nucleate boiling and convective effects under pre-dryout conditions, while another approach is adopted for post-dryout conditions. It is important to clarify that these authors consider that dryout corresponds to the onset of wall drying. This method has been evaluated by several independent laboratories and showed good agreement with experimental results for heat transfer coefficient during convective boiling in small-scale channels. According to the authors, during liquid-vapor flow in microchannels the surface tension effects are predominant over gravitational effects. Hence, they used the Weber number to account for this effect instead of using only the vapor quality, density ratio, and Prandtl number. The dryout occurrence is modeled based on the vanishing process of a liquid film during annular flow, which is quoted to be the flow pattern that precedes the dryout.

192

5 Flow Boiling

Then, based on experimental results for the heat transfer coefficient during convective boiling in microchannels, the authors identified the vapor quality corresponding to the maximum heat transfer coefficient, whereas successive increments of vapor quality after the peak imply on reduction of h (recall Fig. 5.11ii.c). Additionally, they considering that the heat transfer coefficient during annular flow can be given as a function of the liquid film thickness δ as follows: h¼

kl δ

ð5:41Þ

which assumes liquid film thickness uniform along the channel perimeter, absence of entrainment of liquid within the vapor flow, and solely conduction along the liquid film. Therefore, based on experimental results for the dryout inception, the liquid film thickness for this condition was evaluated as 15μm independent of the channel diameter, mass flux, and heat flux. Hence, the authors adopted the relationship for void fraction based on the slip ratio, given by Eq. (2.34), where the slip ratio s ¼ uv/ul is given based on the minimum momentum of two-phase flow as follows: u s¼ v¼ ul

 0:5 ρl ρv

ð5:42Þ

Based on the geometry of annular flow, and by assuming the uniform liquid film thickness, the void fraction is determined as follows: α¼

π ðd2δÞ2 4 πd2 4

  2δ 2 1 ¼ 1 ¼ ρv ð1xdi Þ d 1 þ s ρ xdi

ð5:43Þ

l

and solving for the vapor quality: 1 1¼ xdi xdi ¼





2δ 1 d



2



 0:5 ρl ρv 1 ρv ρl

1

1

 2δ 2 d

1

 0:5 ρl ρv

ð5:44Þ ð5:45Þ

with the liquid film thickness δ as constant and equal to 15μm. Then, for conditions with x  xdryout, the heat transfer coefficient is given as a combination of convective and nucleate boiling effects, as follows: h ¼ F c  hl þ Snb  hnb

ð5:46Þ

where Fc and Snb are, respectively, the convective enhancement and nucleate boiling suppression factors, and hl and hnb refer to the heat transfer coefficients estimated for forced convection and pool boiling. These enhancement and suppression factors are given as follows:

5.3 Predictive Methods for Convective Flow Boiling

193

b 1:05 X tt Fc ¼ 1 þ 1 þ We0:4 v Snb ¼

1 þ 0:4



ð5:47Þ

1 F 1:25 c Rel

 104

1:4

ð5:48Þ

where the Weber number (Wev ¼ (Gx)2d/σρv) assumes only vapor flow along the channel and accounts for the surface tension effects. The heat transfer coefficient associated to purely convective effects is given according to Dittus and Boelter (1985) for turbulent flow and by a constant Nusselt number for laminar flow as follows:

hl ¼

8 >
: 0:023  k l Re0:8 Pr3 l d l

for

Rel < 1000

for

Rel > 1000

ð5:49Þ

where the liquid Reynolds number is evaluated based on the liquid flow rate (Rel ¼ G(1  x)/μl). Conversely, the heat transfer coefficient for nucleate boiling parcel is estimated according to Stephan and Abdelsalam (1980) correlation for synthetic refrigerants, given by Eqs. (5.22) and (5.23). The Lockhart and Martinelli parameter X^ in Eq. (5.47) is given assuming turbulent regime for both phases given by Eq. (4.55), as follows:    0:5  μ 0:1 1  x 0:9 ρv l b X tt ¼ x ρl μv

ð5:50Þ

Therefore, for pre-dryout conditions the heat transfer coefficient is evaluated based on Eqs. (5.46) to (5.50). For vapor qualities higher than the one corresponding to the dryout, the heat transfer coefficient tends to reduce with additional increments of the vapor quality, until the condition of dryout completion, which was assumed by Saitoh et al. (2007) as happening for vapor quality equal to unity. Nonetheless, instead of just interpolating the heat transfer coefficient between the imminence of dryout until saturated vapor single-phase flow, the authors modelled the fraction of dry perimeter Fd as a function of the normalized vapor quality xnor as follows: F d ¼ x3nor þ x2nor þ xnor  0:03

ð5:51Þ

where the vapor quality is normalized between the dryout inception and completion, assumed as unity (xdc¼1.0), as follows: xnor ¼

x  xdi 1  xdi

ð5:52Þ

Then, the resulting heat transfer coefficient for this region is given as follows:

194

5 Flow Boiling

h ¼ ð1  F d Þhpre dryout þ F d hv

ð5:53Þ

where the pre-dryout heat transfer coefficient hpre dryout is estimated according to Eqs. (5.46) to (5.50) for x ¼ xdi and hv is estimated for vapor turbulent flow as follows: hv ¼ 0:023 

kv 0:8 13 Re Pr d v0 v

ð5:54Þ

with the Reynolds number estimated for only vapor flow (Rev0 ¼ Gd/μv). The correlations of this method were adjusted based on experimental results of the original authors for R134a saturated boiling in channels with diameters ranging from 0.51 to 10.92 mm.

5.3.3

Kandlikar and Co-workers

Kandlikar and co-workers have proposed several predictive methods for heat transfer coefficient for micro and conventional-sized channels. Considering that channel reduction results in eventual transition from turbulent to laminar flow, Kandlikar and co-workers proposed a distinct predictive method accounting for this aspect. Hence, Kandlikar (1990, 1991) proposed a predictive method for heat transfer coefficient during convective boiling inside macroscale channels, which is given as follows: h ¼ max ðhnbd , hcd Þ

ð5:55Þ

where the nucleate boiling dominated heat transfer hnbd coefficient is estimated assuming an approach similar to two-phase multipliers, however accounting for nucleate boiling and convective parcels, it is given as follows: 0:8 0:7 hnbd ¼ 0:6683Cn0:2 Fr0:3 l0 þ 1058:0Bo F fl ð1  xÞ hl0

ð5:56Þ

where the first term inside the square brackets corresponds to the parcel relative to convective effects and the second one is related to nucleate boiling effects. The boiling number is given according to Eq. (5.27), the convection number Cn is given as follows: Cn ¼



1x x

0:8 ρ 0:5 v

ρl

ð5:57Þ

and the Froude number for the mixture flowing as liquid is given as follows: Frl0 ¼

G2 ρ2l gd

ð5:58Þ

5.3 Predictive Methods for Convective Flow Boiling Table 5.2 Fluid-dependent parameter of the Kandlikar model (Kandlikar 1990, 1991)

Fluid Water R11 R12 R13B1 R22 R113 R114 R134a R152a Nitrogen Neon

195 Ffl 1.00 1.30 1.50 1.31 2.20 1.30 1.24 1.63 1.10 4.70 3.50

The term Ffl depends on the type of fluid and is given in Table 5.2. The heat transfer coefficient for the mixture flowing as liquid hl0 is given according to Gnielinski (1976), given as follows: hl0 ¼

ðf l0 =8Þð Re l0  1000ÞPrl kl =d  pffiffiffiffiffiffiffiffiffiffi 2=3 1 þ 12:7 f l0 =8 Pr l  1

ð5:59Þ

or Petukhov (1970), given as follows: hl0 ¼

ðf l0 =8Þ Re l0 Prl kl =d  pffiffiffiffiffiffiffiffiffiffi 2=3 1:07 þ 12:7 f l0 =8 Pr l  1

ð5:60Þ

where the friction factor f is given as follows: f l0 ¼ ð1:82 log 10 Re l0  1:64Þ2

ð5:61Þ

The heat transfer coefficient for a condition dominated by convective effects hcd is given as follows: 0:8 0:7 hcd ¼ 1:136Cn0:9 Fr0:3 l0 þ 667:2Bo F fl ð1  xÞ hl0

ð5:62Þ

According to Kandlikar and Balasubramanian (2004), the heat transfer coefficient given by Eqs. (5.55) to (5.62) is valid for macroscale channels, which were characterized for Rel0 higher than 3000 (Rel0  3000). For minichannels, which were characterized for 1600 < Rel0 < 3000, corresponding to a transitional condition, the heat transfer coefficient is estimated based on linear interpolation based on the Reynolds number for heat transfer coefficient evaluated at Rel0 ¼ 3000, given by Eq. (5.55), and the heat transfer

196

5 Flow Boiling

coefficient estimated for Rel0 ¼ 1600, also given by Eq. (5.55); however, the nucleated boiling and convective dominated heat transfer coefficients are given, respectively, by Eqs. (5.56) and (5.62) assuming Froude number as unitary (Frl0 ¼ 1.0), because the gravitational effects vanish with reduction of channel diameter. For laminar flow (Rel0 < 1600), the heat transfer coefficient for the mixture flowing as liquid in circular channel is given as follows:

hl0 ¼

4:364kl =d

for

ϕ ¼ constant

3:657kl =d

for

T s ¼ constant

ð5:63Þ

Notice that for non-circular channels, the constant Nusselt number must be adjusted, such as described by Shah and London (1978). In the case of very small channels, characterized by Kandlikar and Balasubramanian (2004) as operational conditions corresponding to Rel0  100, the nucleate boiling mechanism is dominant; hence, the heat transfer coefficient is given by Eq. (5.56) with unitary Froude number, and the heat transfer coefficient for liquid single-phase flow is estimated based on the constant Nusselt number. The predictive methods proposed by Kandlikar and co-workers were compared and validated with more than 10,000 independent experimental results, comprising results for flow boiling of refrigerants in circular and non-circular channels with hydraulic diameter as small as 0.19 mm.

5.3.4

Wojtan et al. (2005a, b)

We have seen in Sect. 3.3.1 a flow pattern map proposed by Wojtan et al. (2005a), and in the same study (Wojtan et al. 2005b), a predictive method was proposed for heat transfer coefficient for each flow pattern. Considering that the flow pattern predictive method is valid for channels of conventional size, the heat transfer predictive method is also valid for macrochannels. The proposed method is based on the proposal of Kattan et al. (1998), which accounts for the stratification angle γ, such as schematically depicted in Fig. 3.4, as a main geometrical parameter. Based on this parameter, the heat transfer coefficient for stratified, intermittent, and annular flows are given as weighted averaged values as follows: h¼

γhv þ ðπ  γ Þhliq π

ð5:64Þ

where the stratification angle is given as a function of the void fraction estimated value, rather than by solving the momentum balance as performed by Taitel and Dukler (1976). Wojtan et al. (2005b) adopted the void fraction predictive method proposed by Rouhani (1969), given by Eqs. (2.46) to (2.48) and based on the drift

5.3 Predictive Methods for Convective Flow Boiling

197

flux model proposed by Zuber and Findlay (1965), and repeated here to facilitate the implementation of the method: ( α2 ¼

ρv x

("

 1 # 

1 ))1 gdρ2l 4 x 1x 1:18 gσ ðρl  ρv Þ 4 þ 1 þ 0:2ð1  xÞ þ ρv ρl G ρ2l G2 ð5:65Þ

The stratification angle for the smooth stratified flow is evaluated based on the area-averaged void fraction as proposed by Biberg (1999) as follows: 1 h i 1 1 3π 3 1 1  2α2 þ α32  ð1  α2 Þ3 þ α ð1  α2 Þ 2 200 2 h  i

ð1  2α2 Þ 1 þ 4 α22 þ ð1  α2 Þ2 þ α2

γ ss ¼ π  πα2 



ð5:66Þ

The heat transfer coefficient for the dry region of the channel wall, represented by hv in Eq. (5.64), is given as follows:  0:8 kv hv ¼ 0:023 Re v Pr 0:4 v d

ð5:67Þ

where the modified Reynolds number of the vapor flow is given as follows: Re v ¼

Gxd μv α2

ð5:68Þ

The heat transfer coefficient corresponding to the wetted region of the channel wall, given by hliq, is given as an asymptotic combination of convective and nucleate boiling effects as follows: hliq ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 h3cb þ ð0:8hnb Þ3

ð5:69Þ

The heat transfer coefficient for nucleate boiling parcel hnb is estimated according to Cooper (1984) correlation, given by Eq. (5.12), and the heat transfer coefficient for convective boiling parcel hcb is given as a function of the liquid film thickness δ as follows: 0:4 hcb ¼ 0:0133 Re 0:69 δ Pr l

kl δ

ð5:70Þ

and the liquid film thickness is given as a function of void fraction and stratification angle as follows:

198

5 Flow Boiling

δ¼

πd ð1  α2 Þ 4ð π  γ Þ

ð5:71Þ

The Reynolds number based on the liquid film thickness of Eq. (5.70) is given as follows: Re δ ¼

4Gδð1  xÞ μl ð1  α2 Þ

ð5:72Þ

Considering that the stratified flow pattern was divided into three flow patterns, namely, smooth stratified, stratified wavy, and slug+stratified flow, and the stratification angle given by Eq. (5.66), Wojtan et al. (2005b) proposed corrections of stratification angle as a function of the mass velocity. Hence, considering the flow patterns classified according to the methods described in Sect. 3.3.1, during smooth stratified flow, the stratification angle is given by Eq. (5.66) (γ ¼ γ ss). On the other hand, during intermittent and annular flow patterns the stratification angle should be null (γ ¼ 0). During stratified wavy flow, the stratification angle is corrected as follows: γ¼

Gwavy  G Gwavy  Gss

0:61 γ ss

ð5:73Þ

where Gwavy and Gss correspond to the mass fluxes for transition to stratified wavy and smooth stratified, respectively, given by Eqs. (3.126) and (3.125), respectively. and G is the current mass flux. Similarly, γ is also corrected for the condition of slug+stratified wavy flow, however, also accounting for the effect of vapor quality as follows: γ¼



0:61 x Gwavy  G γ ss xia Gwavy  Gss

ð5:74Þ

where xia corresponds to the vapor quality for transition between intermittent and annular flow, which is assumed as a constant as proposed by Taitel and Dukler (1976) and is given by Eq. (3.129). Therefore, by estimating the stratification angle γ, it is possible to estimate the heat transfer coefficient by Eq. (5.64) for smooth stratified, stratified wavy, slug +stratified wavy and slug flow. For mist flow, the authors adjusted the correlation proposed by Groeneveld (1973), resulting in the following relationship: ( h¼

1:06 0:0117 Re 0:79 h Pr v



0:4 )1:83 ρl kv  1 ð 1  xÞ ρv d

 1  0:1

ð5:75Þ

5.3 Predictive Methods for Convective Flow Boiling

199

where the term inside the brackets corresponds to a correction for the Reynolds number for homogeneous flow proposed by Groeneveld (1973), which is given as follows: Re h ¼

  ρv Gd xþ μv ρ l ð 1  xÞ

ð5:76Þ

In the condition of dryout flow, limited by the transitions given by Eqs. (3.130) to (3.132), the heat transfer coefficient reduces sharply with the increment of vapor quality. In this condition, Wojtan et al. (2005b) proposed to interpolate the heat transfer coefficient conditions of dryout inception and completion as follows: h¼

hðxdc Þ  hðxdi Þ ðx  xdi Þ þ hðxdi Þ xdc  xdi

ð5:77Þ

where xdi and xdc correspond to the vapor quality for dryout inception and completion, respectively, proposed by Mori et al. (2000) and adjusted by Wojtan et al. (2005a), given by Eqs. (5.31) and (5.32), respectively. The implementation of this method, compared to the correlation based on the superposition of effects, requires a more complex programming and the resulting predictions are not significantly more accurate than the prediction provided by simpler methods when compared with experimental results, such as the one proposed by Saitoh et al. (2007). Nonetheless, the method cares with the involved phenomena and, therefore, is suitable because it tries to capture different dominant heat transfer mechanisms. Moreover, the method considers the conditions of dryout and mist flow patterns, which were addressed by few investigations.

5.3.5

Thome and Co-workers

Thome et al. (2004) were pioneers to propose a phenomenological model that takes into account the two-phase flow topology to predict the heat transfer coefficient during convective boiling in microchannels. This model is referred in literature as the 3-zone model, since it is based on the assumption that two-phase flow inside a microchannel can be represented by weighing the contributions of liquid slug, elongated bubble (or annular), and a dry region, as depicted schematically in Fig. 5.14. Recall that during two-phase flows inside microchannels, stratified-like flow patterns are not expected. The model was developed for conditions of uniform heat flux and assumes vapor and liquid flow at the same velocities (no-slip), no superheating; hence, the liquid and vapor are kept at Tsat and the energy supplied to the fluid promotes its vaporization. Moreover, it is also assumed that the ONB occurs for null vapor

5 Flow Boiling

Dry region

δmax

Elongated bubble

δmin

Flow direction

Ldry

Lfilm

Liquid slug

d

200

Ll

Fig. 5.14 Schematics of the three-zone model for heat transfer during convective boiling inside microchannels. (Adapted from Thome et al. 2004)

quality (x ¼ 0.0). It must be mentioned that this model aims to capture the evolution of the flow along the channel, rather than just estimating the heat transfer coefficient at a given position. Hence, considering the point that the fluid is under saturated liquid condition, corresponding to the assumed ONB, bubbles start to form and grow until the conditions that drag and buoyance forces overcome the surface tension forces. The authors adopted the bubble growth and detachment period as proposed by Plesset and Zwick (1954), given as follows: Δt b ¼

 2 πρl cpl ρvbilv d 48k l ρl cpl ðT w  T sat Þ

ð5:78Þ

where cpl is the specific heat for constant pressure of the saturated liquid, and (Tw Tsat) corresponds to the wall superheating. Notice that the term inside the brackets corresponds to the inverse of the Jakob number multiplied by the density ratio. Therefore, during the bubble growing period Δtb, the liquid velocity is kept constant, and based on this assumption, it is possible to estimate the initial liquid slug length Ll, shown in Figure 5.14, assuming that the mass transfer to the bubble is negligible (ρl  ρv), as follows: Ll ¼

G Δt ρl b

ð5:79Þ

Similarly, assuming that the bubble will detach when it blocks the channel crosssection, and that the liquid film thickness is much smaller than the channel diameter (δ d ), it is possible to estimate the initial bubble length as follows: 2 Lfilm ¼ d 3

ð5:80Þ

In the initial condition, the dry region would be null (Ldry ¼ 0), and the vapor quality just after the bubble detachment (t ¼ 0 + Δtb) is estimated based on the mass of each phase as follows:

5.3 Predictive Methods for Convective Flow Boiling



201

mv 1 1 ¼ ¼ 3GΔt b mv þ ml 1 þ Ll ρl 1 þ 2ρv d Lfilm ρ

ð5:81Þ

v

Between the phases, no-slip has been assumed, and, consequently, the homogeneous model for void fraction was adopted. Considering the boundary condition of uniform heat flux and that all the energy supplied to the fluid is used for its vaporization, the vapor quality downstream the ONB can be estimated based on the energy balance. Hence, recognizing that the frequency of bubble passage does vary along the bubble length, the total length of the group of liquid slug, film region, and dry region can be evaluated as follows: 

