Multiphase Flow Research [1 ed.] 9781608764792, 9781606924488

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Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

MULTIPHASE FLOW RESEARCH

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Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved. Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

MULTIPHASE FLOW RESEARCH

S. MARTIN AND

J.R. WILLIAMS

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

EDITORS

Nova Science Publishers, Inc. New York

Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Copyright © 2009 by Nova Science Publishers, Inc.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Available upon request.

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CONTENTS Preface

vii

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Research and Review Studies

1

Chapter 1

Numerical Simulation of Multiphase Flow Systems Steffen Schütz, Kathrin Kissling, Martin Schilling and Cornelia Seyfert

Chapter 2

Modeling Multiphase Flow with Phase Change and Heat Transfer: Status Review, Challenges and Future Research Direction R. Panneer Selvam, Suranjan Sarkar, Mita Sarkar and Joseph Johnston

147

Chapter 3

Incorporation of In-Situ Flow Parameters and Flow Pattern Phenomena into a Mathematical Model of an Air-Water Mixture Using Concomitancy Criteria Based on Experimental Studies of Advanced Micro Cooling Modules with Phase Transition Jerry K. Keska and William E. Simon

209

Chapter 4

Influence of the Interfacial Drag on Pressure Loss for Two Phase Flow and Coolability in Porous Media Werner Schmidt

247

Chapter 5

Hydrodynamic Studies on Two-Phase Gas-Liquid Flow in an Ejector Induced Downflow Bubble Column Ajay Mandal

293

Chapter 6

Modeling Explicitly the Flow of Three Dimensional Deformable Particles and Droplets, at High Concentration Michael M. Dupin

359

Chapter 7

A WIMF Scheme for the Drift-Flux Two-Phase Flow Model Steinar Evje, Tore Flåtten and Svend Tollak Munkejord

397

Chapter 8

Modelling of the Interaction of the Dispersed and the Continuous Phases in a Liquid-Liquid Extraction Column A.-H. Meniai, A. Hasseine, O. Saouli and A. Kabouche

437

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vi

Contents

Chapter 9

Two-Phase Gas-Liquid Flow Properties in the Hydraulic Jump: Review and Perspectives Frédéric Murzyn and Hubert Chanson

497

Chapter 10

Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2 in Macro- and Micro-channels Lixin Cheng and John R. Thome

543

Chapter 11

Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design of Bipolar Plates Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

621

Chapter 12

Two-Phase Flow and Transport in Porous Media Including FluidFluid Interfacial Area Jennifer Niessner and S. Majid Hassanizadeh

709

Chapter 13

Pneumatic Conveying of Coarse Particles in Vertical Pipes M.C. Ferreira and J.T. Freire

731

Chapter 14

CFD Modelling of Multiphase Flow Application to Air-Breathing PEM Fuel Cells Maher A.R. Sadiq Al-Baghdadi

769

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Short Communications

813

An Overview of Experimental Observations of Turbulence Modulation in Dilute Multiphase Flows R.N. Parthasarathy

815

Turbulent Viscosity Controlling Time-averaged Velocity Field in Bubble Columns Korekazu Ueyama

827

Index

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PREFACE In fluid mechanics, multiphase flow is a generalisation of the modelling used in twophase flow to cases where the two phases are not chemically related or where more than two phases are present. Each of the phases is considered to have a separately defined volume fraction , and velocity field. Conservation equations for the flow of each species, can then be written down straightforwardly.The momentum equation for each phase is less straightforward. It can be shown that a common pressure field can be defined, and that each phase is subject to the gradient of this field, weighted by its volume fraction. Transfer of momentum between the phases is sometimes less straightforward to determine, and in addition, a very light phase in bubble form has a virtual mass associated with its acceleration. This new book presents the latest research in the field. Within Chapter 1 simulation models to describe multiphase flow systems with are presented and discussed with respect to their physical background, their application range and their numerical implementation. The models comprise multiphase flow systems with different phase combinations and for different phase concentrations. Depending on the models multiphase flow systems can be described in an integral character neglecting details in the transport of single particles but also in a very detailed, microscopic manner with a close look on the single particle behavior. Besides the discussion of well-established models (EulerLagrange and Euler-Euler models), new developments are described which allow the simulation of volumetrically resolved particles, the calculation of the dynamic of phase interfaces and the inclusion of particle-particle interaction mechanisms like agglomeration, coalescence or particle breakup. Multiphase flow is used to refer any fluid flow consisting of more than one phase. The constituent fluids usually have different physical properties separated by an interface that evolves with the flow. Application of multiphase flows in different industries and research fields are enormous. A good understanding of underlying physics of multiphase flows is the need to model and predict the detailed behavior of those flows. Although efforts to compute the motion of multiphase flows are as old as computational fluid dynamics, the difficulty in solving the full Navier–Stokes equations in the presence of a deforming interface and recognition of the interface at every instant of time is a real challenge. The phase change heat transfer associated with multiphase flow is very common in several engineering applications and adds more complexity in the modeling of multiphase flow. In Chapter 2 an overview of multiphase flow modeling techniques for different applications are reviewed. Macroscopic, mesoscopic and microscopic modeling techniques for multiphase flow are discussed with

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respect to grid based and particle based methods. A special attention has been given to the grid based interface capturing methods which are best suited for modeling multiphase flow with phase change heat transfer. The computational modeling efforts so far in phase change heat transfer applications are reviewed in this article in great details. Some of them include bubble dynamics in nucleate boiling, flow boiling and film boiling; droplet and surface interaction and droplet and bubble dynamics in thin liquid film model for spray cooling applications. The numerical and computational issues and challenges involved in modeling these phenomena are identified. Implementation of multiphase flow modeling in parallel computing environment is also discussed. The current understanding and lack of understanding of underlying physics of bubble nucleation and phase change heat transfer from multiphase flow modeling are summarized and future directions of research areas are identified. The need for high-performance thermal protection and fluid management techniques for systems ranging from cryogenic reactant storage devices to primary structures and propulsion systems exposed to extreme high temperatures, and other space systems such as cooling or environmental control for advanced space suits and integrated electronic circuits, requires an effective cooling system to accommodate the compact nature and high heat fluxes associated with these applications. A two-phase forced-convection, phase-transition system can accommodate such requirements through the use of the concept of Advanced Micro Cooling Modules (AMCMs), which are essentially compact two-phase heat exchangers constructed of microchannels and designed to remove large amounts of heat rapidly from critical systems by incorporating phase transition. It is still recognized today that two-phase flow is scientifically one of most challenging fluid dynamic problems to be explored since the 1940s, and this is further complicated when the fluid flows through channels of sub millimeter dimension. Consequently, to successfully design, analyze, and control such systems, it is necessary to first obtain a fundamental understanding of the two-phase flow and heat transfer in the microchannels. This is accomplished through the development of a quantitative model incorporating in situ concentration/void fraction, velocities, and flow patterns. Realizing the significance of research in Chapter 3, this paper presents the results of experimental research on two-phase flow in microchannels with verification and identification of data using concomitant measurement systems. Based on the results of the experimental research conducted on air-water mixture flows in the entire range of concentration and flow patterns in a horizontal square microchannel, a mathematical model based on in situ parameters is developed and presented, which describes pressure losses in two-phase flow incorporating flow pattern phenomena. Validation of the model is accomplished using five concomitant measurement methods (capacitive and conductive concentration measurements, differential and static pressure, and optical signals). A hypothetical model for the two-phase heat transfer coefficient is also presented, which incorporates the flow patterns through the use of a flow pattern coefficient. Chapter 4 investigates the influence of the interfacial drag on the pressure loss of combined liquid/vapour flow through porous media. This is motivated by the coolability of fragmented corium with internal heat sources, which are expected during a severe accident in a nuclear power plant. Due to the decay heat in the particles cooling water is evaporated. To reach steady states the out-flowing steam must be replaced by in-flowing water. The pressure

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field inside the porous structure determines the water ingression, and in effect the overall coolability. Typically, correlations for the dryout heat flux of porous media are adjusted to measurements where water ingression is from a pool positioned above. In these models the nature of the two-phase flow is included in corrections of the permeability and passability, achieved by simple functions of the void fraction. However, already configurations with possible water ingression from below demonstrate that this treatment is insufficient. The drag between the liquid and the vapour phase supports the co-current water inflow from below, and hinders the counter-current flow from above. Thus, the interfacial drag must be explicitly included in the modeling. This necessity is already seen in the measured pressure loss of simple isothermal air/water flow in porous media as well as for boiling particle beds with water ingression from below. Based on such experiments, two models with explicit consideration of the interfacial drag from the literature are discussed. Different flow patterns for the drag coefficients are included with the most advanced of these models. This article proposes some modifications on this model with respect to the small particle sizes that are expected in the reactor application. Furthermore, modifications to the formulation in the annular flow regime are necessary, as here the original model yields unreliable results. Based on the friction laws referred, the models are applied to typical reactor invessel and ex-vessel configurations in two-dimensional geometry. An enhanced overall coolability of the particle bed is already reached by the 2-D nature of the arrangement. Furthermore, the supporting influence of the realistic modeling by explicit consideration of the interfacial drag in the enhanced models is shown. Fine dispersion of gas into liquid is one of the most important criteria for momentum, mass and energy transfer between the phases. It not only provides an intense mixing but also creates increased interfacial area and high mass transfer coefficient. Various types of contacting devices have been developed for achieving effective gas-liquid mixing. These may be broadly classified as co-current, counter-current and cross current systems. Among these, increased interests on co-current contacting systems have been shown because of their ability to handle high fluid flowrate without flooding, low pressure drop, higher interfacial area and mass transfer coefficients. A review of literature on design and development of fluid-fluid cocurrent contacting equipment shows that use of liquid jet ejectors as a gas-liquid distributor in downflow bubble column is gaining in importance as it functions both as a sparger and gas entrainment device. Ejectors are devices that utilize the kinetic energy of a high velocity liquid jet in order to entrain and disperse the gas phase. The bubble column with ejector system is very simple in design and no extra energy is required for gas dispersion as the gas phase is sucked and dispersed by the high velocity liquid jet. Thus, cocurrent down flow bubble column with ejector type gas distributor possesses the following distinct advantages over more conventional devices, such as (i) lower power consumption (ii) almost complete gas utilization (iii) higher overall mass transfer coefficient and (iv) tolerance to particulates and therefore useful for chemical reaction with slurries. In Chapter 5 some important hydrodynamic characteristics on gas entrainment by a high velocity liquid jet; gas-holdup and bubble size distribution; two-phase frictional pressure drop; energy dissipation in ejector and contactor; interfacial area and mass transfer coefficient in an ejector induced downflow bubble column have been discussed. Further since the

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processing media for many processes such as sewage sludge, microbiological culture, polymer solutions etc. are non-Newtonian in nature; an attention has been paid to study with non-Newtonian fluid using Carboxy Methyl Cellulose (CMC) solution at different concentrations. The experimental set-up consists of a column, an ejector at the top of the column and a gas-liquid separator at the bottom along with other accessories like pumps, flow meters, valves and a number of manometers connected at different heights of the column. The column and ejector assembly were fitted with perfect alignment to obtain an axially symmetric jet. Experimentally it has been found that the rate of gas entrainment is strongly dependent on the motive fluid flowrate and the pressure in the separator. A precise idea of gas holdup and bubble size distribution was obtained by measuring the column pressure readings at different points of the column. The interfacial mass transfer area and liquid side volumetric mass transfer co-efficient were measured by both physical and chemical methods. Significantly higher values of interfacial area and volumetric mass transfer coefficient are obtained at low gas flowrate in the present system and the results are compared with the other reported works. Chapter 6 considers what happens when a multi-component system is subjected to a flow. More precisely, how to model it to better understand it. Two state of the art techniques for modeling multi-component flow are presented, based on the type of multi-component is considered. The first technique is to model a large amount of non-coalescing fluid droplets in flow, all of which have possibly different parameters (surface tensions, viscosities, radius, etc.). The second technique is to model a large amount of deformable cells or vesicles in flow, all of which have individual sets of parameters such as shape, rest curvature, elasticity, elasticity response, volume,etc. The underlying fluid dynamics of these two techniques is based on the lattice Boltzmann method, which was chosen for its efficiency and great versatility compared to other techniques. The chapter begins by deriving it, showing how it relates to more commonly known methods. The bulk of the chapter consists of proof of concepts and engineering applications for each technique, in order to demonstrate their efficiency, accuracy, versatility and potential. Two main approaches exist for numerical computation of multiphase flow models. The implicit methods are efficient, yet inaccurate. Better accuracy is achieved by the explicit methods, which on the other hand are time-consuming. In Chapter 7 the authors investigate generalizations of a class of hybrid explicit-implicit numerical schemes [SIAM J. Sci. Comput., 26 (2005), pp. 1449–1484], originally proposed for a two-fluid two-phase flow model. The authors here outline a framework for extending this class of schemes, denoted as WIMF (weakly implicit mixture flux), to other systems of conservation laws. They apply the strategy to a different two-phase flow model, the drift-flux model suitable for describing bubbly two-phase mixtures. Our analysis is based on a simplified formulation of the model, structurally similar to the Euler equations. The main underlying building block is a pressure-based implicit central scheme. Explicit upwind fluxes are incorporated, in a manner ensuring that upwind-type resolution is recovered for a simple contact discontinuity. The derived scheme is then applied to the general drift-flux model. Numerical simulations demonstrate accuracy, efficiency and a satisfactory level of robustness. Particularly, it is demonstrated that the scheme outperforms an explicit Roe scheme in terms of efficiency and accuracy on slow mass-transport dynamics.

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Liquid-liquid extraction columns are a quite good illustration of a two-phase flow system. They are used for contacting two immiscible liquid phases, flowing co or counter currently, and are simply vertical tubes having often, at their base, a distributor which can be a perforated plate or a series of nozzles, enabling the dispersion of one phase (the dispersed) into the other one (the continuous). The key parameter for this type of systems is the drop sizes distribution. This latter affects directly both the hydrodynamic and the mass transfer taking place, and depends greatly on various factors particularly the influence of the flow conditions of the continuous phase on the behaviour of the dispersed phase where the dynamic of the drops is often accompanied by the breakage and/or coalescence phenomena. In fact the study of the behaviour of the dispersed phase in a continuous flow field, generally turbulent is not easy. This is due to the nature of the macroscopic interactions generated by the breakage and coalescence phenomena resulting in a randomly distributed population of drops with respect to their sizes, concentrations and ages. Consequently, it is necessary to consider detailed mathematical models capable of describing the events generated by the interaction between the turbulent continuous phase and the dispersed one (drops), including both phenomena i.e. breakage and coalescence. Generally, the drop breakage term considers the interaction of a simple drop with the turbulent continuous phase continue turbulent, where it effectively undergoes a breakage if the transmitted turbulent kinetic energy exceeds its surface energy. Similarly, drop coalescence can take place due to the interaction between two colliding drops in the turbulent continuous phase, and it is effective if the interstitial liquid film disposes of sufficient time to drain out . As a first part, Chapter 8 considers the modelling of the continuous phase flow, under turbulent conditions, using the two equations model (k-ε) which has proven its ability for this type of problems. The influence of the continuous phase hydrodynamic on drop breakage and coalescence is examined, relying on the link between these phenomena and the dissipation rate of turbulent energy which can be determined from the model resolution and used for the calculation of important parameters like the maximal stable diameter, the breakage and coalescence efficiencies, etc. via computer experiments. In the second part, this two-phase flow system is solved by the drop population balance approach where the development of many experimental research programs has greatly contributed to the elaboration of realistic models which, globally, include the drop breakage and coalescence phenomena as well as their transport. However at this level, mathematical complexities are induced and analytical solutions are not easily obtained for the drop population balance equation. From this study one can see how complex is the modelling of the behaviour of a dispersed phase in a continuous one. The main difficulty is due to the interaction of the involved various factors as well as the nature of the problem which is rather stochastic. Research on multiphase flows has been strongly improved over the last decades. Because of their large fields of interests and applications for chemical, hydraulic, coastal and environmental engineers and researchers, these flows have been strongly investigated. Although they are some promising and powerful numerical models and new computing tools, computations can not always solve all actual practical problems (weather forecast, wave breaking on sandy beach…). The recent and significant developments of experimental techniques such as Particle Imagery Velocimetry (PIV) and conductivity or optical probes

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have particularly led scientists to physical modeling that provide series of data used to calibrate numerical models. Flows with time and length scales that were not achievable in the past are now studied leading to a better description of physical mechanisms involved in mixing, diffusion and turbulence. Nevertheless, turbulence is still not well understood, particularly in two-phase flows. In Chapter 9, the authors focus on a classical multiphase flow, the hydraulic jump. It occurs in bedrock rivers, downstream of spillways, weirs and dams, and in industrial plants. It characterizes the transition from a supercritical open-channel flow (low-depth and high velocity) to a subcritical motion (deep flow and low velocities). Experimentally, this twophase flow can be easily studied. Furthermore, it involves fundamental physical processes such as air/water mixing and the interaction between turbulence and free surface. This flow contributes to some dissipation of the flow kinetic energy downstream of the impingement point, in a relatively short distance making it useful to minimize flood damages. It is also associated with an increase of turbulence levels and the development of large eddies with implications in terms of scour, erosion and sediment transport. These are some of the reasons that make studies on this flow particularly relevant. Although numerical and analytical studies exist, experimental investigations are still considered as the best way to improve our knowledge. After a brief description of the hydraulic jumps, the first part of this chapter aims to review some historical developments with special regards to the experimental techniques and physical modeling (similitude). In the second part, the authors describe and discuss the basic properties of the two-phase flow including void fraction, bubble frequency, bubble velocity and bubble size. The free surface and turbulence properties are presented as well. In the last part, the authors develop some conclusions, perspectives and further measurements that should be undertaken in the future. Chapter 10 addresses various issues related to sub- and super-critical CO2 two-phase flow and heat transfer in macro- and micro-channels such as flow boiling, two-phase flow patterns and pressure drops without and with lubricating oil effect, supercritical gas-lubricating oil two-phase flow, two-phase flow and supercritical flow distribution in heat exchanger headers. Emphasis is given to our newly developed two-phase flow pattern map for CO2 evaporation, flow pattern based flow boiling heat transfer and pressure drop models. Furthermore, some simulation results for electronic chips cooling using CO2 evaporation are presented and discussed. So far, little information on CO2 condensation is available in the literature. Therefore, this chapter does not include CO2 condensation but it is recommended that future research be conducted in this aspect. The future research needs in CO2 two-phase flow and heat transfer have been identified. The proton exchange membrane (PEM) fuel cell is considered to be one of the most promising alternative clean power generators for portable, mobile and stationary applications because of its low to zero emissions, low-temperature operation, high power density and fast start-up. Fuel (humidified hydrogen) and oxidant (humidified air) are fed into the cathode and anode, respectively. The water vapor in the cell introduced by humidification and produced by the cathode electrochemical reaction will condense to liquid water when the partial pressure of the vapor exceeds its saturation pressure. The resulting two-phase transport of water significantly affects the transport of reactants to the porous electrode and the membrane ohmic resistance, and then the cell performance. For example, liquid water accumulating in

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the channels and in the pores of the gas diffusion layer (GDL) increases the transport resistance of oxygen to the catalyst layer (CL), even resulting in cathode flooding at high current densities. In addition, the membrane must remain well-hydrated because the proton transport capability in the membrane is proportional to the membrane water content. Therefore, understanding and control of the water transport in the cell are important for improving cell performance. At present, the local distributions and transport of reactants in the cell are difficult to measure due to the small cell sizes; thus, numerical models (including the two-phase transport) have become an important means for understanding the reactant transport, electrochemical reactions, current density distributions, and cell performance. Chapter 11 presents a complete three-dimensional, two-phase transport model for a PEM fuel cell based on the two-fluid method, in which the two-phase flow of the multi-component reactants and liquid water are coupled with the species transport, electrochemical reactions, proton and electron transport, electro-osmosis and back diffusion of water. Different two liquid water transport equations were developed for the various cell units based on the different liquid water transport mechanisms in the fuel cell. The model is used to investigate the effects of the various flow field designs in the bipolar plates on the cell performance. The improved understanding of the two-phase transport is then used to develop a series of new flow field designs, including the tapered flow field, contracted flow field, and partially blocked or blocked flow fields. The predictions show that the improved flow field designs significantly increase the liquid water removal from the GDL, which enhances the reactant transport rates and improves cell performance. It is evident that in multiphase porous media, exchange of momentum, mass, and energy among the phases occurs through interfaces separating the phases. This important feature, however, is not accounted for in standard models. In fact, interfacial area does not even appear as a parameter in classical models for flow and transport in porous media. In these standard models, interphase mass transfer is most often estimated using empirical relationships. Recently, however, new thermodynamically motivated model concepts for flow and transport in porous media have been developed which overcome this shortcoming and include interfacial areas. Preliminary modeling studies have been performed based on this new theory. They have been shown to overcome also other shortcomings of classical models like describing capillary hysteresis by employing a single function relating interfacial area, capillary pressure and saturation instead of using myriads of capillary pressure - saturation scanning curves as is the strategy of standard models. In Chapter 12, the authors review recent experimental and numerical studies on twophase flow in porous media that stress the need for determining interfacial areas. Also, the authors derive and propose a full set of macroscopic equations that describes two-phase flow without and with interphase mass transfer including interfacial area as a parameter. They show results of a numerical study of mass transfer between water and gas phase comparing the model including interfacial area to a standard model. It turns out that only the model with interfacial area is able to account for interphase mass transfer in a physically-based way. In Chapter 13, an overview on experimental techniques and models applied to describe fluid dynamics in vertical conveying of coarse particles is presented, focusing on the particular characteristics obtained with the use of a spouted bed feeder as a solids feeding device. Influential factors and procedures for the prediction of regime transitions, pressure gradients and solids concentration for mechanical and non-mechanical devices are evaluated

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through the analysis of experimental data obtained for a wide range of experimental conditions. Finally, the experimental techniques for measuring the variables of interest and the fluid dynamic models applied to describe the flow are evaluated with particular emphasis on discussing the drawbacks and advantages of using the two-phase flow model, in analogy to single phase flow, to describe pneumatic conveying. The results were contextualized within this field in the literature, and an analysis of trends and requirements for future research is presented. Fuel cell system is an advanced power system for the future that is sustainable, clean and environmental friendly. Small fuel cells have provided significant advantages in portable electronic applications over conventional battery systems. Competitive costs, instant recharge, and high energy density make fuel cells ideal for supplanting batteries in portable electronic devices. However, the typical proton exchange membrane (PEM) fuel cell system with its heavy reliance on subsystems for cooling, humidification and air supply would not be practical in small applications. The air-breathing PEM fuel cells without moving parts (external humidification instrument, fans or pumps) are one of the most competitive candidates for future portable-power applications. As explained in Chapter 14, multiphase flow is a central issue in air-breathing PEM fuel cell technology because while water is essential for membrane ionic conductivity, excess liquid water leads to flooding of catalyst layers and gas diffusion layers. Understanding the flow of gas/liquid flows is therefore of major technological as well as scientific interest. Computational fuel cell engineering tools requires the robust integration of models representing a variety of complex multi-physics transport processes characterized by a broad spectrum of length and time scales. These processes include a fascinating, but not always well understood, array of phenomena involving multiphase flow, ionic, electronic and thermal transport in concert with electrochemical reactions. The development of physically representative models that allow reliable simulation of the processes under realistic conditions is essential to the development and optimization of fuel cells, improve long-term performance and lifetime, the introduction of cheaper materials and fabrication techniques, and the design and development of novel architectures. The difficult experimental environment of fuel cell systems has stimulated efforts to develop models that could simulate and predict multidimensional coupled transport of reactants, multiphase flow, heat and charged species using computational fluid dynamic (CFD) methods. The strength of the CFD numerical approach is in providing detailed insight into the various flow and transport mechanisms and their interaction, and in the possibility of performing parameters sensitivity analyses. Three-dimensional, multiphase, multi-component flow, non-isothermal CFD model of an ambient air-breathing PEM fuel cell which work in still or slowly moving air has been developed in this Chapter. The model was developed to improve fundamental understanding of the multiphase flow in a porous cathode and anode diffusion layers as well as the transport phenomena in ambient air-breathing PEM fuel cells and to investigate the impact of various operation parameters on performance. In addition to the multiphase flow and phase change in the air-breathing PEM fuel cell, the new feature of the present model is to incorporate the effect of hygro and thermal stresses into actual three-dimensional air-breathing PEM fuel cell model. Fully three-dimensional results of the multiphase velocity flow field, species profiles, liquid water saturation, temperature distribution, potential distribution, water content in the membrane, stresses distribution in the membrane and gas diffusion layers, and local current density distribution are presented and analyzed with a focus on the physical insight and

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fundamental understanding. They can provide a solid basis for optimizing the geometry of the PEM fuel cell stack running with a passive mode. An overview of experimental observations of turbulence modulation in dilute multiphase flows is presented in the first Short Communication. The presence of the dispersed phase can augment or attenuate the turbulence of the continuous phase. The ratio of the dispersed phase size to the continuous phase turbulence length scale, the relative velocity of the dispersed phase with respect to the continuous phase and the Kolmogorov Stokes number affect the addition/dissipation of the continuous phase turbulence. The Kolmogorov Stokes number plays a significant role in the attenuation of turbulence. More controlled experiments are needed to isolate the effects of the dispersed phase on the turbulence energy spectrum of the continuous phase. The main purpose of the second Short Communication is to show that the simple correlation of the turbulent viscosity as a function of column diameter only is to be understood as a correlation of eddy diffusivity in free turbulence as a function of scale only. The time-averaging procedure of Navier-Stoks equations for multiphase flow, and derivation of the theoretical distribution of time-averaged velocity, will also be briefly explained.

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RESEARCH AND REVIEW STUDIES

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In: Multiphase Flow Research Editors: S. Martin and J.R. Williams, pp. 3-146

ISBN: 978-1-60692-448-8 © 2009 Nova Science Publishers, Inc.

Chapter 1

NUMERICAL SIMULATION OF MULTIPHASE FLOW SYSTEMS Steffen Schütz, Kathrin Kissling, Martin Schilling and Cornelia Seyfert Institute of Mechanical Process Engineering, University of Stuttgart, Germany Böblinger Strasse 72, D-70199 Stuttgart*

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Abstract Within this chapter simulation models to describe multiphase flow systems with are presented and discussed with respect to their physical background, their application range and their numerical implementation. The models comprise multiphase flow systems with different phase combinations and for different phase concentrations. Depending on the models multiphase flow systems can be described in an integral character neglecting details in the transport of single particles but also in a very detailed, microscopic manner with a close look on the single particle behavior. Besides the discussion of well-established models (EulerLagrange and Euler-Euler models), new developments are described which allow the simulation of volumetrically resolved particles, the calculation of the dynamic of phase interfaces and the inclusion of particle-particle interaction mechanisms like agglomeration, coalescence or particle breakup.