L ¼ Ll þ Lfilm þ Ldry

x 1x ¼ Δt b G þ ρv ρl

 ð5:82Þ

where the vapor quality depends on the axial position. Based on this parameter, it is possible to estimate the overall vapor length based on the void fraction as follows: Lv ¼ Lfilm þ Ldry ¼ Lαh ¼

Δt b Gx ρv

ð5:83Þ

and, therefore, the corresponding period is estimated as follows: L Δt b Δt v ¼  v  ¼ ρv 1x x 1x 1 þ G ρ þ ρ ρl x v

ð5:84Þ

l

Similarly, the period of the liquid slug passage can be evaluated as follows: Lð1  αh Þ L Δt ¼   ¼ ρl x b Δt l ¼  l x 1x x 1x ρv 1x þ 1 G ρ þ ρ G ρ þ ρ v

l

v

ð5:85Þ

l

The period of time corresponding to vapor flow, given by Eq. (5.84), comprises the period of film flow and dry wall, and to infer both parcels it is necessary to access the liquid film thickness profile. As shown in Fig. 5.14, the liquid film is thickest close to the liquid slug, and thinnest close to the dry region, and for the maximum value, Thome et al. (2004) adjusted a correlation initially proposed by Moriyama and Inoue (1996), which resulted in the following relationship: 0

10:84 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i18 8 μ δmax B  l C 0:07Bd 0:41 þ 0:18 ¼ Cδ, max @3 A d ρl dG ρx þ 1x ρ v

l

ð5:86Þ

202

5 Flow Boiling

where Cδ,max corresponds to an empirical parameter obtained based on the authors’ experimental results (Thome et al. 2004) given by a constant value of 0.29, and the modified Bond number Bd* is given as follows:   2 ρl d x 1x Bd ¼ þ G ρv ρl σ

ð5:87Þ

Considering that the energy supplied to the fluid results only in its vaporization, it is possible to estimate the period of time required for the liquid film to vaporize from δmax to δmin. For an infinitesimal channel length dz, the heat transfer rate can be related to the film thickness variation and with the imposed heat flux as follows: q_ ¼ ρl 2π

  d dδ b δ dz ilv ¼ ϕπd 2 dt

ð5:88Þ

Therefore, solving for the liquid film thickness, assuming that initial thickness is δmax, that the minimum thickness is δmin, and that d/2  δ, the following relationship is obtained: δmin ¼ δmax 

ϕ Δt δ, min ρlbilv

ð5:89Þ

where the minimum liquid film thickness was adjusted by Thome et al. (2004) based on their experimental database as equal to 0.3106 m. Solving for the period of time for the liquid film to reduce from δmax to δmin, the following relationship is obtained: Δt δ, min ¼ ðδmax  δmin Þ

ρlbilv ϕ

ð5:90Þ

Hence, if the required time to vaporize the liquid film until the minimum value is lower than the total time of the vapor bubble passage in a given position (Δtδ,min  Δtv), there should be no dry region in the bubble tale (Ldry ¼ 0). Otherwise, if Δtδ,min > Δtv there should be a dry region, which length is given as follows:   x 1x þ ðΔt v  Δt δ, min Þ Ldry ¼ G ρv ρl

ð5:91Þ

Therefore, up to this point the corresponding geometrical characteristics of the set of liquid slug, annular-like film flow, and vapor bubble have been addressed, and based on this characterization it is possible to estimate the time-averaged heat transfer coefficient. The reader should notice that the objective of this model is inferring the geometrical parameters for a given axial location of the channel, for which it is possible to estimate the vapor quality based on energy balance and all the other parameters.

5.3 Predictive Methods for Convective Flow Boiling

203

The heat transfer coefficient for liquid slug and dry region are evaluated based on predictive methods for single-phase flow, assuming flow developing conditions. Thome et al. (2004) adopted the method proposed by Churchill and Usagi (1972) that combines the heat transfer coefficient for laminar and turbulent regimes under flow-developing conditions, which is given as follows: hs ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h4s,lam þ h4s,turb

ð5:92Þ

where the averaged heat transfer coefficient for the laminar parcel is given by the Shah and London (1978) correlation as follows: hs,lam

rffiffiffiffiffiffiffiffiffiffi Res ks ¼ 2  0:455 Pr Ls =d d 1 3

ð5:93Þ

and the averaged heat transfer coefficient for the turbulent regime is given based on the proposal of Gnielinski (1976) for developing flow as follows: hs,turb

"  2=3 # ðf s =8ÞðRes  1000ÞPrs d ks pffiffiffiffiffiffiffiffiffi 2=3 ¼  1þ L d s 1 þ 12:7 f s =8 Prs  1

ð5:94Þ

where the friction factor f is given by Eq. (5.61), and the subscript s corresponds to each single-phase flow, either liquid slug l or vapor bubble flow dry. The Reynolds number is evaluated based on the classic definition (Res ¼ G d/μs). The heat transfer coefficient during film liquid flow is estimated assuming stagnant liquid film, hence corresponding to a local thermal resistance of conduction. Considering that the thickness reduces from δmax to δmin, Thome et al. (2004) proposed to adopt the averaged thickness value, and the resulting heat transfer coefficient is given as follows: hfilm ¼

2kl δmax þ δmin

ð5:95Þ

The resulting heat transfer coefficient for convective boiling is given as average heat transfer coefficient, weighted according to the periods of time cooresponding to the different heat transfer mechanisms as follows: h¼

hl Δt l þ hfilm Δt film þ hdry Δt dry Δt b

ð5:96Þ

where Δtdry corresponds to max(0, Δtv – Δtδ,min) and Δtfilm corresponds to min (Δtmin, Δtv).

204

5 Flow Boiling

As the reader might have noticed, the evaluation of the wall temperature in Eq. (5.78) depends on the heat transfer coefficient, which in turn depends on Δtb, hence an iterative method is needed for estimation of h. It is suggested to use a guess value for wall temperature, and after evaluating the heat transfer coefficient, estimate the new wall temperature value as Tw,new ¼ Tsat + ϕ/h, and then update the corresponding Tw as Tw ¼ ((factor-1)Tw + Tw,new)/factor, where factor > 1, to reduce the number of required iterations, otherwise the wall temperature and heat transfer coefficient might vary significantly. It should be highlighted that annular flow is dominant in microchannels, and, therefore, this model should be restricted to a narrow range of vapor quality. However, the authors neglected the flow pattern transitions when adjusting the empirical constants of their model by using a database involving vapor qualities from 0 to 1. Later, Cioncolini and Thome (2011) proposed a method to predict the heat transfer coefficient for annular flows. Their method is based on an algebraic model of turbulence applied to the liquid film. Most recently, Costa-Patry et al. (2012) combined the 3-zone model proposed by Thome et al. (2004), which is valid for elongated bubbles, and the model of Cioncolini and Thome (2011), which is valid for annular flow, into a single model for microscale flow boiling, where the flow pattern transition is predicted according to the method proposed by Ong and Thome (2011). Their new method provided reasonable predictions of the independent results of Costa-Patry et al. (2012) for flow boiling in a multichannel configuration. They assumed the minimum film thickness in the 3-zone model equal to the surface roughness. Schweizer et al. (2010) obtained simultaneous images of the flow for a single microchannel of an elongated bubble and the associated temperature distribution on the heated surface. Flow images were obtained in the upper region of the channel with a high-speed camera and the temperature field in the lower region using infrared thermography. From the temperature field along the thin metal surface used as heater, local heat fluxes were estimated. On contrary to the 3-zone model that predicts high heat fluxes at the terminal region of an elongated bubble, Schweizer et al. (2010) observed two regions with high heat transfer rates corresponding to the frontal and end regions of the elongated bubble. The authors suggest that the observed behavior is related to intense evaporation in the triple contact line established in the frontal and terminal regions of the elongated bubble. Such a hypothesis would imply on a dry region between the triple contact lines associated with an elongated bubble. This fact was not corroborated by the results provided by the authors because according to them the heat flux in the regions comprised by the bubble and in the presence of the liquid piston are similar. It seems more feasible that the high heat transfer coefficient close to the bubble front is promoted by the displacement of liquid from the central region of the channel toward the heating surface. This hypothesis is corroborated by the three-dimensional numerical simulations performed by Liu et al. (2012), according to which lower wall temperatures are observed just after the bubble passage due to micro-convection effects resulting from its displacement. They also reported regions of lower wall temperatures at the intermediary region of the bubble, just downstream its front nose.

5.3 Predictive Methods for Convective Flow Boiling

5.3.6

205

Ribatski and Co-workers (Kanizawa et al. 2016; Sempertegui-Tapia and Ribatski 2017)

Kanizawa et al. (2016) presented an extensive analysis of a database comprising experimental results for convective boiling of R134a, R245fa, and R600a in microchannels with internal diameters from 0.38 to 2.6 mm. Based on this analysis, they concluded that the method of Saitoh et al. (2007) provided the best predictions of their experimental results. Nonetheless, this method did not adequately capture most of the identified trends identified displayed by the experimental results. Therefore, these authors proposed an adjustment of the method of Saitoh and co-workers according to their database. For conditions prior to dryout occurrence, the heat transfer coefficient is evaluated according to Kutateladze (1961)–Chen (1966) approach, consisting in combining the convective and nucleate boiling contributions with corresponding enhancement and suppression factors. In the method of Kanizawa et al. (2016), the heat transfer coefficient is given as follows: h ¼ F c  hl þ Snb  hnb

ð5:97Þ

where the heat transfer coefficient associated to convective effects hl is evaluated by assuming only liquid flow according to Eq. (5.49), as per the procedure adopted by Saitoh et al. (2007), and the heat transfer coefficient for nucleate boiling effects is evaluated according to Stephan and Abdelsalam (1980) that for synthetic refrigerants is given by Eqs. (5.22) and (5.23). Considering that the database used in this study includes hydrocarbons, the predictive method for nucleate boiling parcel for these fluids is given by Eq. (5.20) and (5.23). The enhancement factor of convective parcel is given as follows: F c ¼ 1 þ 2:50 

b 1:32 X lv 1 þ We0:24 uv

ð5:98Þ

where the Weber number is based on the vapor in situ velocity, given as follows: Weuv ¼

ρv u2v d σ

ð5:99Þ

and the in situ velocity of the vapor phase uv is given as function of the void fraction α, which in turn is evaluated based on the Kanizawa and Ribatski (2016) method, given by Eqs. (2.56) and (2.58). The Lockhart and Martinelli (1949) parameter X^lv is evaluated assuming turbulent flow for liquid phase, as recommended by Da Riva et al. (2012), as follows:

206

b tv ¼ X

5 Flow Boiling

8 > >
1  x 0:9 ρv l > b tl ¼ 1 Re0:4 :X 18:7 v x ρl μv

for

Rev > 1000

for

Rev  1000

ð5:100Þ

with the vapor Reynolds number equal to Rev ¼ G(1  x)d/μv. The suppression factor of nucleate boiling effects is given as follows: 3

Snb ¼ 1:06 

Bd 810  0:86 Rel0 F 1:25 c 1 þ 0:12 10000

ð5:101Þ

where the Bond number Bd is introduced in order to capture bubble confinement effects and is defined as follows: Bd ¼

ðρl  ρv Þd2 g σ

ð5:102Þ

The dryout inception is estimated based on the method of Zhang et al. (2006) for CHF, from which, based on an energy balance, it is possible to obtain the vapor quality for dryout inception xdi. In a similar way to Saitoh et al. (2007), it is assumed that the dryout completion occurs for vapor quality equal to unity, and for x ¼ 1 the heat transfer coefficient should correspond to the saturated vapor single-phase flow, which is given by the Dittus and Boelter (1985) correlation as follows: hv0 ¼ 0:023 

kv 0:8 13 Re Pr d v0 v

ð5:103Þ

where the Reynolds number is evaluated assuming the mixture flowing as vapor (Rev0 ¼ Gd/μv). Therefore, for the dryout condition (xdi  x  1) the heat transfer coefficient is given as a linear interpolation between the imminence of dryout inception and completion as follows: h ¼ hpre ðxdi Þ 

1x x  xdi þ hv0  1  xdi 1  xdi

ð5:104Þ

Subsequently, Sempertégui-Tapia and Ribatski (2017) complemented the database with results for R1234ze(E) and R1234yf for channels with 1.1 mm of internal diameter. The evaluation of dryout inception and completion are like the approach presented by Kanizawa et al. (2016). Additionally, the Chen (1966) approach is also adopted for evaluation of the heat transfer coefficient before dryout, however it adopted an asymptotic exponent equal to 2 as adopted by Liu and Winterton (1991). Therefore, the heat transfer coefficient in their method is given as follows:

5.3 Predictive Methods for Convective Flow Boiling



207

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðF c  hl Þ2 þ ðSnb  hnb Þ2

ð5:105Þ

where the heat transfer coefficients for convective and nucleate boiling parcels are given by the same methods adopted by Kanizawa et al. (2017), and the enhancement and suppression factors for convective and nucleate boiling parcels are given as follows: F c ¼ 1 þ 2:55  Snb ¼ 1:427 

b 1:04 X tv 1 þ We0:194 uv

ð5:106Þ

Bd 0:032   4 0:981 1 þ 0:1086 Rel F 1:25 c 10

ð5:107Þ

Figure 5.15 depicts the variation of heat transfer coefficient with vapor quality according to the predictive methods presented in this chapter for small diameter channels, except for the methods developed by Wojtan et al. (2005a, b) and Liu and Winterton (1991) which were mainly developed for macroscale channels. It can be observed from this figure that, as expected, the heat transfer coefficient tends to increase with x for reduced and intermediate vapor qualities, and after dryout inception tends to reduce, except for the Liu and Winterton (1991) method that does not predict dryout occurrence. Additionally, it can be observed in this figure that the predictive method of Saitoh et al. (2007) presents a discontinuity in the variation of h with x for vapor quality lower than the one corresponding to the dryout inception. This behavior is associated with the transition from laminar to turbulent flow (and vice versa) of the phases. With the increment of vapor quality, the liquid flow rate reduces with consequent reduction of inertial effects of liquid phase; conversely, the vapor flow rate increases with vapor quality with consequent increment of inertial effects. It can also be observed according to this figure that for reduced and intermediate vapor qualities, the methods predict similar trends, indicating a certain degree of

a) 12000

h [W/m²K]

8000

b) 25000 Liu and Winterton (1991) Saitoh, Daiguji and Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Sempertegui-Tápia and Ribatski (2017) Kandlikar and Balasubramanian (2004) Thome, Dupont and Jacobi (2004a,b) (2004a,b)

6000 4000 2000 0 0.0

Liu and Winterton (1991) Saitoh, Daiguji and Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Sempertegui-Tápia and Ribatski (2017) Kandlikar and Balasubramanian (2004) Thome, Dupont and Jacobi (2004a,b) (2004a,b)

20000

h [W/m²K]

10000

15000

10000

5000 R134a, d = 1.5 mm, G = 300 kg/m²s, φ = 10 kW/m², T sat = 30 °C

0.2

0.4

x [-]

0.6

0.8

1.0

0 0.0

R134a, d = 1.5 mm, G = 500 kg/m²s, φ = 20 kW/m², T sat = 30 °C

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 5.15 Variation of the heat transfer coefficient with vapor quality for R134a convective boiling in 1.5 mm channel

208

5 Flow Boiling

coherence among them, which in turn indicates that the databases used in the adjustment of their empirical constants displays similar heat transfer trends. However, with the increment of the fluid enthalpy (or vapor quality), the deviations among the predictions can be as high as 100%, indicating that previous to the adoption of a predictive method, the reader must check the reference data used for the development of the method, otherwise the errors during the design stage of heat exchangers can lead to over or under sizing. It must be mentioned that the significant deviation among the distinct predictive method emphasizes the need for additional experimental and theoretical studies with the objective of providing higher agreement between predicted and actual heat transfer coefficient values.

5.3.7

Heat Transfer Coefficient Under Transient Heating

Thermal management applications such as the cooling of electronic components and photovoltaic panels deals with transient heating and hot spots. In this context, Aguiar and Ribatski (2019) were pioneers to develop a heat transfer prediction method for flow boiling under conditions of transient heat fluxes and hot spots. In their method, the temperature difference between the channel wall and the fluid ΔTw is estimated along time as a function of the variable heat flux. They assumed that the instantaneous wall superheating and heat flux are given as the sum of their mean values ΔT w and ϕ , and the respective fluctuations as ΔT 0 w(t) and ϕ0 (t) as follows: ΔT w ðt Þ ¼ ΔT w þ ΔT 0w ðt Þ

ð5:108Þ

ϕðt Þ ¼ ϕ þ ϕ0 ðt Þ

ð5:109Þ

The averaged heat flux and temperature difference are related through Newton’s Cooling Law, given by Eq. (5.10), by adopting a proper predictive method for heat transfer coefficient developed for steady-state conditions such those described in this chapter, in Sects. 5.3.1 to 5.3.5. Hence, for a given ϕ it is possible to estimate the corresponding ΔT w , and the steps depicted in Fig. 5.16 are performed for determination of the time varying ΔTw(t). The second step of the procedure described in Fig. 5.16 corresponds to the estimation of the effect of variation of heat flux on the wall superheating and, according to the original authors, can be estimated by a central finite difference scheme, as follows:     0 0 ∂ΔT w ΔT w T sat , G, d, x, ϕ þ Δϕ  ΔT w T sat , G, d, x, ϕ  Δϕ 0 ¼ 2Δϕ0 ∂ϕ

ð5:110Þ

5.3 Predictive Methods for Convective Flow Boiling

209

Fig. 5.16 Summary of the methodology for estimation of wall superheating during transient heat flux conditions. (Adapted from Aguiar and Ribatski 2019)

Start

Evaluate ΔTw based on ϕ, Tsat, G, d and x

Estimate ∂ΔTw/∂ϕ' at ϕ

Based on ϕ'(t), estimate ΔTw': ΔTw'(t)=TF ϕ'(t)

Compose the wall superheating fluctuation with the averaged value: ΔTw(t)= ΔTw+ΔTw'(t)

End

where the wall superheatings are estimated based on a predictive method proposed for steady state conditions, and Δϕ0 consists of an infinitesimal increment of the heat flux. The transfer function TF of the third step presented in the diagram of Fig. 5.16, which gives ΔT 0w ðt Þ as a function of ϕ0 (t), is a first-order lag, given as follows: TF ¼

∂ΔTw =∂ϕ bs þ 1 Τb

ð5:111Þ

^ where ŝ is the Laplace variable, and the time constant T can be determined experimentally, or by assuming a lumped capacitance model for the channel wall, as follows:   b ¼ ðρcÞw Acs Τ P h

ð5:112Þ

where (ρc)w is the product of density and specific heat of the wall material, h corresponds to the mean heat transfer coefficient estimated for ϕ, Acs to the crosssection of the channel wall, and P to the wetted perimeter. It is important to highlight that knowing the transient wall superheating ΔTw(t) is extremely important to evaluate the possibility of transient wall temperature

210

5 Flow Boiling

incursion that could damage the cooled device. Based on the time varying wall superheating ΔTw(t), and by knowing the heat flux ϕ(t), it is possible to estimate the instantaneous heat transfer coefficient h(t) as follows: hð t Þ ¼

ϕð t Þ ΔT w ðt Þ

ð5:113Þ

Aguiar and Ribatski (2019) validated their method based on experimental results for transient heat transfer coefficient during convective boiling of R134a inside round channels with internal diameter between 0.5 and 1.1 mm. The test sections used in their study were 90 mm long channels with a 10 mm central region subjected to variable heat flux with sinusoidal, square, and sawtooth waveforms, with frequencies ranging between 0.5 and 2.0 Hz, mean values between 80 and 120 kW/m2, and half amplitude of 20 and 40 kW/m2. In addition to the predictive methods presented in this chapter, several other methods are being developed and presented in the open literature, and the reader is encouraged to keep up to date with the recent publications. It must be emphasized that most of the predictive methods for heat transfer coefficient during convective boiling are developed based on the adjustment of empirical constants based on experimental databases, which are finite and, generally, comprise a limited number of fluids, range of channel diameter, heat flux, etc. Hence, previous to the adoption of a predictive method it is advisable to check its original publication and confirm whether the dataset used for its development comprises the conditions that will be simulated, since deviations as high as 100% can be verified among methods even for similar operational conditions.