List of Symbols Latin Symbols

*

a A b

[1] [m2] [1]

model parameter area, surface model parameter

b B

[-] [-]

point force birth term

www.imvt.uni-stuttgart.de

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B c c c c C C d D D D D e e E f f f

[Vsm-2] [Jkg-1K-1] [1] [1] [ms-1] [1] [1] [m] [m] [m2s-1] [-]

magnetic induction vector specific heat capacity model parameter volume fraction velocity vector mass flow ratio coefficient diameter diameter diffusion coefficient death term

[kgm2s-1] [1] [-] [J] [Nm-3] [-] [s-1]

angular momentum restitution coefficient property vector energy volumetric force number density function frequency

f f f

[Nm-2]

stress vector

-2 -2

[kgm s ] -2

external volume related force

F F

[ms ] [m-3] [-]

external mass related force volumetric number concentration general tensor

F g g g g

[N] [s-1] [1] [1] [ms-2]

force breakage kernel voidage function radial distribution function gravitational acceleration vector

G

[m-1]

Green’s function for the velocity

G h h H Hα I j J J

[-] [m3s-1] [m] [m] [-] [1] [kgm-2s-1] [Nms2] [-]

velocity vector (inner convection) collision kernel film thickness height Heavyside jump function unity matrix mass flux angular moment of inertia differential operator

J J k k k

[-] [kgm2] [m2s-2] [Wm-1K-1] [1]

molecular flux moment of inertia tensor turbulent kinetic energy thermal conductivity diffusivity coefficient

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Numerical Simulation of Multiphase Flow Systems K kB L

[kgm-3s-1] [JK-1] [m]

drag coefficient Boltzmann constant distance, length

L m m M

[m] [kg] [kgs-1] [kgmol-1]

position vector mass mass flow rate molar mass

M n n N NA p P P q Q Q r r r r r R ℜ s s S S S t t t T T T

[Nm] [1] [1] [1] [mol-1] [Nm-2] [1] [m-2] [Wm-2] [C] [-] [m] [kgm-3s-1] [-] [1] [m] [Jkg-1K-1] [Jmol-1K-1] [m] [m] [-] [m2] [Nm-2] [s] [m] [m] [K] [s] [m-2]

moment vector number normal vector on a surface number Avogadro constant pressure probability Green’s function for the pressure thermal heat flux electrical charge source term radius reaction kinetics kinetic term particle-particle interaction random number moment arm specific gas constant universal gas constant tangential coordinate, curvature parameter tangential vector general source term surface strain rate tensor time tangential coordinate tangential vector temperature turbulent time scale, averaging time Green’s function of the stress tensor

T T v v V V w x

[kgm2s-2] [1] [ms-1] [ms-1] [m3]

torque transformation tensor velocity velocity vector volume

[-] [ms-1] [m]

vector field velocity vector x-coordinate

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x X X y z z

[m] [1]

position vector step function (phase indicator)

[m] [m] [m] [m-3s-1]

position vector y-coordinate z-coordinate collision rate per time and volume

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Greek Symbols α α α α β β β γ γ γ

[rad] [m] [K-1] [1] [rad] [kg-1] [1] [1] [Nm-1] [s-1]

angle distance function volume expansion coefficient volume fraction angle daughter particle size distribution model parameter model parameter surface tension deformation velocity

δ δ δ Δ ε η

[1] [-] [-] [1] [m2s-3] [m]

model parameter delta function Dirac function difference turbulent energy dissipation local coordinate vector

κ λ λ λ λ λ μ μs

[kgs-2] [1] [1] [1] [m] [Wm-1K-1] [kg m-1 s-1] [1]

spring constant diameter ratio viscosity ratio coalescence efficiency mean free path length thermal conductivity dynamic viscosity friction coefficient

μ0

[1]

adhesion coefficient

θ θ ν ν ρ Φ Φ χ

[rad] [1] [1] [m2s] [kg m-3] [-] [m] [1]

angle loss of property due to collision number of daughter particles kinematic viscosity density general state variable Level-Set variable transport of property due to collision

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Numerical Simulation of Multiphase Flow Systems Ψ Ψ σ τ τ τ κ κ ω ω ω ξ ξ

[1] [1] [Nm-2] [Nm-2] [Nm-2] [s] [1] [1] [s-1] [1] [s-1] [m2s-1] [kgm-1s-1]

sphericity general property stress tensor viscous stress viscous stress tensor time scale volume flow ratio model parameter angular velocity mass fraction angular velocity vector thermal surface diffusivity bulk viscosity

ξ

[m]

local coordinate vector

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Dimensionless Numbers CD Ca Co Cu Ma Mo Pr Re St We

[1] [1] [1] [1] [1] [1] [1] [1] [1] [1]

Drag coefficient Capillary number Courant number Cunningham factor Marangoni number Morton number Prandtl number Reynolds number Stokes number Weber number

Indices add app A b b b br B Bass c c coal coll

added mass apparent component A body force bubble boundary breakage buoyancy Basset volume fraction collisional coalescence collision, contact

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al. crit cross CV D e e e el eq ex f fluc g G h i I j k l lift LS m max mag mol Mag n ncoll p p proj q rel rot R Re RPY s sus S t vol V VOF x

critical crossing time control volume drag properties turbulent eddy ensemble averaged electrical equilibrium external fluid phase fluctuation gas phase gravity heat general local coordinate, number index, particle size class interface general local coordinate, number index, particle size class general local coordinate, number index, particle size class liquid phase, particle size class lift force Level-Set mixture maximum magnetic molecular Magnus time step index non-collision particle property projection phase relative rotation roughness Reynolds averaged roll pitch yaw single suspension shear turbulence volume related volume Volume-of-Fluid method physical space

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x-coordinate phase averaged y-coordinate z-coordinate

α σ

Level-Set surface tension

+ 0 1 2 ∞

increase decrease reference value, starting point, pole, steady state solution state 1 state 2 far field flow

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1. Introduction Computational Fluid Dynamics (CFD) has become important in research and industrial applications during the last two decades. On the one hand side this is due to the increasing power of personal computers and supercomputers. On the other hand side the increasing number of applications for CFD is caused by the development of complex mathematically and physically based simulation models and numerical solution algorithms which allow for realistic simulation runs regarding complex flow processes. Commercial CFD packages are available to handle tasks in the automotive and aerospace industry as well as in process engineering and turbo machinery. In these fields modern technical improvements would not be possible anymore without modern methods of Computational Fluid Dynamics. Flow and transport phenomena are resolved by accounting for momentum, heat and mass transfer. Process analysis, design and optimization by means of CFD therefore allow for a significant shortening of development cycles as well as a reduced time to market. Nonetheless the validity of simulation models has to be proven by experimental validation. A lot of CFD models are applied to multiphase flow systems. These systems cover a broad range of technical applications starting with low concentrations of the dispersed phase in terms of single solid particles, droplets or gas bubbles in a continuous gaseous or liquid phase (e. g. dust loaded flue gases, dispersed catalytic particles or droplets in liquids) up to highly concentrated multiphase systems such as waste water slurry or fluidized beds of coal particles in combustion chambers of power plants. In many technical applications the dynamics of free phase interfaces is important, e. g. in mold filling processes where air entrapment must be avoided or in spray technology with droplet formation. This broad range of multiphase flow phenomena can only be covered with multiple mathematical-physical simulation models and specific numerical algorithms. The following sections describe the basis of various multiphase modeling approaches. By giving a short introduction to single phase flows and the relevant balance equations in section 2 the basis for the simulation of multiphase systems is provided. A characterization and a physical description of the broad range of multiphase flow systems and the relevant transport phenomena are given with section 3. The well-established Euler-Lagrange and Euler-Euler

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models are introduced in sections 4 and 5 and evaluated with respect to their physical fundamentals and the appropriate solution strategies. Sections 6, 7 and 8 describe new simulation methods for multiphase flow systems. In section 6 the simulation of flows involving arbitrarily shaped volume expansion particles is introduced which allows for a very detailed calculation of the transport behavior of single particles relying only on the fundamental equations of mechanics and coupled then with modern solution strategies of CFD. The problems of the description and the simulation of free boundaries are elucidated in section 7. Various simulation models are discussed and methods to couple these models with CFD tools were developed recently. In section 8 population balances are introduced describing particle-particle interaction effects due to agglomeration, coalescence or breakup. Though the basic ideas of population models were developed more than 40 years ago, a full coupling of population balances and fluid dynamic equations has only become possible with the availability of powerful computation hardware within the last two decades. At the end of each section some important literature concerning the section contents is given. The choice of the literature is based on the intention to provide basic papers and books on the topics but not to discuss the latest research details which are often very case specific. For an improved understanding of the simulation models and the model equations a short introduction into vector and tensor operations is given. The following mathematical notation is used concerning the general vectors

⎛ ax ⎞ ⎛ bx ⎞ ⎜ ⎟ ⎜ ⎟ a = ⎜ a y ⎟ ; b = ⎜ by ⎟ ⎜a ⎟ ⎜b ⎟ ⎝ z⎠ ⎝ z⎠

(1.1)

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and the general tensors

⎛ a xx ⎜ A = ⎜ a yx ⎜⎜ ⎝ a zx

a xy a yy a zy

⎛ b xx a xz ⎞ ⎟ ⎜ a yz ⎟ ; B = ⎜ b yx ⎟ ⎜⎜ a zz ⎟⎠ ⎝ bzx

b xy b yy bzy

b xz ⎞ ⎟ b yz ⎟ ⎟ bzz ⎟⎠

(1.2)

A scalar product between two vectors is denoted with a single dot “⋅”:

a ⋅ b = a x bx + a y b y + a z bz

(1.3)

A scalar product between two tensors is denoted with a double dot “:”:

A : B = a xx bxx + a xy b yx + a xz bzx + a yx bxy + a yy b yy + a yz bzy + a zx bxz + a zy b yz + a zz bzz

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A vector product between two vectors is denoted with a cross “x”:

⎛ a y bz − a z b y ⎞ ⎜ ⎟ axb = ⎜ a z b x − a x bz ⎟ ⎜ a x by − a y bx ⎟ ⎝ ⎠

(1.5)

A tensor product between two vectors is denoted without a symbol:

⎛ a x bx a x b y a x bz ⎞ ⎜ ⎟ ab = ⎜ a y bx a y b y a y bz ⎟ ⎜⎜ ⎟⎟ ⎝ a z bx a z b y a z bz ⎠ A product between a vector and a tensor in the form a ⊗ B is defined as: ⎛ a x Bxx + a y Byx + a z Bzx ⎞ ⎜ ⎟ a ⊗ B = ⎜ a x Bxy + a y Byy + a z Bzy ⎟ ⎜⎜ ⎟⎟ ⎝ a x Bxz + a y Byz + a z Bzz ⎠

(1.6)

(1.7)

The nabla operator ∇ replaces the vector

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⎛ ∂ ∂ ∂ ⎞ ∇=⎜ ; ; ⎟ ⎝ ∂x ∂y ∂z ⎠

(1.8)

With a dot symbol the nabla operator is combined with the corresponding vector or tensor as a scalar product. Without a dot, the nabla operator is combined with the succeeding scalar or vector to describe a vector or a tensor product, such as the pressure gradient vector or the tensor of the velocity gradients:

⎛ ∂p / ∂x ⎞ ∇p = ⎜⎜ ∂p / ∂y ⎟⎟ ⎜ ⎟ ⎝ ∂p / ∂z ⎠ ⎛ ∂v x ⎜ ⎜ ∂x ⎜ ∂v ∇v = ⎜ x ⎜ ∂y ⎜ ∂v ⎜ x ⎝ ∂z

∂v y ∂x ∂v y ∂y ∂v y ∂z

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(1.9)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ∂v z ⎟ ⎟ ∂z ⎠ ∂v z ∂x ∂v z ∂y

(1.10)

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A further formalism often used with the basic equations of fluid dynamics is the Einstein summation rule to give a short description of vector and tensor operations. The vector and the tensor components in an equation are written with indices replacing the local coordinates. Each term in which one and the same index letter is used twice is added for all three coordinates x, y and z. The term ∂v i / ∂x i means

∂v i ∂v x ∂v y ∂v z = + + ∂x i ∂x ∂y ∂z

(1.11)

∂v i =0 ∂x i

(1.12)

and the equation

is an abbreviation for the mass balance of an incompressible fluid, see section 2.1. The Einstein summation rule is often used in the context of turbulence modeling, see section 2.2, as some mathematical expressions become quite complex using the usual coordinate formulation. So the turbulent dissipation rate used in turbulence modeling is written as

ε=−

μ ∂v i′ ∂v i′ μ ⎛ ∂v ′ ∂v ′ ∂v ′ ∂v ′ ∂v ′ ∂v ′ =− ⎜ x x + x x + x x ρ ∂x j ∂x j ρ ⎝ ∂x ∂x ∂y ∂y ∂z ∂z

∂v ′ ∂v ′ ∂v ′ ∂v ′ ∂v ′ ∂v ′ ⎞ + + + + z z + z z + z z⎟ ∂x ∂x ∂y ∂y ∂z ∂z ∂x ∂x ∂y ∂y ∂z ∂z ⎠⎟

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∂v ′y ∂v ′y

∂v ′y ∂v ′y

∂v ′y ∂v ′y

(1.13)

The indices i and j are each used twice in the Einstein notation. This means that the Einstein summation rule requires a summation for i = ( x, y,z ) and for j = ( x, y,z ) and the turbulent dissipation rate is defined with nine summation terms altogether. The Kronecker delta, which is used as a mathematical sieving function, is in the following defined as: ⎧1 for i = j δij = ⎨ (1.14) ⎩0 for i ≠ j Gauss’ theorem is used for integral transformation to combine volume and surface integrals. For a scalar, vector and tensor function the Gauss’ theorem is defined by Scalar function:

∫V ∇fdV = ∫A fndA

(1.15)

∫V ∇ ⋅ f dV = ∫A f ⋅ n dA

(1.16)

Vector function:

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Tensor function:

∫V ∇ ⋅ FdV = ∫A F ⋅ n dA

(1.17)

Finally the Leibniz’ integral theorem is given by

d dt

∂f

∫V(t) f (x, t)dV = ∫V(t) ∂t dV + ∫A(t) f (v A ⋅ n )dA

(1.18)

with the velocity v A of the surrounding surface A of a closed volume element V.

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2. Basics of Computational Fluid Dynamics with Single Phase Flows Within this section the basic transport equations of fluid mechanics are summarized briefly. In single phase systems the balance equations for total mass, momentum, thermal energy and the quantity of various species in a multicomponent system describe the transport characteristics completely. These transport equations are non-linear partial differential equations. Due to computational limitations for turbulent flow systems the basic equations cannot be solved directly as the turbulent structures are of small-scale size. Therefore the common statistical approach to turbulence resulting in the Reynolds equations and in turbulence models is presented. The transport equations have to be transformed to algebraic equations through discretization. The most important numerical discretization scheme used in CFD, the finite volume method, is introduced. These equations for single phase flows provide the basis for the computation of multiphase flow systems.

2.1. Basic Equations of Fluid Mechanics The derivation of the basic equations of fluid mechanics is founded in the physical principle of mass, momentum and energy conservation.

2.1.1. Total Mass Balance The total mass conservation in a fluid system is described by the continuity equation in a cartesian coordinate system as

(

)

∂ ( ρv z ) ∂ρ ∂ ( ρv x ) ∂ ρv y + + + =0 ∂t ∂x ∂y ∂z

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The time derivative with the fluid density ρ characterizes the rate of mass accumulation in a balance element. The three convective terms consider the mass transport due to the fluid

(

)

velocity v = v x , v y , v z .

2.1.2. Momentum Balance The momentum balance is derived from accounting for all forces acting upon a balance element. As the momentum is a vectorial variable, the momentum balance is written in three dimensions according to

∂ ( ρv x )

(

∂t

∂ ρv y ∂t

∂ ( ρv x v x ) ∂x

+

(

∂ ρv y v x ∂y

) + ∂ (ρv z v x ) = − ∂p + ∂τxx + ∂τyx + ∂τzx + f ∂z

∂x

∂x

∂y

∂z

x

∂z

∂y

∂x

∂y

∂z

y

∂z

∂z

∂x

∂y

∂z

) + ∂ (ρv x v y ) + ∂ (ρv y v y ) + ∂ (ρv z v y ) = − ∂p + ∂τxy + ∂τyy + ∂τzy + f

∂ ( ρv z ) ∂t

+

∂x

+

∂ ( ρv x v z ) ∂x

+

(

∂y

∂ ρv y v z ∂y

) + ∂ (ρv z v z ) = − ∂p + ∂τxz + ∂τyz + ∂τzz + f

(2.2)

z

This is the conservative form of the momentum balance. On the left hand side of eq. (2.2) the local rate of momentum accumulation and the convective momentum transport are considered. The right hand side describes the forces which act on a fluid element and lead to a momentum transport due to pressure p, viscous stresses τij and volumetric forces f j , e. g.

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gravity, centrifugal forces, electrical forces, etc. For a fluid with Newtonian behavior the components τij of the viscous stress tensor are written according to Stokes as

⎛ ∂v ∂v j 2 ⎛ ∂v x ∂v y ∂v z τij = μ ⎜ i + − + + ⎜ ∂x j ∂x i 3 ⎜ ∂x ∂ y ∂z ⎝ ⎝

⎞ ⎞ ⎟ δij ⎟⎟ ⎠ ⎠

(2.3)

The general position indices i and j denote the x-, y- and z-direction. To preserve the equilibrium of the moment of torque at a fluid element the viscous stress tensor has to be symmetric resulting in τij = τ ji . For Newtonian fluids, the dynamic viscosity μ only depends on pressure and temperature whereas for non-Newtonian fluids the viscosity is a function of the velocity gradient. The Kronecker symbol δij is equal to one with i = j , i. e. for the normal stress components, otherwise it is zero. By introducing the viscous stress components in eq. (2.2) the Navier-Stokes equations are obtained.

2.1.3. Thermal Energy Balance For fluid systems one can derive various kinds of energy balances, a total energy balance comprising the parts of mechanical and thermal energy, a pure mechanical energy balance

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and a pure thermal energy balance. For non isothermal flow systems usually the thermal energy balance is solved to compute the temperature distribution dependent on time and position. With the conservation equation of thermal energy for a single component and a single phase system written in terms of the fluid temperature one obtains

∂ (ρc V T ) ∂t

+

∂ (ρc V v x T ) ∂x

+

(

∂ ρc V v y T ∂y

) + ∂ ( ρc V v z T ) = ∂z

∂q y ∂q z ⎞ ⎛ ∂q ⎛ ∂p ⎞ ⎛ ∂v x ∂v y ∂v z ⎞ −⎜ x + + + + ⎟ − T⎜ ⎟ − ( τ : ∇v ) ⎟ ⎜ ∂y ∂z ⎠ ∂y ∂z ⎠ ⎝ ∂T ⎠ v ⎝ ∂x ⎝ ∂x

(2.4)

On the left side the accumulation rate of inner energy and the convective transport of inner energy are calculated. The first term in brackets on the right side gives the molecular energy transport due to thermal conduction. According to Fourier’s law the molecular heat flux is written as

q i = −λ

∂T ∂x i

(2.5)

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The second term on the right side describes the reversible rate of change of inner energy ∂v y ∂v z ∂v y ∂v z ⎛ ∂v ⎞ ⎛ ∂v ⎞ + < 0 ⎟ or due to dilution ⎜ x + due to compression ⎜ x + + > 0⎟ . ∂y ∂z ∂y ∂z ⎝ ∂x ⎠ ⎝ ∂x ⎠ The last term including the viscous stress tensor τ describes the irreversible rate of increase of inner energy due to viscous dissipation. For a Newtonian fluid this dissipation term becomes 2 2 2⎤ ⎧ ⎡ ⎫ ⎛ ∂v ⎞ ⎪ 2 ⎢ ⎛ ∂v x ⎞ + ⎜ y ⎟ + ⎛ ∂v z ⎞ ⎥ ⎪ ⎪⎪ ⎢ ⎜⎝ ∂x ⎟⎠ ⎝ ∂y ⎠ ⎜⎝ ∂z ⎟⎠ ⎥ ⎪⎪ ⎦ ( τ : ∇v ) = μ ⎨ ⎣ ⎬ 2 2 2 ⎪ ⎛ ∂v v v v ∂ ∂ ∂ ∂v ⎞ ⎪ 2 ⎛ ∂v x ⎛ ∂v x ∂v z ⎞ ⎛ ∂v z y ⎞ y ⎞ y ⎪+ ⎜ x + + +⎜ + + + z ⎟⎪ ⎟ +⎜ ⎟ − ⎜ ⎟ ∂x ⎠ ⎝ ∂z ∂x ⎠ ⎝ ∂y ∂z ⎠ ∂y ∂z ⎠ ⎪⎭ 3 ⎝ ∂x ⎪⎩ ⎝ ∂y (2.6)

2.1.4. Species Balance The species balance for a component A within a multi-component system is written in close analogy to the thermal energy equation with the partial density ρ A of species A

(

)

∂ ( v zρA ) ⎛ ∂jA,x ∂jA,y ∂jA,z ∂ρ A ∂ ( v x ρ A ) ∂ v yρ A + + + = −⎜ + + ∂t ∂x ∂y ∂z ∂y ∂z ⎝ ∂x

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⎞ ⎟ + rA ⎠

(2.7)

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The terms on the left describe the rate of accumulation of component A and the convective transport. On the right hand side the molecular flux jA is defined by Fick’s law with the binary diffusion coefficient D AB assuming that component A is a diluted component in a carrier fluid component denoted with B:

jA,i = −ρD AB

∂ωA ∂x i

(2.8)

Here, ωA is the mass fraction of component A and ρ is the total density. The term rA denotes the reaction rate. Apart from Fick’s diffusion law more complex transport approaches are available for multi-component systems. In a single phase fluid system with N material components, N-1 species balance equations according to eq. (2.7) are solved together with the total mass balance (2.1). With all balance equations (2.1), (2.2), (2.4) and (2.7) the time-dependent and local functions of the pressure p, the velocity components v x , v y , v z , the temperature T, the species concentrations

ρi and the (mean) density ρ are described. This system of N+4 equations and N+5 unknowns requires a closing equation which is provided by the thermodynamic equation of state. A simple state equation is the thermal law for ideal gases in the limiting case of low pressures:

p = ρRT

(2.9)

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with the specific gas constant R. With a temperature dependent liquid phase the thermodynamic state can be described simply with

ρ = ρ0 (1 − α ( T − T0 ) )

(2.10)

Here, ρ0 is the reference density at reference temperature T0 and α is the volume expansion coefficient. More complex state equations are provided with the thermodynamical theory of mixtures.

2.2. Turbulent Flow The balance equations derived in the previous sections are valid both for laminar and turbulent flow. From a mathematical viewpoint, these equations are non-linear partial differential equations which can only be solved with adequate numerical methods. This numerical solution requires a time and a space discretization with a certain number of discrete time steps and an amount of discrete computational cells for which the state variables are solved by numerical methods. In laminar flows both discretization procedures can be realized in such a way that the resulting algebraic equations can be solved on common personal computers for many technical applications.

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The smallest flow structures in turbulent flow are defined by the turbulent eddies in the dissipation range. A characteristic size measure is the Kolmogorov microscale λ which is calculated from

λ=4

μ3

(2.11)

ρ3ε

with the dynamic viscosity μ and the density ρ of the fluid phase and the dissipation rate ε of turbulent kinetic energy. For turbulent flows with μ ∝ 10−3 Pas , ρ ∝ 1000 kg / m3 and

ε ∝ 1m 2 / s3 one obtains λ ≈ 60 μm . To resolve the correspondent turbulence structures by a computational solution a calculation grid with a cell length of 30 μm or less is necessary. Assuming a cubic simulation domain with a dimension of 60 cm x 60 cm x 60 cm this would require an amount of 8 ⋅ 1012 grid cells – a dimension which cannot be treated even with supercomputers. In consequence a direct numerical simulation (DNS) of turbulent flows is not possible. But with simulation runs of technical relevance the detailed resolution of each turbulent fluctuation velocity is not necessary at all. So the concept of the Reynolds decomposition of

(

)

state variables is applied: Each state variable Φ Φ = v x , v y , v z , p,T,… is written as the sum of a time-averaged mean value Φ and a fluctuation value Φ ′ :

Φ = Φ + Φ′

(2.12)

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The mean value is defined according to

1 Φ= τt

τt

∫ Φdt

(2.13)

0

with a large value of τ t compared to a characteristic time scale of the turbulent fluctuations. In fact one has to consider two time scales. Besides the turbulent time scale τ t a turbulent flow can be unsteady in the macroscopic sense with a characteristic time scale which is much bigger than the turbulent time scale and which is not averaged with eq. (2.13). An important averaging rule is the fact that the time-average of a fluctuation variable is zero

Φ′ =

1 τt

τt

∫ Φ ′dt = 0

(2.14)

0

whereas the product of two fluctuation variables has a non-vanishing, unknown finite value:

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Φ1′ Φ ′2 =

1 τt

τt

∫ Φ1′Φ ′2dt ≠ 0

(2.15)

0

This kind of averaging is called Reynolds averaging.

2.2.1. Reynolds Equations The Reynolds decomposition and the Reynolds averaging procedure according to eq. (2.12) to (2.15) are applied to each transport equation from section 2.1. The form of the continuity equation remains identical with eq. (2.1)

(

)

∂ ( ρv z ) ∂ρ ∂ ( ρv x ) ∂ ρv y + + + =0 ∂t ∂x ∂y ∂z

(2.16)

The time-averaging of the momentum balance equations leads to

∂ ( ρv x ) ∂t

+

(

∂ ρvx vx ∂x

=−

(

∂ ρv y

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∂t

∂y

∂z

(

) (

) (

)

) (

) (

)

) (

) (

)

∂ ρ v ′y v ′x ∂ ρ v z′ v ′x ∂p ∂τxx ∂τyx ∂τzx ∂ ρ v ′x v ′x + + + − − − + fx ∂x ∂x ∂y ∂z ∂x ∂y ∂z

) + ∂ (ρ vx v y ) + ∂ (ρv y v y ) + ∂ (ρ vz v y ) ∂x

=− ∂ ( ρv z ) ∂t

) + ∂ (ρv y v x ) + ∂ (ρ vz v x )

+

∂z

(

∂ ρ v ′y v ′y ∂ ρ v z′ v ′y ∂p ∂τxy ∂τyy ∂τzy ∂ ρ v ′x v ′y + + + − − − + fy ∂y ∂x ∂y ∂z ∂x ∂y ∂z

(

∂ ρvx vz

=−

∂y

∂x

) + ∂ (ρ v y vz ) + ∂ (ρvz vz ) ∂y

∂z

(

∂ ρ v ′y v z′ ∂ ρ v z′ v ′z ∂p ∂τxz ∂τyz ∂τzz ∂ ρ v ′x v z′ + + + − − − + fz ∂z ∂x ∂y ∂z ∂x ∂y ∂z (2.17)

These equations are called Reynolds equations. With the time-averaging process there appear additional terms ρ v i′ v ′ j describing the time-averaged product of two fluctuation velocities. These nine additional unknowns are derived from the time-averaging of the convective terms and in fact they describe the impact of additional inert forces due to turbulence on the momentum transport. As they act like the viscous stresses they are called Reynolds stresses. In highly turbulent flows the Reynolds stresses exceed the viscous stress terms by decades.

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Due to the increased momentum transport on the microscale turbulent tube flows show a very smooth velocity profile compared to laminar tube flows. To preserve the equilibrium of the moment of torque at a fluid element the Reynolds

(

)

stress tensor must be symmetric ρ v i′ v ′j = ρ v ′j v i′ and the six unknown Reynolds stresses are calculated from turbulence models. This statistical approach to turbulence allows the calculation of the transport effects of turbulence without describing each turbulent fluctuation in detail. To solve the Reynolds equations no highly resolved spatial discretization is necessary and the equations can be computed with common personal computers or supercomputers. The averaging process leads to additional transport terms also in the thermal energy balance and in the species balance equations. The time-averaged thermal energy balance is written as

∂ ( ρc V T ) ∂t

+

(

∂ ρc v v x T ∂x

) + ∂ ( ρcv v y T ) + ∂ ( ρcv v z T ) = ∂y

∂z

(

) (

) (

∂q y ∂qz ⎞ ⎛ ∂ ρc V v ′x T ′ ∂ ρc V v ′y T ′ ∂ ρc V v z′ T ′ ⎛ ∂q −⎜ x + + + + ⎟−⎜ ∂y ∂z ⎠ ⎜ ∂x ∂y ∂z ⎝ ∂x ⎝ ⎛ ∂p ⎞ ⎛ ∂v x ∂v y ∂v z ⎞ −T⎜ + + ⎟ − τ : ∇v ⎟ ⎜ ∂y ∂z ⎠ ⎝ ∂T ⎠ v ⎝ ∂x

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(

) ⎞⎟ ⎟ ⎠

(2.18)

)

The temperature gradients with the molecular heat flux according to Fourier’s law (eq. (2.5)) are written with the time-averaged temperature. With high Reynolds number flows, the turbulent transport of thermal energy described by the second term on the right hand side of eq. (2.18), outnumbers the molecular heat flux. According to the Reynolds equation for thermal energy the turbulent mass transfer amplifies the molecular species transport and the time-averaged species balance is written as

(

) (

) (

)

∂ v yρA ∂ v zρA ∂ρA ∂ v x ρ A + + + = ∂t ∂x ∂y ∂z ⎛ ∂ jA,x ∂ jA,y ∂ jA,z −⎜ + + ⎜ ∂y ∂z ⎝ ∂x

(

) (

) (

∂ v ′yρ ′A ∂ v z′ ρ ′A ⎞ ⎛ ∂ v ′x ρ′A + + ⎟−⎜ ⎟ ∂x ∂y ∂z ⎠ ⎜⎝

The additional turbulent transport terms

(

∂ ρc V v i′T ′ ∂x i

)

and

) ⎞⎟ + r

A

⎟ ⎠

(

∂ v i′ρ ′A ∂x i

(2.19)

)

are also

determined with the aid of turbulence models. In the following the cross bar on the time-averaged state variables is neglected for the sake of simplicity and the transport equations are written in the usual manner. The state variables are time-averaged with respect to the turbulent time scale. The cross bar is only kept for the Reynolds transport terms.

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

2.2.2. Turbulence Models Turbulence models are used to describe the unknown Reynolds stress terms in the balance equations to determine the increased, turbulence induced momentum, heat and mass transfer. Turbulence models are classified according to the number of differential transport equations. Zero-equation models comprise only algebraic equations. Algebraic turbulence models assume implicitly turbulent equilibrium. This means that the local generation and dissipation rates of turbulence are equal as transport and redistribution mechanisms are mathematically ignored with algebraic model equations. Using high-order turbulence models the convective and turbulence induced transport of turbulent kinetic energy is respected in the characteristic differential transport equations. 2.2.2.1. k-ε-Model The k-ε-model is the most famous turbulence model. Two transport differential equations for the turbulent kinetic energy k and the turbulent dissipation rate ε are deduced and solved numerically with the Reynolds transport equations. The turbulent kinetic energy is defined from the sum of the three Reynolds normal stress values:

k=

1 1 v ′x v ′x + v ′y v ′y + v ′z v ′z = v i′ v i′ 2 2

(2.20)

using the Einstein summation rule according to section 1. The dissipation rate ε is the sink term of the turbulent kinetic energy,

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ε=−

μ ∂v i′ ∂v i′ ρ ∂x j ∂x j

(2.21)

caused by the decay of small turbulence structures in the Kolmogorov range due to viscous dissipation. The transport equations for k and ε are written as

⎧⎪ ⎛ ⎛ ∂ (ρ k ) ⎞ μ ⎞ ⎪⎫ + ∇ ⋅ ( ρ v k ) = ∇ ⋅ ⎨ ⎜ μ + t ⎟ ( ∇k ) ⎬ ⎜⎜ ∂t ⎟⎟ σk ⎠ ⎪⎩ ⎝ ⎝ ⎠ convection ⎭⎪ unsteady

diffusion

(

+ μ t ∇ v : ∇ v + ∇v : ( ∇ v )

T

)−

production

∂ ( ρ ε) ∂t unsteady

(2.22)

ρε dissipation

⎧⎪ ⎛ ⎫⎪ μ ⎞ + ∇ ⋅ ( ρ v ε ) = ∇ ⋅ ⎨ ⎜ μ + t ⎟ ( ∇ε ) ⎬ σε ⎠ ⎪⎩ ⎝ ⎪⎭ convection diffusion

)

(

ε ε2 T + Cε1μ t ∇v : ∇v + ∇v : ( ∇v ) − Cε 2 ρ k k production

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dissipation

(2.23)

Numerical Simulation of Multiphase Flow Systems

21

The first term regards the time dependency of the variables k and ε with respect to the macroscopic time scale of the flow. The convective transport is expressed with the second terms. The third term in each equation denotes the redistribution of k and ε due to the molecular viscosity μ and due to turbulent velocity and pressure fluctuations. These turbulence-induced small-scale transport mechanisms are expressed with the turbulent viscosity μ t in many turbulence models. According to Boussinesq μ t is a pure turbulence parameter and no material specific transport coefficient. The turbulent viscosity is defined

μ t = Cμ ρ

k2 ε

(2.24)

and all turbulence models using this concept of turbulent viscosity assume an isotropic turbulence structure. This implies the equality of the turbulent fluctuation velocities v ′x = v ′y = v z′ at each position and for each time step which is physically speaking only valid in the limiting case of very high Reynolds numbers Re → ∞ . With the Boussinesq approach

⎛ ∂v ∂v j ⎞ 2 ⎛ ∂v y ∂v z ⎞ ⎞ ⎛ ∂v −ρ v 'i v 'j = μ t ⎜ i + − ⎜ ρ k + μt ⎜ x + + ⎟ ⎟ ⎟ δij ⎜ ∂x j ∂x i ⎟ 3 ⎜ ∂y ∂z ⎠ ⎟⎠ ⎝ ∂x ⎝ ⎝ ⎠

(2.25)

a direct coupling between the k-ε-model and the unknown Reynolds stresses is given using the turbulent viscosity μ t . The Kronecker symbol δij is equal to one for Reynolds normal stresses with i = j , otherwise it is zero. The model parameters in the k-ε-model are obtained Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

from experiments and optimization calculations, see table 2.1. Table 2.1. Model parameters of the k-ε-model. Cμ 0.09

Cε1 1.44

σk 1.0

Cε2 1.92

The turbulence induced transport terms

(

∂ v i′T ′

)

and

(

∂ v i′ρ′A

σε 1.3

)

in the thermal energy ∂x i ∂x i balance and in the material balance are defined with a similar transportation law as used by Boussinesq:

1 μ t ∂T − v i ′T ′ = − ρ σ t ∂x i − v i′ρ ′A = −

1 μ t ∂ρ A ρ σ t ∂x i

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(2.26)

(2.27)

22

Steffen Schütz, Kathrin Kissling, Martin Schilling et al. The transport parameter σ t is called turbulent Prandtl-Schmidt number. For free flows

σ t = 0.6 can often be assumed. With wall bounded flows σ t = 0.9 is usually valid. According to the model assumption of isotropic turbulence the k-ε-model is not valid for vortex or impingement flows when a strong anisotropy of turbulence is given. In such flow systems transport equations for each Reynolds stress must be solved to preserve the vectorial character of turbulence. 2.2.2.2. Reynolds Stress Models Reynolds stress models are applied to turbulent flows with a complex anisotropic turbulence structure. These models do not base on the concept of turbulent viscosity as six transport equations for the Reynolds stress terms are solved numerically. A further equation to determine the turbulent dissipation rate ε is necessary. Anisotropic turbulence structures are resolved completely. The transport equation for the Reynolds stress ρ v i′ v ′j is written as

(

∂ ρ v 'i v 'j ∂t unsteady

) + v ∂ (ρ v v ) = − ' ' i j

k

∂ xk

convection

( ) ( ) ⎞⎟

⎛ ∂ v ' p' ∂ v 'i p' j ⎜ ∂ ρ v 'k v 'i v 'j − ⎜ + ∂ xk ∂xj ⎜ ∂ xi ⎝

(

)

⎟ ⎟ ⎠

turbulent diffusion

−ρ v 'i v 'k

∂vj ∂ xk

−ρ v 'j v 'k

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production

+μ

⎛ ∂ v' ∂ v 'j ⎞ ∂ vi ⎟ + p' ⎜ i + ⎜ ∂ x j ∂ xi ⎟ ∂ xk ⎝ ⎠

( ) − 2μ ∂ v

∂ 2 v 'i v 'j

∂ xk ∂ xk

molecular diffusion

(2.28)

pressure strain ' i

∂ xk

∂ v 'i ∂ xk

dissipation

The terms on the left describe again the time-dependency and the convective transport of the Reynolds stress. The turbulent diffusion term considers the redistribution of the Reynolds stress due to velocity and pressure fluctuations with a balancing effect. The pressure strain term describes additionally the redistribution of turbulence but with respect to anisotropic effects e.g. the attenuation of turbulence perpendicular to solid walls and the turbulence reinforcement parallel to solid walls. The production and dissipation rates are expressed similarly to the k-ε-model and the molecular viscous diffusion is considered separately as the Reynolds stress model is not restricted to high Reynolds numbers where the turbulence induced transport outweighs the molecular effects. For the transport terms additional approaching equations and models are necessary.