5.4

Solved Examples

Consider R134a flowing at a temperature of 20  C and mass flux of 200 kg/m2s, for uniform heat flux of 5000 W/m2. Evaluate the local heat transfer coefficient for: (a) Subcooled liquid and saturated vapor single-phase flows inside a channel of 10 mm of internal diameter. (b) Subcooled liquid and saturated vapor single-phase flows inside a channel of 1 mm of internal diameter. (c) Convective boiling inside a channel with 10 mm of internal diameter for vapor quality of 0.1. (d) Convective boiling inside a channel with 1 mm of internal diameter for vapor quality of 0.1.

5.4 Solved Examples

211

Solution: Based on the defined Tsat ¼ 20  C, and considering that liquid properties are mainly a function of the temperature, it is possible to determine the following properties: ρl ¼ 1225 kg/m3 ρv ¼ 27.8 kg/m3 μl ¼ 2.068104 kg/ms μv ¼ 1.175105 kg/ms σ ¼ 8.688103 N/m kl ¼ 8.562102 W/mK kv ¼ 1.406102 W/mK Prl ¼ 3.393 Prv ¼ 0.836 In any of the above items, it is interesting to evaluate the Reynolds number for liquid and vapor single-phase flows as follows: Re l0 ¼

Gd μl

Re v0 ¼

Gd μv

where for liquid it is 967 and 9673, and for vapor it is 17,026 and 170,258 for channel diameters of 1 and 10 mm, respectively. Hence, the regime is laminar only for liquid flowing in 1 mm channel, and turbulent, or in transitional regime, for other conditions. (a) As discussed before, the flow regime for both phases flowing in 10 mm ID tube is turbulent. The Dittus and Boelter (1985) correlation is valid for Reynolds numbers higher than approximately 20,000. Therefore, let us adopt the Gnielinski (1976) predictive method, which is valid for Reynolds number in the range of 2500 to 510 (Churchill, 2000), and is given as follows: Nud ¼

ð f =8ÞðRed  1000ÞPr hd pffiffiffiffiffiffiffiffiffi ¼  k 1 þ 12:7 f =8 Pr2=3  1

where the friction factor is estimated as follows: f ¼ ð1:82 log 10 Red  1:64Þ2 Hence, the heat transfer coefficient for subcooled liquid and saturated vapor flowing in a 10 mm ID tube is 498.1 and 428.0 W/m2K, respectively.

212

5 Flow Boiling

(b) In the case of single-phase flow in channel with 1 mm of internal diameter, the procedure of an item can be repeated for the vapor phase, which results in heat transfer coefficient of 699.5 W/m2K. Considering that the subcooled liquid single-phase flow corresponds to laminar regime, and the boundary condition for the heat transfer problem corresponds to uniform heat flux, the Nusselt number is constant and equal to 4.364; hence: Nud ¼

hd ¼ 4:364 k

resulting in 373.6 W/m2K. Notice that, even considering the difference in flow regime, the heat transfer coefficient for laminar flow is approximately 25% lower than the condition for turbulent flow only because of the channel reduction. (c) In the case of convective boiling, it is necessary to check which predictive method is more appropriate for these conditions. The predictive method proposed by Saitoh et al. (2007) was validated for refrigerant flow inside channels with a diameter between 0.5 and 10.9 mm, hence it is within both diameters evaluated in this exercise. Therefore, by implementing Eqs. (5.41) to (5.54), recalling that the nucleate boiling parcel is estimated according to Stephan and Abdelsalam (1980) correlations (Eq. (5.22)), the following parameters are obtained: • Heat transfer coefficient for convective parcel: hl ¼ 419.7 W/m2K • Heat transfer coefficient for nucleate boiling parcel: hnb ¼ 1279 W/m2K • Vapor quality for dryout inception: xdi ¼ 0.9615, hence, the current condition is previous to dryout inception. • The suppression factor of nucleate boiling mechanism is Snb ¼ 0.596, and the enhancement factor of convective effects is Fc ¼ 1.511. • Therefore, the resulting heat transfer coefficient is h ¼ 1396 W/m2K. Notice that the heat transfer coefficient, even for small vapor quality value, is approximately three times higher than the value for single-phase flow. Additionally, notice that the heat transfer coefficient for convective parcel is different from that evaluated for single-phase flow, because it is assumed only for the liquid parcel, corresponding to 90% of the mass flow rate. (d) Repeating the procedure of the previous item, but now assuming internal diameter of 1 mm, the following results are obtained: • Heat transfer coefficient for convective parcel: hl ¼ 665.2 W/m2K • Heat transfer coefficient for nucleate boiling parcel: hnb ¼ 1279 W/m2K • Vapor quality for dryout inception: xdi ¼ 0.7057, hence, the current condition is also previous to dryout inception. • The suppression factor of nucleate boiling mechanism is Snb ¼ 0.977, and the enhancement factor of convective effects is Fc ¼ 1.373. • Therefore, the resulting heat transfer coefficient is h ¼ 2164 W/m2K.

5.5 Problems

213

Notice that the heat transfer coefficient for nucleate boiling parcel is independent of the channel diameter, however, the suppression factor increases with diameter reduction. Conversely, even though the heat transfer coefficient for the convective parcel increases substantially with channel reduction, the enhancement factor presents marginal reduction due to increment of viscous effects. Additionally, notice the effect of channel diameter on the vapor quality for dryout inception, which reduced from 96% for d ¼ 10 mm to 71% for d ¼ 1 mm.

5.5

Problems

A list of problems comprising the concepts of the present and previous chapters is proposed. 1. Derive Eq. (5.41). 2. Discuss the relationship between the variation of heat flux and wall superheating for pool boiling conditions. Start with Eq. (5.11) to obtain a relationship between ΔT and heat flux. 3. Consider annular flow of R134a inside a channel with diameter d, with liquid film thickness δ, at given vapor quality x, mass flux G, and imposed heat flux from the channel surface. Assuming laminar regime for both phases, and properties of saturated liquid and vapor, evaluate the heat transfer coefficient. It is suggested to derive the velocity profiles of both phases by imposing non-slip in the interface as well as force balance. Compare the result for liquid single-phase flow for a similar mass flow rate. 4. Assume R600a two-phase flow in a horizontal channel of 10 mm ID at mass flux of 50 kg/m2s, heat flux of 5000 W/m2, and saturation temperature of 0  C. In a given position, the flow is stratified with void fraction of 50% (vapor quality of approximately 1.94%). Adopting an approach similar to that of Lockhart and Martinelli (1949) to estimate the heat transfer coefficient (evaluate the heat transfer coefficient for each phase assuming single-phase flow in a smaller channel, with the corresponding heat transfer coefficient), estimate the mean heat transfer coefficient in this cross-section. Assume that the perimeter for liquid phase comprises only the contact with channel wall, while for the vapor phase include the contribution of the interface. Compare with the predictions according to Liu and Winterton’s (1991) and Saitoh et al.’s (2007) predictive methods. 5. Now, consider R134a flowing in a 5 mm ID tube at saturation temperature of 5  C, heat flux of 10,000 W/m (Bergles & Rohsenow, 1964), and mass flux of 500 kg/m2s. Assuming annular flow with void fraction of 95% (vapor quality of approximately 60.2%), and uniform liquid film thickness, estimate the corresponding heat transfer coefficient based on the thermal resistance imposed by the liquid film, as given by Eq. (5.41). Compare with the predictions according to Liu and Winterton (1991) and Saitoh et al. (2007).

214

5 Flow Boiling

References Aguiar, G. M., & Ribatski, G. (2019). An experimental study on flow boiling in microchannels under heating pulses and a methodology for predicting the wall temperature fluctuations. Applied Thermal Engineering, 159, 113851. Bergles, A. E., & Rohsenow, W. M. (1964). The determination of forced-convection surfaceboiling heat transfer. Biberg, D. (1999). An explicit approximation for the wetted angle in two-Phase stratified pipe flow. The Canadian Journal of Chemical Engineering, 77(6), 1221–1224. Blander, M., & Katz, J. L. (1975). Bubble nucleation in liquids. AIChE Journal, 21(5), 833–848. Chen, J. C. (1966). Correlation for boiling heat transfer to saturated fluids in convective flow. Industrial & engineering chemistry process design and development, 5(3), 322–329. Churchill, S. W. (2000). The art of correlation. Industrial & Engineering Chemistry Research, 39 (6), 1850–1877. Churchill, S. W., & Usagi, R. (1972). A general expression for the correlation of rates of transfer and other phenomena. AIChE Journal, 18(6), 1121–1128. Cioncolini, A., & Thome, J. R. (2011). Algebraic turbulence modeling in adiabatic and evaporating annular two-phase flow. International Journal of Heat and Fluid Flow, 32(4), 805–817. Collier, J. G., & Thome, J. R. (1994). Convective boiling and condensation. Clarendon Press. Cooper, M. G. (1969). The microlayer and bubble growth in nucleate pool boiling. International Journal of Heat and Mass Transfer, 12(8), 915–933. Cooper, M. G. (1984). Heat flow rates in saturated nucleate pool boiling-a wide-ranging examination using reduced properties. Advances in heat transfer, 16, 157–239. Cooper, M. G., & Lloyd, A. J. P. (1969). The microlayer in nucleate pool boiling. International Journal of Heat and Mass Transfer, 12(8), 895–913. Costa-Patry, E., Olivier, J., & Thome, J. (2012). Heat transfer charcacteristics in a copper microevaporator and flow pattern-based prediction method for flow boiling in microchannels. Frontiers in Heat and Mass Transfer (FHMT), 3(1). Da Riva, E., Del Col, D., Garimella, S. V., & Cavallini, A. (2012). The importance of turbulence during condensation in a horizontal circular minichannel. International Journal of Heat and Mass Transfer, 55(13–14), 3470–3481. Dittus, F. W., & Boelter, L. M. K. (1985). Heat transfer in automobile radiators of the tubular type. International Communications in Heat and Mass Transfer, 12(1), 3–22. Forster, H. K., & Zuber, N. (1955). Dynamics of vapor bubbles and boiling heat transfer. AIChE Journal, 1(4), 531–535. Gerardi, C., Buongiorno, J., Hu, L. W., & McKrell, T. (2010). Study of bubble growth in water pool boiling through synchronized, infrared thermometry and high-speed video. International Journal of Heat and Mass Transfer, 53(19–20), 4185–4192. Gnielinski, V. (1976). New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng., 16(2), 359–368. Gorenflo, D., & Kenning, D. B. R. (2010). Pool boiling (Chapter H2). Berlin/Heidelberg: VDI Heat Atlas. Springer-Verlag. Groeneveld, D. C. (1973). Post-dryout heat transfer at reactor operating conditions (No. AECL-4513). Atomic Energy of Canada Ltd. Gungor, K. E., & Winterton, R. H. S. (1986). A general correlation for flow boiling in tubes and annuli. International Journal of Heat and Mass Transfer, 29(3), 351–358. Gungor, K. E., & Winterton, R. H. S. (1987). Simplified general correlation for saturated flow boiling and comparisons of correlations with data. Chemical Engineering Research and Design, 65(2), 148–156. Han, C. H., & Griffith, P. (1965). The mechanism of heat transfer in nucleate pool boiling—part II: the heat flux-temperature difference relation. International Journal of Heat and Mass Transfer, 8(6), 905–914.

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Hetsroni, G., Mosyak, A., & Pogrebnyak, E. (2015). Effect of Marangoni flow on subcooled pool boiling on micro-scale and macro-scale heaters in water and surfactant solutions. International Journal of Heat and Mass Transfer, 89, 425–432. Jabardo, J. M., Silva, E., Ribatski, G., & de Barros, S. F. (2004). Evaluation of the Rohsenow correlation through experimental pool boiling of halocarbon refrigerants on cylindrical surfaces. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 26(2), 218–230. Judd, R. L., & Hwang, K. S. (1976, November). A comprehensive model for nucleate pool boiling heat transfer including microlayer evaporation. Journal of Heat Transfer, 98(4), 623–629. Kandlikar, S. G. (1990). A general correlation for saturated two-phase flow boiling heat transfer inside horizontal and vertical tubes. ASME Journal of Heat Transfer, 112, 219–228. Kandlikar, S. G. (1991). A model for predicting the two-phase flow boiling heat transfer coefficient in augmented tube and compact heat exchanger geometries. ASME Journal of Heat Transfer, 113, 966–972. Kandlikar, S. G., & Balasubramanian, P. (2004). An extension of the flow boiling correlation to transition, laminar, and deep laminar flows in minichannels and microchannels. Heat Transfer Engineering, 25(3), 86–93. Kanizawa, F. T., & Ribatski, G. (2016). Void fraction predictive method based on the minimum kinetic energy. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(1), 209–225. Kanizawa, F. T., Tibiriçá, C. B., & Ribatski, G. (2016). Heat transfer during convective boiling inside microchannels. International Journal of Heat and Mass Transfer, 93, 566–583. https:// doi.org/10.1016/j.ijheatmasstransfer.2015.09.083 Kattan, N., Thome, J. R., & Favrat, D. (1998). Flow boiling in horizontal tubes: Part 1—Development of a diabatic two-phase flow pattern map. Journal of Heat Transfer, 120(1), 140–147. Kim, J. (2009). Review of nucleate pool boiling bubble heat transfer mechanisms. International Journal of Multiphase Flow, 35(12), 1067–1076. Kutateladze, S. S. (1961). Boiling heat transfer. International Journal of Heat and Mass Transfer, 4, 31–45. Kutateladze, S. S. (1948). On the transition to film boiling under natural convection. Kotloturbostroenie, 3, 10–12. Lienhard IV, J. H., & Lienhard V, J. H. (2020). A heat transfer textbook. 5th Edition. Phlogiston press. Liu, Q., Palm, B., & Anglart, H. (2012, July). Simulation on the flow and heat transfer characteristics of confined bubbles in micro-channels. In International conference on nanochannels, microchannels, and minichannels (Vol. 44793, pp. 63-70). American Society of Mechanical Engineers. Liu, Z., & Winterton, R. H. S. (1991). A general correlation for saturated and subcooled flow boiling in tubes and annuli, based on a nucleate pool boiling equation. International Journal of Heat and Mass Transfer, 34(11), 2759–2766. Lockhart, R. W. & Martinelli, R. C. (1949). Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chemical Engineering Progress, Vol. 45–1, 39–48. Mikic, B. B., & Rohsenow, W. M. (1969, May). A new correlation of pool-boiling data including the effect of heating surface characteristics. Journal of Heat Transfer, 91(2), 245–250. Mori, H., Yoshida, S., Ohishi, K., & Kakimoto, Y. (2000). Dryout quality and post-dryout heat transfer coefficient in horizontal evaporator tubes. In 3rd European thermal sciences conference (Heidelberg, 10–13 September 2000) (pp. 839–844). Moriyama, K., & Inoue, A. (1996). Thickness of the liquid film formed by a growing bubble in a narrow gap between two horizontal plates. Journal of Heat Transfer, 118(1), 132–139. Nukiyama, S. (1934). Film boiling water on thin wires. Society of Mechanical Engineering, 37. Ong, C. L., & Thome, J. R. (2011). Macro-to-microchannel transition in two-phase flow: Part 1– Two-phase flow patterns and film thickness measurements. Experimental Thermal and Fluid Science, 35(1), 37–47.

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Petukhov, B. S. (1970). Heat transfer and friction in turbulent pipe flow with variable physical properties. Advances in heat transfer, 6(503), i565. Plesset, M. S. & Zwick, S. A. (1954). The growth of vapor bubbles in superheated liquids. Journal of applied physics, 25(4), 493–500. Ribatski, G. (2002). Theoretical and experimental analysis of pool boiling of halocarbon refrigerants (Doctoral Thesis, University of São Paulo). Ribatski, G. (2013). A critical overview on the recent literature concerning flow boiling and two-phase flows inside micro-scale channels. Experimental Heat Transfer, 26(2-3), 198–246. Ribatski, G., & Saiz-Jabardo, J. M. (2003). Experimental study of nucleate boiling of halocarbon refrigerants on cylindrical surfaces. International Journal of Heat and Mass Transfer, 46(23), 4439–4451. https://doi.org/10.1016/S0017-9310(03)00252-7 Rohsenow, W. M. (1971). Boiling. Annual Review of Fluid Mechanics, 3(1), 211–236. Rosenhow, W. M. (1952). A method of correlating heat transfer data for surfaceboiling of liquids. Transactions of ASME, 4, 969–975. Rouhani, Z. (1969). Modified correlations for void and two-phase presssure drop. Nyköping: Aktiebolaget Atomenergi. Saitoh, S., Daiguji, H., & Hihara, E. (2007). Correlation for boiling heat transfer of R-134a in horizontal tubes including effect of tube diameter. International Journal of Heat and Mass Transfer, 50(25–26), 5215–5225. https://doi.org/10.1016/j.ijheatmasstransfer.2007.06.019 Schweizer, N., Freystein, M., & Stephan, P. (2010, January). High resolution measurement of wall temperature distribution during forced convective boiling in a single minichannel. In International conference on nanochannels, microchannels, and minichannels (Vol. 54501, pp. 101–108). Sempértegui-Tapia, D. F., & Ribatski, G. (2017). Flow boiling heat transfer of R134a and low GWP refrigerants in a horizontal micro-scale channel. International Journal of Heat and Mass Transfer, 108, 2417–2432. Shah, R. K., & London, A. L. (1978). Laminar flow forced convection in ducts (Advances in Heat Transfer, supplement 1). Stephan, K., & Abdelsalam, M. (1980). Heat-transfer correlations for natural convection boiling. International Journal of Heat and Mass Transfer, 23(1), 73–87. Stephan, P., & Hammer, J. (1994). A new model for nucleate boiling heat transfer. Heat and Mass Transfer, 30(2), 119–125. Taitel, Y., & Dukler, A. E. (1976). A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AIChE Journal, 22(1), 47–55. Thome, J. R., Dupont, V., & Jacobi, A. M. (2004). Heat transfer model for evaporation in microchannels. Part I: Presentation of the model. International Journal of Heat and Mass Transfer, 47(14-16), 3375–3385. Tibirica, C. B., & Ribatski, G. (2014). Flow patterns and bubble departure fundamental characteristics during flow boiling in microscale channels. Experimental Thermal and Fluid Science, 59, 152–165. Tibirica, C. B., Czelusniak, L. E., & Ribatski, G. (2015). Critical heat flux in a 0.38 mm microchannel and actions for suppression of flow boiling instabilities. Experimental Thermal and Fluid Science, 67, 48–56. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005a). Investigation of flow boiling in horizontal tubes: Part I—A new diabatic two-phase flow pattern map. International Journal of Heat and Mass Transfer, 48(14), 2955–2969. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005b). Investigation of flow boiling in horizontal tubes: Part II—Development of a new heat transfer model for stratified-wavy, dryout and mist flow regimes. International Journal of Heat and Mass Transfer, 48(14), 2970–2985. Zhang, W., Hibiki, T., Mishima, K., & Mi, Y. (2006). Correlation of critical heat flux for flow boiling of water in mini-channels. International Journal of Heat and Mass Transfer, 49(5–6), 1058–1072. https://doi.org/10.1016/j.ijheatmasstransfer.2005.09.004 Zuber, N., & Findlay, J. (1965). Average volumetric concentration in two-phase flow systems. Journal of Heat Transfer, 87(4), 453–468.

Chapter 6

Critical Heat Flux and Dryout

6.1

Introduction

The critical heat flux (CHF) is an undesired operational condition, resulting from a deficient rewetting of the heating surface accompanied by a sharp increase of the wall superheating as a consequence of the heat transfer coefficient h reduction under a condition of imposed heat flux. In the 1950s, the expansion of the nuclear industry implied on the need of a better understanding of the CHF and an enormous growth in the number of studies concerning this subject, as pointed out by Groeneveld et al. (2007). In the same paper, these authors highlighted the proposal of more than thousand correlations for the prediction of CHF of water flowing inside ducts. Groeneveld et al. (2007) also indicated the complexity of the mechanisms associated to the CHF, concluding that a theory for CHF capable of being generalized to all applications was not available until that time. It can be certainly mentioned that, despite the recent advances, the same status quo remains nowadays and a unique theory is still not available. Figure 5.6 schematically depicts the boiling curve, whereas the CHF condition (point 5) corresponds to an abrupt increment of the wall temperature Tw in case of imposed heat flux, or by a sharp reduction of heat flux ϕ in case of controlled wall temperature. Under conditions of controlled heat flux, such as electronics and nuclear applications, the abrupt increment of surface temperature is usually catastrophic for the heated surface and may cause irreversible damages to the equipment. Therefore, CHF is also referred in literature as burnout and boiling crisis (Hewitt 1981). The term departure from nucleate boiling (DNB) is also common in literature, such as in Collier and Thome (1994). This last nomenclature derives from the process of ceasing of vapor generation on the heated surface by nucleate boiling mechanism (getting away from the nucleate boiling condition of Fig. 5.6). The CHF condition corresponds to the formation of vapor blankets on the heater surface, implying on a much lower heat transfer coefficient compared to a condition under which the liquid phase is in almost permanent contact with the heated surface. © The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_6

217

218

6

Critical Heat Flux and Dryout

Consequently, the heat transfer coefficient presents a drastic reduction, and energy is transferred by convection through the vapor film. In the case of imposed heat flux, subsequent heat flux increments beyond the occurrence of CHF lead to a continuous vapor blanket along a large portion of the surface. Under this condition, film boiling is established, and thermal energy is transferred from the wall to the liquid phase by convection and conduction through the vapor film, with radiation becoming relevant as the wall temperature increases. The term film boiling comes from the fact that vaporization occurs in the interface between the vapor film and neighboring liquid. Some studies, such as Yagov et al. (2016), investigate the film boiling condition focused on the required surface temperature for its occurrence, which is related to the Leidenfrost phenomenon which corresponds to the condition that the liquid hovers over the heated surface without touching it. Conversely, under conditions of controlled surface temperature, successive increments of temperature beyond the CHF results in increment of surface area occupied by vapor blankets, with consequent reduction of heat transfer coefficient and heat flux. It is speculated that the vapor blanket is unstable and large portions of vapor are released periodically from the film, and this condition is denominated as transition boiling, such as detailed by Collier and Thome (1994). Heat transfer predictive methods for film boiling and transition boiling are available in the open literature, such as presented by Collier and Thome (1994) and Hewitt (1981). These methods are not described in the present text because the occurrence of CHF is undesirable for the operation of heat dissipation systems due to low heat transfer coefficient, and high probability of damaging the cooling device. Therefore, the above-mentioned literature is recommended to those interested on further details. On the other hand, for several applications it is desirable to predict the conditions of occurrence of CHF, so the engineer can design the equipment to avoid its achievement. In this context, several aspects should be taken into account when dealing with CHF, such as boiling under quiescent or forced convection condition, heater geometry, gravity level, among others (Katto 1994), with specific complication for each case. Therefore, it is not surprising that most of the CHF predictive methods are developed based either on an adjustment of a simple model or by direct correlation of the relevant non-dimensional parameters based on a broad database. Even considering the phenomenological models, an adjustment of empirical parameters due to the phenomenon complexity is also performed to provide good agreement with experimental results. According to Katto (1994), in general, the predictive models can be categorized as hydrodynamic and microlayer models. Both approaches were originally developed for pool boiling conditions, and subsequently adjusted for convective flow. In this context, it is interesting to discuss the effect of surface wettability, such as introduced in Chap. 5. Hydrophobic surfaces favor bubble nucleation due to the repelling characteristic of the liquid phase, and consequently implies on lower surface superheating for nucleate boiling, and provides higher heat transfer

6.2 Hydrodynamic Model

219

coefficient during nucleate boiling conditions because of the higher number of active cavities. However, the advantage of vapor formation can anticipate the CHF exactly due to excessive vapor formation. On the other hand, hydrophilic surfaces require higher surface superheating for the onset of nucleate boiling and tend to present lower heat transfer coefficient during nucleate boiling conditions in comparison with hydrophobic surfaces, related to higher difficulty of vapor formation and tendency to maintain contact with liquid phase. However, after the proper establishment of active nuclei, the CHF tends to be delayed exactly due to tendency of the surface to be wetted. Hence, even though these aspects are not always accounted for in predictive methods, the reader must be aware of the possible impact on the definition of operational range of heat transfer devices, and surface wettability characteristics. Recently, by assuming that the CHF is a near wall phenomenon associated to fluid–solid interactions on the heating surface instead of a macroscale hydrodynamic instability, Zhang et al. (2019) proposed a Monte Carlo model based on continuum percolation that describes the near-wall stochastic interaction of bubbles. According to the authors, this model is capable of capturing the effect of the surface characteristics on the CHF because it follows the evolution of the bubble footprint area as the heat flux increases with the CHF coinciding with a critical condition in the percolation process. In the foregoing subsections, the most widely accepted models for CHF are addressed.

6.2

Hydrodynamic Model

The hydrodynamic model has been under development since the 1950s and was firtly introduced by the Kutateladze’s research group, and was subsequently considered and adjusted by other groups, such as Zuber and Tribus (1958) that provided one of the most classical presentations. This approach considers that during nucleate boiling under pool boiling conditions, vapor is generated along the heater surface and flows away from the surface, with simultaneous liquid flow toward the surface. With successive increment of vapor flow rate by the increment of heat flux, eventually the counter current flow will not be sustained, and the liquid refeeding to the surface becomes insufficient to maintain the surface wet. The impossibility of simultaneous flow of vapor away from the surface and liquid toward the surface, such as depicted in Fig. 6.1, is limited by interfacial instabilities, which can cause increment of interfacial waves with consequent partial blocking of the available cross-section for liquid flow. Zuber and Tribus (1958) reproduced pictures obtained during pool boiling experiments, depicting characteristics of vapor and liquid flow during CHF, showing that vapor detaches from the surface as columns with approximately uniform spacing among each of them, such as schematically depicted by Λcrit in Fig. 6.1.

220

6

Critical Heat Flux and Dryout

Λcrit

Liquid

Λcrit

Vapor

Heater

Λcrit

Vapor

Λcrit

Heater

Front view Λcrit

Λcrit

Top view Fig. 6.1 Schematics of the hydrodynamic model for critical heat flux

Additionally, and according to the visual observations and measurements, the authors found out that the spacings among the vapor columns are approximately similar to the critical wavelength proposed by Bellman and Pennington (1954), which characterizes the interfacial perturbation length that causes exponential growth or suppression of waves, and is dominated by surface tension and buoyancy effects. The critical wavelength is given according to the Kelvin equation, which provides two values as follows: 

Λcrit,1 Λcrit,2

12 σ ¼ 2π gð ρl  ρv Þ  12 3σ ¼ 2π gð ρl  ρv Þ

ð6:1Þ ð6:2Þ

Additionally, these authors observed the formation of vapor bubbles/slugs shortly downstream the surface and concluded that the spacing between the vapor columns is independent of the heat flux. Therefore, the increment of heat flux causes increment of the frequency of bubble detachment, keeping the spacings among their departure points fixed. Zuber and Tribus (1958) modelled the condition of CHF assuming that the formed vapor slugs are spherical and present a diameter of Λcrit/2. Therefore, the vapor mass per bubble is given as follows: msphere ¼

1 πΛ3 ρ 48 crit v

ð6:3Þ

6.2 Hydrodynamic Model

221

Additionally, the mass flux of vapor Gv can be given as a function of the heat flux ϕ and enthalpy of vaporization ilv as follows: ϕ ¼ Gvbilv

ð6:4Þ

Therefore, based on these aspects and considering that one bubble is generated per each area of Λcrit  Λcrit, and that an oscillating interface presents a double critical frequency, as pointed out by Rayleigh (1896), it is possible to relate the bubble frequency ωb and heat flux by equating mass fluxes Gv according to both approaches as follows: πΛ3crit ρv ϕ 2ωb ¼ bilv 48Λ2crit π ϕ ¼ Λcrit ρv^ilv ωb 24

ð6:5Þ ð6:6Þ

where the critical wavelength Λcrit is given either by Eq. (6.1) or (6.2). According to the discussion presented by Zuber and Tribus (1958), as well as in previous studies, the bubble frequency for CHF is evaluated based on the Kelvin and Helmholtz instability theory as a function of phase properties and critical wavelength as follows: 1 ωb ¼ Λcrit

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 2π ρl þ ρv Λcrit

ð6:7Þ

And combining with Eqs. (6.1) and (6.6), we obtain the following relationship for the CHF: ϕcrit

π ¼ ρvbilv 24

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 gσ ðρl  ρv Þ ð ρl þ ρv Þ 2

ð6:8Þ

and by considering that ρl  ρv, the above equation can be simplified to: ϕcrit

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 gσ ðρl  ρv Þ ¼ 0:131  ρvbilv ρ2l

ð6:9Þ

which is valid for low-pressure systems. For higher pressure systems, the dominant phenomenon that leads to instabilities of the countercurrent flow are related to Taylor instabilities rather than Kelvin instabilities. According to Zuber and Tribus (1958), the square of propagation velocity of small interfacial disturbances is given as follows:

222

6

Γ2 ¼

Critical Heat Flux and Dryout

ρl ρv 4σ ð j þ j l Þ2  Λðρl þ ρv Þ ðρl þ ρv Þ2 v

ð6:10Þ

where Λ is the perturbation wavelength, and jv and jl are, respectively, the liquid and vapor superficial velocities. It is required that Γ should be real for stability, therefore in the limit the right-hand side (RHS) of Eq. (6.10) should be null. The vapor and liquid velocities can be related by continuity equation, because for steady-state condition, the mass flow rate of vapor leaving the surface should equate the liquid flow rate toward the surface as follows: jv ρv ¼ jl ρl

ð6:11Þ

Therefore, by equating the RHS of Eq. (6.10) to zero, and substituting the liquid superficial velocity according to Eq. (6.11), it is possible to infer the critical velocity of vapor phase as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρl 4σ jv ¼ Λρv ρl þ ρv

ð6:12Þ

Now, returning to the relationship between bubble volume and frequency ωb, and the flow rate by assuming that the region of reference for the vapor flow rate is circular with the radius of Λ /4, it is possible to relate the estimated superficial velocity according to Eq. (6.4), assuming a circular area of influence with a diameter of Λ/2 as follows:    2 4 Λ 3 Λ ρv π 2ωb ¼ ρv π jv 3 4 4

ð6:13Þ

from which, it is possible to determine a relationship for the bubble frequency as follows: ωb ¼

3 2Λ

rffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρl 4σ Λρv ρl þ ρv

ð6:14Þ

Again, the mass flux of vapor can be related to the heat flux and enthalpy of vaporization as follows:   ϕ 4 Λ 3 2ωb ¼ ρv π 3 4 bilv Λ2 π ϕ ¼ ^ilv ρv Λωb 24

Gv ¼

ð6:15Þ ð6:16Þ

6.2 Hydrodynamic Model

223

π 3^ i ρ ϕ¼ 24 2 lv v

rffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρl 4σ Λρv ρl þ ρv

ð6:17Þ

where, under critical conditions the wavelength Λ is given either by Eqs. (6.1) and (6.2). Hence, the CHF is given as follows: ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffi ρl π 3 ^ pffiffiffiffiffip pffiffiffiffiffi ilv ρv 4 σgðρl  ρv Þ 24 2π ρl þ ρv r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρl π 3 ^ pffiffiffiffiffip 4 p ffiffiffiffiffi p ffiffi ffi ilv ρv σgðρl  ρv Þ ¼ 4 ρ þ ρv 24 3 2π l

ϕcrit,1 ¼ ϕcrit,2

ð6:18Þ ð6:19Þ

which corresponds to the limits for the occurrence of CHF. Zuber and Tribus (1958) suggested to simplify the above equations by assuming that the liquid phase density is much higher than the vapor phase density (ρl  ρv). Additionally, they 3 ffiffiffiffi recommended to adopt the maximum estimated value, considering p4 ffiffi3p 1, 2π given as follows: ϕcrit ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π ^ 12 p  ilv ρv 4 gðρl  ρv Þσ ¼ 0:131  ^ilv ρ2v 4 gðρl  ρv Þσ 24

ð6:20Þ

This relationship was compared with experimental data by Zuber and Tribus (1958) and provided satisfactory agreement. Subsequently, Lienhard and Dhir (1973) argued that Taylor instabilities limit the countercurrent flow rather than Kelvin instabilities. Therefore, Lienhard and Tribus (1973) concluded that the constant of Eq. (6.20) should be increased by 14%, resulting in a coefficient of 0.149 rather than 0.131. It is important to highlight that Kutateladze (1951), based on experimental results and on an analogy with the liquid-vapor hydrodynamics in a distillation column and the two-phase flow at the CHF, has correlated experimental results for CHF with an expression similar to Eq. (6.20), however, with a multiplier constant of 0.16, also corroborating the validity of the modelling approach proposed by Zuber and Tribus (1958). It must be remarked that the predictions of CHF according to Eq. (6.20) is valid for horizontal infinite plate, and Lienhard and Dhir (1973) proposed correction factors for finite geometries, and the reader is encouraged to check this paper for further and specific methods. Carey (1992), supported by evidences in literature, pointed out the following main drawbacks of the hydrodynamic models: (i) a logical reason that justifies the radius of the vapor columns as a fraction of Λcrit and independent of heat flux was not given; (ii) vapor columns apart of fixed spacings are typical of film boiling conditions, however, it is not clear if this behavior persists throughout the transition boiling regime until the CHF; (iii) as mentioned above, results from literature revealed that the CHF is affected by the surface wettability whose effect is not

224

6

Critical Heat Flux and Dryout

taken into account by the hydrodynamic models; and (iv) results for boiling of metal liquids revealed CHF values from 2 to 4 times higher than the predictions given by the hydrodynamic models. Such a behavior may be associated with stronger effects of conduction and convection occurring for these fluids which are not taken into account in the models that consider only vapor transport mechanisms.