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2.3. Numerical Solution Procedures The basic balance equations of fluid mechanics constitute non-linear partial differential equations of second order. Apart from some special cases an analytic solution is not possible and the velocity, pressure, etc. distribution is determined with numerical methods. This requires the discretization of the computational domain with small but finite-size cells or discrete node points and the discretization of the time and spatial derivatives in the transport equations which must be transformed to algebraic expressions. The most important discretization techniques are finite-difference, finite-volume and finite-element schemes and spectral methods. Historically, finite-difference schemes were used first due to their simple mathematical background. Finite-volume methods are state-ofthe-art nowadays in computational fluid dynamics. Finite-element methods are applied with complex grid structures and strong deformation gradients e.g. with simulation tasks in polymer processing. Spectral methods are very accurate but difficult to apply with complex geometries of the flow domain. So the first two methods mentioned above are explained with more details.

2.3.1. Finite-Difference Scheme The discretization principle of finite-difference schemes relies on the approximation of time and spatial derivatives by an approach with Taylor series with a known truncation error. The description of a variable Φ i+1 at position x i+1 dependent on a variable Φ i at position x i with x i+1 = x i + Δx is defined by the Taylor series

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1 ⎛ ∂2 Φ ⎞ 2 ⎛ ∂Φ ⎞ Φ i+1 ≈ Φ í + ⎜ ⎟ Δx + ⎜⎜ 2 ⎟⎟ ( Δx ) with x i ≤ ξ ≤ x i+1 2! ⎝ ∂x ⎠ ⎝ ∂x ⎠i ξ

(2.29)

From this the first spatial derivative is approximated by 2 1 ⎛ ∂Φ ⎞ Φ i+1 − Φ i 1 ⎛ ∂ Φ ⎞ − ⎜ 2 ⎟ (Δ x) ⎜ ⎟ ≈ ⎜ ⎟ Δx 2! ⎝ ∂x ⎠ ⎝ ∂x ⎠i ξ

(2.30)

truncation error

This procedure is called forward differencing scheme of first order accuracy because the truncation error, the second term on the right side of eq. (2.30) is proportional to ( Δx ) . This 1

term is neglected in the numerical solution procedure and the derivative is approximated by the first summand on the right side of eq. (2.30). Alternatively a backward differencing scheme of first order accuracy can be used, basing on the Taylor series

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

1 ⎛ ∂2 Φ ⎞ 2 ⎛ ∂Φ ⎞ Φ i−1 ≈ Φ i − ⎜ Δ + x ⎜⎜ 2 ⎟⎟ ( Δx ) with x i−1 ≤ ξ ≤ x i ⎟ ∂ x 2! ⎝ ⎠i ⎝ ∂x ⎠ ξ

(2.31)

which leads to the expression 2 ⎛ ∂Φ ⎞ Φ i − Φ i−1 1 ⎛ ∂ Φ ⎞ ≈ + ( Δ x )1 ⎜ ⎜ ⎟ 2 ⎟ ⎜ ⎟ ∂ Δ x x 2! ⎝ ⎠i ⎝ ∂x ⎠ ξ

(2.32)

truncation error

and the first derivative is approximated by the first term on the right side. The analogous discretization schemes are mostly applied to the time derivative terms in differential equations. Finite difference approximations of second order accuracy are possible with a Taylor series regarding the terms up to fourth order accuracy. The forward and the backward Taylor series provide

1 ⎛ ∂2 Φ ⎞ 1 ⎛ ∂3 Φ ⎞ 1 ⎛ ∂4 Φ ⎞ 2 3 4 ⎛ ∂Φ ⎞ x x x Φ i+1 ≈ Φ í + ⎜ Δ + Δ + Δ + ⎜⎜ 2 ⎟⎟ ( ) ⎜⎜ 3 ⎟⎟ ( ) ⎜⎜ 4 ⎟⎟ ( Δx ) ⎟ x 2! 3! 4! ∂ ⎝ ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠ξ (2.33)

1 ⎛ ∂2 Φ ⎞ 1 ⎛ ∂3 Φ ⎞ 1 ⎛ ∂4 Φ ⎞ 2 3 4 ⎛ ∂Φ ⎞ Φ i−1 ≈ Φ í − ⎜ Δ + Δ − Δ + x x x ( ) ( ) ⎜ ⎟ ⎜ ⎟ ⎜⎜ 4 ⎟⎟ ( Δx ) ⎟ 2 3 ⎜ ⎟ ⎜ ⎟ 2! ⎝ ∂x ⎠ 3! ⎝ ∂x ⎠ 4! ⎝ ∂x ⎠ ⎝ ∂x ⎠i ξ i i Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

(2.34) The subtraction (2.33) – (2.34) provides an expression for the first order derivative with 3 2 ⎛ ∂Φ ⎞ Φ i +1 − Φ i−1 1 ⎛ ∂ Φ ⎞ ≈ − ⎜ ⎟⎟ ( Δ x ) ⎜ ⎟ 3 ⎜ 2Δx 6 ⎝ ∂x ⎠ ⎝ ∂x ⎠i ξ

(2.35)

truncation error

This approximation is called central differencing scheme with a truncation error of order two, proportional to

( Δx )2 .

This error term is neglected in the numerical solution. The

addition of equations (2.33) and (2.34) results in an approximation for the second order derivative with a second order truncation error:

⎛ ∂ 2 Φ ⎞ Φ i +1 − 2Φ i + Φ i−1 1 ⎛ ∂ 4 Φ ⎞ 2 − ⎜ 4 ⎟ (Δ x) ⎜⎜ 2 ⎟⎟ ≈ 2 ⎜ ⎟ 12 ⎝ ∂x ⎠ (Δ x) ⎝ ∂x ⎠i ξ truncation error

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(2.36)

Numerical Simulation of Multiphase Flow Systems

25

As the derivatives are simple to calculate and to implement in a numerical software tool the finite difference scheme is often used for simple simulation problems. With nonequidistant calculation grids, the spatial discretization value Δx must be replaced by local values ( Δx )i . The disadvantage is the lacking conservation character of the balance equations in the discretized description. This means that the conservation of mass, momentum and energy is not guaranteed for the discretized equations when finite differencing schemes are used. The resulting numerically induced momentum, mass and energy sinks or sources can distort the physical contents of the simulation results.

2.3.2. Finite-Volume Scheme

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As the simple finite-difference scheme shows some difficulties with respect to physical consistency the finite-volume scheme is the most widely spread discretization scheme used for the numerical solution of fluid dynamic equations. For the finite-volume scheme the computational domain is discretized into small but finite-volume elements called control volumes CV or cells as shown with Fig. 2.1. This notation is also used in 2D-applications when the “volumes” are constituted by planar or curved faces.

Figure 2.1. Finite-volume discretization scheme with discrete control volumes.

Each discrete control volume is defined by its midpoint and the limiting faces (or limiting lines in a two-dimensional model). The state variables are calculated formally for the cell midpoints and the state at the cell midpoint is representative for the whole cell. Further developments of finite-volume techniques use a staggered grid technique where two grids are defined and shifted mutually with a half cell dimension. Various state variables e. g. the velocity components and the pressure are defined on the one or on the other grid. This is done for the sake of an easy calculation during approximation routines for state variables and to

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

improve the numerical stability with respect to the convergence behavior of the numerical solution algorithm. The basic idea of the finite-volume discretization scheme is the integration of the balance equations with respect to each discrete finite-volume grid cell. The balance equations discussed in section 2.1 can be written in a general way according to

f [Φ ] =

∂ (ρΦ ) ∂t

+ ∇ ⋅ ( ρ vΦ ) + ∇ ⋅ J Φ + SΦ = 0

(2.37)

The variable Φ is a placeholder for the velocity, the thermal energy or the mass concentration of a material component. The first and second term describe the rate of accumulation and the convective transport of Φ. The third term describes the molecular transport with the flux J Φ for example the molecular momentum transport due to viscosity or the molecular heat transport. The source and sink terms are denoted with SΦ . Applying the finite-volume method the transport equation (2.37) is integrated with respect to a discrete volume element V

F [Φ ] =

∂

∫∫∫ f [Φ ] dV = ∂t ∫∫∫ ρ Φ dV + ∫∫∫ ∇ ⋅ (ρ v Φ ) dV + ∫∫∫ ∇ ⋅ JΦ dV + ∫∫∫ SΦ dV = 0 V

V

V

V

V

(2.38) Now the integration theorem of Gauß is applied transforming the volume integral of the

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divergence ∇ ⋅ V ( x ) of an arbitrary vector field V ( x ) into a flux integral

∫∫∫ ∇ ⋅ V ( x )

dV =

V

∫∫ V ( x ) ⋅ n dS

(2.39)

S

On the right side of eq. (2.39) all fluxes of the vector field V ( x ) are integrated with respect to all surfaces S with normal vector n comprising the considered balance volume element. With this integral transformation, eq. (2.38) is written as

F [Φ ] =

∂

∫∫∫ f [Φ ] dV = ∂t ∫∫∫ ρ Φ dV + ∫∫ (ρ v Φ ) ⋅ n dS + ∫∫ JΦ ⋅ n dS + ∫∫∫ SΦ dV = 0 V

V

S

S

V

(2.40) This is a flux-oriented formulation of the general transport equation. The convective and molecular fluxes across the surfaces of each finite volume element are considered. The conservation principle is assured with the transport equation (2.40) because the convective or molecular flux of Φ out of one finite volume element across any surface is identical to the flux into the neighbor element across the same shared surface.

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Following the idea of a constant state variable Φ in a single finite volume element, the partial time derivative term can be written as total time derivative. Also the source term is constant for any finite volume element and eq. (2.40) is transformed to

F [Φ ] =

∫∫∫ f [Φ ] dV =

d ( ρΦ )

V

dt

V+

∫∫ (ρ v Φ ) ⋅ n dS + ∫∫ JΦ ⋅ n dS +SΦ V = 0 S

(2.41)

S

This transformation can be applied to the mass, momentum, energy and species balance and for each discrete finite volume element a corresponding balance equation is derived. The flux terms are approximated by the sum of fluxes across the N surfaces ( i = 1,…, N ) of a finite volume element, each surface with its local normal vector n i .

∫∫ S

(ρ v Φ ) ⋅ n dS +

∫∫ S

J Φ ⋅ n dS ≈

N

∑ i =1

(ρvΦ )i ⋅ n iSi +

N

∑ JΦ,i ⋅ niSi

(2.42)

i =1

This procedure requires the values of the state variables Φ i on the cell surfaces.

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According to the basic ideas of the finite-volume scheme, the variables are only known at the midpoints of the cells. Therefore an adequate approximation is necessary to determine the variables on the cell surfaces. This problem is shown with Fig. 2.2:

Figure 2.2. Two finite-volume cells with the midpoints x P and x P +1 and the surface x e between.

A quite simple approximation scheme is the UPWIND-scheme of first order accuracy. The value Φ e of the state variable on the cell surface at x e is set identical to the value at the midpoint of the cell from which the advection comes:

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

Φ e = Φ P if Φ e = Φ P+1

( v ⋅ n )e ≥ 0 if ( v ⋅ n )e < 0

(2.43)

and the variable Φ is described by a stepwise approximation. This scheme is appropriate for convective-dominated transport processes. As steep gradients of state variables are smoothed with this algorithm of first order accuracy, transport processes with a higher impact of molecular transport terms are computed with low precision. Therefore a second-order UPWIND scheme according to

( v ⋅ n )e ≥ 0 Φ e = Φ P+1 + ∇Φ P+1 ⋅ Δx P+1 if ( v ⋅ n )e < 0 Φ e = Φ P + ∇Φ P ⋅ Δx P if

(2.44)

can be applied. The gradient of the state variable is calculated with

∇Φ P =

1 V

N

∑ Φ i ⋅ ( ni ⋅ Si )

(2.45)

i =1

with the sum over all surfaces Si ( i = 1,…, N ) of the discrete cell P. V is the volume of cell P. The vectors Δx P and Δx P+1 denote the vectors between the cell midpoints of the cells P and P+1 and the midpoint of the common face of these cells, see Fig. 2.3. The analogue calculation is done for the gradient ∇Φ P+1 . With this procedure the values

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Φ i are determined by linear interpolation from the midpoint values of both adjacent cells at each cell surface.

Figure 2.3. Two adjacent cells with the vectors between the cell midpoints and the face midpoint.

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Further approximation schemes as the power law or the QUICK scheme can be found in literature.

2.3.3. Time-Marching Schemes The finite-volume method is used for the spatial discretization of the balance equations. With time-dependent flow processes we also need a time discretization scheme. Starting from eq. (2.41) the balance equation is written as

d (ρΦ ) dt

V=−

∫∫ (ρ v Φ ) ⋅ n dS − ∫∫ JΦ ⋅ n dS −SΦ V S

(2.46)

S

with the right hand side discretized according to the finite-volume scheme. For the time discretization a finite-difference scheme is usually applied according to

(ρΦ )n+1 − (ρΦ )n Δt

V=−

∫∫ (ρvΦ ) ⋅ ndS − ∫∫ JΦ ⋅ ndS −SΦ V S

(2.47)

S

The superscript n+1 denotes the new time step, the superscript n indicates the old time step. Dependent on whether the right side of eq. (2.47) is evaluated at time t n or at time t n +1 the time integration scheme is called explicit or implicit. Especially for extremely non-linear problems the implicit integration scheme is more accurate and numerically absolutely stable.

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References [1] Anderson, J. D.: Computational Fluid Dynamics: The Basics with Applications. 1st Edition, McGraw Hill International Editions, 1995. [2] Ferzinger, J. and Peric, M.: Computational Methods for Fluid Dynamics. New York, Berlin: Springer, 1996. [3] Bird, R. B., Stewart, W. E. and Lightfoot, E. N.: Transport Phenomena. New York: John Wiley & Sons, 1960. [4] Drikakis, D. and Geurts, B. J.: Turbulent Flow Computation. Dordrecht, Boston, London: Kluwer Academic Publishers. [5] Fletcher, C. A. J.: Computational Techniques for Fluid Dynamics, Vol. 1: Fundamental and General Techniques. 2nd edition, New York, Berlin: Springer, 1991. [6] Fletcher, C. A. J.: Computational Techniques for Fluid Dynamics, Vol. 2: Specific Techniques for Different Flow Categories. 2nd edition, New York, Berlin: Springer, 1991. [7] Gatski, T. B., Hussaini, M. Y. and Lumley, J. L.: Simulation and Modeling of Turbulent Flows. Oxford University Press, 1996. [8] Hinze, J. O.: Turbulence. 2nd Edition, New York: McGraw-Hill, 1975. [9] Launder, B. E. and Spalding, D. B.: Lectures in Mathematical Models of Turbulence. London: Academic Press, 1972.

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[10] Launder, B. E. and Spalding, D. B.: The Numerical Computation of Turbulent Flows. Computational Methods in Applied Mechanical Engineering 3 (1974), 269–289. [11] Launder, B. E., Reece, G. J. and Rodi, W.: Progress in the Development of a Reynoldsstress Turbulence Closure. Journal of Fluid Mechanics 68 (1975), 537-566. [12] McComb, W. D.: The Physics of Fluid Turbulence. Oxford Science Publications, 1996. [13] Mott, R. L.: Applied Fluid Mechanics. New Jersey, 2006. [14] Patankar, S. V.: Numerical Heat Transfer and Fluid Flow. Taylor and Francis, 1980. [15] Rodi, W.: Turbulence Models and their Application in Hydraulics. Delft: IAHR, 1984. [16] Schlichting, H.: Boundary-Layer Theory. 7th Edition (McGraw-Hill Classic Textbook Reissue), McGraw-Hill Publishing Company, 1987. [17] Wendt, J. F.: Computational Fluid Dynamics. New York: Springer, 1991.

3. Basics of Multiphase Flow Systems

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Multiphase flow systems are at first characterized by the phases and their dispersity state which define the basic physical characteristics. Within multiphase systems each combination of two or more solid, liquid and gaseous phases is possible. But there can also exist multiphase systems consisting of a single condition of aggregation with immiscible materials and different chemical properties, e.g. a non-soluble mixture of water and oil or a mixture of physically different solid particles. Mixtures of two or more gases can usually be treated as a single phase flow as gases mix ideally in relevant technical applications. Only in the supercritical state there can be a segregation of some gas components in a gas mixture. According to the dispersity state one has to distinguish multiphase systems with dispersed particles and with quasi-continuous phases. A summary of systems with dispersed particles is given with Fig. 3.1.

Figure 3.1. Examples for dispersed multiphase systems.

In the simplest case a multiphase system comprises one continuous and one dispersed phase. With liquid droplets or solid particles in a continuous gas phase we talk of fog or smoke. A packing is also a multiphase system with a gas or liquid volume flow through a quiescent amount of solid particles. Gas bubbles in a liquid phase are generally called a gasliquid flow and with high bubble concentrations we talk of foam. In this case the gas volume fraction can be higher than the liquid volume fraction but the gas bubbles are separated one from the other and the liquid bridges between show a connected structure. Having liquid droplets in another liquid phase the system is called emulsion and with solid particles in a liquid there is a suspension. With gas bubbles or liquid droplets in a continuous solid material the system is called porous medium.

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Besides the dispersed systems there are flow patterns consisting of two or more phases with the phases completely separated one from the other. An example is the flow of an oilwater mixture with almost the same volume fractions in a horizontal pipe. Usually the oil with lower specific weight is transported in the upper half of the pipe and the water with higher density is in the lower part of the tube cross section. An example of the multitude of flow phenomena in a simple system with air and water in a horizontal pipe is shown in Fig. 3.2.

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Figure 3.2. Multiphase flow phenomena in a gas-liquid (air-water) tube flow and flow chart dependent on the mass fluxes of the two phases (Brennen 2005).

At low gas concentrations there is a bubble flow with an increased gas concentration in the upper half of the tube. With increasing concentrations and increasing momentum forces the bubbles coalesce forming gas plugs and resulting in a stratified flow with a gas layer and a liquid layer. When the inert forces of the liquid phase increase the interface becomes unstable and a wavy structure develops. With increasing inert forces liquid droplets are disrupted from the liquid film into the gas phase. A higher gas volume flow rate leads to the slug flow and furtheron the flow pattern changes to an annular flow with a liquid annulus close to the tube wall and a gas flow with liquid droplets along the tube axis. Finally at high gas and liquid fluxes the liquid is atomized resulting in a dispersed flow state with liquid droplets in a continuous gas phase. With this example one can see the complexity of possible multiphase flow patterns within a simple geometry and material combination. Similar phenomena can be detected in gas-solid transport processes with various fluxes of the phases. For the development of simulation models for multiphase flow systems and their numerical solution a basic distinction between dilute and dense multiphase flow systems is performed: •

A multiphase system is assumed to be diluted with a volume phase fraction up to five or ten percent concerning the phase with the lower concentration. In this case the

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•

phase with the lower volume phase fraction is often described as consisting of single, dispersed particles and the transport behavior is calculated by an Euler-Lagrange model (see section 4) as a common model approach. With this model the transport behavior of each discrete particle is described with a force balance. A multiphase system is assumed to be dense with a volume phase fraction higher than ten percent concerning each phase. In this case each phase is assumed to be quasi-continuous and the transport behavior is described by an Euler-Euler model (see section 6). The basic equations of single phase flows as introduced in section 2 are applied to each phase and the detailed information on the transport of single particles is lost. Usually the application of Euler-Euler models is combined with a strong interphase coupling and the coupling effects are described with (semi-) empirical model approaches. The different transport characteristics of particles or gas bubbles with different sizes can be considered by defining discrete size classes and each class is assumed as a separate Eulerian phase.

This distinction between dilute and dense multiphase flows is a first, very rough classification. In section 4, further quantitative criteria are given to better determine the kind of multiphase flow systems. The definition of a dilute or dense multiphase flow is the basic step when choosing the adequate simulation model. But there are further important questions in modeling multiphase flow systems which influence the choice of the appropriate multiphase model: • •

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•

•

•

•

•

What is the goal of the simulation run: Particle transport behavior, phase distribution, particle separation, pressure drop? Which characteristic material properties of the phases must be considered: solid particles, deformable and compressible droplets or gas bubbles, etc? What is the particle size range of interest in a dispersed system? The relevant forces acting on particles with some millimeters in size can be completely different from those relevant for particles in the micron or the submicron range. What is the impact of the particle-wall contact and the particle shape on the transport behavior? For small particles in wide flow cross sections the particles can be assumed as volumeless mass points and the impact of the real particle shape is considered with shape factors (see section 4). But with particles in narrow flow cross sections this assumption is not longer valid. Regarding non-spherical particles all six degrees of freedom (translation and rotation in three directions) must be considered and new models are necessary for the detailed particle transport. These models are described in section 5. When there is a transient flow regime due to varying volume phase fractions (see Fig. 3.2) the simulation models change dependent on the kind of phase distribution. This can occur in gas-liquid flows with boiling or condensation. Different transport effects occur in laminar or turbulent flows. Especially the turbulence-induced fluctuating transport of small particles described as a direct numerical simulation is not possible, see section 2.2. How strong is the force coupling between the different phases? With gas bubbles at low concentrations in a liquid phase the coupling is unidirectional as the fluid forces

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•

33

caused by the liquid phase have strong impact on the transport behavior of the gas phase. Any reaction of the gas bubbles on the liquid phase can be neglected and the bubble dynamics is described with an Euler-Lagrange model combined with a socalled uncoupled solution algorithm (see section 4 and especially section 4.5.3). With the transport of solid particles at high concentrations in a liquid phase the mutually acting forces lead to a strong bidirectional phase coupling requiring advanced modeling and solution algorithms e. g. as provided by an Euler-Euler model. How important are interfacial effects concerning the behavior of free surfaces within a multiphase system? Regarding small droplets in a gas phase or small gas bubbles in a liquid phase, the dispersed particles can be assumed to be spherical and rigid without a deformable phase interface. With increasing bubble and droplet diameters deformable and fluctuating interfaces due to interacting viscous or inertia and surface tension forces lead to different particle transport behavior. This is of particular importance when heat or mass transfer effects across a phase interface are determined by the time-dependent shape of the interface. The stability of interfaces is impacted by fluid forces and the transport behavior of surfactants when the break-up of gas bubbles or droplets is described. Furthermore a macroscopic wavy structure at a free surface shows great impact on the pressure drop in a gas-liquid tube flow or – with a real macroscopic technical application - the drag force affecting on a moving ship. Simulation models especially describing the dynamics of deformable interfaces by additional force balances and particular interface reconstruction techniques are described in section 7.

Multiphase flow systems occur in many technical applications. In the following some multiphase transport phenomena in process engineering are discussed. A wide application range can be found in mechanical separation technology. Characteristic processes like sedimentation, filtration and centrifugation are usually described by Euler-Lagrange models often without reaction effects from the dispersed particles on the continuous phase at low particle concentrations. By following the pathlines of single particles their transport and separation behavior is described leading to detailed information to improve separation apparatuses. For example, the simulation of metal or ceramic diesel soot filters gives detailed information on the particle deposition characteristics, on the increasing pressure difference across the filter elements and on the regeneration behavior of the filter elements when the soot particles are burned off the filter surface. Apart from this macroscopic look on filtration processes a microscopic look allows the description of filtration effects on single filter fibers in filter media. Regarding the particle deposition on single fibers the blocking of filter media can be predicted. Detailed filtration mechanisms as inertia, diffusion, surface adhesion, thermophoresis can be investigated using detailed models when following the impact of single particles. When single particles are followed in a detailed manner the classical Euler-Lagrange model regarding particles as volumeless mass points must be abandoned and with new model approaches the individual particle transport is described explicitly with six degrees of freedom concerning translation and rotation. These new model approaches combined with particular dynamic meshing techniques are described in section 5. Separation processes with solid particles of high volume-specific weight in gaseous or liquid flow systems should often be modeled considering the reaction forces of the particles

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

on the continuous phase. E. g. this is necessary with transport processes of metal particles in an oily slurry or with plastic particles in pneumatic conveying. Regarding processes with high particle concentrations like slurry dewatering or the mixing of solid particles in fluidized beds there is no possibility to follow the transport of single particles as the particle concentration is too high. It is the same with gas-liquid reactors often used in organic or biological reaction systems or in rectification columns with similar concentrations of the gas and liquid volume fractions at high velocities. A three-phase system consisting of solid, liquid and gaseous components at increased volume fractions can be found in wet scrubbers applied to the cleaning of flue gases. In these cases the behavior of the solid particles or gas bubbles is described by Euler-Euler models. The description of free surfaces can be coupled with an Euler-Lagrange or an Euler-Euler model. The surface dynamics itself is described by separate force balances including surface tension and further surface effects by the temperature-gradient driven movement of surfaces, which is known as Marangoni convection. Usually the description of free surfaces is a twostep process, comprising the surface motion and the surface reconstruction. This two-step procedure is realized in the volume-of-fluid model or newly in first applications of the levelset method. Both methods can be basically described as Euler-Euler methods with some model amendments to describe the surface dynamics. A direct simulation of free surfaces in Stokes flows at low Reynolds numbers is possible using the boundary-integral method. By this method the transport of phase interfaces is computed directly without using reconstruction techniques. The various modeling approaches for free phase interfaces mentioned above are described in detail in section 7. When the transport of bubbles, drops or solid particles is combined with agglomeration, coalescence or particle breakage resulting in time- and position-dependent particle size distributions further models must be used combined with Euler-Lagrange or Euler-Euler approaches. These models are called population models describing the shift in particle property distributions like particle size distributions, distributions of particle shape or concentration etc. The particle-particle interaction kinetics are usually modeled using statistical approaches concerning the collision kinetics, the breakage rates, etc. The resulting integro-differential equations require specific numerical techniques for their solution. It is possible to combine detailed and user-defined population balances with multiphase models in commercial CFD software. A short introduction to this modeling and simulation technique is given in section 8.

References [1] Sadhal, S. S., Ayyaswamy, P. S. and Chung, J. N.: Transport Phenomena with Drops and Bubbles. New York: Springer, 1997. [2] Crowe, C., Sommerfeld, M. and Tsuji, Y.: Multiphase Flows with Droplets and Particles. CRC Press, 1998. [3] Joseph, D. D. and Renardy, Y. Y.: Fundamentals of Two-Fluid Dynamics, Part I: Mathematical Theory and Applications. New York: Springer, 1992. [4] Joseph, D. D. and Renardy, Y. Y.: Fundamentals of Two-Fluid Dynamics, Part II: Lubricated Transport, Drops and Miscible Liquids. New York: Springer, 1993.

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35

[5] Brauer, H.: Grundlagen der Einphasen- und Mehrphasenströmungen. Aarau and Frankfurt/Main: Sauerländer, 1971. [6] Brennen, C. E.: Fundamentals of Multiphase Flow. Cambridge: Cambridge University Press, 2005.