6.3

Macrolayer Model

Another classical approach reported in literature to model the CHF was proposed by Haramura and Katto (1983), which is referred in the literature as macrolayer model. This term derives from the fact that the model assumes a thin liquid film laying on the heating surface under the CHF condition, referred to as macrolayer, and the vapor bubble (or blanket) hovers over the liquid film in the condition of CHF. It must be mentioned that reported a thin liquid film between the bubble and heater surface close to the CHF condition was previously reported in literature. Therefore, vapor is generated by the liquid film to the bubble vapor, which grows attached to the macrolayer due to the balance of buoyancy and inertial forces, where the last parcel is related to the displacement of liquid portion around the bubble. It is considered that the CHF occurs if the liquid film beneath the bubble evaporates away before the bubble detachment from the surface. Figure 6.2 presents

Vapor bubble

δc

δc

uv

Liquid film

Λd

Λd

ul

Λd

Fig. 6.2 Schematics of the vapor boiling during CHF according to macrolayer model, adapted from Haramura and Katto (1983)

6.3 Macrolayer Model

225

schematically the model adopted by Haramura and Katto (1983) to develop the predictive method for CHF, where the most important geometrical parameters are the spacing between bubbles Λd and the liquid film thickness δc. Initially, it is considered the propagation velocity of interfacial wave, given by: Γ2 ¼

ρl ρv 2πσ ð u þ ul Þ 2  Λ ð ρl þ ρv Þ ð ρl þ ρv Þ 2 v

ð6:21Þ

which should be a real value to be stable. Based on mass conservation in the interface between solid and fluid, it is possible to relate the heat flux and vapor velocity as follows: ϕAw ¼ uv ρv Av bilv

ð6:22Þ

where the vapor velocity is considered to be much higher than liquid velocity due to continuity and density difference. Therefore, by assuming the limiting value for the wave propagation velocity to be real, that is, stable, and substituting the vapor velocity in Eq. (6.21), the following relationship for the wavelength in the imminence of instability can be found: Λinst

ðρ þ ρv Þρv ¼ 2πσ l ρl

 2  2 bilv Av ϕ Aw

ð6:23Þ

where Av is the cross-sectional area of the bubble stem, and Aw is the corresponding area of influence of each bubble/stem along the heater surface. Previous to the instability/dryout, the liquid film thickness δc depends on the wavelength and should be between 0 and Λinst / 2, therefore Haramura and Katto (1983) adopted the mean arithmetic value between these limits as follows: 0 þ Λ2inst π ðρl þ ρv Þρv δc ¼ ¼ σ 2 2 ρl

 2  2 bilv Av ϕ Aw

ð6:24Þ

which relates the heat flux and liquid film thickness, or interfacial wavelength. Therefore, Haramura and Katto (1983) suggested to adopt the most critical (most dangerous) wavelength for the spacing between bubbles Λd, which is given as a function of Taylor instabilities, as follows: Λd ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi σ 3  2π gð ρl  ρv Þ

ð6:25Þ

226

6

Critical Heat Flux and Dryout

Therefore, by assuming that the influence region of one bubble as Λd (Berenson 1961), it is possible to estimate the bubble volumetric growing ratio as follows: Qb ¼

ϕΛ2d bilv ρv

ð6:26Þ

As mentioned before, the stability of the bubble over the liquid film depends on the balance between buoyance forces and inertial forces of the liquid displaced by bubble growth, which is inherently a transient process. Therefore, based on the modelling of the bubble growth process performed by Haramura and Katto (1983), the period of bubble growth during its cycle process is given as follows: 

3 Δt b ¼ 4π

15 4 11 ρ þ ρ 35 1 v 16 l Q5b gð ρl  ρv Þ

ð6:27Þ

Now, by assuming that the CHF occurs when the liquid refeeding is deficient during the hovering period, and that the liquid film evaporates during this period, it is possible to infer a relationship between the CHF as follows: Δt b ϕAw ¼ ρl δc ðAw  Av Þbilv

ð6:28Þ

and substituting the previously derived parameters and solving for the heat flux as follows: 

3 4π

15

 !35     ϕΛ2 15  ρl þ ρv ρv  ^i 2  A 2  4 11 16 ρl þ ρv π v lv d Aw  Av ^ilv ϕAw ¼ ρl σ   ilv ρv ρl ϕ Aw 2 g ρl  ρv ð6:29Þ

16

ϕ5 ¼

 1  1 6 π 4π 5 b165 1 5 σ ilv 2 ðρl þ ρv Þρ5v 2 3 Λd 0

gð ρl  ρv Þ

4 11 16 ρl þ ρv

Av Aw

2   A 1 v ð6:30Þ Aw

115

C  1 B 6 π 4π 5 b165 B 1 C ϕ ¼ σ ilv B ðρ þ ρv Þρ5v  2 qffiffiffiffiffiffiffiffiffiffiffiffiffi C 2 3 A l @ pffiffiffi 3  2π gðρlσρv Þ 16 5

!35 

gðρl  ρv Þ

4 11 16 ρl þ ρv

!35 

Av Aw

2  1

Av Aw



ð6:31Þ

6.3 Macrolayer Model

227

1  1  6 π 4π 5 b165 gðρl  ρv Þ 5 ϕ ¼ σ ilv ðρl þ ρv Þρ5v 2 2 3 12π σ 16 5

gðρl  ρv Þ

4 11 16 ρl þ ρv

!35 

Av Aw

2  1

Av Aw



ð6:32Þ  1 6 4 π 1 5 45b165 ϕ ¼ σ ilv ðgðρl  ρv ÞÞ5 ðρl þ ρv Þρ5v 2 9π 16 5

1

ρ 4 11 16 l þ ρv

!35 

Av Aw

2   A 1 v Aw ð6:33Þ

 1 6 4 π 1 5 45b165 ϕ ¼ σ ilv ðgðρl  ρv ÞÞ5 ðρl þ ρv Þρ5v 2 9π 16 5

ϕ

16

 5 π 1 4b16 ¼ σ i ðgðρl  ρv ÞÞ4 ðρl þ ρv Þ5 ρ6v 2 9π lv

1

ρ 4 11 16 l þ ρv

!35 

1 11

4 16 ρl þ ρv

Av Aw

!3 

2   A 1 v Aw ð6:34Þ

Av Aw

10 

A 1 v Aw

5

ð6:35Þ  ϕcrit ¼

π

4

211 32

161

0  B @

ρl ρv

þ1

11 ρl 16 ρv

5 1

þ1

C 3 A

1 16



Av Aw

2



Av  Aw

3 !165 1 1 bilv ρ2v ðσgðρl  ρv ÞÞ4 ð6:36Þ

Now, the only unknown variable in Eq. (6.36) is the vapor and heater area ratio Av / Aw. The reader should notice that Eq. (6.36) obtained through the macrolayer model is somewhat similar to Eq. (6.20) obtained by Zuber and Tribus (1958) based on the hydrodynamic model. Therefore, Haramura and Katto (1983) equated the CHF given by Eq. (6.36) with the estimation according to Eq. (6.20), to obtain the relationship for the area ratio. Additionally, these authors assumed that Av / Aw  1 and that ρl  ρv, obtaining the following relationship for the area ratio: Av ¼ Aw



3

3

π 4 1116 7 49 38  216

85  15  15 ρv ρ ¼ 0:0584 v ρl ρl

ð6:37Þ

Recall that the prediction according to the hydrodynamic model, Eq. (6.20), provided reasonable predictions of the experimental results for large horizontal flat plates and high pressure fluids, therefore, it is also expected that the prediction according to Eq. (6.36) are also satisfactory for such conditions. Moreover, Kutateladze (1951) correlated the CHF with a relationship similar to Eq. (6.36). Haramura and Katto (1983) also presented predictive methods for other geometries, such as finite plates and cylindrical surfaces.

228

6

Critical Heat Flux and Dryout

Haramura and Katto (1983) also proposed an analytical approach for the prediction of CHF during forced convection over flat plates. In their method, it is considered that the thermal energy supplied along a length L to the fluid is responsible for the vaporization of the liquid of the macrolayer with velocity u as follows: ϕL ¼ ρlbilv uδc

ð6:38Þ

Therefore, by substituting the liquid film thickness δc given by Eq. (6.24) in the above equation, adopting the area ratio simplification of Eq. (6.37), and assuming mass flux G ¼ uρl, it is possible to estimate the CHF as follows: ϕcrit

 157  1  1 ρv ρv 3 ρl σ 3 b ¼ 0:1749  ilv G 1þ ρl ρl G2 L

ð6:39Þ

where the last term of the RHS can be recognized as inverse of the Weber number, which relates the ratio between inertial and surface tension forces. Haramura and Katto (1983) also proposed predictive methods for CHF during cross flow through cylinders. Remark that CHF might occur even for saturated or subcooled condition, but the above described methods should be evaluated for properties estimated at saturated conditions for the operational pressure.

6.4

Critical Heat Flux During In-Tube Flow

Katto (1994), in his wide review, indicated that the CHF during internal flow in channels is more probable to occur close to the outlet of the channel, even though some researches have reported the occurrence of hot spots upstream the outlet. Katto (1994) argued that downstream the point of CHF incipience, the heat exchanger probably operates also with CHF due to increment of vapor fraction, and considering that CHF is usually accompanied by sharp increment of heater temperature with possible damage of the heat transfer surface, it is unlikely that CHF occurs upstream the outlet. It is also important to characterize the difference between saturated and subcooled CHF, because this distinction has been considered by the authors when developing their prediction methods. The last occurs under vapor quality at the test section outlet lower than 0 while the first is characterized by vapor qualities higher than 0. Recall that usually CHF is considered as different from dryout, because dryout during convective boiling results in reduction of heat transfer coefficient but within safe limits, due to partial and progressive drying of the channel wall. In the case of convective boiling, the flow inertia also tends to push liquid droplets periodically to the surface, settling the reduction of heat transfer coefficient. On the other hand,

6.4 Critical Heat Flux During In-Tube Flow

229

Onset of annular flow

b)

d)

G Droplets deposition Dryout a) Evaporation Liquid droplets detachment

Subcooling

Saturated

0

x

c)

Fig. 6.3 Critical heat flux mechanisms during flow boiling, adapted from Semeria and Hewitt (1972)

during the CHF the rewetting of the surface is deteriorated due to inertial forces resulting from the vapor flow away from the surface or from the liquid–vapor interface. Nonetheless, some authors also refer to CHF and dryout as synonyms, such as Hewitt (1981), but in this book we keep them as distinct phenomena. Figure 6.3 modified from Semeria and Hewitt (1972) illustrates the different mechanisms associated to the CHF during flow boiling inside vertical tubes as follows: (a) dryout under a vapor clot; (b) bubble crowding and vapor blanketing; (c) evaporation of liquid film surrounding a slug bubble; and (d) lack of enough liquid to sustain and stable liquid film on the heating surface. As shown in these figures, under subcooled conditions and high mass velocities, turbulence effects, responsible for breaking the bubbles in smaller ones, plus drag effects, responsible for detaching the bubbles from the wall, both act to avoid the formation of vapor clots close to the tube wall. Therefore, intense nucleation and detachment of small bubbles become necessary to avoid the tube wall rewetting and the establishment of CHF. On the other hand, under saturated conditions the amount of liquid entrained within the vapor phase in the channel core increases with increasing the mass velocity due to the increment of shear effects of the vapor on the liquid–vapor interface, intensifying the detachment of liquid droplets from the crest region of

230

6

Critical Heat Flux and Dryout

the disturbance waves. This behavior implies on the reduction of the vapor quality dryout as the mass velocity increases under annular flow conditions. Under conditions of extremely high heat flux and annular flows, bubble nucleation followed by its growth and detachment may cause the liquid film rupture promoting also a lower dryout vapor quality. Under low mass velocities and vapor qualities close to zero, as illustrated by Semeria and Hewitt (1972), the presence of elongated bubbles with a dry region at the bubble tale may result on the establishment of CHF if the transient wall temperature evolves up to such a value that its rewetting is not possible. For high liquid subcooling and low and intermediary mass velocities, the mechanisms described in Sect. 6.3 are expected as being associated to the CHF under flow boiling conditions. Katto and Ohno (1984) proposed a predictive method for CHF during convective boiling inside vertical and uniformly heated tube, resulting in a relationship similar to Eq. (6.39), but also including the effect of the ratio between the channel diameter and the heated length. Additionally, the authors included the possible effect of subcooling of the fluid at the channel inlet, by considering the subcooling enthalpy. Although, their method is not recent, it provides reasonable predictions of independent CHF data obtained for flow boiling under micro and macroscale conditions and is one of the most used prediction methods for CHF, as pointed out by Shah (1987). Their method was adjusted based on their own data and on experimental data from literature for water, halocarbon refrigerants, and helium and tube diameters down to 1 mm. In this method, the CHF is given as follows:   Δ^i ϕcrit ¼ ϕco 1 þ K i ^ilv

ð6:40Þ

where Δii is the inlet subcooling enthalpy, bilv is the latent heat of evaporation, and ϕco is the basic heat flux as nominated by them, which can be calculated according to the following five equations: ϕco ð1Þ 1 ¼ CðWedh Þ0:043 b L=d h Gilv  0:133 ϕco ð2Þ ρ 1 ¼ 0:10 v ðWedh Þ1=3 ρl 1 þ 0:0031ðL=dh Þ Gbilv  0:133 ϕco ð3Þ ðL=dh Þ0:27 ρ ¼ 0:098 v ðWedh Þ0:433 ρl 1 þ 0:0031ðL=d h Þ Gbilv  0:60 ϕco ð4Þ ρ 1 ¼ 0:0384 v ðWedh Þ0:173 ρ 1 þ 0:280ðWedh Þ0:233 ðL=dh Þ l Gbilv

ð6:41Þ ð6:42Þ ð6:43Þ

ð6:44Þ

6.4 Critical Heat Flux During In-Tube Flow

231

 0:513 ϕco ð5Þ ðL=dh Þ0:27 ρ ¼ 0:234 v ðWedh Þ0:433 ρl 1 þ 0:0031ðL=d h Þ Gbilv

ð6:45Þ

where the Weber number is given as follows: Wedh ¼

G2 dh σρl

ð6:46Þ

and the dimensionless parameter C of Eq. (6.41) is obtained as follows:



8 > < > :

0:25

for

L=d h < 50

0:25 þ 0:0009½ðL=d h Þ  50 0:34

for for

50 L=dh 150 L=d h > 150

ð6:47Þ

The inlet subcooling parameter K in Eq. (6.40) is giving by the following three equations: 1:043 4C ðWedh Þ0:043

ð6:48Þ

0:0124 þ ðdh =LÞ 5  6 ðρ =ρ Þ0:133 ðWed Þ1=3

ð6:49Þ

Kð1Þ ¼ Kð2Þ ¼

v

Kð3Þ ¼ 1:12 

l

h

1:52ðWedh Þ0:233 þ ðd h =LÞ ðρv =ρl Þ0:60 ðWedh Þ0:173

ð6:50Þ

The values of ϕco and K in Eq. (6.40) are determined according to the criteria presented in Fig. 6.4. Shah (1987) developed a correlation based on a broad database gathered from 16 independent studies involving 23 fluids, tube diameters from 0.315 to 37.5 mm, Start

ρv / ρl < 0.15

Yes

No ϕco(1) < ϕco(5)

No

K(1) > K(2)

Yes ϕco = ϕco(1)

No

ϕco = min{ϕco(1), ϕco(2), ϕco(3)} K = max{K(1), K(2)}

Yes ϕco = max{ϕco(4), ϕco(5)}

K = K(1)

K = min{K(2), K(3)}

End

Fig. 6.4 Algorithm for determination of parameters ϕco and K of Eq. (6.40)

232

6

Critical Heat Flux and Dryout

reduced pressures from 0.0014 to 0.96, inlet vapor quality from 4.00 to 0.85, mass velocities from 3.9 to 29.051 kg/m2s, and critical vapor quality (corresponding to the CHF) from 2.6 to 1. In his method, two different calculation procedures are proposed, considering the following parameter as the transitional criterion: Y ¼ Pe  Fe0:4

 μ 0:6 l

μv

ð6:51Þ

where Pe and Fe are, respectively, the Péclet number and the entrance effect factor given by Pe ¼

Gd h cp,l kl

ð6:52Þ

  L Fe ¼ 1:54  0:032 dh

ð6:53Þ

where L is the axial distance between the tube entrance and the CHF location, comprising the total tube length. If Eq. (6.53) gives Fe less than 1, Fe ¼ 1 should be adopted. Besides the length L, the method of Shah (1987) also defines the effective length Le equal to L when xeq,in 0 and equal to the boiling length when xin > 0 given as follows for uniform heated channels1: 8   > < L þ Gxeq,inbilv d =ð4ϕcrit Þ   Lb ¼ > : L þ Gxeq,inbilv At =ðϕcrit PÞ

for

round channels

for

other geometries

ð6:54Þ

where At is the channel cross-sectional area and P its perimeter. For Helium or when Y 106 for other fluids or when the effective tube length is higher than 160=p1:14 r , the following procedure should be adopted: Bocrit ¼ 0:124

 0:89  4 n dh 10 ð1  xIE Þ Le Y

ð6:55Þ

where xIE ¼ xeq,in when the fluid is subcooled (xeq,in 0) at the test section inlet, otherwise xIE ¼ 0. The critical Boiling number in Eq. (6.55) is given as follows:

1 In the present study the method of Shah (1987) was extended also to non-circular channels since Tibiriçá et al. (2008) observed reasonable prediction by this method for non-circular channels.

6.4 Critical Heat Flux During In-Tube Flow

233

Bocrit ¼

ϕcrit bilv G

ð6:56Þ

The value of n in Eq. (6.55) is estimated as follows:



8 > < > :

ðd h =Le Þ0:33 ðd h =Le Þ

0:54 0:5

0:12=ð1  xIE Þ

for

Helium

for

Y 106

for

Y > 106

ð6:57Þ

Otherwise the critical Boiling number is calculated as follows: Bocrit ¼ Fe  Fx  Bo0

ð6:58Þ

and the lowest value for the critical Boiling number given by Eqs. (6.55) and (6.58) is adopted for the estimative of the CHF through Eq. (6.56). In Eq. (6.58), Bo0 is the highest of the values given by the following three equations: Bo0 ð1Þ ¼ 15ðY Þ0:612

Bo0 ð2Þ ¼ 0:082ðY Þ0:3 1 þ 1:45p4:03 r

Bo0 ð3Þ ¼ 0:0024ðY Þ0:105 1 þ 1:15p3:39 r

ð6:59Þ ð6:60Þ ð6:61Þ

and Fx is estimated as follows: For xcrit > 0

Fx ¼

8  2 3n

0:24157xcrit  > 1:25105   > 1 ð p 0:6 Þ 0:833x r crit Y > 1:25105 > 41 þ 5 < Y 0:35

for

xcrit > 0

> > > > :

for

xcrit < 0

h in Þðpr 0:6Þ F 1 1  ð1F 20:35

ð6:62Þ where n¼ and

0 1

for for

pr 0:6 pr > 0:6

ð6:63Þ

234

6

( F1 ¼

1 þ 0:0052ðxcrit Þ0:88 Y 0:41 1 þ 4:452ðxcrit Þ

0:88

Critical Heat Flux and Dryout

for

Y 1:4  107

for

Y > 1:4  107

ð6:64Þ

and F1 ¼

F 0:42 1

for

F1 4

0:55

for

F1 > 4

ð6:65Þ

Most recently, Zhang et al. (2006) proposed a predictive method for the CHF inside microchannels. These authors gathered a wide experimental database from the open literature and performed an analysis using artificial neural network to identify the dominant non-dimensional parameters. Based on this analysis, these authors found that the boiling number for CHF, Bocrit, can be presented as a function of Weber number, density ratio, inlet vapor quality, and channel length to diameter ratio. Subsequently, Zhang et al. (2006) performed the regression based on artificial neural network and on the mentioned database to determine the constants and exponents. The resulting predictive method is given as follows: "

Bocrit

#  2:31  0:361 #0:295  0:311 "  0:170 ρv ρv L L ¼ 0:0352  Wedh þ 0:0119 2:05  xeq,in dh dh ρl ρl

ð6:66Þ The sub index h in the diameter term for the above equations refers to the hydraulic diameter because the predictive method is also suitable to non-circular channels. Additionally, the vapor quality at the section inlet xeq,in in Eq. (6.66) is 108

10-1

L/d = 10 L/d = 50

107

L/d = 10 L/d = 50 L/d = 100

L/d = 100

Bocrit [-]

φcrit [W/m²]

10-2

10-3

Zuber and Tribus (1958)

106

105

104

Zuber and Tribus (1958) 10-4 10-3

10-2

10-1

100

101

Wed [-]

102

103

104

103 100

101

102

103

104

G [kg/m²s]

Fig. 6.5 Variation of (a) boiling number with Weber number, and (b) heat flux with mass velocity, for CHF conditions for water convective boiling, d ¼ 3 mm, pin ¼ 1 MPa, xeq,in ¼ 0.20 according to the model of Zhang et al. (2006)

6.5 Solved Example

235

evaluated assuming thermal equilibrium and is given as a function of the inlet enthalpy. Figure 6.5 depicts the variation of the critical heat flux with mass flow rate in non-dimensional and dimensional form, showing that the CHF increases with mass flow rate, as expected. These pictures also depict the estimation according to the Zuber and Tribus (1958) model, which was developed for stagnant conditions. The increment of working pressure and reduction of inlet enthalpy increase marginally the CHF, but as not as pronounced as the reduction of the length to diameter ratio. A comparison performed by Tibiriça et al. (2008) of a microscale CHF database from literature against nine prediction methods revealed that the best predictions were provided by Shah (1987) as a general method (saturated and subcooled) and Zhang et al. (2006) for saturated conditions. In the case of the database containing only subcooled CHF data, the method of Hall and Mudawar (2000) provided the most accurate predictions. Their database comprised more than 1000 data points and includes both subcooled and saturated CHF results, covering 7 fluids (water, R12, R113, R134a, R123, CO2, and Helium), mass velocities from 10.5 to 134,000 kg/m2s, and experimental CHFs up to 276 MW/m2 for both multi- and single-microchannel configurations. Most recently, predictive methods for CHF inside microchannels were also proposed in the literature, such as Ong and Thome (2011) and Tibiriçá et al. (2012), which were also developed for convective boiling subjected to uniform heat fluxes. Based on the previous discussion, it is expected that CHF might occur in the channel outlet, therefore at the point L downstream the channel inlet. Therefore, Eq. (6.66) can be seen as an energy balance for determination of vapor quality at the CHF. On the other hand, some researchers, such as Kanizawa et al. (2016), adopted the predictive method for CHF to model the dryout occurrence, which corresponds to the maximum heat transfer coefficient possible for fixed mass and heat fluxes, as well as channel diameter and fluid properties. According to this approach, it is considered that the operational heat flux ϕ is equal to the CHF given by Eq. (6.66), and the channel point of CHF (L/d) is determined by an iterative method. Then, based on energy balance it is possible to estimate the vapor quality for dryout.

6.5

Solved Example

Consider a horizontal surface facing upward submersed in stagnant water at 1 atm of absolute pressure. For this condition, evaluate the CHF value according to the hydrodynamic model proposed by Zuber and Tribus (1958) and evaluate the heat transfer coefficient and wall superheating in the imminence of CHF assuming the predictive method proposed by Stephan and Abdelsalam (1980). Then, evaluate the heat transfer coefficient during film boiling assuming the heat flux equal to CHF based on Berenson (1961), which is given as follows:

236

6

2

hfb ¼

Critical Heat Flux and Dryout

31=4

3 0 6 gðρl  ρv Þρv kvbilv 7 0:4254 h i1=2 5 μv ΔT w,fb gðρ σρ Þ l v

where the modified enthalpy of vaporization is given as follows: h  i bi0lv ¼ bilv 1 þ 0:5 cpv ΔT w,fb =bilv and is combined with radiation by assuming the equivalent heat transfer coefficient for radiation for infinite parallel black plates:

hrad ¼

  σ SB T 4w,fb  T 4sat T w,fb  T sat

where σ SB corresponds to the Stephan and Boltzmann constant equal to 5.67108 W/m2K4, and the corresponding wall superheating. Finally, assuming a stagnant vapor film with thickness δ of 0.2 mm, and heat transfer by conduction combined with thermal radiation from the surface to the liquid at Tsat, evaluate the overall heat transfer coefficient. Assume vapor properties evaluated at saturation conditions and black surfaces. Solution: For water at 1 atm, the following properties can be found in thermodynamic tables: Tsat ¼ 100 C ¼ 373.15 K. ρl ¼ 958.4 kg/m3, ρv ¼ 0.5975 kg/m3, μl ¼ 2.819104 kg/ms, μv ¼ 1.227105 kg/ms, σ ¼ 0.05891 N/m, îlv ¼ 2.257106 J/kg, cpl ¼ 4217 J/kgK, cpv ¼ 2044 J/kgK, kl ¼ 0.6651 W/mK, kv ¼ 0.02508 W/mK. Hence, by solving Eq. (6.20), it is possible to estimate the CHF as follows: 1

ϕcrit ¼ 0:131  ilv ρ2v

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 gðρl  ρv Þσ ¼ 1:108  106 W=m2

Therefore, considering the imminence of the CHF, let us consider the Stephan and Abdelsalam (1980) predictive method for heat transfer coefficient during pool boiling, given by Eq. (5.19) as follows:

6.5 Solved Example

237

!1:58 !1:26   0:673 5:22 bilv d2b cpl T sat d2b ρl  ρv hnb d b ϕd b 6 ¼ 2:46  10

2

2 kl k l T sat ρl kl =ρl cpl k l =ρl cpl where db is the bubble departure diameter given by Eq. (5.23) as follows: d b ¼ 0:146

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2σ ¼ 0:5171 mm gð ρl  ρv Þ

and the corresponding hnb for ϕ ¼ ϕcrit is equal to 188,285 W/m2K. The corresponding wall superheating can be evaluated based on the Newton cooling law as follows: ϕ ¼ hnb ΔT w,nb ¼ hnb ðT w,nb  T sat Þ which results in superheating of 5.89 K, that can be considered relatively low for the heat flux value. But the corresponding heat transfer coefficient is of the order of hundreds of kW/m2K, hence, the result is coherent. Subsequently, let us evaluate the system performance after the occurrence of the CHF, initially accounting for the heat transfer coefficient predictive method given in this exercise. Considering that the predictive method for heat transfer coefficient and for the thermal radiation parcel depend on the wall superheating, which in turn depends on the heat transfer coefficient, an iterative method is required for the solution. Moreover, considering the combination of heat transfer by convection during film boiling and radiation, the problem is non-linear and the use of at least a spreadsheet is advisable. The following scheme can be adopted: • Guess the surface temperature, for example, assuming overall heat transfer coefficient as 1000 W/m2K, the resulting surface superheating ΔTw,fb would be 1108 K for ϕcrit. • Tolerance ¼ small value. • Error ¼ large value. • ΔTw,fb,aux ¼ ΔTw,fb. • Repeat: – – – – – –

Evaluate hfb and hrad based on ΔTw,fb. Evaluate the overall heat transfer coefficient h ¼ hfb + hrad. Update the wall superheating ΔTw,fb ¼ ϕcrit / h. Error ¼ | ΔTw,fb.- ΔTw,fb,aux|. ΔTw,fb,aux ¼ ΔTw,fb. While (Error > Tolerance).

238

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By performing the above approach, the ΔTw,fb is obtained as 1628 K, and the corresponding heat transfer coefficient as hfb ¼ 122.6 W/m2K, hrad ¼ 558 W/m2K, and h ¼ 680.7 W/m2K, which is 276 times lower than the estimation for nucleate boiling condition. It must be emphasized that 82% of the heat transfer rate occurs by thermal radiation due to the high temperature difference and relatively low heat transfer coefficient for film boiling. Nonetheless, considering that melting temperature of 1010 carbon steel is approximately 1516 C, a heat exchanger made of this material would fail during CHF with imposed heat flux and appropriate material must be selected. This example justifies the usage of burnout as a synonym for CHF since the surface would be damaged. Additionally, if the radiation parcel was neglected, the temperature difference would rise to 10,682 K, and if only radiation was considered, neglecting the film boiling parcel, the temperature difference would be 1730 K, emphasizing the importance of accounting for thermal radiation. Finally, let us consider the simplification of a vapor film formed in contact with the surface that corresponds to a thermal resistance of conduction in parallel with heat transfer by radiation. The thermal radiation parcel can be estimated based on the same approach of the previous case. In the case of the thermal resistance due to conduction through the vapor film, the following equivalent heat transfer coefficient can be obtained: hcond ¼

kv δ

Hence, by repeating the algorithm described in the previous case, it is possible to estimate the overall heat transfer coefficient and solve the problem. In this case, the temperature difference is 1626 K, with hcond ¼ 125.4 W/m2K and hrad ¼ 556.3 W/ m2K, with overall heat transfer coefficient of 681.7 W/m2K. This result is pretty similar with the analysis of film boiling due to the definition of the vapor film thickness, with 82% of the heat transfer rate being transferred from the surface to the liquid via radiation. Considering a vapor film thickness of 1.0 mm instead of 0.2 mm, the temperature difference raises to 1709 K, and the radiative parcel to 96% of the total heat transfer rate, which again emphasizes the importance of accounting for radiative heat transfer. In conclusion, the reader must bear in mind that, in general, CHF should be avoided otherwise the heat transfer surface might be damaged in case of imposed heat flux, and evaluating the heat transfer coefficient value after CHF might be pointless.

References

6.6

239

Problems

1. Consider a vapor layer with thickness δ in contact with a flat horizontal surface with imposed heat flux ϕ, with liquid above it, which can be considered at Tsat in the interface. Assuming the vapor as stagnant: (a) Derive a relationship for the heat transfer coefficient assuming fluid reference temperature as Tsat. (b) Considering water at atmospheric pressure, evaluate the variation of heat transfer coefficient with δ between 0.1 to 5.0 mm. (c) Compare the estimated heat transfer coefficient with conditions of liquid single-phase flow, and nucleate boiling according to Stephan and Abdelsalam (1980). (d) Include the effect of thermal radiation by assuming infinite black bodies

4 4 (ϕrad ¼ σ SB T surf  T sat , σ SB as the Stephan–Boltzmann constant, and equal to 5.607108 W/m2K4), and evaluate the overall equivalent heat transfer coefficient. 2. Estimate the CHF for R134a flow boiling inside a 2 mm ID channel, 50 mm long, at 200 kg/m2s, and saturation temperature of 30 C. Assume saturated liquid in the inlet. 3. Repeat the previous exercise assuming subcooling degree of 2 C in the inlet. 4. Compare the heat transfer coefficient in the imminence of CHF during pool boiling conditions for water at 101.3 C, R134a at 770.6 kPa, and R600a at 1683 kPa. Assume the predictive method proposed by Stephan and Abdelsalam (1980) to estimate the heat transfer coefficient, paying attention that the corresponding correlation depends on the fluid type, and consider either the hydrodynamic model or the macrolayer model for CHF. 5. Determine the variation of the CHF with reduced pressure for flow boiling of water and R134a inside a 5 mm ID tube, 150 mm long, at 500 kg/m2s, and subcooling degree at the channel inlet of 20 C according to Katto and Ohno (1984). 6. Repeat the previous exercise considering the method of Shah (1987).

References Bellman, R., & Pennington, R. H. (1954). Effects of surface tension and viscosity on Taylor instability. Quarterly of Applied Mathematics, 12(2), 151–162. Berenson, P. J. (1961). Film-boiling heat transfer from a horizontal surface. Journal of Heat Transfer, 83, 351–358. Carey, V. P. (1992). Liquid-vapor phase-change phenomena: An introduction to the thermophysics of vaporization and condensation processes in heat transfer equipment. New York: Taylor & Francis. Collier, J. G., & Thome, J. R. (1994). Convective boiling and condensation. Oxford: Clarendon Press.

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Groeneveld, D. C., Shan, J. Q., Vasić, A. Z., Leung, L. K. H., Durmayaz, A., Yang, J., et al. (2007). The 2006 CHF look-up table. Nuclear Engineering and Design, 237(15–17), 1909–1922. Hall, D. D., & Mudawar, I. (2000). Critical heat flux (CHF) for water flow in tubes—II: Subcooled CHF correlations. International Journal of Heat and Mass Transfer, 43(14), 2605–2640. https:// doi.org/10.1016/S0017-9310(99)00192-1 Haramura, Y., & Katto, Y. (1983). A new hydrodynamic model of critical heat flux, applicable widely to both pool and forced convection boiling on submerged bodies in saturated liquids. International Journal of Heat and Mass Transfer, 26(3), 389–399. Hewitt, G. F. (1981). Chapter 9: Burnout. In A. E. Bergles, J. G. Collier, J. M. Delhaye, G. F. Hewitt, & F. Mayinger (Eds.), Two-phase flow and heat transfer in the power and process industries (pp. 545–560). New York: Hemisphere. Kanizawa, F. T., Tibiriçá, C. B., & Ribatski, G. (2016). Heat transfer during convective boiling inside microchannels. International Journal of Heat and Mass Transfer, 93, 566–583. Katto, Y. (1994). Critical heat flux. International Journal of Multiphase Flow, 20, 53–90. Katto, Y., & Ohno, H. (1984). An improved version of the generalized correlation of critical heat flux for the forced convective boiling in uniformly heated vertical tubes. International Journal of Heat and Mass Transfer, 27(9), 1641–1648. Kutateladze, S. S. (1951). A hydrodynamic theory of changes in a boiling process under free convection. Izvestia Akademia Nauk Otdelenie Tekhnicheski Nauk, 4, 529–536. Linehard, J. H., & Dhir, V. K. (1973). Extended hydrodynamic theory of the peak and minimum pool boiling heat fluxes. Washington, DC: National Aeronautics and Space Administration. Lienhard, J. H., & Dhir, V. K. (1973). Hydrodynamic prediction of peak pool-boiling heat fluxes from finite bodies. https://doi.org/10.1115/1.3450013 Ong, C. L., & Thome, J. R. (2011). Macro-to-microchannel transition in two-phase flow: Part 2– flow boiling heat transfer and critical heat flux. Experimental Thermal and Fluid Science, 35(6), 873–886. Rayleigh, J. W. S. B. (1896). The theory of sound (Vol. 2). London: Macmillan. Semeria, R., & Hewitt, G. F. (1972). Aspects of gas–liquid flow. In Seminar on recent developments in Heat Exchangers, International Centre for Heat and Mass Transfer, Trogir, Croatia. Shah, M. M. (1987). Improved general correlation for critical heat flux during upflow in uniformly heated vertical tubes. International Journal of Heat and Fluid Flow, 8(4), 326–335. Stephan, K., & Abdelsalam, M. (1980). Heat-transfer correlations for natural convection boiling. International Journal of Heat and Mass Transfer, 23(1), 73–87. Tibiriçá, C. B., Felcar, H. O. M., & Ribatski, G. (2008, May). An analysis of experimental data and prediction methods for critical heat fluxes in micro-scale channels. In 5th European ThermalSciences Conference, Eindhoven. Tibiriçá, C. B., Ribatski, G., & Thome, J. R. (2012). Saturated flow boiling heat transfer and critical heat flux in small horizontal flattened tubes. International Journal of Heat and Mass Transfer, 55(25–26), 7873–7883. Yagov, V. V., Lexin, M. A., Zabirov, A. R., & Kaban’kov, O. N. (2016). Film boiling of subcooled liquids. Part I: Leidenfrost phenomenon and experimental results for subcooled water. International Journal of Heat and Mass Transfer, 100, 908–917. Zhang, W., Hibiki, T., Mishima, K., & Mi, Y. (2006). Correlation of critical heat flux for flow boiling of water in mini-channels. International Journal of Heat and Mass Transfer, 49(5–6), 1058–1072. Zhang, L., Seong, J. H., & Bucci, M. (2019). Percolative scale-free behavior in the boiling crisis. Physical Review Letters, 122(13), 134501. Zuber, N., & Tribus, M. (1958). Further remarks on the stability of boiling heat transfer. Report 58-5 (No. AECU-3631). California of University, Los Angeles. Department of Engineering.

Chapter 7

Condensation

In Chap. 5, a discussion concerning homogeneous nucleate boiling is presented, which is modelled by accounting for the required energy to generate a vapor bubble within a liquid media that in turn depends on the probability of an amount of highly energetic particles to collide forming a high energetic region, which propitiates the formation of a vapor nuclei, with subsequent bubble growth. Considering the case of condensation, homogeneous transition from vapor to liquid phase can also occur and can be modelled by considering a similar approach as in the case of homogeneous vapor nucleation. However, this subject is not the focus of the present study because the phenomenon of homogeneous condensation is rarely observed in industrial applications. For those interested, the books of Collier and Thome (1994) and Carey (2007) are suggested as complementary material. In the case of heterogeneous nucleation of liquid droplets, it is worth mentioning that certain degrees of subcooling of the surface and of the vapor contacting the surface are required to form the initial liquid portion in a similar way to bubble formation phenomenon, and several approaches are found in literature to estimate the required subcooling, such as the one proposed by Hill et al. (1963). As discussed in Chap. 5, the boiling phenomenon might present critical conditions, such as critical heat flux or dryout. However, under condensation conditions no such thing is present and consequently the analysis is a bit simpler, unless the problem is extended until solidification with consequent formation of a solid thermal resistance. Additionally, and as discussed in Chap. 2, the heat transfer coefficient during convective condensation can be modelled by combination of gravitational and convective effects, in a similar way of the approach presented by Chen (1966) for convective boiling, which considers a combination of nucleate boiling and convective effects, such as schematically depicted in Fig. 2.17. Therefore, in this chapter, initially the modelling approach for film-wise condensation is described in a similar way to the classical study presented by Nusselt (1916)

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_7

241

242

7 Condensation

for condensation in a stagnant vapor medium contacting a vertical flat surface, and then this mechanism is combined with convection to model the heat transfer process inside channels. Finally, the sublimation might also be present with direct transition from vapor to solid phase in some engineering and natural conditions. Hence, the frost formation, which is common on finned evaporators for refrigeration and air-conditioning application, for example, is also a subject of studies and the interested reader is encouraged to check the studies of Hermes and coworkers (Hermes et al. 2009) for further information.