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4. Euler-Lagrange Models Euler-Lagrange models describe the transport behavior of dispersed single particles in a multiphase flow in a very detailed manner. The basis of Euler-Lagrange models is a force balance defined for each single particle. With this force balance, various transport effects can be regarded with the particle drag force showing the greatest impact. Dynamic forces, electric, magnetic or thermophoretic forces are considered in particular applications. Also the simulation of phase transition effects due to evaporation, condensation or surface reaction is possible (Durst et al. 1984, Lu et al. 1993, Sommerfeld 1996). In section 4.1 the basic ideas of Euler-Lagrange models are introduced. The definition of various forces is the content of section 4.2. In each transport model the relevance of the various forces must be estimated and the transport equation must be defined equivalently. Generally the term “particle” denotes solid particles, liquid droplets and gas bubbles. As long as the particles can be modeled as rigid bodies the model equations derived in sections 4.1 and 4.2 are valid. More detailed information on the transport behavior of gas bubbles with deformable phase interfaces are given in section 4.3. A detailed and comprehensive discussion of the mathematical description and the transient simulation of moving free interfaces is given in section 7. Regarding the history of multiphase flow models the Euler-Lagrange approach was the first one and various types of transport equations were derived. They are discussed in section 4.4. The numerical solution procedure is of great importance due to the accuracy of the calculated particle tracks. In section 4.5 some information is given concerning the spatial discretization of the flow field and on the choice of the time steps and the time discretization scheme. Special topics are discussed in sections 4.6, 4.7 and 4.8 concerning the turbulenceinfluenced particle transport, particle-wall collisions and particle-particle collisions.

4.1. Basics of Euler-Lagrange Models The detailed description of the transport behavior of single particles based on force balances can only be applied with low particle concentrations in diluted multiphase flows. In section 3 the volume concentration of the phases was introduced to classify multiphase flow systems. A further characteristic parameter to distinguish a diluted and a dense multiphase system is the L of the mean particle distance L and the characteristic particle diameter d p . This ratio dp ratio is

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al. 1/ 3

L ⎛ π 1+ κ ⎞ = ⋅ d p ⎜⎝ 6 κ ⎟⎠

(4.1)

with the volume flow ratio

κ=

Vp Vf

=

m p ρf ⋅ mf ρ p

of the particle phase (index “p”) and the fluid phase (index “f”). In the case of

(4.2)

L > 10 dp

particles can be treated as isolated. This means that the transport of single particles is determined by interaction forces with the fluid phase and particle-particle collisions play no important role concerning the particle transport. Therefore the Euler-Lagrange model is mp ≈ 1 and a density applied in this case. E.g. in gas-solid systems with a mass flow ratio mf

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ratio

ρp ρf

≈ 103 one obtains

L ≈ 10 . dp

The denomination of the Euler-Lagrange model is based on the different points of view by which the transport phenomena in the continuous fluid phase and the discrete particle phase are described. The calculation of the time-dependent or the steady flow field of the continuous phase is performed from the viewpoint of an outstanding, quiescent Eulerian observer. In contrast, the simulation of the transport of single particles is defined from the viewpoint of a Lagrangian observer who is moving with a particle. The most important basic idea of an Euler-Lagrange model is the conception of a volumeless mass point. Each real particle independent of its size or shape is assumed as a mass point loosing its individual particle properties, see Figure 4.1

Figure 4.1. Reduction of real particles to mass points.

Deriving the force balance for arbitrary, non-spherical particles, the impact of the real shape can be considered with appropriate model parameters, see section 4.2.2.3. The particle trajectory within a flow field is calculated from a force balance basing on Newton’s law:

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mp

dv p dt

=

37

∑ Fi

(4.3)

i

A particle with a constant mass m p and with the absolute velocity v p is accelerated by the impact of external forces Fi . As the particles are not resolved volumetrically, the pressure and the viscous forces acting from a fluid phase on a particle at its surface cannot be determined directly from the flow field around the particle by a surface integration. In fact one needs empirical or semiempirical models to calculate the relevant forces affecting the particle transport.

4.2. Forces In this section a range of external forces acting on single particles and the appropriate calculation models are discussed. To describe the particle motion within a real multiphase transport process, the relevant forces must be selected and included into the force balance.

4.2.1. Volume Forces Volume forces are usually caused by a static or dynamic force field around a particle. The gravitational force

FG = m p g

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is

defined

with

the

(

particle

)

mass

mp

and

(4.4) the

gravity

acceleration

vector

g = 0;0; −9.81ms −2 . With a particle moving in an imposed centrifugal field the gravity

(

)

acceleration is replaced by the centrifugal acceleration g = ω2x r; ω2y r; ω2z r with the angular

(

)

velocity vector ω = ωx ; ωy ; ωz and the radius r of the curved particle trajectory. The buoyancy force in the gravity or centrifugal field is calculated

FB = ρf Vp g

(4.5)

with the density ρf of the fluid phase and the particle volume Vp . Further volume forces can be impacted by electric and magnetic fields. For a spherical particle with an electrical charge Q in an electric field with the field strength E the electric force is given with

Fel = QE

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(4.6)

38

Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

and in a magnetic field the relation

(

Fmag = Q v p xB

)

(4.7)

holds with the magnetic induction B .

4.2.2. Drag Force The drag force, resulting from pressure and viscous effects of the continuous phase at the surface of a particle is the most important dynamic force in almost all kinds of multiphase flows which determines the transport behavior of particles. A broad range of work was performed within the last century describing the drag force for spherical and non-spherical particles within various flow regimes. 4.2.2.1. Drag Force According to Stokes for Spherical Particles The first analysis of drag forces acting on spherical particles was performed by Stokes. Assuming a pure viscous flow around a sphere which moves at low velocity and negligible inert forces in a quiescent fluid phase, Stokes derived the expression (Stokes 1851, Lord Rayleigh 1911)

FD = −3πμf d p v p

(4.8)

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from an analytical solution of the simplified Navier-Stokes equations in the limiting case Re → 0 . This drag force is valid in the so called Stokes-regime (“creeping flow”) with a particle Reynolds number

Re p =

ρf v p d p μf

< 0.5

(4.9)

where the norm of the particle velocity vector v p is the characteristic dynamic parameter. Including the drag force (eq. (4.8)) in the basic transport equation (4.3) assuming a spherical π particle with mass m p = ρ p d 3p one obtains the time-dependent particle velocity 6

dv p dt

=−

18 μf 1 vp = − vp 2 ρ τp dp p

(4.10)

Here τ p is the relaxation time, a dynamic parameter describing how fast a moving particle reaches a new steady state when the boundary conditions alter. This parameter has a great importance with respect to the choice of discrete time steps for the numerical solution of the particle transport equation, see section 4.5.2.

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An amendment to the Stokes law eq. (4.8) was given by Oseen (Oseen 1910). From a semi-analytical solution of the Navier-Stokes equations for the flow field around a moving spherical particle using a series expansion for the velocity, he derived the drag force

3 ⎛ ⎞ FD = −3πμ f d p v p ⎜ 1 + Re p ⎟ 16 ⎝ ⎠

(4.11)

considering the impact of convective momentum transport in the flow field around the particle with a first approach. 4.2.2.2. General Approach of the Drag Force for Spherical Particles in the Continuum Regime Without the restricting assumptions of the Stokes regime, the drag force of a moving spherical particle in an arbitrary flow of an advecting fluid phase is described by the empirical approach

FD = C D

ρf v rel v rel A proj 2

(4.12)

with the drag coefficient C D which is dependent on the particle Reynolds number

Re p =

ρf v rel d p μf

(4.13)

The relative particle velocity v rel is the difference between the absolute fluid velocity Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

v f and the absolute particle velocity v p according to Fig. 4.2. The projection cross section A proj is perpendicular to the relative velocity. The correlations for the drag coefficient C D for a spherical particle are given in Fig. 4.3.

Figure 4.2. Relative velocity of a particle in an advecting fluid.

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

Figure 4.3. Drag coefficient of a spherical particle dependent on the particle Reynolds number.

Table 4.1 shows the analytic correlations described by the diagram in Fig. 4.3.

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Table 4.1. Analytical correlations for the drag coefficient of spherical particles in different flow regimes. Stokes regime

transition regime

Newton regime

turbulent boundary layer

Re p < 0.5

0.5 < Re p < 103

103 < Re p < 3 ⋅ 105

Re p > 3 ⋅ 105

CD =

24 Re p

CD =

24 4 + + 0.4 Rep Rep

C D ≈ 0.5

The correlations for the drag coefficient presented above are valid when the spherical particles move in an infinite flow domain so wall impacts on the particle motion are neglected. For spherical particles moving in narrow channels the wall impact becomes of great importance leading to an increase in the drag coefficients about one or two decades. The characteristic dimensionless number is the wall distance parameter

λ=

dp D

(4.14)

with the channel diameter D. The drag coefficient is given according to Fig. 4.4 (McNown and Newlin 1951, McNown and McPherson 1936) for a spherical particle with diameter d p moving centrically in a cylindrical tube with diameter D.

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4.2.2.3. General Approach of the Drag Force for Non-Spherical Particles in the Continuum Regime Regarding non-spherical particles, the drag coefficient is not only dependent on the particle size but also on the particle shape. A characteristic shape parameter is the sphericity

Ψ=

surface of volume equivalent sphere ≤1 surface of real particle

(4.15)

For spherical particles there is Ψ = 1 but for fibrous particles Ψ = 0.1 or less can occur. The dependence of the drag coefficient on the sphericity is shown in Fig. 4.5 for particles moving in an infinite flow domain (Haider and Levenspiel 1989, Michaelides 2004). The curves in the diagram are approximated by the correlation (Morsi and Alexander 1972)

CD =

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+

24 Re p

(

)

⎡1 + exp 2.3288 − 6.4581Ψ + 2.4486Ψ 2 ⋅ Re ( 0.0964+0.5565Ψ ) ⎤ p ⎢⎣ ⎥⎦

(

) + 15.8855Ψ ) + Re

(4.16)

exp 4.905 − 13.8944Ψ + 18.4222Ψ 2 − 10.2599Ψ 3 ⋅ Re p

(

exp 1.4681 + 12.2584Ψ − 20.7322Ψ 2

3

p

Figure 4.4. Drag coefficient of a spherical particle dependent on the particle Reynolds number and the wall distance parameter λ = dp/D.

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Steffen Schütz, Kathrin Kissling, Martin Schilling et al.

Figure 4.5. Drag coefficient of a non-spherical particle dependent on the particle Reynolds number and the sphericity ψ.

4.2.2.4. Drag Force for Spherical Particles in the Molecular Regime

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For small particles with diameters in the range d p < 10 μm moving in an ambient gas the assumption of a continuum phase at the surface cannot be kept any more because the drag force is reduced dependent on molecular momentum impacts. So the classical adhesion boundary condition at the particle surface with zero relative velocity is no longer valid and molecular slip effects must be considered. In this case the approach for the drag force is taken as

FD =

ρg 1 CD v rel v rel A proj Cu 2

(4.17)

with the Cunningham factor Cu ≥ 1. The Cunningham factor is calculated according to

Cu = 1 +

⎛ δd p ⎞ ⎤ 2λ ⎡ ⎢β + γ exp ⎜ − ⎟⎥ d p ⎣⎢ ⎝ 2λ ⎠ ⎦⎥

(4.18)

In this equation some molecular parameters have to be determined:

ƒ

the mean free path length of the gas molecules

λ=

μg κρg v g

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(4.19)

Numerical Simulation of Multiphase Flow Systems

ƒ

the mean molecular velocity of the gas phase

vg =

ƒ

43

8k BT πm mol

(4.20)

the molecular mass

m mol =

Mg

(4.21)

NA

Within this calculation procedure some model parameters and physical constants are used:

ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ

Model parameter β = 1.155 Model parameter γ = 0.471 for dp 6 mm independent of the liquid phase material properties. Table 4.2. Drag coefficients for air bubbles in deionized water for 20°C Range of Re and We

Re < 0.49

0.49 < Re < 33

33 < Re < 661

661 < Re < 1237 and We ≈ 4

Re > 1237 and We < 8

CD = f (Re, We, Mo)

CD = CD =

20.68 Re0.643

CD = CD =

16 Re

72 Re

Re 4 Mo 48

CD =

We 3

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Table 4.3. Drag coefficients for air bubbles in tap water 20°C Range of Re and We

Re < 0.49

CD = f (Re, We, Mo)

CD =

0.49 < Re < 100

CD =

100 < Re < 717

CD =

Re > 717 and We < 8

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16 Re

20.68 Re0.643 6.3 Re0.385

CD =

We 3

Numerical Simulation of Multiphase Flow Systems

49

Figure 4.8. Rising velocity of single air bubbles dependent on the bubble diameter in pure water, tap water and an aqueous saline solution.

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4.4. Transport Equations Basing on Newton’s law and regarding the relevant pressure forces acting on a moving particle one of the first “comprehensive” transport equations was derived by Basset, Boussinesq and Oseen, the so-called BBO-equation.

ρp

d vp ( t ) π 3 d vp ( t ) π 1 = − 3d p πμf v p ( t ) − ρf d 3p + dp 6 dt 2 6 dt 3 π + d 3p ρ p − ρf g − d 2p πρf μ f 6 2

(

)

t

∫

t0

d vp ( τ) dτ

1 dτ t−τ

(4.34)

The terms on the right hand side of eq. (4.34) describe the Stokesian drag force, the added-mass force, the difference between gravity and buoyancy force and the Basset force. Strictly speaking, this equation is valid for an unsteady tracking of a spherical particle in a quiescent fluid phase. A first enhancement of the BBO-equation was given by Tchen (Tchen 1947) by replacing the absolute particle velocity by the relative velocity between particle and fluid phase

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v rel = v f ( t ) − v p ( t ) in an advecting fluid phase. However this simple substitution is only valid for a uniformly moved particle. A mathematically correct transformation of eq. (4.34) to the particle transport in an arbitrarily moving fluid phase was performed by Buevich (Buevich 1966) with the following expression

ρp

Dv f ( t ) π 3 d vp ( t ) π = 3d p πμf v f ( t ) − v p ( t ) + ρf d 3p dp 6 dt 6 Dt

(

+

)

π 3 ⎛ Dv f ( t ) dv p ( t ) ⎞ π 3 1 ρf − dp ⎜ ⎟ + d p ρ p − ρf g ⎜ Dt 2 6 d t ⎟⎠ 6 ⎝

3 + d 2p πρf μ f 2

(

t

∫

t0

⎛ Dv f ( τ ) dv p ( τ ) ⎞ − ⎜⎜ ⎟ d τ ⎠⎟ ⎝ Dτ

)

(4.35)

1 dτ . t−τ

With the equations (4.34) and (4.35) we have two transport equations for discrete particles assumed as mass points which are strictly speaking only valid for creeping flow

( Rep < 0.5)

around the particles. This is due to the fact that the different flow phenomena

and the resulting forces are superimposed by simple addition which is only valid for linear problems with the Stokesian formulation of the drag force.

(

)

For increasing particle Reynolds numbers Re p > 0.5 the transport phenomena become

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non-linear. Additional non-linear effects are considered implicitly by using the semiempirical approach for the drag force according to eq. (4.12) with the drag coefficient C D instead of the Stokesian formulation. The main difficulty is based on the fact that force interactions between two arbitrary forces in the transport equations, e. g. between the drag force and the added-mass force are not known. So coupling effects between the forces of higher order cannot be described explicitly. The most important parameter in the particle transport equations is the drag coefficient C D which is deduced from experiments anyway in the flow regimes

( Rep > 0.5) . With these experiments some non-linear effects are considered implicitly.

First of all the solution of the particle transport equations is the particle velocity dependent on time and position. A further integration of the particle velocity with respect to time gives the particle position and the particle trajectories in a flow field can be calculated. In reality, the simulation results based on the previous transport equations are in good agreement with experimental data. So the simplifying model assumptions in the transport equations are justified from practice. Another topic which outweighs the lack in the knowledge of force interactions is the application of a suitable numerical procedure of high order to integrate the ordinary differential equations (4.34) or (4.35). So the quality of the results depends strongly on the numerical solution procedure.

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4.5. Numerical Solution Procedure For the numerical solution of the particle transport equations the state variables (pressure, velocity components) of the continuous phase are assumed to be known from CFD simulation. To determine the particle trajectories a usually non-linear force balance for each particle in each discrete mesh cell has to be solved by time-integration and the velocity of each discrete particle is calculated. A further integration of the particle velocity with respect to a discrete time step Δt results in the new local particle coordinate. Subsequently some notes are given with respect to the appropriate choice of the mesh cell size Δx and the time step Δt .

4.5.1. Spatial Discretization

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The spatial discretization is performed by defining a mesh with discrete cells covering the physical space in which the flow process occurs. For Euler-Lagrangian models where the particles are treated as volumeless mass points the grid cells should be bigger than the finitevolume particle size, see Fig. 4.9. With a particle-cell ratio according to the left part of Fig. 4.9 the transport state of a single particle can be described by the clearly defined state variables of the grid cell where the particle belongs to. The characteristic fluid velocity determining the dynamic forces is either known at the cell midpoint or it can be interpolated from the node values of the corresponding grid cell. In the case of large particles compared to the cell dimensions as shown in the right part of Fig. 4.9 this clearly defined allocation of state variables of a single grid cell to the particle transport equation is not possible with a classical Euler-Lagrange scheme when the particle diameter exceeds the cell dimensions.

Figure 4.9. Comparison of particle size and grid cell dimensions, left: particle within a clearly assigned computational cell, right: particle covering multiple computational cells with no distinct cell allocation.

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No more than some years ago simulation and interpolation techniques to describe single particles exceeding the cell dimensions were developed to deal with larger particles on static simulation grids, see section 5.

4.5.2. Choice of Time Steps In section 4.2.2.1 the particle relaxation time τ p was defined as a characteristic time scale to describe the dynamic behavior of a single particle. For a full resolution of the particle dynamics in a simulation run the discrete time step Δt must be smaller than τ p , a good choice is

Δt ≈ 0.25τ p ÷ 0.4τ p

(4.36)

If the time step is chosen bigger than τ p the particle is always calculated in a quasi-

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steady state. In practice the user must decide if the simulation run should provide detailed data on the particle transport dynamics or not. E. g. if a particle in a flow system experiences a slow surface reaction and if only the reaction progress is the interesting result of the simulation run the time-dependent transport behavior need not to be resolved. Generally speaking the particle relaxation time becomes smaller with lower ratios between the particle density and the fluid viscosity according to the definition of τ p , see eq. (4.10). This means that the time steps must be chosen smaller in solid-liquid systems than in solid-gas systems. But one must consider that a time step reduction cannot be performed in any order. With a small time step Δt and simultaneously “big” grid cell dimensions Δx one obtains a high number of discrete particle transport steps within each single grid cell. This procedure is not reasonable because it is always the same set of state variables of the fluid phase in the respected grid cell by which the particle transport behavior is determined. Hence small time steps require a fine discretization of the physical space. A characteristic parameter describing the ratio of Δt and Δx is the (grid cell specific) Courant number

Co =

v f Δt Δx

(4.37)

giving the ratio of the distance v f Δt a fluid element passes within one time step Δt compared to the cell dimension Δx . For a good time resolution of a transport process the Courant number should comprise values in the range 0.25 ≤ Co ≤ 0.4 . This means that two to four time steps are used to determine the transport through a single grid cell.

4.5.3. Uncoupled and Coupled Solution Algorithm and Numerical Integration of the Particle Transport Equation The numerical solution of the transport equations of the fluid phase and the discrete particles can be performed in an uncoupled or in a coupled manner. With the uncoupled solution

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procedure the transport equations of the continuous phase are solved first and the distribution of pressure, velocity, temperature,… is determined. Basing on these field variables the force balances for the particle trajectories are solved subsequently and the particle positions are calculated dependent on time. The impact of the particles on the fluid motion can be neglected with low particle volume concentrations (less than five volume percent) and low particle densities. This consecutive solution procedure shows a one-way coupling as the particle transport is affected by the fluid motion but actually it is denoted uncoupled algorithm. The solution algorithm is shown in Fig. 4.10.

Figure 4.10. Solution algorithm for the uncoupled simulation procedure.

An example for a simple particle transport equation is given with

d vp dt

(

= C D Re p

(

= C D Re p

) 43 ρ ρfd p

)

p

v p,rel v p,rel +

ρ p − ρf ρp

g

ρ p − ρf 3 ρf vf − v p vf − v p + g 4 ρp d p ρp

(

)

(4.38)

In this equation, the impact of the drag force, gravity force and the buoyancy force on the particle transport is considered. Form the mathematical viewpoint, eq. (4.38) is an ordinary non-linear differential equation. The non-linearity is caused by the combination of the absolute and the relative particle velocity but also implicitly via the drag coefficient C D

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which depends on the relative velocity. So the characteristic drag coefficient defining the particle transport in a discrete time step Δt = t n +1 − t n must be determined from a mean value +1 of the relative particle velocity v np,rel at time t n and the unknown (!) value vnp,rel at time +1 t n +1 . As we have to estimate a first value vnp,rel to determine the drag coefficient C D , the +1 and C D have to be improved iteratively in each time step Δt which is values of vnp,rel

symbolized by the inner iteration loop in Fig. 4.10. n+1

The key topic in these algorithms is the calculation of the new particle velocity vp

at

time t n +1 basing on the values v np at time t n e.g. on the basis of eq. (4.38). This step is referred to with the bottom box in Fig. 4.10. For the integration procedure of this ordinary, non-linear differential equation, various numerical methods can be applied. With a simple discretization scheme the differential quotient is approximated by a simple differencing scheme:

dv p dt

(

where f t, v p

)

≈

v np +1 − v np Δt

(

= f t, v p

)

(4.39)

denotes the right hand side of eq. (4.38). With this differencing scheme of

first order accuracy the truncation error in the numerical procedure is proportional to ( Δt ) . 1

When all variables on the right hand side of eq. (4.38) are evaluated with the known values at

(

)

the time point t n with f t n , v np , the numerical procedure is called explicit Euler scheme.

(

)

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The computation of all state variables on the right hand side of eq. (4.38) as f t n +1 , v np +1 at time t n +1 leads to an implicit Euler scheme. As the application of a first order integration scheme results in modest accuracy also with very small time steps Δt one usually prefers higher order integration methods. A very popular method is the explicit Runge-Kutta scheme (Hairer et al. 1990) of fourth order accuracy with a truncation error dependent on ( Δt ) . The integration procedure is described 4

by

v np +1 = v pn +

1 ( k1 + 2k 2 + 2k3 + k 4 ) Δt 6

(

k1 = f t n , v pn

)

1 1 ⎛ ⎞ k 2 = f ⎜ t n + Δt, v pn + k1Δt ⎟ 2 2 ⎝ ⎠

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(4.40)

(4.41)

(4.42)

Numerical Simulation of Multiphase Flow Systems

1 1 ⎛ ⎞ k 3 = f ⎜ t n + Δt, v np + k 2 Δt ⎟ 2 2 ⎝ ⎠

(

k 4 = f t n + Δt, v pn + k 3 Δt

)

55 (4.43)

(4.44)

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In many commercial CFD software packages this integration procedure is available. Otherwise it can be implemented via programming interfaces by user defined subroutines. With increasing particle concentrations the reaction of the particles on the fluid phase cannot be neglected any more. Then the transport equations of the continuous fluid phase are only solved until a certain degree of convergence is reached. The particle transport and the interaction forces are determined and the flow field of the continuous phase is calculated again. The appropriate algorithmic scheme of this coupled solution procedure is shown with Fig. 4.11. The second box in Fig. 4.11 comprises the complete algorithmic scheme for the noncoupled simulation according to Fig. 4.10.

Figure 4.11. Solution algorithm for the coupled simulation procedure.

4.6. Particles in Turbulent Flow Regimes The particle transport in turbulent flows is not only affected by the macroscopic velocity field but also by the microscopic fluctuation velocities. So the small scale momentum forces due to turbulence especially acting on small particles have great impact on the resulting particle trajectories. As a direct numerical simulation is not possible the turbulent particle transport is described by statistic models (Berlemont et al. 1990, Calabrese and Middleman 1979, Chen and Wood 1984, Corrsin and Lumley 1956, Desjonqueres et al. 1988, Deutsch and Simon 1991, Ebert 1992, Elgobashi and Abou-Arab 1983, Gore and Crowe 1989, Hall 1975, Hetsroni 1989, Sommerfeld 1990).

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4.6.1. Stochastic Particle Tracking One of the most popular simulation models to describe the turbulent transport of single particles is the Discrete Random Walk (DRW) model. The basic idea is the linear superposition of a macroscopic particle velocity as it is calculated from a force balance and a microscopic particle fluctuation velocity determined from a stochastic particle tracking model. With these two characteristic velocities, particle trajectories are calculated. In this section the DRW model which is used commonly in CFD simulations is presented to determine the particle fluctuation velocity. For a basic understanding of this model two characteristic turbulent time scales must be introduced. The first one is the time scale of dissipation. This is equivalent to the life-time of small eddies in the range of the turbulent microscale. It is calculated from

τL = CL

k ε

(4.45)

The turbulent kinetic energy k and the turbulent dissipation rate ε are taken from the results obtained from solving turbulence models according to section 2.2.2. The parameter C L is assumed to be C L = 0.15 using a k-ε-model and C L = 0.3 with a Reynolds stress model. From this time scale the mean life-time τe of turbulent eddies is calculated either by

τe = 2 τ L

(4.46)

τe = τ L log ( r )

(4.47)

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or by the stochastic approach

with the random number 0 ≤ r ≤ 1 . The second time-scale is the crossing-time a particle needs to pass through a turbulent eddy. This crossing-time is estimated as

⎡ ⎛ Le τcross = −τ p ln ⎢1 − ⎜ ⎢ ⎜ τp v f − v p ⎣ ⎝

⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦

(4.48)

The turbulent macro-scale Le is calculated as

L e = cμ

k 3/ 2 ≈ 0.07D ε

(4.49)

and often approximated with 7% of a characteristic length scale D of the flow geometry, e. g. the diameter of a tube or a vessel.

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The components v ′p,i ( i = x, y,z ) of the particle fluctuation velocity are determined by the fluid fluctuation velocity scaled by a Gaussian distributed random number r:

v ′p,i = r

( v f′ ,i )

2

(4.50)

The cross bar denotes a time-averaged value. Using a turbulence model for the continuous phase which describes the turbulence structure as isotropic like the k-ε-model all components v ′p,i of the fluctuation velocity vector v ′p are identical and can be evaluated as

v ′p,i =

2 k 3

(4.51)

from the definition of the turbulent kinetic energy. With a turbulence model considering the anisotropic character of turbulence like the Reynolds stress model, one can calculate the different components of the particle fluctuation velocity. The turbulent particle fluctuation is a highly unsteady transport process. The time interval Δt fluc for which the particle fluctuation velocity is assumed with a specific value v ′p is taken as the minimum one comparing the eddy life time τe and the time τcross meanwhile a particle crosses a turbulent eddy:

Δt fluc = min( τe , τcross )

(4.52)

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After one time interval Δt fluc the turbulent particle fluctuation velocity is re-calculated. Usually this time interval Δt fluc is longer than a discrete time step Δt in the simulation run. Another model in which the turbulent fluctuation velocity of the particles is calculated at each numerical time step is the continuous random walk model (CRW model) which is used scarcely due to the necessary calculation effort. Only with highly fluctuating turbulent velocities with an extreme anisotropy in the turbulence structure the CRW model shows better results than the DRW approach.

4.6.2. Cloud Model With the particle cloud model a mean particle trajectory is calculated for an ensemble of particles. Along this mean trajectory a particle cloud consisting of multiple particles is defined. The concentration distribution of the particles around the mean trajectory is described by a Gaussian probability density function. The width of the concentration distribution is described with the variance of the Gaussian distribution function which depends on the turbulent fluctuation velocity.

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4.7. Particle-Wall Collisions Particle-wall collisions play an important role in many multiphase transport processes where the impact of solid walls on the particle transport behavior is of importance e. g. in pneumatic conveying or in particle-fluid separation where the particle separation occurs on solid walls. The most important parameters which determine particle-wall interactions are the particle size and shape, the turbulence intensity of the fluid phase, the collision angle and the wall roughness. The material data of wall and particle determine the type of collision (Durst and Rasziller 1989, Sommerfeld 1992, Sommerfeld et al. 1993, Tsuji et al. 1989).

4.7.1. Elastic and Plastic Collision With the detailed description of a particle-wall collision the restitution coefficient characterizes whether the collision is elastic or plastic. With the y-coordinate perpendicular to a solid wall, the restitution coefficient is defined as

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e=−

v p2,y

(4.53)

v p1,y

with the indices 1 and 2 describing the particle velocities before and after the collision. For a value e = 1 the collision is completely elastic, with e = 0 the collision is plastic. The value of the restitution coefficient is determined by the material combination of particle and wall. The collision process itself is subdivided into the two phases of compression and rebound. When the particle stops slipping parallel to a surface during the compression phase the collision is defined as non-slipping. With a slipping collision the particle stops slipping only in the rebound phase or does not stop slipping at all. For all kinds of particle-wall collisions the result of a detailed particle transport simulation are the translational and rotational velocity components after the collision. In the following two subsections velocity correlations are given for a slipping and for a non-slipping collision. The indices 1 and 2 denote again the state variables before and after collision. 4.7.1.1. Slipping Collision For a slipping collision the translational and the angular velocity components of a particle after the collision are calculated from

v p2,x = v p1,x + μs

v p1,x +

dp

ωp1,z

2

v p1,rel

(1 + e ) v p1,y

v p2,y = −ev p1,y v p2,z = v p1,z + μs

(4.54)

v p1,z −

dp 2

ωp1,x

v p1,rel

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(1 + e ) v p1,y

Numerical Simulation of Multiphase Flow Systems

ωp2,x = ωp1,x − 5μs

v p1,z −

dp

ωp1,x

2

v p1,rel

(1 + e )

59

v p1,y dp

ωp2,y = ωp1,y

(4.55)

ωp2,z = ωp1,z + 5μs

v p1,x +

dp 2

ωp1,z

v p1,rel

(1 + e )

v p1,y dp

where μS is the material dependent friction coefficient. The relative velocity between particle and wall before the collision is calculated from 2

v p1,rel

dp dp ⎛ ⎞ ⎛ ⎞ = ⎜ v p1,x + ωp1,z ⎟ + ⎜ v p1,z − ωp1,x ⎟ 2 2 ⎝ ⎠ ⎝ ⎠

2

(4.56)

regarding the translational and the rotational component. The limiting condition for a slipping collision is 7 v p1,rel ≥ μ 0 (1 + e ) v p1,y (4.57) 2 with the adhesion coefficient μ0 .

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4.7.1.2. Non-slipping Collision For a non-slipping collision the translational and the angular velocity components of a particle after the collision are calculated from

dp ⎞ 5⎛ v p2,x = ⎜ v p1,x − ωp1,z ⎟ 7⎝ 5 ⎠ v p2,y = −ev p1,y

(4.58)

dp ⎞ 5⎛ v p2,z = ⎜ v p1,z + ωp1,x ⎟ 7⎝ 5 ⎠ ωp2,x =

2v p2,z dp

ωp2,y = ωp1,y ωp2,z = −

2v p2,x dp

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(4.59)

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4.7.2. Wall Roughness Wall roughness has a great impact on the particle transport direction after a wall collision (Sommerfeld 1996). In Fig. 4.12 two limiting cases are shown with two particles whose diameters are smaller or bigger than a characteristic geometric length of the wall structure. In a simple model, the wall roughness is assumed to be periodical with a characteristic height H R and with a characteristic length L R . The maximum reflexion angle for a small spherical particle with d p < L R is

⎛ HR ⎞ ϕ max = arctan ⎜ ⎟ ⎝ LR / 2 ⎠

(4.60)

assuming that the particle penetrates completely the roughness structure. For a big particle d p > L R the maximum reflexion angle is calculated from

⎛ H /2 ⎞ ϕ max = arctan ⎜ R ⎟ ⎝ LR ⎠

(4.61)

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when the particle “lies” upon the roughness and penetrates the structure with half the roughness height.

Figure 4.12. Particles and wall roughness.

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For the numerical solution of the particle-wall collision model the particle trajectory is extrapolated for each new time step. If within one time step a particle-wall collision occurs the components of the translational and the rotational particle velocity according to sections 4.7.1.1 or 4.7.1.2 are computed and the new particle positions are determined.

4.8. Particle-Particle Collisions At the beginning of this section the ratio

L of the mean particle distance and the particle dp

diameter was defined to distinguish diluted and dense multiphase flow systems. Another criterion can be derived regarding the frequency of particle-particle collisions (Abrahamson 1975, Tanaka and Tsuji 1991). Comparing two characteristic time scales, the relaxation time τ p according to eq. (4.10) and the collision time τcoll as the reciprocal of the collision frequency f coll , a multiphase system is assumed to be diluted for

τp τcoll

< 1 and to be dense

τp

> 1 . In the second case the transport behavior of single particles is more τcoll determined by particle-particle collisions than by the dynamics of the continuous phase. The collision frequency for spherical particles can be derived on kinetic gas theory with

with

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f coll =

1 τcoll

=

z1,2 N p,1,vol

=

π d p,1 + d p,2 4

(

)

2

v p,1 − v p,2 N p,2,vol

(4.62)

In this equation z1,2 is the number of collisions per time and volume among all particles of the two particle size classes with the mean diameters d p,1 and d p,2 . The variable N p,vol denotes the volumetric number density of these particles. Very detailed collision models base on a direct numerical simulation of collision events. With these models a multitude of particle trajectories must be calculated by solving force balances and the particle positions must be stored. The particles themselves are treated as dimensionless mass points but the particle size must be known. For each time step the new positions of the particles are determined and all particle positions are compared reciprocally to determine if there occur particle collisions. The criterion for a collision event is defined by

rcoll,i, j
8 mm

slug flow

α2m =



π/6 5 (dp −8 mm)+α2 α2

: dp < 8 mm : dp > 8 mm

α3m =



π/6 5 (dp −8 mm)+α3 α3

: dp < 8 mm : dp > 8 mm

α4m =



π/6 5 (dp −6 mm)+α4 α4

: dp < 6 mm : dp > 6 mm

1

original modified

0.9 0.8 0.7 0.6

void α

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Db = 3.75 mm. Thus, the applied bubble diameter become larger than the pores for small particles, in contradiction to the geometric ideas of the model. Additionally, the assumption of gas bubbles in the pores becomes questionable for small particles. This requires some modifications of the original Tung/Dhir model to extend it to smaller particle diameters. The first point to be modified, is the diameter of the gas bubbles or the slugs. This diameter strongly influences the interfacial drag in bubbly and slug flow, as could be seen in equation (18). Based on the assumption of a cubic arrangement of spherical particles √ a maximum diameter of Dbmax = dp ( 2 − 1) may be deduced for a bubble. To obtain a connection to the original Tung/Dhir model, a modified bubble diameter is simply defined by:   r σ Dbm = min 1.35 , 0.41 dp . (24) g (̺l − ̺g )

0.5 0.4 0.3 0.2

bubbly flow 0.1 0

0

2

4

6

8

10

12

14

particle diameter dp [mm]

Figure 2. Flow pattern map for the modified Tung/Dhir model. Additionally, further modifications to the interfacial drag in the annular flow regime are Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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necessary. As already discussed above, for smaller particles, channel flow is established. Compared to the classical picture of Tung and Dhir, the interfacial area between gas and liquid is reduced. This motivates a decrease of the drag for decreasing particle diameter. By adaptation to the experimental results of Tutu et al. [19], discussed in detail in the next
2 chapter, a multiplicative factor of dp /6x10−3 is proposed for particles smaller than 6 mm in this work. Additionally, as can be seen in the formulation of the friction term for annular flow in equation (23), this drag decreases linearly to zero when the void fraction reaches the limit α → 1. This decent seems to be too weak compared to usual correlations. So, in this work an additional multiplier of (1−α)2 is proposed to get a more realistic decrease of the interfacial drag in the annular flow regime for increasing void fraction. This leads to the following modified formulation of the interfacial friction in the annular flow regime:   ̺g µg m (1 − α) jr + (1 − α) α |jr | jr ∗ Fi = K Krg η ηrg  (25) 2  dp : d < 6 mm p −3 2 6x10 (1 − α) 1 : d > 6 mm p

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In general, all modifications proposed above influence mainly the formulation of the interfacial drag. As will be seen in the next chapter, in general it is not easy to separate the different friction contributions in the experiments. Interfacial and particle drag are superimposed, but can be separated for specific conditions with no net water flow. On the other hand, for boiling beds - as in the reactor application - this splitting is not possible. One has to rely on the assumption, that the friction laws are valid. Comparisons to experiments with boiling beds are presented in the next chapter. Especially the DEBRIS experiments described in chapter 4.2.1. support the above proposed modifications of the interfacial drag.

4. 4.1.

Application of the Models Comparison to Isothermal Air/Water Flow Experiments

Isothermal air/water experiments are a sufficient method to investigate the friction laws of two-phase flow in porous media with a simple experimental setup. By fixing the water and air flow rates through a vertical test column filled with particles, defined steady state conditions, either for co- or for counter-current flow, may be established. As the gas flow rate from the bottom to the top is fixed, a constant void fraction establishes over almost the whole bed height. Therefore, the capillary pressure is constant and its gradient is zero. So, the same pressure gradient acts on both fluids. This pressure gradient in the test column can easily be measured. Additionally, the void fraction in the bed can be determined, either by visual observation through transparent vessel walls, or by a change of the water level on the top. The experimental data can be compared to theoretical results of the models to verify the friction laws. For this, the momentum conservation equations (9) and (10) are used in a dimensionless form. Dividing by gε(̺l −̺g ) yields:

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258

Werner Schmidt ̺l g + Fpl⋆ − Fi⋆ g (̺l − ̺g ) ̺g g ⋆ αP⋆ = α + Fpg + Fi⋆ g (̺l − ̺g )

(1 − α) P ⋆ = (1 − α)

(26) (27)

with P⋆ =

−∇p , g(̺l − ̺g )

F⋆ =

F . g ε (̺l − ̺g )

(28)

The two friction terms of the particle-fluid drag are combined in the force Fp . Eliminating the normed pressure gradient P ⋆ yields: ⋆ α (1 − α) + α Fpl⋆ − (1 − α) Fpg − Fi⋆ = 0

(29)

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Inserting the friction laws presented in the previous chapter provides an equation with the three unknowns jg , jl and α. Fixing one velocity, usually liquid, and varying the void fraction, the other velocity can be calculated easily by selecting the physically relevant root. Additionally, adding the two equations (26) and (27) yields an equation for the pressure gradient: (1 − α)̺l + α̺g ⋆ P⋆ = + Fpl⋆ + Fpg (30) ̺l − ̺g

So, sets of jg , jl , P ⋆ and α may be calculated that can directly be compared with the experimental values. A comparison of calculated results with experimental air/water data measured by Tutu et al. [19] is shown in Figure 3 and 4. Air was injected into the bottom of a water filled test column of stainless steel spheres (dp = 6.35 mm). The superficial velocity of the air was varied, corresponding to the mass flux through the test column. For the established steady states the pressure gradient was measured by the difference of two pressure taps. Additionally, the void fraction in the bed was determined by the increase of the liquid level above the bed. In Figure 3 the dimensionless pressure gradient is plotted as function of the superficial velocity of the gas. The Figure shows that the pressure gradient becomes less than one, indicating a pressure loss due to the gas flow. The classical models, which do not explicitly consider the interfacial friction, cannot reproduce this behaviour. This can already be seen in equation (26). As there is no net water flow the superficial velocity jl = 0 leads to Fpl⋆ = 0. With Fi⋆ = 0 this yields P ⋆ = ̺l /(̺l −̺g ) ≈ 1. In these models the pressure field must always be equal to the hydrostatic pressure, uninfluenced by the gas flux. This behaviour is not verified by the experimental data. The measured pressure first decreases with increasing gas flow rate, although the water is still the continuous phase. This can only be explained by including the drag of the up-flowing gas on the liquid. The models of Schulenberg and Tung/Dhir, which include an explicit formulation for the interfacial friction, show this characteristic qualitatively. As already mentioned in the last chapter, the interfacial friction law in the annular flow regime of the Tung/Dhir model must be modified to obtain a reliable trend for large voids up to one. This can also be seen in Figure 3. With increasing gas flow rate, the pressure gradient first strongly decreases below the hydrostatic one due to the drag of the up-flowing

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1

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*

p [-]

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Lipinski (p*≈1) * Reed (p ≈1) Hu/Theofanous (p*≈1) Tung/Dhir mod.Tung/Dhir Schulenberg p* Exp. Tutu (1984)

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Figure 3. Dimensionless pressure gradient P ⋆ for an isothermal air/water experiment with no net water flow. gas bubbles and slugs on the liquid. Later, with further increased gas flow, the friction between the phases decreases again for void fractions greater than 0.5, as the interfacial area between the gas and the water decreases. This explains the increase in the normed pressure gradient for higher gas fluxes. A limit is reached when the gas replaces all the water. This is the case when the pressure loss due to the gas flow is equal to the hydrostatic head of the water, independent of the model. This limit is not reached for the original Tung/Dhir model. To compensate this, it is proposed in the previous chapter to multiply the interfacial friction by (1−α)2 in the annular flow regime, to obtain a cubic decrease for large void fractions. The comparison with the experimental data, especially the development for high gas fluxes in Figure 3, strongly supports this modification of the Tung/Dhir model. Similar conclusions are drawn from the corresponding void data in Figure 4. Although the experimental data points are not up to the annular flow regime, the modified Tung/Dhir model fits the experimental points better. This is due to the decreased bubble diameter in the slug and bubbly flow regime (Db = 2.6 mm modified instead of 3.75 mm in the original form), yielding an increased interfacial friction. Additionally, as already described for the pressure gradient, the tendency in the annular flow regime for the original Tung/Dhir formulation is again not reliable. The one-phase flow limit with α = 1 should be reached at the same gas velocity for all models, independent of the interfacial drag. The modified formulation, with decreased interfacial friction in the annular flow regime, fulfills this condition. For the special case without net water flow in the porous medium (jl = 0), an enhanced analysis is possible. The interfacial friction can be deduced directly from the measured

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Werner Schmidt 1 0.9 0.8

void α [-]

0.7 0.6 0.5 0.4 Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg α Exp. Tutu (1984)

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Figure 4. Void fraction α in the case of Figure 3. pressure gradient and void fraction via the liquid momentum equation given in (26). The results for the described data, together with the results of the various models, are given in Figure 5 as function of the void fraction. While the interfacial friction is always zero for the classical models, a principle agreement of the enhanced models with the experimental data is seen. But, especially the local maximum at higher void (higher jg ) in the original Tung/Dhir formulation seems to be not verified by the experiment. This hump originates in the annular flow regime. Furthermore, as already mentioned, the linear decrease of Fi⋆ for α → 1 is unreliable. These faults are eliminated by the modifications of the Tung/Dhir model proposed in chapter 3.2.3., which then yields the best estimate for the experimental data. Measurements for larger particles with a diameter of dp = 9.9 mm (Chu et al. [3]) and dp = 12.7 mm (Tutu et al. [19]) yield similar results. A comparison of the experimental data with the different models is given in Schmidt [14]. For smaller particle diameters, which are to be expected during a severe accident in a nuclear power plant, the models show greater deviations compared to the experimental data. Results of Tutu et al. for dp = 3.18 mm are presented in Figures 6 and 7. The described modifications of the original Tung/Dhir model - especially the modifications of the flow pattern limits and the modified slug size - are of decisive influence for smaller particle sizes. E.g. the annular flow regime starts at a void fraction of 0.45, while the pure bubbly flow disappears, obviously because the pores are too small for the bubbles. The influence of the interfacial friction is plotted in Figure 6. The decrease in the pressure gradient with increasing gas flow rate is only reproduced by the enhanced models including the interfacial friction explicitly. But the further measured increase for higher gas

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0.18

dp = 6.35 mm ε = 0.38 j0L = 0.0 mm/s

0.16

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Lipinski (Fi =0) * Reed (Fi =0) * Hu/Theofanous (Fi =0) Tung/Dhir mod.Tung/Dhir Schulenberg F*i Exp. Tutu (1984)

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Fi [-]

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0.08 0.06 0.04 0.02 0 0

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Figure 5. Experimentally and theoretically deduced dimensionless interfacial friction in the case of Figure 3. fluxes is steeper than expected from these models. Additionally, at the void development, the modified Tung/Dhir model only yields satisfying results. This is seen even better for the interfacial friction, given in Figure 8. Especially for voids greater then α = 0.5 the measured interfacial drag is zero. This can be explained in the flow pattern picture with the transition to a channel-like configuration with minor contact of gas and water, as described by Haga et al. [7]. The interfacial friction is significantly reduced because of the smaller interfacial area in this flow pattern. In the modifications to the Tung/Dhir model proposed in the previous chapter, this reduction is included in the redefined lower limit for the annular flow regime and in the reduction of the interfacial friction with decreasing particle diameter. As such small particulate debris has to be expected during a severe accidents in a nuclear power plant, the modifications gain importance for these applications. Chu et al. [3] also performed corresponding experiments with a net water flow in cocurrent as well as in counter-current configuration. The co-current results, with water and air injection into the bottom, confirm the above results in principle. As an example, results for a fixed water superficial velocity are given in Figure 9. As, per definition, the net water flow is fixed, the void limit of α = 1 can not be reached in this case. Increasing the gas mass flux yields an increasing pressure gradient, that presses the fluids through the porous structures, as can be seen in Figure 9. More interesting with respect to debris coolability is the counter-current flow configuration, where water is added to the top and exhausted from the bottom of the test section, yielding a top-to-bottom flow. Then, the bottom injected gas and exhausted water are in counter-current mode, similar to the case of a boiling bed with a coolant pool on the top.

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1

dp = 3.18 mm ε = 0.39 j0L = 0.0 mm/s

0.8

*

p [-]

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Lipinski (p*≈1) Reed (p*≈1) Hu/Theofanous (p*≈1) Tung/Dhir mod.Tung/Dhir Schulenberg * p Exp. Tutu (1984)

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Figure 6. Dimensionless pressure gradient P ⋆ for an isothermal experiment filled with smaller particles (dp = 3.18 mm). Again, for fixed water mass flux, the air inlet rate is varied. Results of the measured pressure gradient and void fraction, as well as the corresponding theoretical results are given in Figure 10. In contrast to the co-current case, an upper limit for the gas superficial velocity exists, corresponding to a maximum gas mass flux. This limit is the counter-current flooding limit. No larger gas fluxes than this limit are possible in such a configuration. From Figure 10 it can be seen that this value corresponds to a maximum void fraction. This fraction cannot be one, because a certain amount of the cross-section or volume is necessary for the water flow. A further increase of the gas flow rate is not possible because this would hinder the water flow.

4.2.

Comparison to Experiments with Boiling Debris Beds

In contrast to the isothermal air/water experiments, the local flow rates vary inside a boiling bed, even in a one dimensional configuration. The local gas flux, driven by buoyancy forces, is determined by the integrated steam flux, and thus by the bed power integrated from the bottom. For a homogeneous power distribution the steam mass flux is given by ̺g jg (z) = Q z/LH, where Q is the volumetric power density and LH = hg − hl is the specific latent heat of the evaporation. In a steady state, the corresponding local water flux is directly calculated by the mass conservation equation. Assuming water inflow just from a pool above the bed, the liquid mass flux follows from ̺l jl (z) = −̺g jg (z). The dryout heat flux of 1-D boiling debris corresponds to the counter-current flooding

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1

void α [-]

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jG [m/s]

Figure 7. Void fraction α in the case of Figure 6.

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limit. This limit is reached near the top, where the steam flux as well as the down-flowing water flux are highest. The friction at this bottleneck - the location of highest steam fraction - is decisive for the coolability of the whole bed. 4.2.1. The DEBRIS Experiment For specific investigations of the exchange terms, the friction laws, as well as the heat transfers in boiling particulate beds, experiments have been conducted in the DEBRIS facility at IKE, University of Stuttgart [15]. The experimental setup is sketched in Figure 11. A ceramic cylinder of 12.5 cm diameter and a height of 60 cm is filled with oxidised steel spheres. These particles are heated inductively to represent the decay heat. The power distribution is nearly homogeneous. 64 thermoelements are distributed in the test column to detect local dryout. Along the bed height, 8 pressure tubes are connected to differential pressure transducers to allow pressure gradient measurements at 7 different levels. The coolant flowing into the porous region comes from a water pool above the bed. Optionally, an adjusted water inflow rate from below can be injected. With this facility dryout experiment, as well as experiments on quenching of dry, hot particles are possible. Besides the direct measurements of the dryout heat flux by increasing the bed power until the first local temperature raise is detected, experiments with steady states in boiling beds can be used to deduce the friction laws. The produced up-flowing steam accumulates from bottom to top while the evaporated water must be replaced by water inflow. So, for a steady state the water and steam fluxes at each level are defined by the total power below. The measured pressure gradients can again be used to compare the various friction

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models with the experimental data. All the measured values for different heating power can be collected in one plot. Figure 12 shows the experimental data for 6 mm particles at a pressure of 1 bar. The pressure gradient, adjusted by the hydrostatic head of water, is shown versus the superficial velocity of the steam jg . Because of the statistical character of the measurements, an error in the order of 500 to 1000 Pa/m must be assumed for the experimental data. Additional to the superficial velocity of the steam jg the corresponding superficial liquid velocity jl is provided as second x-axis in the plots. In the top part of Figure 12 the results for a pure top-fed configuration are shown. The steam and the water fluxes are always in counter-current mode, as can be seen by comparing the different x-axises. With fixed water injection from the bottom, co-current mode occurs in lower particle bed regions, as can be seen on the jl -axis in the lower part of the Figure. In addition to the experimental data, results of calculations based on the different models for the given conditions are also plotted in Figure 12. In principle, the development of the pressure gradient shows the same behaviour as in the counter-current air/water flow of Figure 10. Again, it can be seen that the classical models, without interfacial friction, cannot reproduce the experimental data, and must be rejected. Only the enhanced models, including an explicit formulation of the interfacial drag, fit qualitatively to the experimental results. Numerous data points have been measured for small vapour velocities. In this range, the enhanced models show no significant difference. For higher gas fluxes, where stronger

0.14

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*

Fi [-]

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dp = 3.18 mm ε = 0.39 0 jL = 0.0 mm/s

*

Lipinski (Fi =0) * Reed (F i =0) * Hu/Theofanous (F i =0) Tung/Dhir mod.Tung/Dhir Schulenberg F *i Exp. Tutu (1984)

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Figure 8. Experimentally and theoretically deduced dimensionless interfacial friction in the case of Figure 6. Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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1.2

p* , α [-]

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Figure 9. P ⋆ and α for co-current air/water flow with constant water flow rate.

1 0.9

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*

p , α [-]

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0.8

0.4 Lipinski Reed Hu/Theofanous Tung/Dhir mod.Tung/Dhir Schulenberg p* Exp. Chu (1983) α Exp. Chu (1983)

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Figure 10. P ⋆ and α for counter-current air/water flow with constant water flow rate. difference between the models are observed, it was difficult to establish steady states. Only a few data points with a larger spread could have been measured in this range. Based on this

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R e f lu x C o n d e n s e r

O v e r f lo w s

P L 7

d p 7

5 2 4 9

P L 6 d p 6 P L 5 d p 5 P L 4

d p 4

P L 3 d p 3

C o V o P re H =

n tr lu m s s u 1 0 0

o l e re , m m

a m b ie n t r e s p p l7

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W

P L 2

d p 2 d p 1

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4 5 4 1 3 7 3 3 2 9 2 5 2 1 1 7 13 9

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1 9 0

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1 4 0 9 0 4 0 0

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o l e ., m m

a t e r In je c t io n

Figure 11. DEBRIS experimental setup.

data alone, no final conclusion can yet be drawn to finally prove the model formulations. The development of the classical models to obtain the measured dryout heat flux, corresponding to the maximum jg , can also be seen in Figure 12. The first approach by Lipinski yields a dryout heat flux higher than measured for top cooled particulate beds. By increasing the friction in the relative passability, as done by Reed, the measured critical heat flux is better reproduced. In Figure 12 this is seen by the smaller maximum reachable gas velocity. As mentioned, Hu and Theofanous [9] criticised the published dryout heat flux data due to the measuring procedure, and consequently introduced even stronger particle fluid friction. This adaptation process can also be seen in the isothermal air/water experiments shown in Figure 10. By inspecting the pressure gradient development, it can be seen that this adjustment does not represent the measured pressure losses. These can only be explained by including the interfacial friction, similar to the case without net water flow. In principle, the results with additional water feeding from below show the same behaviour, as can be seen in the lower plot. The pressure field obtained by the enhanced models again shows the basic behaviour. These models fit well in both configurations, the

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-∇p-ρLg [Pa/m]

0

+ +++ -1000 + + + + -2000 + + +++ + ++ -3000 + -4000

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+ ++ ++

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Figure 12. Pressure gradient in steady states in the DEBRIS experiments for dp = 6 mm (top: without water injection from below; bottom: with water injection from below). counter-current (jl < 0)) as well as the co-current flow (jl ≥ 0)). Data of analogous measurements with smaller particles with dp = 3 mm are provided in Figure 13 together with the results of the models. The sub-figure on the top represents a pure

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-∇p-ρLg [Pa/m]

0

+ + + + + ++ + + -1000 + +++ + + + + + ++ ++ +++ + + ++ + + + + + -2000 + + + +

++ + +

+ + + + + ++ + +

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-∇p-ρLg [Pa/m]

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-1000

+ +

+ +

+ + + + + + + +

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Figure 13. Pressure gradient in steady states in the DEBRIS experiments for dp = 3 mm (top: without water injection from below; bottom: with water injection from below).

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top-fed configuration, while the lower plot shows the results with a small water inflow of jl = 0.2 mm/s from below. This yields data points in co-current as well as in counter-current mode. As was already seen for the isothermal experiments in the previous section, the modifications to the Tung/Dhir model proposed in chapter 3.2.3. gain importance for such small particles. This is observed when comparing the calculated results of the modified with the ones of the original formulation. While the original Tung/Dhir model yields a steady decrease of the pressure gradient with increasing vapour flux, the modified formulation also shows the interim increase due to the reduced interfacial drag in the annular flow regime and fits the measured data well. This behaviour is also predicted by the Schulenberg model. But again, due to the spread in the experimental data, no final conclusion can be drawn on which of the these two models is the better one. 4.2.2. Application to DHF Experiments Various experimental programs for direct investigation of the dryout heat flux for volumetric heated porous structures were performed, especially in the early 1980’s. The majority of these experiments the particulate debris consists of inductively heated metallic spheres in a one-dimensional test column. Some authors used real granular particle beds and applied direct electrical resistance heating (e.g. Hu and Theofanous [9]) or heated the particles by irradiation (e.g. DCC experiments, Reed et al. [13]). The coolant, mostly water, infiltrates from a pool at the top. Under these conditions the first dryout has to be expected after a very slow transient slightly above the limiting bed power near the bottom of the test column. 4 Lipinski Reed Theofanous Tung/Dhir

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2

dryout heat flux Q dry [MW/m ]

3.5

mod. T/D

3

Schulenberg Exp. Barleon/Werle (1981) Exp. Stevens/Trenberth (1982)

2.5

Exp. Squarer et al. (1982) Exp. Catton et al. (1983) Exp. Hofmann (1984)

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Figure 14. Dryout heat flux in top-fed beds for different particle diameters. Results of measurements for the commonly used conditions described above can be compared to one-another for different parameters. The dryout heat flux versus the particle

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diameter measured by various authors for a system pressure of 1 bar is shown in Figure 14 together with results of the different friction models. In general, it is observed that the different experiments yield some spread in published data, due to different experimental setups and measurement procedures. In the models, the dryout heat flux corresponds to the counter-current flooding limit and, therefore, depends on the drag at this limit, as already discussed. A general statement on the validity of the models only based on these values is not possible. However, the tendency to yield smaller dryout heat fluxes can be seen with increasing friction in the classical model formulations from Lipinski over Reed to Theofanous. While the commonly used Reed model appears sufficient for small particles, the measured DHF values are higher than predicted for larger particles. A good representation of the data over the whole range is again given by the modified Tung/Dhir model. So, these results support again the modifications proposed in chapter 3.2.3.. For configurations with enhanced coolant inflow possibilities, the dryout heat flux will increase. In a pure 1-D setup, as sketched in Figure 1, inflow from below is favoured, because the small void fractions there yield a reduced coolant-particle friction. Additionally the coolability is supported by the drag of the up-flowing steam. In contrast to the top-fed case, the dryout does not occur at the bottom of the particle bed. The evaporated water for a main part of the bed is replaced by inflow from below, and only some fraction may infiltrate from the top pool. The up-flowing water is evaporated on its way up to a level where all is spent. Only the region above this level is cooled from above and may be regarded as a top-fed configuration with gas inflow from below. The inflow rate from below is determined by the pressure gradient at the bottom and thus by the pressure in the porous media. This pressure field strongly depends on the drag terms and thus on the model formulation. To catch both situations, the co- and the countercurrent mode the interfacial drag must be considered explicitly in the model. Only two experiments with such inflow conditions form below are known the author. Stevens and Trenberth [17] used spheres with a diameter between 0.126 mm and 5 mm while Hofmann [8] used a particle diameter of 3 mm. These experiments show an increase of the dryout heat flux by a factor between 1.5 and 3. The measured values are plotted in Figure 15 together with results of the different models. Compared to the top-fed case the dryout heat flux is increased because of the facilitated water inflow from below, as expected. Hofmann’s data, in particular, shows a significant stronger increase than calculated by the classical models. The interfacial drag in the enhanced models additionally favours the water inflow from below, while hindering inflow from the top, here the influence of the bottom inflow is much stronger. A summary of calculated results of the different models for a particle diameter of dp = 3 mm, corresponding to the Hofmann experiment, is given in Table 3. As can be seen, the ratio of bottom to top dryout heat flux (DHFbottom / DHFtop ) is more pronounced for the enhanced models, where almost all evaporated water is replaced by inflow via the bottom into particulate bed. This can be seen in the Table at the level that is fed from below. The gain in coolability is most significant for the original Tung/Dhir formulation, but is also affected by the very low top-fed value. For the top-fed configuration, Hofmann reported a dryout heat flux of qdry = 92 W/m2 . Comparing this with Table 3, it should be noted that Reed for the classical models, and Schulenberg as well as the modified Tung/Dhir formulation yield satisfying results. But the value for additional bottom inflow is better reached

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Exp. Hofmann (1984) Exp. Stevens/Trenberth (1982)

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particle diameter dp [mm]

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Figure 15. Dryout heat flux in particle beds with bottom water inflow driven a hydrostatic head for different particle diameters.

for the enhanced models. In general, it is not possible to define the particle-fluid drags in the models without interfacial friction to fit the top as well as the bottom-fed configuration. So, the interfacial drag must be modeled explicitly. A better experimental basis of such investigations is necessary for the validation of the enhanced models.

Table 3. Theoretical dryout heat flux for top- and bottom-fed configurations with dp = 3 mm

dryout heat flux (top-fed) dryout heat flux (top-/bottom-fed) DHFbottom / DHFtop flow rate from below (jl [mm/s]) level fed from below [%]

L 1.22 1.48 1.21 0.564 0.825

R 0.90 1.22 1.35 0.454 0.807

T 0.71 1.01 1.42 0.374 0.801

TD 0.60 2.00 3.34 0.925 0.999

L: Lipinski; R: Reed; T: Theofanous; TD: original Tung/Dhir; mTD: modified Tung/Dhir; S: Schulenberg

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mTD 0.93 1.42 1.53 0.642 0.976

S 0.82 1.58 1.94 0.725 0.990

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272

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5.

Application of the Models to Typical Reactor Cases

5.1.