7.1

Film Condensation on an Isothermal Surface

The modelling of film condensation on a flat isothermal surface was originally presented by Nusselt (1916), focusing on conditions of a surface exposed to saturated vapor of a pure substance. Figure 7.1 depicts schematically the liquid film flowing along the surface, which can be inclined or vertical. The heat transfer process is based on the following simplifying assumptions: (i) (ii) (iii) (iv) (v)

Steady state condition. Two-dimensional problem. Transport properties variations in each phase are negligible. Newtonian fluid. Isothermal surface at temperature Ts. y

z

y δ δ

y

y

Tsat

u Ts

dṁ

Tsat Ts

δ

T

δ u

Ts

z

dz z+dz

g

θ

Fig. 7.1 Schematics of film condensation on isothermal flat surface

u

Control volume

7.1 Film Condensation on an Isothermal Surface

243

(vi) Vapor at saturation temperature at Tsat. (vii) The liquid at vapor–liquid interface is at Tsat. (viii) The shear stress in the liquid vapor interface is negligible because of the relatively low viscosity of vapor phase, and because of the low flow velocity of the liquid film. (ix) Convective parcels of momentum and energy transfer are negligible because the film flow velocity is considerably low. Hence, the momentum and heat transfer are governed by diffusive and buoyancy forces. (x) Variation of velocity and temperature in the direction perpendicular to the surface is considerably higher than parallel to it, similarly to the analysis of the boundary layer during forced convection parallel to a flat plate. (xi) Uniform pressure along the horizontal position. (xii) Based on the previous assumption, it is possible to estimate the pressure variation based on the hydrostatic problem along the vapor side. (xiii) No viscous dissipation along the liquid flow. (xiv) The sensible heat for liquid subcooling is negligible in comparison with the latent heat. (xv) The fluid properties are evaluated at the mean liquid film temperature corresponding to the average temperature between the wall and the saturation temperatures. Before starting with the development of velocity and temperature profiles, it is interesting to perform a dimensional analysis of the relevant parameters in this problem. The heat transfer coefficient should be given as a function of liquid specific heat, liquid density, fluid vaporization enthalpy, buoyance force, liquid thermal conductivity, liquid viscosity, temperature difference between the surface and vapor, and axial position or length, therefore: h ¼ f nðρl , ^ilv , gz ðρl  ρv Þ, kl , μl , cp,l , ðT sat  T s Þ, L or zÞ

ð7:1Þ

where units can be written as a combination of kg, J, m, s, and K. Therefore, the problem is governed by 9 variables that can be written according to 5 dimensional groups, resulting in 4 non-dimensional parameters, as follows: μ cp,l cp,l ðT sat  T s Þ ρl gz ðρl  ρv Þ^ilv ðL or zÞ3 hðL or zÞ ¼ fn l , , ^ilv kl kl μl k l ðT sat  T s Þ

! ð7:2Þ

or: NuL or z ¼ f nðPrl , Ja, ΠÞ with:

ð7:3Þ

244

7 Condensation

hðL or zÞ kl μ cp,l Prl ¼ l kl

NuL or z ¼

ð7:5Þ

cp,l ðT sat  T s Þ bilv

ð7:6Þ

ρl gz ðρl  ρv Þ^ilv ðL or zÞ3 μl kl ðT sat  T s Þ

ð7:7Þ

Ja ¼ Π¼

ð7:4Þ

where the first two non-dimensional parameters are Nusselt and liquid Prandtl numbers, respectively. The parameter Ja corresponds to the Jakob number and represents the ratio between sensible and latent heat of a fluid for a given temperature difference, and its value is usually small. The fourth term has no specific denomination and will be derived in the foregoing analysis. In this analysis, the contribution of the gravitational acceleration gz is taken according to the flow direction, therefore given by gsin(θ), where θ is the inclination of the plate in relation to the horizontal plane. Therefore, starting with the z component of Navier–Stokes equation for liquid phase as follows:  2   2  ∂u ∂u ∂u ∂p ∂ u ∂ u ρl þ þu þv þ ρl gz þ μ l ¼ ∂t ∂z ∂y ∂z ∂z2 ∂y2

ð7:8Þ

where the parcel relative to the perpendicular direction of the paper was already simplified, and by accounting for the aforementioned assumptions, the left-hand side (LHS) is null by hypothesis i and ix. Similarly, the first term inside the bracket of the right-hand side (RHS) is simplified by hypothesis x (∂2u/∂y2  ∂2u/∂z2), resulting in the following relationship: 2

0¼

∂p ∂ u þ ρl gz þ μ l 2 ∂z ∂y

ð7:9Þ

Based on hypotheses xi and xii, the pressure variation along z direction can be estimated as follows: ∂p ¼ ρv gz ∂z

ð7:10Þ

and substituting into Eq. (7.9), we obtain the following relationship: ðρ  ρv Þgz ∂ u ¼ l μl ∂y2 2

ð7:11Þ

7.1 Film Condensation on an Isothermal Surface

245

We may try to find a velocity profile as a function of y coordinate only, whereas the influence of the z coordinate will appear in the liquid film thickness δ. Therefore, by integrating the previous equation twice along the y direction, we obtain the following relationship for the velocity profile: u¼

ðρl  ρv Þgz y2 þ C1 y þ C2 μl 2

ð7:12Þ

The integrating constants can be determined from the boundary conditions, where the first one is related to non-slip condition in the solid–liquid interface, therefore: uðy ¼ 0Þ ¼ 0

ð7:13Þ

then C2 is null. The second boundary condition can be determined from hypothesis viii, related to null shear stress between the liquid and vapor. Therefore, based on Newton’s Viscosity Law for a Newtonian fluid, this boundary condition is given as follows: μl

 ∂u  ¼0 ∂y y¼δ

ð7:14Þ

where δ is the liquid film thickness. Hence, the constant C1 is given as follows: 

 ρg  ρv gz C1 ¼ δ μl

ð7:15Þ

and the velocity profile is given as follows: u¼

 ðρl  ρv Þgz  y2  þ δy μl 2

ð7:16Þ

where the unknown parameter is the liquid film thickness δ, which will be determined based on the mass and energy balance along the flat surface. The mass flow rate of the liquid in a given position can be evaluated based on the integration of velocity profile as follows: Z m_ l ¼

δ

ρl udA ¼

0

m_ l ¼ where W is the surface width.

ρl ðρl  ρv Þgz  y3 δy2 δ W  þ μl 6 2 0

ð7:17Þ

ρl ðρl  ρv Þgz δ3 W μl 3

ð7:18Þ

246

7 Condensation

By evaluating the mass balance for an infinitesimal control volume with length dz, such as depicted in Fig. 7.1, the mass variation between two positions is related to condensation rate ṁcond, as follows: d m_ ¼ m_ cond

ð7:19Þ

Additionally, by considering the energy balance over this control volume accounting for hypothesis xiv, we obtain the following relationship for steadystate condition, null net work, null viscous energy dissipation (hypothesis xiii), and negligible contribution of kinetic and potential energy variations: Z q_ cv ¼

!

!

biρV  dA ¼ bil d m_  biv m_ cond ¼ bilv dm_

ð7:20Þ

cs

The mass flow rate is given by Eq. (7.18). In this equation, the only variable term is the liquid film thickness. Therefore, by differentiating Eq. (7.18) and combining with Eq. (7.20) we obtain the following relationship: q_ cv ¼ bilv

ρl ðρl  ρv Þgz 1 3 W dδ 3 μl

ð7:21Þ

According to hypothesis vi and vii, the vapor and the liquid–vapor interface are under saturated conditions, and the heat transferred associated to the liquid subcooling is neglected according to hypothesis xiv; therefore, the control volume of Fig. 7.1 exchanges heat only with the solid surface, which can be evaluated based on the temperature profile of the liquid film, through the Fourier law. Therefore, starting with the energy equation in differential form for an incompressible and Newtonian fluid, we have:  ∇  ðkl ∇T Þ þ g_ ¼ ρcp l

∂T ! þ V  ∇T ∂t

 ð7:22Þ

which can be simplified to liquid film based on the listed hypothesis (i, ii, iii, ix, x, xiii) to the following relationship: 2

∇2 T ¼

∂ T ¼0 ∂y2

ð7:23Þ

Integrating the above equation twice, and considering the boundary conditions given by assumptions v and vi, we obtain the temperature profile along the liquid film as follows: y T ¼ ðT sat  T s Þ þ T s δ

ð7:24Þ

7.1 Film Condensation on an Isothermal Surface

247

Then, by accounting for no-slip condition in the solid–liquid interface, it is possible to evaluate the heat flux based on the Fourier law for heat conduction as follows: ϕs ¼ kl

 k ðT  T s Þ ∂T  ¼  l sat  δ ∂y y¼0

ð7:25Þ

Therefore, by evaluating the heat transfer rate in the control volume with the heat flux (qcv ¼ ϕs W dz), we obtain the following ordinary differential equation for the liquid film thickness from Eq. (7.21): 

ρ ðρ  ρv Þgz 1 3 k l ðT sat  T s Þ Wdz ¼ ilv l l W dδ δ 3 μl dz ¼

ρl gz ðρl  ρv Þilv 3 δ dδ μl kl ðT sat  T s Þ

ð7:26Þ ð7:27Þ

Therefore, integrating the above equation from the starting point of the liquid film thickness, assuming δ (z ¼ 0) ¼ 0, we obtain the following relationship: μ k ðT  T s Þ δ4 ¼ 4 l l sat z4 ρl gz ðρl  ρv Þilv z3

ð7:28Þ

The liquid film thickness increases along the flow direction proportional to the distance of the starting point at ¼. Since we are interested in a relationship for the heat transfer coefficient, we can equate the heat flux given by Fourier law, Eq. (7.25), with the heat flux estimated through the Newton’s Cooling Law, as follows: ϕs ¼ 

kl ðT sat  T s Þ ¼ hðT s  T sat Þ δ

ð7:29Þ

then, by solving for h and considering the liquid film thickness of Eq. (7.28), we determine a relationship for the heat transfer coefficient as follows: k k 1 ffiffiffi h¼ l¼ l p δ z 44

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4 ρ g ðρ  ρ Þ^ l z l v ilv z μl kl ðT sat  T s Þ

ð7:30Þ

and in non-dimensional form as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4 ρ g ðρ  ρ Þ^ hz l z l v ilv z Nuz ¼ ¼ 0:707 kl μl k l ðT sat  T s Þ

ð7:31Þ

248

7 Condensation

Recall that the term inside the root is the non-dimensional parameter determined from the dimensional analysis, and given by Eq. (7.7). Additionally, neither Jakob and liquid Prandtl numbers appear in the resulting equation. In fact, both parameters account for the sensible parcel of the liquid energy variation, which was neglected during the modelling. Therefore, Sadasivan and Lienhard (1987) proposed a correction for the latent heat accounting for Prl and Ja based on experimental results for film condensation, which is valid for Prl  0.6 and is given as follows:

  bilv ¼ bilv 1 þ Ja 0:683  0:228 Prl

ð7:32Þ

which should substitute îlv in Eqs. (7.28), (7.30), and (7.31). Notice that the Jakob number shall present small values, since the temperature difference is usually not high enough to compensate for the small specific heat in comparison to the latent heat. Therefore, the correction for the latent heat given by Eq. (7.32) is close to unity. Similarly, Bejan (2013) presented a modelling approach that accounted for the sensible heat of cooling, and, therefore, without taking into account the hypothesis xiv, he obtained an equivalent latent heat of vaporization given as follows: h i bilv ¼ bilv 1 þ 3 Ja 8

ð7:33Þ

However, the expression (7.32) is preferable because it was validated with experimental results. Additionally, the reader might have noticed that the heat transfer coefficient reduces along the flow direction, being proportional to the inverse of the distance at ¼, and this aspect is related to the increase of thermal resistance imposed by the liquid film. In this context, it is interesting to evaluate the conditions that correspond to laminar flow, which is given based on the Reynolds number. For this application, the Reynolds number is defined based on the liquid film thickness, rather than on the distance from the upper board of the plate, as follows: Reδ ¼

ρl uδ m_ ¼ Wμl μl

ð7:34Þ

and by combining this equation with Eq. (7.18), it is possible to estimate the Reynolds number with more readily known parameters, as follows: Reδ ¼

ρl gz ð ρl  ρv Þ δ 3 3 μ2l

ð7:35Þ

where the liquid film thickness is given by Eqs. (7.28) and (7.32). According to Gregorig et al. (1974), the modelled approach presents negligible deviation from experimental results for Reynolds numbers up to approximately 7, and acceptable

7.1 Film Condensation on an Isothermal Surface

249

predictions for Reynolds numbers up to 400. According to these authors, for Reδ of approximately 8, ripples are noted along the liquid film, indicating the incipience of instabilities, and for Reδ higher than approximately 400, the flow is completely turbulent. For design purposes, or for simplified analysis, it is interesting to evaluate the mean heat transfer coefficient, which can be determined by its definition as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 0 1 L kl 1 4 ρl gz ðρl  ρv Þ^ilv z3 ffiffi ffi p hdz ¼ dz L 0z 44 μl kl ðT sat  T s Þ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 4 kl 3 4 ρl gz ðρl  ρv Þ^ilv L3 ffiffi ffi ¼ p L 44 μl kl ðT sat  T s Þ

h ¼ 1 L

Z

L

ð7:36Þ

and an equivalent Nusselt number is given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 3 4 ρ g ðρ  ρ Þb hL l z l v ilv L NuL ¼ ¼ 0:943 kl μl kl ðT sat  T s Þ

ð7:37Þ

The average heat transfer coefficient for condensation on a horizontal tube with uniform wall temperature is obtained following an almost similar procedure, except by the fact that the angle θ varies from 0  at the upper region of the tube down to 180  at its bottom. Therefore, the perimeter-averaged heat transfer coefficient on a horizontal tube is given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 3 4 ρ gðρ  ρ Þb hD l l v ilv D Nu ¼ ¼ 0:728 kl μl kl ðT sat  T s Þ

ð7:38Þ

In the above equation, as well as in Eq. (7.37), the fluid properties are evaluated based on the film temperature that is the arithmetic average between the wall and saturation temperature. In this context, it is interesting to mention that drop-wise condensation might also occur in conditions of stagnant vapor. This process corresponds to the nucleation of liquid droplet in a cooled surface exposed to vapor, which grows statically due to successive condensation or coalescence with neighboring droplets, and then falls on the surface due to gravity or drag forces when it reaches a critical size without the formation of laminar film flow. This process is more likely to occur in systems where the contact angle is high (hydrophobic surfaces), and the adhesion forces of the liquid to the solid surface is not high enough to propitiate a liquid formation, and it is of difficult modelling and experimental investigation because the droplet nucleation site is usually unknown. Jakob (1936) proposed a modelling approach for the dropwise condensation, which considers that the condensation actually starts as a liquid film, and due to surface tension forces the film contracts forming a droplet.

250

7 Condensation

Subsequently, several authors have reinforced this model by adding details or by comparing with experimental results. However, even today this mechanism is not completely understood, and will not be detailed in this book.

7.2

Predictive Methods for In-Tube Convective Condensation

This section describes predictive methods for heat transfer coefficient during convective condensation inside channels of conventional and reduced sizes. As a general comment, we might expect that the heat transfer coefficient during condensation increases with mass velocity and vapor quality and reduces with increasing channel diameter and temperature difference between the surface and fluid. The last aspect is more pronounced under conditions dominated by gravitational effects, as can be inferred from Eq. (7.36), and is related to the increase of the thermal resistance due to liquid film thickening. Additionally, during condensation the vapor quality reduces along the flow path, therefore the starting point of condensation presents the highest heat transfer coefficient with successive reduction until liquid single-phase flow. In general, the methods for prediction of the heat transfer coefficient during condensation can be classified as follows: (i) purely empirical; (ii) based on the Colburn analogy; and (iii) flow-pattern based. The first group is based on the curve fitting of an empirical correlation to a database. The purely empirical predictive methods are based on curve fitting of a correlation to a database, and most of the proposals are focused on conditions of low mass velocities, when gravitational effects are dominant. In these cases, Eq. (7.38), derived based on the Nusselt model for film condensation on a round tube, is modified to account for the region of the tube internal wall along which the liquid film flows due to gravity. In this group, the methods developed for conditions of low mass velocities when gravitational effects are dominant. Equation () based on the model for film condensation on a round tube is modified in order to take into account the stratified perimeter, corresponding to the internal region of the tube along which the liquid film flows due to gravity. Among the purely empirical methods the ones proposed by Chato (1962) and Jaster and Kosky (1976) are the most commonly cited in the literature. In these methods, the heat transfer is neglected along the region over which the liquid flows according to the main flow direction, corresponding to the flooded part of the tube. Chato (1962) based on his experimental results proposed that the heat transfer coefficient is given simply through the product between the heat transfer coefficient for condensation on horizontal tubes, given by Eq. (7.38) and 0.76, where this multiplier is associated to the parcel of the tube perimeter corresponding to the mechanism of film-wise condensation. A different approach is adopted when inertial effects are dominant corresponding to conditions of high two-phase flow velocities. In this case, a combination of a

7.2 Predictive Methods for In-Tube Convective Condensation

251

single-phase correlation for heat transfer coefficient for in-tube turbulent flow and either a two-phase multiplier as proposed by Shah (1979) or an equivalent Reynolds Number is adopted (Akers et al. 1959; Cavallini and Zechin 1974). In the methods classified within group ii, generally developed for annular flows, as the model proposed by Moser et al. (1998), the momentum-heat analogy is considered and the Nusselt number is equationed based on the velocity and temperature profiles along the liquid film thickness. Analogous to flow-pattern-based methods for the heat transfer coefficient during flow boiling, as the one proposed by Wojtan et al. (2005a, b), the methods for in-tube condensation of group iii are also developed based on flow pattern predictive methods, and then the heat transfer process is modelled according to the main heat transfer mechanism corresponding to each flow pattern. Among the prediction methods belonging to this group, Dobson and Chato (1998) were the pioneers to propose a flow-pattern-based method for the heat transfer coefficient during in-tube condensation. A similar approach was employed by Thome et al. (2003) and Cavallini et al. (2006). Mostly recently, artificial intelligence tools have been successfully employed as methods for prediction of the heat transfer coefficient for in-tube condensation. Furlan and Ribatski (2020), based on a broad database from literature and using dimensionless numbers as input parameters, found that machine learning methods (multi-layer perceptron with backpropagation and gradient boosted decision tree) provided satisfactory predictions of independent databases and they were also able to accurately predict the effects on the heat transfer coefficient of the vapor quality, mass velocity, saturation temperature, channel diameter, and flow pattern when compared to the experimental results, overperforming the most accurate end up-todate correlations and models from literature. Moreover, the machine learning methods presented 90% lower processing time than the model of Cavallini et al. (2006) when applied to a simulation of a tube-in-tube condenser.

7.2.1

Dobson and Chato (1998)

Dobson and Chato (1998) proposed their method segregating the flow as gravitydominated and shear-dominated corresponding approximately to stratifiedwavy and annular flow patterns, respectively. In their method, the flow pattern transitions are defined as the criteria proposed by Soliman (1982) modified by them according to their database that comprises heat transfer and flow pattern results for horizontal flows in smooth tubes with diameters ranging from 3.14 mm and 7.04 mm for the refrigerants R12, R22, R134a, and near-azeotropic blends of R32/R125 in 50 percent/50 percent and 60 percent/40 percent. According to their method, annular flow is observed for either G  500 kg/m2s or FrSo  20 while stratified flow occurs for G < 500 kg/m2s and FrSo < 20. The Froud number based on the definitions of Soliman (1982) is given as follows:

252

7 Condensation

For Rel  1250: 0:025 Re 1:59 l

b tt 1 þ 1:09X b X tt

1:26 Re 1:04 l

b tt 1 þ 1:09X b tt X

0:039

Fr So ¼

!1:5 1 Ga0:5

ð7:39Þ

1 Ga0:5

ð7:40Þ

For Rel > 1250: 0:039

Fr So ¼

!1:5

where Rel is the Reynolds number considering only the flow of the liquid parcel and Ga is the Galileo number relating buoyance and viscous effects given as follows: Ga ¼

ρl ðρl  ρv Þd 3 μ2l

ð7:41Þ

The heat transfer for the annular flow pattern was modelled based on the twophase multiplier approach, with the liquid Nusselt number given as follows: " # hd 2:22 0:4 ¼ 0:023 Re 0:8 1 þ 0:89 Nul ¼ l Pr l kl b X

ð7:42Þ

tt

In contrast to the hypothesis adopted by Chato (1962), in this study the authors observed that as the vapor velocity is increased, the heat transfer contribution at the bottom of the tube becomes relevant, and assuming only film-wise condensation along a parcel of the tube perimeter, neglecting forced convection in the bottom region, implies on underestimating the perimeter-averaged heat transfer coefficient. Moreover, as the vapor velocity increases, interfacial shear effects on the liquid film are also increased implying on an axial film velocity component. As the relative importance of gravitational force decreases with increasing vapor velocity, the film-wise solution becomes inappropriate and at a certain limit only the annular flow solution becomes satisfactory. Therefore, the method proposed by Dobson and Chato (1998) considers the addition of the contributions of the forced convection heat transfer in the lower part of the horizontal tube and the film-wise condensation in its upper part weighted according to the stratified angle γ, defined as illustrate in Fig. 3.4. They suggested the approximation proposed by Jaster and Kosky (1976) to estimate the stratification angle as a function of the superficial void fraction, given as follows: γ ffi1

cos 1 ð2α  1Þ π

ð7:43Þ

with the void fraction, α, given according to the Zivi (1964) correlation, Eq. (2.52).