Development of Dryout in a Pure 1-D Top-fed Configuration

All results presented thus far are for steady state conditions. Such steady states depend exclusively on the friction forces for the adjusted fluid velocities. In the case of boiling beds, the local gas flux is directly defined by the accumulated steam flux from below, and thus by the integrated power from the bottom of the bed to the respective local height. The corresponding coolant flux is then determined by the criterion that all evaporated coolant must be replaced. This is not valid for non-steady state conditions. Here a more sophisticated analysis, also including the mass conservation equations, is necessary. These investigations are conducted with the WABE-2D code (see [14]). The system of mass, momentum and energy conservation equations is solved numerically in a quasi continuum approach. Additionally, a formulation for the capillary pressure is used in the code to include effects due to capillary forces, especially at the boundaries of the porous bed to the water pool. Based on the results of the previous chapter, only the Reed model, as the commonly used classical formulation, and the improved models of Schulenberg and Tung/Dhir, including the modifications proposed in chapter 3.2.3., are applied in the following calculations. The first results presented here will investigate the transient development to a local dryout in a simple top-fed 1-D configuration, as sketched in Figure 1 a. A homogeneous column of particulate debris with porosity ε = 0.4 and particle diameter dp = 3 mm at a system pressure of p = 5 bar is assumed. To simplify the comparison with the in-vessel calculations shown in the next section, a bed height of H = 1.6 m is chosen. By varying the specific power in the particle bed, the maximum bed power still leading to a steady state may be determined. Bed powers less than this value yield steady, cooled states, while higher powers result in transients that lead to a local dryout. The location of the first dryout is dependent on the overpower and the bed conditions. Due to the assumed homogeneity of the power distribution, this maximum specific power Q in W/kg can easily be converted into the dryout heat flux by multiplying it with the density, the volume fraction of the solid material and the total bed height. For simplification, the WABE-2D calculations are performed using the Reed friction model with saturated bed conditions initially assumed. The establishment of steady states, as well as the transient behaviour for bed powers beyond the dryout heat flux, are shown in the first calculation. Initially, a specific power of Q = 150 W/kg is chosen. As can be seen in Figure 16, showing the development of the local void at different levels in the bed, a steady distribution is quickly established. The corresponding void profile is given in Figure 17 by the black curve. The void increases from bottom to top. Due to capillary forces, the liquid fraction slightly increases at the very top, as the water sucked into this region. Therefore the profile has a maximum void, corresponding to a minimum liquid saturation, slightly below the top of the bed. Increasing the bed power yields an increased steam flux, and by this an increase of the void fraction. After a short transient, a steady state is established again, as long as the power is below a critical value. This is also shown in Figures 16 and 17, where the specific power is raised to Q = 225 W/kg, a value slightly below the limit. Here a steady state can still be established, but the maximum void of the counter-current flooding limit is nearly reached in the top region. Additionally the void profile in Figure 17 shows a broader nose region

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273

Q = 240 W/kg Q=223W/kg

1 0.9 0.8 0.7

void α / -

0.6 0.5

z = 0.025 m z = 0.175 m z = 0.3.25 m z = 0.475 m z = 0.625 m z = 0.775 m z = 0.925 m z = 1.08 m z = 1.23 m z = 1.38 m z = 1.52 m

0.4 0.3 0.2 0.1 0

0

500

1000

1500 time t / s

2000

2500

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Figure 16. Development of the void fraction in a top-fed column of 3 mm particles and height 1.6 m for a power history given on the top of the Figure.

near the top. For a further increase of power, no more steady states are possible and a transient behaviour begins. The transient development to local dryout after an increase of the specific power beyond the critical value is also shown in Figure 16 and in the corresponding void profiles in Figure 17. Here the specific power is raised to a value about 8 % above the critical conditions. Due to the increased power, the steam flux to the top is increased too. This also increases the void, which now hinders the coolant inflow, yielding an undersupply of coolant water. As can be seen in the Figures 16 and 17, the local void first increases in upper regions, while it remains almost stable in the lower bed parts, due to an internal flow of coolant water in the bed itself. The lower bound of the coolant undersupply migrates downwards, superimposed by a decrease of the minimum liquid fraction, until local dryout, seen clearly in the void profile development in Figure 17. The location of this first dryout depends on the overpower magnitude. The more the overpower exceeds the critical one, the higher the location of first dryout. As can be seen in the void profiles in Figure 17, the dryout location then moves towards the bottom during the transient, as water from the top cannot reach the region below the dryout. The existing water blow the dryzone is evaporated, superimposed by a small downwards water flux inside this region. Thus, the lower bound of the dry-zone moves

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274

Werner Schmidt 1.6 Q = 150 W/kg (steady state) Q = 225 W/kg (steady state)

1.4

Q = 240 W/kg (transient) t = 600 s t = 800 s t = 1000 s t = 1500 s (first dryout) t = 2000 s t = 2500 s

height z [m]

1.2

1

0.8

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0

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0.6

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1

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Figure 17. Development of the void profile for the case depicted in Figure 16. downwards. Conversely, the upper bound moves also downwards, as in the dry zone no more steam is produced and the steam flux to the top is reduced. The water flow from the top reaches down to the dry-zone, where it is consumed entirely during the path. This process continues, until the dry zone has reached the bottom of the bed. Finally, a steady liquid distribution with a dry region at the bottom establishes. The particle bed region above the dryzone is cooled by water inflow from above. On its way down the coolant is evaporated and reaches just until the upper bound of the dry-zone. This development can clearly be seen in the contour plots given in Figure 18. The liquid fraction s = (1−α) as well as the steam velocity jg are given in the upper row for different times. The local dryout is displayed by the white colour. Even easier this is seen in the temperature distributions, given in the lower row of Figure 18 together with the velocity of the liquid coolant. Besides the downward motion the absolute overheating of the dry-zone increases with time as can also be seen in Figure 19. The development of the particle temperature is shown for lower bed regions. Before the first dryout occurs the particles are almost at saturation temperature. The increase in temperature at the level z = 0.275 m indicates the first dryout. Subsequently, lower cells also become dry while, due to the now missing steam production in the dry-zone, the regions of the first dryout are quenched again. After reaching the bottom of the bed, the dryout location is stable. As there is now no more heat sink remaining, the temperature will increase unlimited here, as can be seen for the lowest cells at z = 2.5 cm and z = 7.5 cm in Figure 19. The fluctuations of the void after the first dryout in Figure 16, even in upper bed regions, are a direct consequence of this development. The dried out region is marginally heated and is quenched when water reaches this position again.

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t = 1500 s

t = 500 s (steady state)

t = 2000 s

t = 2500 s

1.6

1.6

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t = 1500 s

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450 435 420 405 390 375 360 345 330 315 300 285 270 255 240 225 210 195 180 165 150

0 0

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Figure 18. Development of the liquid volume fraction s = (1−α) and superficial velocity of the steam (upper row), as well as the particle temperature and water velocity (lower row) for the case depicted in Figure 16.

276

Werner Schmidt

500

z = 0.025 m z = 0.075 m z = 0.125 m z = 0.175 m z = 0.225 m z = 0.275 m above z = 0.325 m ( TS = Tsat )

particle temperature TS [oC]

450

400

350

300

250

200

150 0

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1000

1500

2000

2500

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time t [s]

Figure 19. Development of the particle temperature in the lower calculation cells for the case depicted in Figure 16.

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5.2.

In-Vessel Particulate Debris

This yields a stronger evaporation, and by this a small pressure increase, hindering further water to reach this region. The produced steam arises through the bed, yielding temporally higher void fractions in upper parts. This procedure repeats during the movement of the dry-zone. Varying the discretisation in the WABE-2D calculations with half and quarter cell sizes shows no significant differences in the results, especially the calculated frequency of the fluctuations remain the same. Realistic configurations of particulate debris, that are to be expected during severe accidents in a nuclear power plant, are not purely one dimensional. Multidimensional effects yield enhanced flow conditions. Especially coolant flow paths with less resistance to lower bed regions increase the overall coolability of particulate corium in a water pool shown in chapter 4.2.2. for a one dimensional configuration with bottom inflow driven by hydrostatic pressure. For configurations where the coolability must be established passively in a water environment, as in the reactor application, facilitated coolant flow paths define a similar effect. Corresponding to the 1-D configuration of the last chapter, a particulate debris bed is now assumed in the lower head of a reactor pressure vessel. The configuration, as well as the discretisation of the calculations, is presented in Figure 20. As the calculation domain in WABE-2D is fixed to a simple structured grid, the parts outside the particle bed must be modeled. First of all, the boundary of the vessel is mapped to the discretisation grid.

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Hoehe z / m height z [m]

Influence of the Interfacial Drag on Pressure Loss...

277

0110 1010 2.5 1010 1010 water 0000000000000000000000000000 1111111111111111111111111111 2 0000000000000000000000000000 1111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 1.5 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 debris bed 0000000000000000000000000000 1111111111111111111111111111 1 111111111111111111111111111111111 000000000000000000000000000000000 porosity ε =0.4 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 wall 0.5 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 porosity ε =10 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 000000000000000000000000000000000 111111111111111111111111111111111 11111111 000000000000000000000000000000000 111111111111111111111111111111111 000000000 −10

0

0.5

1

1.5 Radius r / m radius r [m]

2

2.5

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Figure 20. Configuration of the particulate debris in the lower head (case1). To avoid any kind of fluid flow in the outer region, a very small porosity of ε = 10−10 is assumed here. Additionally, the local power is set to zero. Above the particulate debris bed, a water area with no local heat sources is assumed. As the friction terms implemented in WABE-2D do not include the inertial and the stress terms of the Navier-Stokes equations, a small amount of residual porosity and particle diameter must be assumed here to represent a resistance against the motion. Variations have shown that a porosity of about ε ≈ 0.8 in this region is sufficient, higher values displayed no more influence on the calculation results. Except for the geometry, the same debris parameters as in the 1-D case of Figure 16 are used in the calculation. Again, a porosity of ε = 0.4, a particle diameter of dp = 3 mm and a system pressure of p = 5 bar is assumed. The particulate debris fills the hemispherical lower plenum of radius 2.5 m to a maximum height of H = 1.6 m. This corresponds to the 1-D configuration of the previous chapter. The total corium mass in this geometry is M = 80 t. As before, a specific power of Q = 240 W/kg (corresponding to a volumetric power of 1.15 MW/m3 ) is applied. This value yields a dryout in a 1-D configuration. As can be seen in Figure 21, in the two dimensional configuration this power is discharged, and a steady state establishes. A maximum void of αmax = 0.72 is reached in the upper centre of the particle bed. The reason for this enhanced coolability can be seen in the contour plots in Figure 22. Water infiltration is easier here due to the lower bed height in the outer parts of the debris bed. Driven by gravity, or correspondingly by the established pressure distribution, the coolant flows along the vessel wall to lower bed regions, as can

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278

Werner Schmidt 1 z = 0.025 m z = 0.075 m z = 0.134 m z = 0.269 m z = 0.448 m z = 0.675 m z = 0.925 m z = 1.225 m z = 1.475 m z = 1.575 m above z = 1.6 m (no particles)

0.9 0.8 0.7

void α

0.6 0.5

void profile

2.5

(1 st cell)

0.4 height z [m]

2

0.3

1.5

1

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0

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100

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300

0

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void α

400

0.75

1

500

time t [s]

Figure 21. Development of the local void fraction and steady void profile in the inner cells.

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be seen in Figure 22 b. Therefor, the evaporated liquid in main parts of the bed is replaced by lateral water ingression over regions of small voids corresponding to high water fraction and reduced particle-liquid drag. The pressure field shown in Figure 22 b is the calculated local pressure, reduced by the local hydrostatic head of a water column. It is obvious from equation (5), that the gradient of this field is the driving force for the water flow when no interfacial friction is considered. The lowest pressure is reached in the centre near the bottom of the bed. The water is pulled towards this region. As can be seen in the Figure, the lower parts of the bed are supplied by lateral water influx driven by this pressure field. This coolant flow increases the overall coolability. A major fraction of the total in-flowing water now is not in counter-current mode against the highest steam flow. But, in the upper parts, the evaporated liquid is still replaced by vertical inflow from above. This holds true especially for the region of smallest void in the top centre of the bed. From the outer parts the excessive water flows to lower central regions and, by this, supports the overall coolability. Local dryout, and thus the limit for overall coolability, is reached for increased limiting bed power. This limit is reached for the in-vessel configuration given in Figure 20 - based on the Reed friction model - at a specific power of Q = 301 W/kg. The resulting distributions of the liquid fraction, the reduced pressure and the velocities for this calculation are plotted in Figure 23. As for the steady state before, the location of the minimum liquid fraction is almost in the centre at the top of the debris part. In contrast to the 1-D configuration, discussed in the previous section, here a local liquid fraction of almost zero may be achieved for still steady states. As discussed before, in the pure top-fed 1-D case, a minimum local

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Q = 240 W/kg (steady state) steam velocity 0.5 m/s

2.5 liquid fraction s = (1-α) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

height z [m]

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radius r [m] Q = 240 W/kg (steady state) water velocity 1 mm/s

2.5 pressure ∆pl [Pa] 0 -15 -30 -45 -60 -75 -90 -105 -120 -135 -150 -165 -180 -195 -210 -225 -240 -255 -270 -285 -300

height z [m]

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2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

radius r [m]

Figure 22. Distribution of liquid fraction and steam velocity (top) as well as local pressure, reduced by the hydrostatic head, and liquid velocity (bottom). saturation of about smin ≈ 0.26, depending on the model formulation, is necessary to assure the water inflow to lower bed regions in counter-current mode. In the two dimensional case

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Werner Schmidt Q = 301 W/kg (steady state) steam velocity 0.5 m/s

2.5 liquid fraction s = (1-α) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

height z [m]

2

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radius r [m] Q = 301 W/kg (steady state) water velocity 2 mm/s

2.5 pressure ∆pl [Pa] 0 -37.5 -75 -112.5 -150 -187.5 -225 -262.5 -300 -337.5 -375 -412.5 -450 -487.5 -525 -562.5 -600 -637.5 -675 -712.5 -750

height z [m]

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2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

radius r [m]

Figure 23. Distribution of liquid fraction and steam velocity (top) as well as local pressure, reduced by the hydrostatic head, and liquid velocity (bottom) for a power slightly below dryout (Reed model).

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this is no more the limiting factor, because lower bed regions are laterally supplied by cooling liquid. This effect is clearly observed in the distribution of the reduced pressure and the liquid velocity in Figure 23 b. The lowest local pressure is now at the location of the smallest liquid saturation, nearly at the top of th particle bed. Due to the pressure gradient in the centre the water coming from the side is even dragged up to this location, yielding a co-current flow here, similar to the bottom-fed configurations discussed in chapter 4.2.2.. Only the very top of the central region is directly supplied from above in counter-current mode.

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A further small increase of the bed power yields a local dryout exactly at the point of the smallest liquid saturation in the central top region. Now no more all the evaporated water can be replaced by the inflow into the bed. The flow paths of the liquid are the same as before in the steady state, but now the in-flowing water is consumed before reaching the dry spot. The size of the dry zone depends on the overpower, but its location will be stable, execpt for some extension due to hot spot heat conduction to the surrounding regions. In general, it is not clear how to define a dryout heat flux in such multidimensional configurations. As can be seen in Figure 23 a the produced steam rises straight to the top, driven by the buoyancy force, same as in the one dimensional case. In contrast now the evaporated coolant is not just replaced by a counter flow downwards. The counter-current flooding limit, determining the dryout heat flux in 1-D configurations, is no more limiting. But, based on the steam flux, the heat flux to the top may be defined for a column where the almost dry spot occurs at the centre, where the bed is highest. For such a configuration a dryout heat flux can be defined by the maximum heat flux from this column before reaching a local dryout. By this definition it is clear, that this dryout heat flux now has no more a comparable value for different configurations, because it strongly depends on the bed geometry. For example, for a particle bed surrounded by a cylindrical grid in a water environment, the dryout heat flux will decrease for increasing cylinder diameter. As a conclusion of the non-generality of the dryout heat flux, multidimensional calculations have to be performed for the specific geometry to investigate the overall coolability. For this purpose, verified codes based on realistic friction laws are necessary. Nevertheless, using the definition of the dryout heat flux based on the maximum bed height, comparisons to a pure 1-D case and to cases with the same geometrical configuration may be conducted. Using the bed parameters for a 1-D configuration of height H = 1.6 m yields a maximum specific power of Q1D max = 228 W/kg, corresponding to a 1D = 1.72 MW/m2 . This value has to be compared with the two didryout heat flux of qdry mensional result of Qmax = 301 W/kg (volumetric power 1.44 MW/m3 ) maximum hating power, or equivalently qdry = 2.3 MW/m2 dryout heat flux. Due to the multidimensionality an increase in coolability of about 32 % is observed. As a consequence, the coolability potential of particulate corium during a severe accident in a nuclear power plants will be in general underestimated when based on one dimensional considerations, as usually done. Similar calculations have been executed for the enhanced friction models of Schulenberg and Tung/Dhir, including the modifications proposed in chapter 3.2.3.. For the Schulenberg model, the distributions for a steady state slightly below the dryout point can be seen in Figure 24. The first point to be discussed, is the smaller maximum specific power

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Werner Schmidt

of Qmax = 281 W/kg (qdry = 2.15 MW/m2 ) compared to the result of the Reed calculation. This difference can already be observed in the 1-D configuration, where the Schulenberg 1D = 1.56 MW/m2 (Q1D = 204 W/kg). Looking at model yields a dryout heat flux of qdry max the friction laws given in chapter 3.2., this difference becomes clear. Schulenberg uses the same exponents in the particle-liquid drag as Reed, and increases the particle-gas drag by increasing the exponent in the relative passability of the gas. Additionally, Schulenberg included the interfacial friction, acting against coolant inflow from above. This explains the worse coolability calculated for Schulenberg’s model. Nevertheless, comparing the 1-D and 2-D results yields a slightly higher gain in cooling potential of 38 % for the Schulenberg model. The interfacial friction hinders the water ingression from the pool, especially in centre regions of the geometry, where the steam flux is highest. Displayed clearly by comparing the water velocities of the Reed results (Figure 23) with the results for the Schulenberg model (Figure 24) a more pronounced lateral water flow for the Schulenberg model is observed. This is also reflected in the pressure fields. Much smaller reduced pressures (≈4000 Pa) are reached inside the bed, leading to higher pressure gradients driving the water flow. The increased friction for the flow is partly compensated by the higher pressure gradients in the bed, as already seen in chapter 4.2.1. for 1-D configurations. Additionally, the lowest pressure now is at the bottom centre of the debris bed. Above this location, the water flows against the pressure gradient, due to the drag force of the up-flowing steam on the liquid. On the other hand, the direct inflow from above is reduced by the interfacial drag at the centre top of the bed. In general it can be concluded that the flow paths of both models are comparable with a stronger tendency to lateral flow for the Schulenberg model. This leads to a comparable gain in coolability. A summary of the results for the different models is given in Table 4. Besides the maximum specific power Qmax the corresponding dryout heat flux qdry and the total discharged power for the whole bed configuration is given. The corresponding results for the modified Tung/Dhir model are shown in Figure 25. In general, all the points discussed before for the Schulenberg model hold here too. Due to the smaller particle fluid drags, better coolability is achieved for this model compared to the Reed formulation, in the 1-D as well as in the 2-D configuration. Comparing the 1-D and the 2-D results, the modified Tung/Dhir formulation yields only a coolability gain of about 26 %, but this is strongly affected by the large 1-D value with a maximum specific power of Q1D max = 252 W/kg. This can be explained by looking at the formulation of the friction terms. In general the particle fluid drag is smaller in the Tung/Dhir formulation than in the Reed model. The interfacial drag is reduced due to the modifications proposed in chapter 3.2.3. for the 3 mm particles of the bed. But outside in the pool, described by larger particles and porosity, this friction is dominant and hinders the water ingression from above into central regions. On the other hand, due to the smaller friction compared with the Schulenberg model, the pressure gradient in the bed is not so pronounced, yielding a weakened lateral water flow. Summarising these results, a strong increase in coolability is observed due to the lateral coolant flow in the two dimensional configuration, compared to a 1-D consideration. This gain in the overall coolability is mainly independent of the friction model. In the enhanced models, including an explicit formulation for the interfacial drag, the increased friction is compensated by increased pressure gradients that establish inside the porous bed.

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Q = 281 W/kg (steady state) steam velocity 0.5 m/s

2.5 liquid fraction s = (1-α) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

height z [m]

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radius r [m] Q = 281 W/kg (steady state) water velocity 1 mm/s

2.5 pressure ∆pl [Pa] 0 -200 -400 -600 -800 -1000 -1200 -1400 -1600 -1800 -2000 -2200 -2400 -2600 -2800 -3000 -3200 -3400 -3600 -3800 -4000

height z [m]

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2

1.5

1

0.5

0

0

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1

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2

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radius r [m]

Figure 24. Distribution of liquid fraction and steam velocity (top) as well as local pressure, reduced by the hydrostatic head, and liquid velocity (bottom) for a power slightly dryout (Schulenberg model). The water velocities outside the particulate debris are ignored.

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Werner Schmidt Q = 324 W/kg (steady state)

Q = 324 W/kg (steady state) steam velocity 0.5 m/s

water velocity 1 mm/s

2.5

2.5

height z [m]

1.5

1

0.5

0

0

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1

1.5

2

pressure ∆pl [Pa] 0 -150 -300 -450 -600 -750 -900 -1050 -1200 -1350 -1500 -1650 -1800 -1950 -2100 -2250 -2400 -2550 -2700 -2850 -3000

2

height z [m]

liquid fraction s = (1-α) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

2

1.5

1

0.5

2.5

0

0

0.5

radius r [m]

1

1.5

2

2.5

radius r [m]

Figure 25. Distribution of liquid fraction and steam velocity (top) as well as local pressure, reduced by the hydrostatic head, and liquid velocity (bottom) for a power slightly below dryout (modified Tung/Dhir model). Table 4. Results for in-vessel calculations for different models (R: Reed, S: Schulenberg, mTD: modified Tung/Dhir)

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DHF [MW/m2 ] max. spec. power [W/kg] total power [MW] gain

5.3.

1-D WABE-2D 1-D WABE-2D 1-D WABE-2D

R 1.74 2.20 228 301 18.34 24.21 32%

S 1.56 2.15 204 281 16.41 22.60 38%

mTD 1.97 2.48 257 324 20.60 26.06 26%

Ex-Vessel Particulate Debris

If the corium can not be cooled inside the vessel, e.g. due to the non availability of cooling water, a melt pool will develop in the lower plenum and attack the reactor pressure vessel. After vessel failure, the melt will flow out into the reactor pit. If the cavity is filled with water, automatically or due to accident management measures, the melt jet will pour into water. Especially for Boiling Water Reactors (BWR) with the great depth of the cavity a good fragmentation of melt jets may be expected under such conditions. These fragments settle as particulate debris. To avoid further destruction and radioactive contamination of the environment, coolability should be reached at least at this stage, before losing the containment integrity as the last safety barrier. Configurations with fragmented corium in a water pool, as described above, have a high potential of coolability, especially when water access from below is possible, shown in the calculations. Following this scenario a large Boiling Water Reactor is assumed where the total core and major parts of core components have been destroyed. The fragmented corium has settled as particulate debris in the reactor pit. The chosen corium consists of about 150 t

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Influence of the Interfacial Drag on Pressure Loss...

285

UO2 , 80 t Zircalloy and 210 t structure material yielding a total corium mass of M = 440 t. These values are typical for reactors of approximately 4000 MW thermal power. As the decay heat after one hour is about 1 % of this value, about 40 MW have to be discharged from the particles in the reactor pit, corresponding to a specific power of Q ≈ 90 W/kg. In the calculations a radius of 4.4 m, typical for BWR reactors, is chosen for the reactor pit. The geometry is sketched in Figure 26 a. At the outer boundary a supporting base of the mounting devices is considered as wall area. The particulate debris is chosen to have a mound shape with a slope angle of 33.7◦ . For a porosity of ε = 0.4 a bed height of Hmax = 2.8 m is obtained. As for the in-vessel calculations a particle diameter of dp = 3 mm is chosen, but now at a smaller system pressure of only p = 3.15 bar is used, typically for the containment. As mentioned in chapter 1. this lower pressure yields less coolability due to the reduced steam density. In addition to this base configuration, a modification shall be considered in the calculations to investigate some supporting features that may be considered in reality. Objective of this modification is to take advantage of enhanced coolability if water inflow from below can be installed. This may be achieved via the sump below the reactor pit. If there is a connection from the water filled containment to the grid covered sump, water at hydrostatic pressure is connected to the bottom of the particle bed. As the pressure inside the bed is less than hydrostatic, a water influx establishes and support the overall coolability. This is considered by defining the hydrostatic pressure of a water column as boundary condition at the bottom in the calculations. As can be seen in the sketch in Figure 26 b, a radius for the inflow region of r = 0.6 m is chosen. Thus, inflow is only possible via a minor part of the lower bound. Calculations utilising the Reed model as well as the enhanced models of Schulenberg and Tung/Dhir, including the modifications proposed in sub-chapter 3.2.3., were carried out for the two configurations. The resulting steady state distributions of the liquid fraction (s = 1 − α), pressure (reduced by the hydrostatic part) and velocities for a specific power just below the overturn are given in the next Figures. Figure 27 shows the results for the Reed friction model. As can be seen in sub-figure 27 a, steady states are reached up to a specific power of Qmax = 146 W/kg in the base configuration. Due to the multidimensional effects of the mound shape, this value again is noticeably higher than for a pure 1-D configuration with the same height of Q1D max = 112 W/kg. This again yields a gain of 30 % in the overall coolability. The reason for this gain can be seen in the water velocities shown in Figure 27 a. Central regions, where the bed is highest, are supported by a lateral water inflow from the lower outer parts of the bed. This is even enhanced by the base with the very shallow bed part above. An almost dry spot is seen near the top of the bed. This spot is laterally shifted from the centre. Here the water infiltration from the top is hindered by the huge steam flux from below, but the lateral water flow mainly supplies central regions. Enhanced coolability is expected by opening the water loop over the reactor sump. The calculation result for the Reed model yielding a maximum specific power of Qmax = 152 W/kg is given in Figure 27 b. To resolve the velocity field in the debris part the vectors of the water velocities in the first axial calculation cell over the bottom inlet are ignored in the Figure. A strong water influx from below is observed. But, comparing the maximum power to the one without bottom inflow, only a slightly higher value is reached. The bottom inflow yields no significant enlargement in the overall coolability. Integrating the liquid

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Werner Schmidt

2.8 m

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 111111111111111111 000000000000000000 11111111111111111111111 00000000000000000000000 33.7°

1m

286

3.3 m

4.2 m

(a) Base configuration

11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 00000 11111 00000000000000000000000 11111111111111111111111 111111111111111111 000000000000000000 11111111111111111111111 00000000000000000000000 33.7°

1m

2.8 m

1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

3.3 m

4.2 m

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(b) Including water loop over sump

Figure 26. Configuration of the particulate debris in the reactor pit. volume flux over the boundaries yields a water volume flux of 6.51 l/s from below, while the main fraction of 26.33 l/s still flows via the top surface into the particle bed. Thus, only about 20 % of the water is coming from below. The given boundary pressure at the inlet also affects the pressure field in the whole bed. As the pressure field shown in the Figures is the local pressure reduced by the hydrostatic one of a water column, a value near to zero establishes above the inlet. The pressure gradient here corresponds to the pressure loss of the in-flowing water. As a consequence, the pressure rises everywhere in the particle bed, leading to smaller pressure gradients. Thus the driving force for the coolant flow is reduced in the bed, yielding only a minor gain in coolability. The interfacial drag, hindering the water inflow from the top, should support the overall coolability if the coolant is supplied from below. Results of the calculations for the different configurations are provided in Figure 28 for the Schulenberg and in figure 29 for the modified Tung/Dhir model. In the base configurations (sub-figures a) the effect of the geometry alone are observed. First of all, again an increased coolability compared to 1-D considerations is seen, due to the geometrical shape. The maximum specific power values

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287

Q = 146 W/kg (steady state)

Q = 146 W/kg (steady state) steam velocity 0.8 m/s

water velocity 2 mm/s

2.5

2.5 liquid fraction s = (1-α)

height z [m]

2

1.5

1

0.5

0

0

1

2

3

pressure ∆p [Pa] 0 -20 -40 -60 -80 -100 -120 -140 -160 -180 -200 -220 -240 -260 -280 -300 -320 -340 -360 -380 -400

2

height z [m]

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

1.5

1

0.5

0

4

0

1

radius r [m]

2

3

4

radius r [m]

(a) case 1; maximum possible specific power 146 W/kg Q = 152 W/kg (steady state)

Q = 152 W/kg (steady state) steam velocity 0.8 m/s

liquid velocity 2 mm/s

2.5

2.5 liquid fraction s = (1-α)

height z [m]

1.5

1

0.5

0

0

1

2

3

4

pressure ∆pl [Pa] 0 -25 -50 -75 -100 -125 -150 -175 -200 -225 -250 -275 -300 -325 -350 -375 -400 -425 -450 -475 -500

2

height z [m]

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

2

1.5

1

0.5

0

0

radius r [m]

1

2

3

4

radius r [m]

(b) case 2; maximum possible specific power 152 W/kg

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Figure 27. Distribution of liquid fraction and steam velocity (left) as well as local pressure reduced, by the hydrostatic head, and liquid velocity (right) for ex-vessel particulate debris according Figure 26 (Reed model). are summarised in Table 5. A gain of 20 % for the Schulenberg model and of only 9 % for the modified Tung/Dhir model is calculated compared to the 1-D results. Again the small gain for the Tung/Dhir model is partly due to the high value for the 1-D configuration. As in the in-vessel cases, the drag of the up-flowing steam hinders the water inflow over the shell of the mound. A pseudo water loop forms in the water area. Some water is drawn out of the ”gusset” above the outer support base, only having minor influence on the overall coolability. Besides this, as for the in-vessel cases, the main difference to the Reed result can be seen in the pressure distribution. Again the enhanced models yield a significant increased pressure gradient with the lowest pressure value at the bottom of the particle configuration. Enabling the water inflow possibility from below over the reactor sump, this smaller pressure is connected to the outer hydrostatic one. Thus coolant is pressed into the debris until the pressure loss due to the inflow compensates the pressure gradient, leading to steady inflow conditions. The enlarged water inflow for the enhanced models, compared to the Reed results, are clearly seen in the Figures and by the integrated coolant influxes from below. While in the Reed model only 20 % of the total coolant inflow is from below, here 80 % for the Schulenberg model and 72 % for the modified Tung/Dhir formulation flows in via the bottom. Comparing the calculated power limits of Table 5 with the expected decay heat of about

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288

Werner Schmidt Q = 121 W/kg (steady state)

Q = 121 W/kg (steady state) steam velocity 0.8 m/s

water velocity 2 mm/s

2.5

2.5 liquid fraction s = (1-α)

height z [m]

2

1.5

1

0.5

0

0

1

2

3

pressure ∆p [Pa] 0 -200 -400 -600 -800 -1000 -1200 -1400 -1600 -1800 -2000 -2200 -2400 -2600 -2800 -3000 -3200 -3400 -3600 -3800 -4000

2

height z [m]

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

1.5

1

0.5

0

4

0

1

radius r [m]

2

3

4

radius r [m]

(a) case 1; maximum possible specific power 121 W/kg Q = 156 W/kg (steady state)

Q = 156 W/kg (steady state) steam velocity 0.8 m/s

water velocity 2 mm/s

2.5

2.5 liquid fraction s = (1-α)

height z [m]

1.5

1

0.5

0

0

1

2

3

4

pressure ∆p [Pa] 0 -200 -400 -600 -800 -1000 -1200 -1400 -1600 -1800 -2000 -2200 -2400 -2600 -2800 -3000 -3200 -3400 -3600 -3800 -4000

2

height z [m]

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

2

1.5

1

0.5

0

0

radius r [m]

1

2

3

4

radius r [m]

(b) case 2; maximum possible specific power 156 W/kg

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Figure 28. Distribution of liquid fraction and steam velocity (left) as well as local pressure, reduced by the hydrostatic head, and liquid velocity (right) for ex-vessel particulate debris according Figure 26 (Schulenberg model).