7.2 Predictive Methods for In-Tube Convective Condensation

253

The final correlation proposed by Dobson and Chato (1998) is given as follows:

0:25 0:23 Re 0:12 hd GaPr l v0 Nul ¼ ¼ þ ð1  γ=π ÞNuFC kl Jal b 0:58 1 þ 1:11X tt

ð7:44Þ

where 0:4 NuFC ¼ 0:0195 Re 0:8 l Pr l

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 þ c12 b X tt

ð7:45Þ

and the constants c1 and c2 for 0  Frl0  0.7 are given as follows: c1 ¼4:172 þ 5:48Frl0  1:564Fr2l0 c2 ¼1:773  0:169Frl0 and for Frl0 > 0.7: c1 ¼7:242 c2 ¼1:655 Frl0 is the Froude number for the two-phase mixture flowing as liquid, Eq. (3.127), Rev0 is the Reynolds number for the two-phase mixture flowing as vapor (Rev0 ¼ Gd/μv), Ga is the Galileo number, Eq. (7.41), Jal is the liquid Jakob number, Eq. (7.6), and Rel is the Reynolds number considering only the liquid parcel (Rel ¼ G(1-x)d/μl). Dobson and Chato (1998) were pioneers in proposing a flow-pattern-based method for the prediction of the heat transfer coefficient for in-tube horizontal condensation. However, some aspects were not properly addressed in their method as follows: (i) the correlation proposed by Zivi (1964) adopted by Dobson and Chato (1998) to estimate the void fraction neglects the effect of the mass velocity on the void fraction, which can be relevant in certain conditions; (ii) the heat transfer coefficient variation is not continuous as the flow pattern changes from stratified to annular. Such a behavior is not realistic according to the experimental studies and may result in convergence problems when this method is incorporated in a software for heat exchanger simulation and optimization; (iii) shear effects and secondary flows close to the tube wall promote an increase in the liquid height near the wall for stratified flows, reducing the effective stratified angle, γ. Later, these aspects were properly addressed by Thome and coworkers (El Hajal et al. (2003) and Thome et al. (2003)) in their flow-pattern-based method for condensation inside horizontal tubes.

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7 Condensation

7.2.2

Cavallini et al. (2006)

As mentioned before, the heat transfer during condensation in conventional channels is governed mainly by gravitational and convective effects, and predictive methods were published based on the combination of these effects. Therefore, in order to simplify the approach, Cavallini et al. (2006) proposed a predictive method for the heat transfer coefficient that identifies conditions dependent and independent of ΔT, with specific corresponding governing mechanisms. Based on this characterization, it is possible to infer the local flow patterns that can be annular, wavy stratified, or intermittent flow. The annular flow pattern is characterized by conditions of high flow velocities, hence, the heat transfer process is dominated by convective effects, and presents small contribution of gravitational effects. Conversely, the heat transfer during stratified wavy flow is dominated by gravitational effects, and during intermittent flows by the combination of gravitational and convective effects, with dominance of convective effects. The conditions dominated by either mechanism is determined based on inertial effects of vapor phase, represented by the non-dimensional gas velocity, described by Wallis (1969), as follows: Gx jv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g d ρv ð ρl  ρv Þ

ð7:46Þ

The transition value of the non-dimensional gas velocity was proposed by Cavallini et al. (2006) based on their experimental results as follows:

jv,trans ¼

8"
jv,trans ): " hann ¼ hl0 1 þ 1:128  x

0:8170

#  0:3685  0:2363  2:144 ρl μl μv 0:100 1 Prl ρv μv μl ð7:48Þ

7.2 Predictive Methods for In-Tube Convective Condensation

255

where hl0 is estimated according to Dittus and Boelter (1985) for the mixture flowing as liquid as follows:  0:8 Gd kl hl0 ¼ 0:023 Pr 0:4 l d μl

ð7:49Þ

For conditions that gravitational effects contribute significantly to the heat transfer process ( jv  jv,trans), the flow might correspond to stratified wavy or intermittent flows. Therefore, to account for these conditions with no discontinuity of the heat transfer coefficient estimative due to flow pattern transition, Cavallini et al. (2006) proposed the following relationship to predict the heat transfer coefficient: "



hsw,slug ¼ hann

jv,trans jv

#

0:8  hsw

jv



jv,trans

ð7:50Þ

þ hsw

where hsw corresponds to the heat transfer coefficient estimated for stratified wavy flow, adjusted by the researchers accordding to their experimental results, and given as follows:

hsw

0:725 ¼  0:3321 1 þ 0:741 1x x

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3   4 k ρ ðρ  ρ Þgb v ilv l l l þ 1  x0:087 hl0 μl d ðT sat  T s Þ

ð7:51Þ

where hl0 is given by Eq. (7.49). Notice that the first term of the RHS is similar to the derived relationship for the heat transfer coefficient during film condensation, and given by Eq. (7.36), emphasizing that this term accounts for gravitational parcel and is dependent on the temperature difference. Figure 7.2 depicts the variation of heat transfer coefficient with vapor quality predicted according to the Cavallini et al. (2006) method for R600a and R1234ze (Z) in 12.7 mm ID tube, showing also the transition between conditions dominated or not by gravitational effects. It can be observed according to this figure that the heat

20000

b) 14000 G = 50 kg/m²s G = 100 kg/m²s G = 200 kg/m²s G = 400 kg/m²s G = 800 kg/m²s G = 1600 kg/m²s

R600a, d = 12.7 mm, Tsat = 40 °C, Ts = 36 °C

12000 10000

h [W/m²K]

h [W/m²K]

a) 30000

10000

8000

G = 50 kg/m²s G = 100 kg/m²s G = 200 kg/m²s G = 400 kg/m²s G = 800 kg/m²s G = 1600 kg/m²s

R1234ze(Z), d = 12.7 mm, Tsat = 40 °C, Ts = 36 °C

6000 4000

j*v,trans

j*v,trans

2000 0 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

0 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 7.2 Variation of the heat transfer coefficient with vapor quality according to Cavallini et al.’s (2006) predictive method for (a) R600a and (b) R1234ze(Z) in a 12.7 mm ID tube

256

7 Condensation

transfer coefficient is significantly impacted by the mass velocity. Moreover, it is also shown in this figure that R600a presents considerably higher heat transfer coefficient than R1234ze(Z). This behavior is associated to the higher vapor-specific volume of R245fa compared to R134a. Furlan (2020) suggests the method of Cavallini et al. (2006) for the prediction of the heat transfer coefficient during condensation in small diameter channels of hydrocarbons based on a comparison of 14 prediction methods and a broad database from the literature.

7.2.3

Shah (2016)

Shah (2016) proposed a new method for prediction of the heat transfer coefficient during condensation in horizontal small diameter tubes (deq  3 mm) using a broad database gathered from literature including experimental results for 13 fluids (water, CO2, halocarbon refrigerants, and hydrocarbons), hydraulic diameters from 0.10 to 2.8 mm; reduced pressures ranging from 0.0055 to 0.94; mass velocities from 20 to 1400 kg/m2s; and channels of rectangular, circular, semi-circular, triangular, and square cross-sectional shapes. The original Shah correlation (Shah 1979) for conventional channels and its updated versions (Shah 2009 and Shah 2013) were considered as the starting point for this new method. The ease of its implementation and the reasonable predictions when compared to independent databases are the main features of the method of Shah (2016). In this method, the heat transfer coefficient is estimated according to heat transfer regimes defined by the author based on heat transfer behaviors instead of different two-phase flow topologies. The regimes are given as follows: Regime I: When Wev0  100 and the dimensionless vapor velocity defined in Eq. (7.46) is: jv  0:98ðZ þ 0:63Þ0,62

ð7:52Þ

where Z was defined by Shah (1979) as follows: Z ¼ ð1=x  1Þ0:8 p0:4 r

ð7:53Þ

the condensation heat transfer coefficient is given by Eq. (7.48) with hl0 estimated through Eq. (7.49). For non-circular channels, the all-vapor Weber number is calculated using the hydraulic diameter. In the method of Shah (2016), all the other equations are based on an equivalent diameter, defined by him as the ratio of four times the cross-sectional flow area and the heat transfer perimeter.

7.2 Predictive Methods for In-Tube Convective Condensation

257

Regime II: If the regime is neither I nor III according to the respective criteria, the heat transfer process corresponds to regime II and the condensation heat transfer coefficient is given as follows: h ¼ hann þ hNu

ð7:54Þ

where hann is given by Eq. (7.48) and hNu is given by the following equation: 1=3

hNu ¼ 1:32 Re l



ρl ðρl  ρv Þgk 3l μ2l

1=3 ð7:55Þ

Regime III: If  1 jv  0:95 1:254 þ 2:27Z 1:249

ð7:56Þ

the heat transfer coefficient is equal to hNu given by Eq. (7.38).

7.2.4

Jige, Inoue, and Koyama (2016)

Jige, Inoue, and Koyama (2016) performed a series of experiments for determination of pressure drop and heat transfer coefficient during condensation inside multiport microchannel heat sink, operating with synthetic refrigerant fluids. During convective condensation inside small-scale channels, the surface tension forces also have significant contribution on flow development and heat transfer process, and this contribution increases as the channel diameter decreases. Conversely, the gravitational effects are suppressed due to reduction of channel size, as discussed in Chap. 3. Based on these aspects, Jige and coworkers proposed a heat transfer coefficient predictive method that accounts for these mechanisms and basically comprises intermittent and annular flow patterns. The heat transfer for annular flow pattern was modelled by the authors by solving a simplified model of the liquid film flow in a rectangular channel accounting for the shear stress and surface tension effects, obtaining different correlations according to the dominant mechanism. For conditions dominated by friction force, which is related mainly to convective effects, the heat transfer coefficient is given as follows: hf ¼

kl Φv0 dh 1  x

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ρ f v0 l Rel 0:6 þ 0:06Re0:4 Pr 0:3 l l ρv

ð7:57Þ

258

7 Condensation

where the term inside the brackets can be considered as intensification factors of convective effects, in a similar way to the proposition of Chen (1966) for convective boiling and discussed in Chap. 5. The term dh corresponds to the hydraulic diameter, and Rel corresponds to the Reynolds number only for the liquid phase based on Þdh hydraulic diameter (Rel ¼ Gð1x ). The two-phase multiplier of the vapor-phase μl Φv0 was proposed by Jige et al. (2016) based on their experimental results and is given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1:25  0:75ffi ρ f μl ρv 1:8 1:43 v l0 Φv0 ¼ x1:8 þ ð1  xÞ þ 0:65x0:68 ð1  xÞ ρl f v0 μv ρl

ð7:58Þ

The friction factor for the mixture flowing as each phase for circular channel are given as follows:

f v0

f l0

8 16 > > < Re for v0 ¼ 0:046 > > for : Re 0:2 v0 8 16 > > < Re for l0 ¼ 0:046 > > for : Re 0:2 l0

Re v0  1500 Re v0 > 1500

ð7:59Þ

Re l0  1500 Re l0 > 1500

ð7:60Þ

with the Reynolds numbers for the mixture flowing as each phase is also evaluated Gd h h based on hydraulic diameter ( Re l0 ¼ Gd μl and Re v0 ¼ μv ). For non-circular channels, it is recommended to check the original publication, or Kays and London (1998), for the corresponding friction factor or Nusselt number relationship. In the same token, the heat transfer coefficient for conditions dominated by surface tension effects is given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ρlbilv σdh kl hs ¼ 0:51 μl kl ðT sat  T s Þ dh

ð7:61Þ

where no suppression factor for the surface tension effects was indicated by the original authors. Based on these aspects, Jige et al. (2016) combined both parcels for estimative of the heat transfer coefficient during annular flow in a canonical way as proposed by Churchill (2000), given as follows: hann ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 h3f þ h3s

ð7:62Þ

7.2 Predictive Methods for In-Tube Convective Condensation

259

Flow direction

Liquid slug

Vapor bubble

Liquid single-phase flow

Annular flow

Liquid single-phase flow

Annular flow

Fig. 7.3 Schematics of intermittent flow, adapted from Jige et al. (2016)

The authors modelled the intermittent flow as a successive passage of liquid slugs and vapor bubble, and the heat transfer coefficient is evaluated assuming liquid single-phase flow, and annular flow, respectively. This approach is somewhat similarly to the three-zone model proposed by Thome et al. (2004) for convective boiling inside microchannels. The model proposed by Jige et al. (2016) is schematically depicted in Fig. 7.3, and the resulting heat transfer coefficient is given as follows: h ¼ ð1  αÞhl þ αhann

ð7:63Þ

where the weighting factor α is the void fraction evaluated according to a homogeh i1 Þ neous model (α ¼ 1 þ ρρv ð1x ). The heat transfer coefficient for liquid singlex l

phase flow hl is evaluated according to the Gnielinski (1976) correlation for turbulent flow, and the equivalent for laminar flow as follows:

hl ¼

8 > > > > > >
rffiffiffiffiffi  > >  d h , Rel > 2000 > 2 > f l > : 1 þ 12:7 Pr 3l  1 2

ð7:64Þ

with the liquid Reynolds evaluated assuming hydraulic diameter and only the Þdh liquid parcel of the two-phase flow (Rel ¼ Gð1x ), and the liquid friction factor μl evaluated according to Eq. (7.60) however with Rel instead of Rel0. In summary, the heat transfer coefficient is estimated according to Eq. (7.63), and the annular flow condition consists of long vapor bubbles, without liquid slugs. Figure 7.4 depicts the variation of the heat transfer coefficient with vapor quality for

260

7 Condensation

14000

20000 G = 50 kg/m²s

h [W/m²K]

10000 8000

G = 100 kg/m²s G = 200 kg/m²s

16000

G = 400 kg/m²s G = 800 kg/m²s

h [W/m²K]

12000

12000

G = 1600 kg/m²s

6000

G = 50 kg/m²s G = 100 kg/m²s G = 200 kg/m²s G = 400 kg/m²s G = 800 kg/m²s G = 1600 kg/m²s

8000

4000 4000 2000 R134a, d = 1 mm, Tsat = 40 °C, Ts = 36 °C

0 0,0

0,2

0,4

0,6

x [-]

0,8

1,0

0 0,0

R245fa, d = 1 mm, Tsat = 40 °C, Ts = 36 °C

0,2

0,4

0,6

0,8

1,0

x [-]

Fig. 7.4 Heat transfer coefficient variation with vapor quality during convective condensation inside microchannels, according to the method of Jige et al. (2016)

R134a and R245fa inside 1 mm ID microchannel. In a similar way to the predictions for macrochannels, the heat transfer coefficient increases with vapor quality and mass velocity, which is related to intensification of convective parcel. Rossato et al. (2017) pointed out that the predictive method proposed by Jige et al. (2016) provided the best agreement of their experimental results for heat transfer coefficient during convective condensation inside multichannel heat exchanger.

7.3

Solved Examples

Examples comprising concepts discussed in this chapter are presented and discussed. Considering R134a at a temperature of 40  C flowing at 250 kg/m2s in horizontal channel, which has uniform wall temperature of 35  C, estimate the heat transfer coefficient for the following: (a) For internal diameter of 10 mm, evaluate the heat transfer coefficient of saturated liquid and superheated vapor, with superheating degree of 5  C. (b) Similar to the previous exercise, however for channel with internal diameter of 1 mm. (c) Now, consider two-phase flow at Tsat ¼ 40  C with vapor quality of 90% in a 10 mm ID channel. (d) Reevaluate for convective condensation with vapor quality of 90% in a 1 mm ID channel. Solution: Based on the saturation temperature, it is possible to obtain the thermophysical properties of the fluid for saturation conditions, as follows: ρl ¼ 1147 kg/m3 ρv ¼ 50.12 kg/m3 μl ¼ 1.612104 kg/ms μv ¼ 1.268105 kg/ms

7.3 Solved Examples

261

Prl ¼ 3.19 Prv ¼ 0.9017 îlv ¼ 163,003 J/kg cpl ¼ 1498 J/kgK cpv ¼ 1145 J/kgK kl ¼ 7.572102 W/mK kv ¼ 1.61102 W/mK In the case of liquid flow, the properties are evaluated for saturation conditions. In the case of vapor, for superheating of 5  C at T ¼ 40  C, the corresponding pressure is 887.5 kPa, with resulting relevant properties: ρv ¼ 42.03 kg/m3 μv ¼ 1.261105 kg/ms kv ¼ 1.592102 W/mK Prv ¼ 0.8506 Then, it is possible to solve the example items. (a) Single-phase flow inside channel with 10 mm of internal diameter, and wall temperature of 35  C. Based on the input data and thermophysical properties, it is possible to estimate the Reynolds number for both phases (Re ¼ Gd/μ), resulting in 15,511 and 198,187 for saturated liquid and superheated vapor, respectively, hence, turbulent regime for both phases. Then, by selecting the predictive method proposed by Gnielinski (1976), which is valid for Reynolds number higher than 2500, it obtained heat transfer coefficient values of 651.8 and 553.2 W/m2K for liquid and vapor, respectively. (b) Now, repeating the previous item, the Reynolds number for liquid and vapor phases are, respectively, 1551 and 19,819. Hence, it corresponds to laminar regime for saturated liquid flow, and turbulent for vapor flow. Then, adopting the Gnielinski (1976) predictive method for superheated vapor for heat transfer coefficient, the obtained value is 900.8 W/m2K, which is 63% higher than the case of flow inside a larger channel despite the Reynolds number reduction, and can be attributed only to channel dimension reduction. In the case of saturated liquid flow, considering that the boundary condition corresponds to uniform wall temperature, the Nusselt number is constant and equal to 3.657, resulting in heat transfer coefficient of 276.9 W/m2K, which is 58% lower than the case of flow inside a larger channel due to flow regime difference. (c) In the case of two-phase flow, since the surface temperature is smaller than the fluid temperature (Tw < Tsat), the heat transfer process occurs by condensation. Hence, a proper predictive method must be selected, and considering the channel size, which is 10 mm, it is possible to infer whether it corresponds to micro or macrochannel. Based on Kew and Cornwell (1997), given by Eq. (2.69), the

262

7 Condensation

transitional channel diameter is 1.508 mm, hence for d ¼ 10 mm the channel can be considered a macroscale case. Therefore, let us consider the predictive method proposed by Cavallini et al. (2006) for heat transfer coefficient during condensation inside macroscale channels. By solving Eqs. (7.46), (7.47), (7.48), (7.49), (7.50), and (7.51), the following main parameters are found: jv* ¼ 3.065 jv,trans* ¼ 2.552 (