Qreactor ≈ 90 W/kg, the high potential for overall coolability in the containment can be seen even for the simpler Reed model. In the calculations presented, only a homogeneous configuration with a large amount of structure material is assumed. Besides the precondition of sufficient fragmentation and quenching of hot particles, configurations including hindering effects (e.g. local conglomerates with less porosity or a layer of smaller particles settled on the top that act like an sponge) have not been investigated here. In general, the emphasised lateral water inflow due to geometric conditions and the interfacial drag supports the coolability potential even in such configurations. A detailed analysis is necessary to investigate the safety margins, which are remarkably higher for the enhanced models. Based on these calculations it is concluded, that the interfacial friction has a significant influence on the overall coolability if water access from below is enabled. Although only a minor fraction of 2 % of the lower bound is directly connected to the coolant pool, a remarkable increase in the maximum coolable bed power is calculated. This shows the importance to include the interfacial drag in the momentum equations, as done by Schulenberg and M¨uller or Tung and Dhir.

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Influence of the Interfacial Drag on Pressure Loss... Q = 135 W/kg (steady state)

289

Q = 135 W/kg (steady state) steam velocity 0.8 m/s

water velocity 2 mm/s

2.5

2.5 liquid fraction s = (1-α) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

1.5

1

0.5

0

0

1

2

3

0 -150 -300 -450 -600 -750 -900 -1050 -1200 -1350 -1500 -1650 -1800 -1950 -2100 -2250 -2400 -2550 -2700 -2850 -3000

2

height z [m]

height z [m]

2

pressure ∆p [Pa]

1.5

1

0.5

0

4

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2

radius r [m]

3

4

radius r [m]

(a) case 1; maximum possible specific power 135 W/kg Q = 152 W/kg (steady state)

Q = 152 W/kg (steady state) steam velocity 0.8 m/s

water velocity 2 mm/s

2.5

2.5 liquid fraction s = (1-α) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

1.5

1

0.5

0

0

1

2

3

pressure ∆p [Pa] 0 -150 -300 -450 -600 -750 -900 -1050 -1200 -1350 -1500 -1650 -1800 -1950 -2100 -2250 -2400 -2550 -2700 -2850 -3000

2

height z [m]

height z [m]

2

1.5

1

0.5

0

4

0

1

2

radius r [m]

3

4

radius r [m]

(b) case 2; maximum possible specific power 152 W/kg

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Figure 29. Distribution of liquid fraction and steam velocity (left) as well as local pressure, reduced by the hydrostatic head, and liquid velocity (right) for ex-vessel particulate debris according Figure 26 (modified Tung/Dhir model). Table 5. Maximum specific powers Qmax (in [W/kg]) and the fraction f of water inflow from below in the ex-vessel calculations for the different models (R: Reed, S: Schulenberg, mTD: modified Tung/Dhir) R 1-D base configuration including inflow via sump

6.

Qmax 112 146 152

S f 20 %

Qmax 101 121 156

f 80 %

mTD Qmax f 124 135 152 72 %

Summary and Conclusion

Within this article two main conclusions have been worked out. The first conclusion is of more general importance for the multiphase flow in porous media. By comparison of the simple models with the pressure loss in iso-thermal air/water flow experiments, the importance of an explicit consideration the interfacial drag is demonstrated. Without this term, as in the simple models, the pressure field inside the porous structure can not be

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Werner Schmidt

calculated. The second central conclusion is more specific for coolability applications investigated within the frame of reactor safety. Here a particle bed configuration with internal heat sources due to the decay heat is expected. These particle beds must be cooled by evaporation of water. Steady and safe states are achieved when all evaporated coolant water is replaced by infiltration through the boundaries. It is shown, that significantly more heat can be removed if direct water inflow into bottom regions is possible, or at least flow paths with less resistance to lower bed regions exist. This effect originates from the geometric configuration. In lower bed regions, the void fraction is less and thus the flow cross section of the water is greater compared to pure inflow from the top, where the counter-current flooding limit limits the coolability. This shows the importance to investigate the overall coolability of the particle bed in realistic geometric configurations. Furthermore, the water inflow rates, especially to lower bed regions, depend in particular on the pressure field in the particle bed. This pressure field is determined by the dynamics of the flowing fluids in the porous media, thus of water and steam in the coolability application. Besides the friction at the solid particles also the infacial drag between the steam and water has to be considered. The water inflow to lower bed regions is supported by the drag of the rising steam. By considering an explicit term for the interfacial drag in the pressure loss correlations, realistic pressure fields and thus realistic flow rates are reached. Conclusively, an enhanced modeling with explicit consierdation of the interfacial drag between the fluids is required.

Acknowlegement

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This work was accomplished at the Institut f¨ur Kernenergetik und Energiesysteme (IKE), University of Stuttgart, Germany.

Nomenclature Latin symbols Db dp Fp Fi F⋆ g j

m m N/m3 N/m3 m/s2 m/s

jr

m/s

K Kr P⋆ p

m2 Pa

diameter off the gas bubbles particle diameter volumetric particle-fluid drag force volumetric gas-liquid drag force dimensionless force gravitational constant superficial velocity  of the fluids  relative velocity jr =

jg α

−

jl 1−α

permeability relative permeability of the fluids dimensionless pressure gradient pressure

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Greek symbols α β ε η m ηr m µ kg/m s σ N/m ̺ kg/m3 Indices g l

void, gas volume fraction in the pores volume fraction porosity passability relative passability of the fluids dynamic viscosity surface tension density of the phase

gas,vapour liquid

References [1] J.S. Andrade, U.M.S. Costa, M.P. Almeida, H.A. Makse, and H.E. Stanley. Inertial Effects on Fluid Flow through Disordered Porous Media. Physical Review Letters, Vol.82(No.26):p. 5249–5252, 1999. [2] R.H. Brooks and A.T. Corey. Hydraulic properties of porous media. Hydrology Papers, Colorado State University, March 1964. No. 3.

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[3] W. Chu, V.K. Dhir, and J. Marshall. Study of pressure drop, void fraction and relative permeabilities of two-phase flow through porous media. In Heat transfer-Seattle, volume Vol.79 of AIChE Symposium series, pages 224–235, 1983. [4] H. Darcy. Les Fontaines Publiques de la Ville de Dijon. Dalamont, Paris, 1856. ´ D´ecossin. Experimental investigations on particulate debris bed coolability in a [5] E. multi-dimensional configuration. In OCDE/CSNI-Workshop on EX-VESSEL DEBRIS COOLABILITY, 16.-18. Nov. 1999. [6] S. Ergun. Fluid flow through packed columns. Chemical Engineering Progress, Vol. 48(No.2):pp. 89–94, Feb. 1952. [7] D. Haga, Y. Niibori, and T. Chida. Tracer responses in gas-liquid two-phase flow through porous media. In Procieedings World Geothermal Congress, Kyushu - Tohoku, Japan, 2000. [8] G. Hofmann. On the location and mechanisms of dryout in top-fed and bottom-fed particulate beds. Nuclear Technology, Vol. 65, April 1984. [9] K. Hu and T.G. Theofanous. On the measurement and mechanism of dryout in volumetrically heated coarse particle beds. Int.J. Multiphase Flow, Vol.17(No.4), 1991. [10] R.J. Lipinski. A one dimensional particle bed dryout model. ANS Transactions, Vol.38:386–387, June 1981. Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

292

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[11] R.J. Lipinski. A coolability model for postaccident nuclear reactor debris. Nulcear Technology, Vol.65:pp. 53–66, April 1984. [12] A.W. Reed. The effect of channeling on the dryout of heated particulate beds immersed in a liquid pool. PhD thesis, Massachusetts Institute of Technology, Cambridge, Feb. 1982. [13] A.W. Reed, K.R. Boldt, E.D. Gorham-Bergeron, R.J. Lipinski, and T.R. Schmidt. DCC-1 / DCC-2 Degrated Core Coolability Analysis. Technical Report NUREG/CR4390, SAND85-1967, Sandia National Laboratory, Oct. 1985. [14] W. Schmidt. Influence of Multidimensionality and Interfacial Friction on the Coolability of Fragmented Corium. Doktorarbeit, Institut f¨ur Kernenergetik und Energiesysteme, Universit¨at Stuttgart, Mai 2004. ISSN-0173-6892. [15] P. Sch¨afer, M. Groll, W. Schmidt, W. Widmann, and M. B¨urger. Coolability of Particle Beds: Examination and Influence of Friction Laws. In Proceedings of ICAPP´04, Pittsburgh, PA; USA, June 13.-17. 2004. [16] T. Schulenberg and U. M¨uller. A refined model for the coolability of core debris with flow entry from the bottom. In Proceedings of the Sixth Information Excange Meeting on Debris Coolability, number EPRI NP-4455. University of California, Los Angeles, March 1986.

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[17] G.F. Stevens and R. Trenberth. Experimental studies of boiling heat transfer and dryout in heat generating particulate beds in water at 1 bar. In Proceedings of the Fifth Post Accident Heat Removal Information Exchange Meeting, pages 108–113. Nuclear Research Center Karlsruhe, 1982. [18] V.X.Tung and V.K.Dhir. A hydrodynamic model for two-phase flow through porous media. International Journal of Multiphase Flow, Vol. 14(No. 1):pp. 47–65, 1988. [19] N.K. Tutu, T. Ginsberg, and J.C. Chen. Interfacial drag for two-phase flow through high permeability porous beds. Journal of Heat Transfer, Vol.106:pp. 865–870, Nov. 1984.

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In: Multiphase Flow Research Editors: S. Martin and J.R. Williams, pp. 293-357

ISBN: 978-1-60692-448-8 © 2009 Nova Science Publishers, Inc.

Chapter 5

HYDRODYNAMIC STUDIES ON TWO-PHASE GAS-LIQUID FLOW IN AN EJECTOR INDUCED DOWNFLOW BUBBLE COLUMN Ajay Mandal1 Dept. of Petroleum Engineering, Indian School of Mines University Dhanbad-826 004, India

Abstract

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Fine dispersion of gas into liquid is one of the most important criteria for momentum, mass and energy transfer between the phases. It not only provides an intense mixing but also creates increased interfacial area and high mass transfer coefficient. Various types of contacting devices have been developed for achieving effective gas-liquid mixing. These may be broadly classified as co-current, counter-current and cross current systems. Among these, increased interests on co-current contacting systems have been shown because of their ability to handle high fluid flowrate without flooding, low pressure drop, higher interfacial area and mass transfer coefficients. A review of literature on design and development of fluid-fluid cocurrent contacting equipment shows that use of liquid jet ejectors as a gas-liquid distributor in downflow bubble column is gaining in importance as it functions both as a sparger and gas entrainment device. Ejectors are devices that utilize the kinetic energy of a high velocity liquid jet in order to entrain and disperse the gas phase. The bubble column with ejector system is very simple in design and no extra energy is required for gas dispersion as the gas phase is sucked and dispersed by the high velocity liquid jet. Thus, cocurrent down flow bubble column with ejector type gas distributor possesses the following distinct advantages over more conventional devices, such as (i) lower power consumption (ii) almost complete gas utilization (iii) higher overall mass transfer coefficient and (iv) tolerance to particulates and therefore useful for chemical reaction with slurries. In this paper some important hydrodynamic characteristics on gas entrainment by a high velocity liquid jet; gas-holdup and bubble size distribution; two-phase frictional pressure drop; energy dissipation in ejector and contactor; interfacial area and mass transfer coefficient in an ejector induced downflow bubble column have been discussed. Further since the processing media for many processes such as sewage sludge, microbiological culture, polymer solutions 1

E-mail address: [email protected]

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Ajay Mandal etc. are non-Newtonian in nature; an attention has been paid to study with non-Newtonian fluid using Carboxy Methyl Cellulose (CMC) solution at different concentrations. The experimental set-up consists of a column, an ejector at the top of the column and a gas-liquid separator at the bottom along with other accessories like pumps, flow meters, valves and a number of manometers connected at different heights of the column. The column and ejector assembly were fitted with perfect alignment to obtain an axially symmetric jet. Experimentally it has been found that the rate of gas entrainment is strongly dependent on the motive fluid flowrate and the pressure in the separator. A precise idea of gas holdup and bubble size distribution was obtained by measuring the column pressure readings at different points of the column. The interfacial mass transfer area and liquid side volumetric mass transfer co-efficient were measured by both physical and chemical methods. Significantly higher values of interfacial area and volumetric mass transfer coefficient are obtained at low gas flowrate in the present system and the results are compared with the other reported works.

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1. Background Efficient dispersion of gases into liquid is of considerable importance for carrying out gasliquid reactions and mass transfer operations. This can be done with a variety of contacting devices such as bubble columns, mechanically agitated tanks, packed columns, plate and tray columns, spray towers etc. Bubble column is a type of unit in which gas is dispersed in the continuous liquid phase as fine bubbles. There are many chemical and biochemical processes, viz. hydrogenation, oxidation, fermentation, petroleum refining, coal liquefaction etc, where the overall production rate is often controlled by gas-liquid interfacial mass transfer. For such types of processes the reaction rate is proportional to the interfacial area. In gas-liquid twophase systems, large interfacial areas are obtained if the gas is dispersed into the liquid phase as fine bubbles. Thus efficient dispersion of gases in liquids is of major importance in many chemical engineering operations and its significance continues to grow with the development of pollution control, fermentation and waste treatment equipment. This has initiated a wide interest amongst researchers and designers to develop efficient systems for the dispersion of gases in liquids. A review of literature on the design and development of fluid-fluid contacting equipment shows that contactors or reactors belonging to the jet-mixing category with cocurrent or countercurrent contacting of phases such as ejectors, venturies and other similar devices for gas-liquid or liquid-liquid contacting are gaining in importance these days, because of high interfacial area and mass transfer coefficients obtained in such systems. These are all cocurrent flow devices, where the kinetic energy of a fluid is used to achieve fine dispersion and mixing between the phases. There are wide verities of commonly used gas-liquid contactors viz. bubble columns, mechanically agitated tanks, packed columns, plate and tray columns, spray towers etc. For selecting an appropriate gas-liquid contactor the following considerations should be taken into account: • • • • • •

Maximum conversion of reactants Greatest selectivity for desired products Minimum environmental impacts Ease of automation and process control Simplicity of scale-up Low capital and operating cost

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Each of the above gas-liquid contactors is suitable for specific services having some advantages and disadvantages. However, recently bubble columns are being used widely in many chemical process industries, e.g. oxidation, hydrogenation, halogenation, fermentation, coal liquefaction etc. due to it’s unique advantages. These are: • • • • • •

simple construction and low capital cost; no moving parts and minimum maintenance; ability to handle solids; ease of temperature control; little floor space required and reasonable interfacial mass transfer area.

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Bubble columns are classified based on flow of phases, e.g. horizontal flow, vertical up or down flow, co-current, counter-current, crosscurrent etc. Significant works have been done on flow-pattern, gas-phase distribution and pressure drop measurements in horizontal flow systems (Ebner at el., 1987; Xu et al., 1998; Andreussi et al., 1999). Extensive studies have also been done on co-current up-flow bubble column. Kantak et al. (1995) investigated effects of gas and liquid properties on gas phase dispersion in a co-current up-flow bubble column with water, aqueous alcohol and aqueous CMC solutions of lower concentration as liquid phase. Chen et al. (1998) have discussed gas holdup distributions of cocurrent two-phase upflow system in a large-diameter bubble column with air-water and air-drakeoil system, measured by computed tomography method. Havelka et al. (1997), Gharat and Joshi (1992), Zahradnik et al. (1997) have also done work on up-flow systems. But relatively little attention has been paid to co-current down flow system compared to that of up-flow, though co-current down flow bubble columns are gaining in importance because of it’s distinct advantages viz. • • • • •

bubbles are finer and more uniform in size, coalescence of bubbles is negligible, homogenization of the two phases in the whole column is possible, a large amount of liquid can be contacted with a small amount of dispersed gas efficiently, and higher residence time of the gas bubbles.

Dispersion of one fluid into another is an important aspect of gas-liquid and liquid/liquid contacting because of its direct influence on transfer operations. Spargers are commonly used gas distributor in bubble column. The design of the distributor has a strong effect on gas dispersion. Literature review reveals that ejector type mixing devices are also good alternative for achieving efficient dispersion of one fluid into other. A wide variety of liquids, gases and vapours may be used as either primary or secondary stream. Based on primary and secondary streams the ejector type contacting devices may broadly be classified as: (i) (ii) (iii) (iv)

gas-gas system, i.e. gases are used as both primary and secondary fluid. gas-liquid system, i.e. gas is used as primary fluid and liquid as secondary fluid. liquid-liquid system, i.e. liquids are used as both primary and secondary fluid. liquid-gas system, i.e. liquid is used as primary fluid and gas as secondary fluid.

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However studies on liquid-gas system with liquid as primary motive fluid are found to be increasing in recent times. Some of the earliest reported investigations using water, as primary fluid and air as the secondary fluid are those of Gosline and O’Brine (1934), Kroll (1947), Silvester (1961), Reddy and Kar (1968) and Swiggett (1969). Liquid jet is considered to be useful for gas-liquid contactors as it generates fine bubbles by impinging the liquid jet on the liquid surface. Another advantage of liquid jet is that, it entrains certain amount of gas into the pool of liquid. Burgess and Molloy (1973) carried out experiments of gas absorption in a plunging liquid jet reactor. They reported that reactor is analogous to a gas sparged stirred tank reactor with plunging jet acting as both the reactor agitator and gas bubble generator. Tojo and Miyanami (1982) reported the oxygen transfer characteristics in a downflow gas-liquid jet mixture system, where air is entrained and dispersed by liquid jet produced through nozzle. They showed that gas-liquid jet mixture with a downflow jet is more efficient for gas-liquid mass transfer than the corresponding upflow jet mixture. Recently downflow bubble columns with ejector type of gas-liquid distributor have been recommended for chemical processes particularly where interfacial mass transfer area is the rate-controlling step. Ejectors are devices that utilize the kinetic energy of a high velocity liquid jet in order to entrain and disperse the gas phase. The bubble column with ejector system is very simple in design and no extra energy is required for gas dispersion as the gas phase is sucked and dispersed by the high velocity liquid jet. So from energy point of view also it is very attractive. Hence, cocurrent down flow bubble column with ejector type gas distributor is getting importance due to its distinctive advantages over more conventional devices. These are:

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• • • •

lower power consumption, almost complete gas utilization, higher overall mass transfer coefficient and tolerance to particulates and therefore viable for slurry chemical reaction.

Considerable works have been reported by different authors on efficient dispersion of gas by liquid jet in gas-liquid two-phase cocurrent contactor with venturies, nozzles and ejectors as gas-liquid mixing devices. Ohkawa et al. (1985) studied the flow characteristics of downflow bubble columns with gas entrainment by a liquid jet and established a correlation for gas holdup and gas entrainment rate. Bando et al. (1988) reported that with simultaneous gas-liquid injection nozzle in down flow bubble column a spouting and a calm section appeared; and gas holdup and interfacial area was higher in spouting section than in calm section. Yamagiwa et al. (1990) experimentally studied the flow behaviour of the cocurrent downflow bubble column with gas entrainment by a liquid jet operating at high liquid feed rate. They have evaluated the gas entraining capacity of their apparatus in terms of energy efficiency, i.e. energy required to supply unit volume of gas. Ohkawa et al. (1986) studied flow behaviour and performance of a vertical liquid jet system having downcomers in an airwater system. They investigated the effect of operating variables such as the nozzle diameter, the jet velocity, the downcomer diameter and height, etc. on the flow characteristics such as the bubble penetration depth, gas entrainment rate, gas holdup, etc. Kundu et al. (1997) reported the performance of ejector for dispersion of gas into liquid in a co-current gas-liquid

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downflow bubble column. Evans et al. (2001) described the performance of confined plunging liquid jet bubble column as a gas-liquid reactor. From the above literature review, it is found that significant works have been reported, but most of them have worked with air-water system. Though in most cases, liquid in two phase gas-liquid system have been considered as Newtonian, there are a lot of situations viz. polymer processing, mineral recovery, food processing, bio-medical engineering, biochemical reactors etc., where this assumption of Newtonian behaviour is not valid. So the nonNewtonian behaviour of the liquid phase like pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic behaviour or one involving yield stress have also to be considered. Mahalingam & Valle (1972) studied two-phase flow in horizontal tubes using various aqueous polymer solutions showing pseudoplastic behaviour. They reported an increase in gas holdup with increasing gas flowrate and with decreasing pseudoplasticity. Eisenberg and Wineberger (1979) developed techniques for the prediction of pressure drop and holdup in an annular two-phase flow of gas-non-Newtonian liquid based on Lockhart-Martinelli correlation. Das et al. (1992) reported holdup for two phase flow of gas-non-Newtonian liquid (CMC solution) mixture in horizontal and vertical pipes. They correlated the void fraction by drift flux model. Godbole et al. (1982) determined gas holdup fraction in a batch bubble column with highly viscous glycerine and CMC solutions using the dynamic gas disengagement method. However, most of the studies are concerned with gas-non-Newtonian two-phase flow in pipe or ordinary bubble column. Akagawa et al. (1989) studied the gas-non-Newtonian liquid (aqueous polyacrilamide solutions) two-phase flow in a cocurrent upflow vertical column. However, works with non-Newtonian liquid in ejector induced cocurrent downflow system are scanty. The performance of a bubble column is best characterized by the available gas-liquid interfacial area (a) especially in the regime in which absorption is accompanied by a fast chemical reaction. Again, for gas-liquid processes that take place in absorption regime with slow reaction the mass transfer properties can be characterized by volumetric mass transfer coefficient (kLa). Thus a precise knowledge of available gas-liquid interfacial area and volumetric mass transfer coefficient is very important for designing and scaling up a gasliquid reactor. Efficient gas dispersion in cocurrent downflow bubble column with ejector type gas distributor leads to an enhanced mass transfer. Investigations on mass transfer in bubble columns with liquid driven ejectors have been reported by several authors. Huynh et al. (1991) studied mass transfer characteristics in an upward venturi/bubble column combination. They reported that gas holdup was increased by 50% to 150% and the overall volumetric mass transfer coefficient was tripled when venturi was used as “gas distributor” instead of a porous distributor. Havelka et al. (2000) measured volumetric liquid-side mass transfer coefficient in the cocurrent ejector-distributor bubble column. Significant works have also been done on mass transfer in downflow system with efficient dispersion, e.g. liquid jet ejector (Dutta and Raghavan, 1987), venturi-bubble column (Briens et al.,1992) and plunging jet bubble column [Evans et al. (2001)]. However, comparison of these different systems is difficult, because of their different ejector-contactor configurations and operating ranges of liquid and gas flowrates. From the above discussion, it may be expected that a cocurrent bubble contactor utilizing a jet ejector for improved gas-liquid mixing, would be highly suitable for many industrial processes like absorption, desorption and scrubbing, gas-liquid reactions, aerobic fermentations, waste treatment etc. Therefore, a precise knowledge of the hydrodynamic and

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mass transfer characteristics of the two-phase flow in an ejector-induced downflow bubble column would be of considerable interest.

2.1. Experimental Apparatus and Procedure

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The schematic diagram of the experimental setup is shown in Fig. 1. It consists of an ejector assembly, E, an extended pipeline contactor, C, a gas liquid separator, SE, and other accessories like centrifugal pump (PU), pressure gauge (PG), manometers (M1-M11), control valves (V1-V5), solenoid valve (SV1-SV3), rotameter (R), gas flowmeter (GM), circulating tank (T) etc. The ejector assembly and extended contactor were made of transparent perspex for visual observation of the flow and mixing patterns. Two different lengths of extended contactors (1.5 m and 1.9 m respectively) were used for studying different hydrodynamic characteristics. The major dimensions of the apparatus are given in Table 1.

Legends: AI, Air inlet; C, Contactor; E, Ejector assembly; GM, Gas flowmeter; hm, hl, liquid level in arm L, Gas-liquid mixture height in the column; M1-M11, Manometers; PG, Pressure Gauge; PU-1, 2, Pump; R, Rotameter; SE, Separator; SV1-SV3, Solenoid Valves; T, Circulating tank; V1-V5, Valves; VS, Collector vessel.

Figure 1. Schematic diagram of experimental set-up.

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The ejector assembly along with the nozzle and extended pipe line contactor was aligned properly to achieve an axially symmetric liquid jet. The nozzle was fixed at the optimum position at a distance of one throat diameter from the entry of the throat. This distance was decided from our earlier investigations with horizontal and vertical liquid-gas ejector system (Dutta, 1976; Mukherjee, 1984). The ejector fitted with the extended pipeline contactor serves as both sparger and plunging jet type of bubble column reactor. The lower end of the contactor projected 450 mm inside the separator. The air –liquid separator is a rectangular mild steel vessel of 320×320 mm size and 910 mm height and was large enough to minimize the effect due to liquid leaving the system or air-liquid separation. The nozzles used in the experiments are straight type. It was found that the liquid jet plunged into the accumulated liquid, which entrained air along with it, and there was a zone of intense mixing of gas and liquid. The gas-liquid mixture moved downward to a certain distance depending on the jet momentum and then gas bubbles moved up and got released from the liquid. This was simply a case of plunging jet system. The manometers attached to the system did not show any change in this case, because the air in the system got recirculated in the separator. These phenomena continued until the liquid level increased and touched bottom part of the extended vertical contactor attached to the ejector. As soon as the liquid level inside the separator touched the bottom part of the column, i.e. point ‘t’ (Fig.2c), there was sudden change in suction characteristics of the secondary air due to arresting of the liquid jet inside the contactor and at this liquid level the suction of air was maximum. Some portion of the intense mixing zone entered the vertical column, thus changing the level in the column from ‘t’ to ‘t1’. When the liquid level was further increased by adjusting the valve V5, the jet plunged in the liquid in the extended contactor, as shown in Fig. 2d. Two distinct zones were observed at this stage: an intense gas-liquid mixing zone followed by a downflow fine bubble zone. At this stage, the intense mixing zone was completely in the extended contactor. The intense mixing zone is the zone where the jet penetrates the liquid, releases its energy and disperses the gas. The gas-liquid mixture finally got separated in the separator; the liquid moved out from the valve, V5, while the air from the valve, V4. It was interesting to note that amount of air entrained decreased by increasing the height from ‘t’ at constant liquid flowrate due to increased back pressure. Table 1. Dimensions of the ejector-contactor assembly

Description Height of the suction chamber, hs Diameter of suction chamber, ds Diameter of throat, dt Secondary inlet of ejector,di Length of the Throat, Lt Length of the Diffuser, Ld Diameter of the contactor, dc Length of the contactor (short) Length of the contactor (long)

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Dimension (in mm) 50.0 60.0 19.0 10.0 184.0 204.0 51.6 1500.0 1900.0

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Ajay Mandal

In the present investigation, the short column (1.5 m) was specifically used to study gas entrainment and holdup characteristics with water and aqueous CMC solutions as primary fluids. Mass transfer experiments were also performed with this column. Whereas, the long column (1.9 m) was used to study the pressure drop characteristics, energy dissipation in the ejector and intense mixing zone with aqueous CMC solutions.

2.2. Fluids In the present study aqueous solutions of carboxy methyl cellulose (CMC) at four different concentrations (1.0, 1.5, 2.0 & 2.5 kg/m3) were used as non-Newtonian liquid. Some experiments were also performed with tap water as Newtonian liquid. In both cases air was used as secondary fluid. However, for the measurement of mass transfer parameters aqueous sodium hydroxide and sodium carbonate-bicarbonate buffer solutions were used as the primary fluid while air-carbon dioxide mixture was used as secondary fluid. The density of liquid was measured with the help of a specific gravity bottle and the surface tension by stalagmometer. The rheological properties of CMC solutions were measured by the tube viscometer and the details are discussed below. The physical properties of the fluids have been presented in Table 2. The Rheological Properties of Carboxy Methyl Cellulose It is well known that CMC solution in water is a time-independent non-Newtonian pseudo-plastic fluid and its rheology is described by Ostwald-Dewaele model or Power Law model,

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⎛ du ⎞ τ = K ⎜⎜ ⎟⎟ ⎝ dy ⎠

n

(1)

where K and n are constants for a particular fluid and the value of n is less than one. The constant K is known as the consistency of the fluid; the higher the value of K, the more viscous the fluid. The constant n is called flow index and is a measure of degree of departure from Newtonian behaviour. The viscosity of non-Newtonian liquids can be defined in number of ways, e.g. apparent viscosity (μa), effective viscosity (μeff) and infinite viscosity (μ∞). In the present analysis effective viscosity, μeff has been used throughout calculations. It is defined as the ratio of shear stress at the wall to the average shear rate at the boundary and for pipe flow, on the basis of Poiseuille’s equation, is given by

μ eff =

τ DΔ P 4 L = w 8V D 8V D

(2)

where, D and L are the diameter and length of the pipe respectively and V is the average velocity through the pipe.

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301

μ eff , for pseudoplastic fluids in pipe can be

derived (Mandal, 2003a) from the following general equation,

−

dV = f (τ ) dr

(3)

The obtained effective viscosity in terms of the fluid properties and dimention of the pipe can be written as

μ eff =

τw 8V D

= K (8V / D)

n −1

⎛ 3n + 1 ⎞ ⎜ ⎟ ⎝ 4n ⎠

n

(4)

or

μeff = 8n-1 Vn-1 D1-n K′

(5)

⎛ 3n + 1 ⎞ K′ = K ⎜ ⎟ ⎝ 4n ⎠

(6)

where

n

Therefore, the value of μeff can be evaluated provided K and n are known. It is clear from Eq. (4) that if a logarithmic plot is made between τw and 8V/D, a linear relationship will result and the slope of the line should give the value of n and intercept K[(3n+1)/4n]n i.e. K′. The values of K and n for different CMC solutions are given in Table 2.

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Table 2. Physical Properties CMC solutions measured at 29±1° C Fluid

CMC-1 CMC-2 CMC-3 CMC-4 Water Air

Concentration Flow behaviour kg/m3 index, n 1.0 1.5 2.0 2.5 − −

0.948 0.910 0.871 0.850 −* − **

Consistency index Density K, Pa sn ρl, kg/m3 0.00218 0.00419 0.00588 0.00692 − −

* viscosity of water : 0.797×10-3 kg/m s ** viscosity of air : 1.863×10-5 kg/m s

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1000.8 1001.2 1001.3 1001.5 999.70 1.1650

Surface tension σl, N/m 0.0735 0.0745 0.0750 0.0755 0.0710 −

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Figure 2. Schematic diagram of experimental procedure.

Hydrodynamic Studies on Two-Phase Gas-Liquid Flow…

303

2.3. Flow Rates and Regimes Flow conditions have significant effect on momentum, mass and energy transfer in gas-liquid contacting systems and hence, it is desirable to maintain a constant flow regime throughout. Depending on the flow conditions there are mainly four types of flow, viz. homogeneous bubbly flow, heterogeneous churn flow, slug flow and annular flow. Other types of flow like froth, mist etc. have also been reported in literature (Heuss, et al., 1965; Aihara & Fu, 1986). Generally flow regimes are largely dependent upon the superficial gas velocity and column diameter. But for the present system of ejector induced downflow bubble column, where gas is sucked by high velocity liquid jet, flow regimes primarily depend on liquid flowrate at constant column diameter. Two distinct zones were observed in the column at steady state: an intense gas-liquid mixing zone at the top followed by a downflow homogeneous bubbly flow zone. In bubbly flow the bubbles are quite uniform in size; and they move in orderly fashion with little collision among bubbles and the liquid is mildly stirred by the bubbles. As liquid flowrate increases, there is a natural tendency of the gas bubbles to coalesce forming large bubbles. Above a certain limit of liquid flowrate (which depends on physical properties of liquid and nozzle diameter), rate of coalescence increases rapidly. Under this condition, large bubbles tend to move upward due to higher buoyant forces and consequently, heterogeneous churn-turbulent or slug flow results. However, in the present work, experimental runs were taken only in the homogeneous bubbly flow zone as it offers the maximum gas liquid contacting area with a stable operation. The operating range of liquid flowrate and the corresponding air entrainment rates have been presented in Table 3. Table 3. Operating range of liquid flowrate and corresponding gas entrainment rate

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Primary fluid flowrate, Ql (m3/s) Short column Water: 1.68×10-4 – 3.02×10-4 CMC-solutions: 1.03×10-4– 2.44×10-4 Long column CMC-solution: 2.00×10-4–3.53×10-4

Corresponding air entrainment rate, Qg (m3/s) 0.083×10-4 – 1.22×10-4 0.15×10-4 – 2.06×10-4 0.40×10-5 –9.0×10-5

Kulkarni and Shah (1984) reported a flow regime map in terms of superficial gas and liquid velocities for possible operation in a cocurrent downflow system. With a sparger type gas distributor they found that operation is possible only in regime A, regime C is practically undesirable and operation is not possible in regime B as shown in Fig. 3. However, from the figure it may be found that in the regime B operation is also possible with efficient gas-liquid distributor like ejector-nozzle system. Fig. 3. also shows that the gas to liquid flow ratio in the present work is much higher than in Bando et al. (1988) and comparable with that in Kundu et al. (1995), and iseven higher with non-Newtonian liquids.

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(1 98 4)

al.(1 988 )

Sh ah ar ni & Ku lk

0.05

Ban do e t

Kun du e t

0.10

al(1 995 )

Sysmbol Fluids dn(mm) Water 5 CMC-1 7 CMC-2 5 CMC-3 4 CMC-4 5

Regime-B operation not possible

Practically undesirable

me regi ition s n T ra Regime-A

Regime C

Superficial gas velocity, Vg [m/s]

0.15

operation possible

0.00 0.0

0.1

0.2

0.3

0.4

Superficial liquid velocity, Vl [m/s] Figure 3. Flow regime map of stable co-current downflow.

3. Results and Discussions

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3.1. Gas Entrainment Gas Entrainment by Liquid Jet Ejector Downflow bubble column with gas entrainment by liquid jet ejector system conforms to a new type of cocurrent bubble column, which can be successfully used in chemical, fermentation and waste treatment processes because of its self aeration characteristics. Entrainment occurs due to the of plunging liquid jet and rate of entrainment is mostly controlled by the jet velocity. However, for a plunging liquid jet with a downcomer, entrainment also depends on the two-phase gas-liquid mixing height inside the downcomer. The entrained gas disperses into liquid as fine bubbles in the intense mixing zone and the dispersed gas bubbles are then carried downward by the high momentum of downflow liquid. A minimum liquid flowrate is required to move the bubbles in the downward direction by prevailing over their upward buoyant force, which depends on the bubble size and physical properties of the fluids. Significant works have been reported on gas entrainment by plunging liquid jet through nozzles, venturies and ejector system with and without downcomers. Mechanism of gas entrainment Gas entrainment is a very complex process and largely controlled by the jet velocity. Bin (1993) made a comprehensive review of literature on the mechanism and characteristics of gas entrainment by plunging liquid jets. According to Van de Sande and Smith (1973, 1974), as liquid jet comes out of a nozzle, a thin layer of gas film is formed around it. They also

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reported that for liquid jets with rough surface, some gas is also entrapped into the outer boundary. When the jet impacts the pool surface, both the entrapped gas and the boundary layer gas are carried under the free liquid surface. The jet itself introduces a flow in the receiving liquid whose streamlines, at the plunging point, are directed parallel to the jet. A hole appears round the jet by the flow of liquid itself. This creates a low-pressure zone, which helps the entrainment of the gas. Shear stresses then break up the captured air into fine bubbles. According to McCarthy et al. (1969), Henderson et al. (1970) and Evans & Jameson (1991) the vertical jet produces an indentation in the pool surface, called induction trumpet, which causes the entrainment at the plunging point. Entrainment from such a trumpet is very regular and the phenomenon is analogous to a jet ejector pump, with the plunging point being the site of a free surface vortex. A mechanism of air entrainment by vertical plunging liquid jet has been investigated by Lin and Donnelly (1966) in both laminar and turbulent regions. They reported that entrainment for turbulent jets results from disturbances on the free surface caused by jet instability while, entrainment by laminar jets occurs due to the formation of a thin shell of gas at the point of entrance, the development of oscillations in the shell and by the subsequent breaking up of bubbles. Lara (1979) reported two regions of air entrainment by a vertical plunging jet, viz. droplet region and continuous jet region. In the first case, entrainment occurs when jet breaks into droplets prior to reaching the pool surface. Whereas in the second case, entrainment takes place by a continuous liquid jet. He illustrated the threshold of these two entrainment regions as a function of jet length and jet velocity. Thomas et al. (1984) proposed another mechanism of air entrainment for higher jet velocities. According to them most of the entrained air enters the main flow via the layer of foam formed on the surface of the receiving fluid. Air enters the interstices in the foam, possibly as a result of wave action and splashing, and is then entrained into the main body of the flow along with the recirculating foam. Several other mechanisms of gas entrainment have been reported by different authors (Burgess and Molloy, 1973; McKeogh and Ervine, 1981). Recently, attempts have been made to clarify the effects of nozzle design on entrainment and other related phenomena, especially in the restricted geometry systems (Ohkawa et al., 1985, 1986; Dutta and Raghavan,1987; Bando et al., 1988; Evans, 1990; Kusabiraki et al. 1990; Yamagiwa et al.,1990), which occur when a vertical liquid jet plunges through the surface of liquid contained in a vertical column. For the last case entrainment is affected by the upward buoyant force of the gas bubbles. Minimum entrainment jet velocity When a liquid jet strikes the pool of a liquid, air entrainment occurs when the jet velocity exceeds a certain critical value. The minimum entrainment jet velocity (Ve) is well characterized for coherent viscous laminar jet and is very complicated for turbulent jets. Lin and Donelly (1966) obtained an empirical correlation for laminar viscous jets,

We j = 10 Re 0j.74

(7)

where Wej and Rej are related to the jet diameter and minimum entrainment jet velocity at the plunging point. McKeogh and Ervine (1981) defined a minimum velocity below which jets will entrain air only intermittently. They proposed four mechanisms characterizing the aerated region in the pool based on type of jet. They also described the minimum jet velocity to

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entrain air as a function of the turbulent intensity in the jet. Bin (1988) studied vertical plunging liquid jets and developed a correlation for minimum air entrainment velocity as a function of jet length and nozzle diameter. However, for liquid jet with a downcomer even after gas entrainment occurrs at the plunging point, there is almost complete recirculation of the entrained gas unless certain momentum is imposed on the gas bubbles to move downward. This is because of upward buoyant force of the gas bubbles, which tend to move the bubbles upwards as stated earlier. Thus a minimum liquid jet velocity is required to move the gas bubbles out of the column, which depends on the length of the column, bubble size and the physical properties of the fluids. Variations of gas entrainment rate As discussed earlier, the gas entrainment rate is dependent on the motive fluid flowrate and the resistance of liquid in the column (hl). Again, hl is dependent on the pressure developed in the separator (Ps), as separator pressure holds the liquid in the column. As the separator pressure increases the entrained gas bubbles face more resistance to move downward and consequently lower entrainment results. The variations of gas entrainment rate (Qg) with separator pressure for air–water and air-CMC solution systems at different liquid flow rates have been presented in Figs. 4-6. It may be seen from these figures that at constant liquid flowrate gas entrainment rate decreases with Ps. Rate of air entrainment pattern is almost same for air-water and air-CMC systems. However, a higher entrainment was observed for the latter system, which is due to formation of a higher gas film thickness around the highly viscous non-Newtonian liquid jet, which is similar to the results reported by Kumagai and Endoh (1982) in the same range of Reynolds numbers. The effects of motive liquid flowrate on air entrainment rate at constant Ps have been presented in Fig. 7. It may be seen from the plots that entrainment rate increases with liquid flow rate for a particular nozzle, due to increase in average velocity of air film around the liquid jet (Evans et al., 1996). Further, on extrapolating the plots of Qg vs QL of Fig. 7 the liquid flowrate corresponding to zero gas flowrate gives the minimum entrainment flowrate for different systems. For the present system minimum entrainment velocity based on nozzle diameter was found to vary between 2 m/s and 7 m/s. Effect of nozzle diameter on gas entrainment is very complicated. There are two contrasting effects of nozzle diameter on gas entrainment by plunging liquid jet. A decrease in nozzle diameter and hence increase in jet velocity at a constant liquid flowrate tends to increase the entrainment by increasing the mean linear velocity of the gas film around the jet surface (Dutta and Raghavan, 1987, Kundu et al., 1994; Zahradnik et al., 1997; Ide et al., 2001,). On the other hand, an increase in nozzle diameter increases the contacting perimeter around the liquid jet surface, which in turn increases the entrainment rate (Ohkawa et al.,1986; Yamagiwa et al., 1990). Moreover, it is often interfered by other parameters like jet flow regime, gas-liquid mixing height for ejector having downcomer and physical properties of liquid. Experimental results of the present system show mixed effects of nozzle diameter on gas entrainment rate for different solutions in the wide range of motive liquid flowrate. The experimental data has been analyzed by expressing entrainment rate as dimensionless air entrainment ratio, Qr (Qg/Ql). The following generalized correlation for both Newtonian and non-Newtonian fluids was obtained by multiple linear regression analysis of the experimental data.

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Qr = 0.3074 × 10 4 Re 0ln.545 Mon0.512 H R−1.34

(8)

The correlation coefficient and standard deviation of the Eq. (8) were found to be 0.931 and 17.7 % respectively. The calculated values of Qr were plotted against the experimental values and are shown in Fig.8. Energy efficiency of gas entrainment The energy efficiency of the present ejector system as gas-entrainment device was evaluated by the kinetic power P of the jet divided by the volumetric gas entrainment rate Qg. The value of P was obtained by the following equation (Bin and Smith, 1982),

πρ l d n2Vln3

(9)

8

Table 4 shows the air entrainment ratio ranges of present work and other authors. On an average the present data are comparable with McKeogh and Ervine (1981), Ciborowski and Bin (1972), Ohyama et al. (1953) and Ohkawa et al. (1986); but considerably smaller than that of Van de Sande and Smith (1975). This is because, Van de Sande and Smith used only plunging jet without any downcomer but in the present system, a minimum height of gasliquid mixture is always maintained in the downcomer, which has a negative effect on air entrainment. The energy efficiency of the present system and other types of gas-liquid contactors with gas entraining or gas sparging have been summarized in Table 5. The gas flowrate Qg per unit liquid volume VL of the apparatus has been often used as one of the measures to express the aeration intensity of the aerator. The values of Qg/VL are also shown in the Table. It may be seen from the Table that the present system showed higher efficiency compared to that of other types of gas-entraining-aeration systems.

-4

1.5x10

System: Air-water dn:6 mm

3

Symbol Ql (m /s) -4

2.18x10

-4

2.35x10

3

Gas entrainment rate, Qg [m /s]

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P =

-4

2.52x10 -4

-4

1.0x10

2.68x10

-4

2.85x10

-5

5.0x10

0.0 3

2.0x10

3

4.0x10

3

6.0x10

3

8.0x10

Separator pressure, Ps [Pa]

Figure 4. Effect of separator pressure on gas entrainment at different liquid flowrates for air-water system.

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1.5x10

3

System:air-CMC-1 dn: 5 mm

-4

Symbol Ql (m /s) -4

1.40x10

Gas entrainment rate, Qg [m /s]

-4

3

1.60x10

-4

1.78x10

-4

1.0x10

2.00x10

-4

-4

2.26x10

5.0x10

-5

0.0 3.00x10

3

4.50x10

3

6.00x10

3

7.50x10

3

Separator pressure, Ps [Pa]

Figure 5. Effect of separator pressure on gas entrainment at different liquid flowrates for air-CMC-1 system. -4

1.5x10

3

Gas entrainment rate, Qg [m /s]

System: Air-CMC2 dn: 6 mm

-4

1.0x10

4

3

Symbol Ql X10 (m /s) 1.03 1.17 1.32 1.48

-5

5.0x10

0.0 3

3

3

6.0x10

4

8.0x10

1.0x10

Separator pressure, Ps [Pa]

Figure 6. Effect of separator pressure on gas entrainment at different liquid flowrates for air-CMC-2 system.

-4

dn: 6 mm, Ps : 4.4 kPa

Gas entrainment rate, Qg [m /s]

2.25x10

Symbol System water 3

CMC-1 (1.0 kg/m )

3

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4.0x10

3

CMC-2 (1.5 kg/m ) -4

1.50x10

-5

7.50x10

0.00 -5

7.50x10

1.50x10

-4

-4

2.25x10

-4

3.00x10

3

Liquid flowrate, Ql [m /s]

Figure 7. Effect of motive liquid flowrate on air entrainment rate.

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1.0

Symbol System Water CMC-1 CMC-2 CMC-3 CMC-4

0.8

Qr-Calculated

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Qr-Experimental

Figure 8. Comparison of experimental value of Qr and that calculated from Eq. (8).

Table 4. Gas entrainment ratio reported by different authors Authors

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Van de Sande & Smith (1975) Mckeogh & Ervine (1981) Ciborowski & Bin (1972) Ohyama et al. (1953) Ohkawa et al. (1986) Present work

Fr =(Vln/ gd n )

Qr

5-30 7-20 8-16 20-50 5-55 15-65

0.19-1.2 0.14-0.45 0.02-0.50 0.10-0.50 0.04-1.0 0.10-0.80

Table 5. Aeration performance of gas-liquid contactors Authors Type of contactor P/Qg[kWs/m3] Matsumura et al.(1982) Tank-type gas entrainer 102-103 Fukuda et al. (1963) Aerated –stirred fermenter 80-140 Topiwala and Hamer (1974) Hollow impeller 300-700 Zundelevich (1979) Turbo aerator 60-800 Ohkawa et al.(1987) Water jet aeration in pool system 15-30 Present Work Ejector induced downflow column 5-200

Qg/VL [1/s] 0.0015-0.03 0.001-0.05 0.002-0.02 0.01-0.002 2×10-4-0.002 0.02-0.20

3.2. Gas Holdup Variations of gas holdup In cocurrent downflow, the gas bubbles move against their high buoyant force, having a higher slip velocity compared to upflow system. Thus the bubbles have a larger residence time and hence higher gas holdup. Briens et al. (1992) studied the venturi-bubble column with both downflow and upflow mode, and obtained much higher gas holdup (0.15-0.40) in

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downflow compared to upflow (0.08-0.12). The gas distributor and physical properties of fluids also have some effect on gas holdup (Deckwer, 1992). The bubble flow is also characterized by the dispersion of the bubbles and the free space available between the bubbles. If the bubbles are small enough (db < 1mm) and there is sufficient free space, bubble population increases with increase in gas flowrate and thus gas holdup increases. But beyond a certain limit there is little effect of gas flowrate on bubble population and coalescence of bubbles increases significantly and hence gas holdup remains constant (Godbole et al., 1982; Schumpe and Deckwer, 1982; Zahradnik et al., 1997). Here bubble void fractions are generally in the range of 0.01-0.30. In the present system of cocurrent downflow the bubbles are comparatively bigger in size (db ~3-5 mm) and spend more time in the column and, thus, higher gas holdup (0.38-0.60) was obtained. Fig. 9 shows variation of overall gas holdup with gas flowrate at constant liquid flowrate. At a constant liquid flowrate, gas holdup increases with increase in gas flowrate because of increased bubble population. However, span of gas holdup variation is very small, as even at very low gas flowrates higher gas holdup was observed due to higher residence time of the bubbles. Fig.10 shows the variation of gas holdup with separator pressure at different liquid flowrates for air-water system. As separator pressure increases the bubbles face more resistance to move downward, which raises the bubble residence time. Thus gas holdup was found to increase with separator pressure at constant liquid flowrate. It may also be seen from the plots that gas holdup decreases with liquid flowrate at the same separator pressure, due to increase in momentum imparted by liquid on the gas bubbles. Fig.11 shows the variation of gas holdup with separator pressure for different solutions at constant liquid flowrate. It was found experimentally that in the homogeneous flow regime gas holdup for CMC solutions are higher than that of water and increases with CMC concentration. This is because, for highly viscous CMC solutions the gas bubbles inside the column face more resistance to move in the downward direction. Thus the residence time of the gas bubbles increase, which in turn increases the gas holdup. However, very complex behaviours of gas holdup variations for different CMC solutions were experimentally observed at different flow regimes due to differences in their rheological behaviours (Mandal et al., 2003a). Schumpe and Deckwer (1982) also obtained an increase in gas holdup with CMC concentration up to 0.8 wt % CMC due to decreasing bubble rise velocity, but found it behaves differently in the heterogeneous flow regime. At constant liquid flowarate gas entrainment rate is hardly dependent on the nozzle diameter. However, nozzles with small diameter produce relatively small size bubbles, which move downward faster due to lower buoyant force. Thus under similar conditions lower gas holdup was observed with smaller nozzles in the present work. Typical plots of variations of gas holdup with separator pressure for different nozzles have been shown in Fig. 12 for air-water system. Bando et al (1988) also obtained similar trend of gas holdup in a cocurrent downflow bubble column with simultaneous gas-liquid injection nozzle system. A comparative picture of gas holdup obtained by different authors over a range of gas and liquid flow rate is given in Table 6. In two-phase gas-liquid flow, gas holdup generally depends on the type of flow and gas to liquid flow ratio (Qr). At comparable Qr, higher gas holdup was observed in cocurrent downflow compared to upflow due higher residence time of gas bubbles. Briens et al. (1992) obtained a maximum gas holdup of 0.40 in a downward venturi-bubble column combination while in upward flow the gas holdup achieved was only 0.12 under similar conditions. Bubble size has opposite effect on gas holdup, depending on whether the system is upflow or downflow. For downflow system, large bubbles spend more

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time in the column and, thus, higher gas holdup is obtained. In the present work of downflow system, a relatively higher gas holdup was obtained compared to other reported works even at low gas flowrates as shown in Table 6. Gas holdup obtained in air-CMC solution systems (0.45-0.61) are slightly higher than those for air-water system (0.38-0.50) due to highly viscous behaviour of CMC solutions. Analysis of holdup by different correlations and models Gas holdup data of the present system was analyzed by different models and correlations, of which modified Lockhart-Martinelli (1949) correlation, slip velocity model and drift flux model of Zuber and Findlay (1965) were found to satisfactorily explain the present data. Analysis by Lockhart-Martinelli correlation: Though the Lockhart-Martinelli correlation predicts holdup for horizontal flow of gas-Newtonian liquid, it has been tried in the present system because the modification proposed by others are based on this correlation. Typical values of the liquid holdup, εL and the corresponding Lockhart-Martinelli parameter, X defined by Eq. 10, have been plotted in Fig. 13 along with the Lockhart-Martinelli curve. However, it may be seen from figure that the experimental data do not conform well to the Lockhart-Martinelli correlation.

X 2 = ⎛⎜ Δp f ⎜ Δz ⎝

⎞ ⎟⎟ ⎠l

⎛ Δp f ⎜⎜ ⎝ Δz

⎞ ⎟⎟ ⎠g

(10)

Analysis has also been done with the proper modification of the Lockhart-Martinelli correlation proposed by Davis (1963) for vertical flow. Thus the modified parameter, X ′ is expressed as:

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X ′ = 0.19 Frtp0.185 X

(11)

Values of the modified parameters X ′ calculated by Eq. (11) are also plotted in Fig 14 along with the Lockhart-Martinelli curve. It may be found that the experimental data are very close to the Lockhart-Martinelli curve. The marginal deviations are probably due to different geometrical system and non-Newtonian behaviour of the liquid phase. Slip Velocity Model: Slip velocity model first introduced by Behringer (1936) has been used to analyze the experimental holdup data for two-phase flow with air-water and air-CMC solutions in the present system. The slip velocity of bubbles relative to surrounding liquid is defined as us = ± [ug – ul] = ±

[

Vg

εg

−

Vl 1− ε g

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]

(12)

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Ajay Mandal

where negative sign is incorporated to consider the downflow system. Thus, gas holdup data can be correlated as a function of Vg and Vl. Slip velocity is a function of the rise velocity ub of a single bubble in still liquid and is expressed as us = ubƒ(εg) (13) Hills (1976) obtained the following correlation for cocurrent upflow system us = 0.24 + 4.0 εg1.72 for Vl ≤ 0.3m s-1

(14)

with ub = 0.24m/s. Clark and Flemmer (1985) performed a regression analysis on their results using the equation proposed by Wallis (1969) us = ub (1-εg)n

(15)

and found best fit for n = 0.702 and ub = 0.25 m/s. The experimental data of the present system was analyzed by plotting the slip velocity against gas holdup and the following equation was obtained us = ub + c1 εg

(16)

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The plots of us vs εg have been drawn as shown in Fig. 15-17. It may be found that the experimental data fall around a straight line. The values of slope of the straight lines, c1 and intercept, ub for different CMC solutions are given in Table 7. A negative value of ub is found due to downflow system, but the magnitude is in good agreement with the results obtained by other authors. Analysis by Zuber and Findlay (1965) Model The model suggested by Zuber and Findlay (1965) is used by several authors for analysis of the holdup data in both gas-Newtonian and gas-non-Newtonian liquid flow in simple pipeline contactors. Chandrakar (1985) has successfully applied this model for the analysis of the holdup in an ejector-induced vertical upflow of air-CMC solution system. Ohkawa et al. (1986) have also used the model for a downflow bubble column with plunging water jet system. This approach has been tried for the present system also with both air-water and airCMC solution systems. The average gas velocity, u g versus gas-liquid mixture velocity, Vm yields a linear relation in accordance with Eq. (17),

u g = CoVm + Vd

(17)

The plots of u g and Vm are straight lines as shown in Figs. 18-19 for different systems. The values of the distributed parameter, Co and the weighted average drift velocity, Vd obtained from the slope and intercept of the straight line respectively have been presented in Table 8. Co values for the present system varies from 1.23 to 1.56 while Vd varies from - 0.61 to – 0.17. Negative value of Vd implies downflow system. Clark and Flemmer (1985) obtained average values of Co as 1.07 and 1.17 for upflow and downflow respectively. They

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also obtained the value of drift velocity as 0.25 m/s for upflow and – 0.25 m/s for downflow of air water system. However these values scatter a little with bubble sizes, physical properties of liquid and geometric parameters of the system. Das et al. (1992) got average values of Co and Vd as 1.50 and 0.53 m/s respectively for vertical upflow system with airCMC system. The values of Co and Vd obtained by Yamagiwa et al. (1990) in cocurrent downflow with air-water system were 1.17 and – 0.19 m/s respectively. Kelkar et al. (1983) made studies on different alcohols in cocurrent upflow system and obtained the values as Co =1.24-2.41 and Vd = 0.06-0.083 m/s. Thus, the model satisfies the expremental data of the present system satisfactorily. The Co value for air water system is higher than that of air-CMC solution system, which indicates that the bubble size distribution is more uniform in CMC solution. Drift velocity in first system is also higher compared to the latter indicating a lower bubble rise velocity in CMC solution at zero gas voidage. Correlation of gas holdup A correlation has been developed by dimensional analysis to predict the dispersed phase holdup in terms of the physical, dynamic and geometric variables of the system. The multiple linear regression analysis of the data yielded the following correlation for gas holdup comprising both Newtonian and non-Newtonian liquid system,

εg =0.617 Rel-0.163 Mo-0.028 Ar-0.031 Hr 0.207

(18)

Numerical analysis was done on 1110 experimental data with different liquids and operating conditions. The correlation coefficient and overall standard deviation of Eq. (18) were calculated and found to be 0.902 and 4.4% respectively. The calculated values of εg from Eq. (18) were plotted against the experimental values and are shown in Fig. 20.

System: Air-CMC-3 dn: 6 mm

0.5

Gas holdup, εg

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0.6

3

Symbol Ql (m /s) -4

2.10x10 0.4

-4

2.45x10

-4

2.62x10

-4

2.79x10

-4

2.95x10 0.3 0.00

-5

2.50x10

-5

5.00x10

-5

7.50x10

Gas entrainment rate, Qg [m /s]

Figure 9. Variation of overall gas holdup with gas flowrate.

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-4

1.00x10

3

314

Ajay Mandal 0.6

System: Air-water dn : 6 mm

3

Symbol Ql (m /s) -4

2.18x10

-4

2.35x10

-4

2.52x10

Gas holdup, εg

0.5

-4

2.68x10

0.4

0.3 2.0x10

3

3

3

4.0x10

6.0x10

Separator pressure, Ps [Pa]

Figure 10. Variation of gas holdup with separator pressure for air-water system.

Gas holdup, εg

0.6

0.5

Symbol System Air-Water Air-CMC-1 Air-CMC-2 dn : 6 mm -4

3

Ql : 2.10X10 m /s 0.4 2.0x10

3

4.0x10

3

6.0x10

3

Figure 11. Variation of gas holdup with separator pressure for different solutions.

Air-Water System

0.5

Gas holdup, εg

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Separator pressure, Ps [Pa]

Symbol dn(mm) 7 6 5

0.4

-4

3

Ql : 2.68x10 m /s 3.00x10

3

3

4.50x10

6.00x10

3

7.50x10

3

Separator Pressure, Ps [Pa]

Figure 12. Effect of nozzle diameter on gas holdup for air-water system.

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315

Lockhart-Martinelli curve

1 0.9 0.8 0.7 0.6 0.5

εl

0.4 0.3

System: Air-CMC solution 3

Symbol CMC conc. (kg/m ) 1.0 2.0 2.5

0.2

0.1 10

20

30

40

50

60

70 80 90 100

X

Figure 13. Comparison of holdup data with the Lockhart-Martinelli correlation. 2

1 0.9 0.8 0.7 0.6

Lockhart-M artinelli curve

εl

0.5 0.4 0.3

System: Air-CM C solution 3

Symbol CM C conc. (kg/m ) 1.0 2.0 2.5

0.2

0.1 1

2

3

4

5

6

7 8 9 10

20

/

Figure 14. Comparison of holdup data with Davis modification of the Lockhart-Martinelli correlation. 0.30

System: Air-Water Symbol dn (mm) 5 6 7 Vl: 0.096-0.150 m/s

0.25

0.20

us [m/s]

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X

0.15

0.10

0.05

0.00 0.35

0.40

0.45

εg

0.50

0.55

Figure 15. Slip velocity for air water system.

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Ajay Mandal 0.24

System: air-CMC-1 Symbol dn(mm) 4 5 Vl: 0.06-0.10 m/s

0.20

us [m/s]

0.16

0.12

0.08

0.04

0.00 0.36

0.40

0.44

0.48

0.52

0.56

0.60

εg

Figure 16. Slip velocity for air-CMC solution system.

0.16

us [m/s]

0.12

System: Air-CMC-2 Symbol dn(mm) 4 5 Vl : 0.05-0.08 m/s

0.08

0.04

0.00 0.45

0.50

0.55

0.60

0.65

0.70

Figure 17. Slip velocity for air-CMC solution system.

0.12

ug [m/s]

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εg

Air-Water system dn = 6 mm 4

3

Symbol Ql x10 (m /s) 2.35 2.52 2.68

0.08

0.04

0.00 0.10

0.12

0.14

0.16

0.18

0.20

Vm [m/s]

Figure 18. Zuber Findlay correlation for air-water system.

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0.20

System:Air-CMC-1 dn= 6 mm 4

_ ug [m/s]

0.16

0.12

317

3

Symbol Ql x10 (m /s) 1.6 1.78 2.0 2.26 2.44

0.08

0.04

0.00

0.06

0.09

0.12

0.15

0.18

0.21

0.24

Vm [m/s]

Figure 19. Zuber Findlay correlation for air-CMC system.

0.7

Symbol

εg-Calculated

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0.6

System air-water air-CMC-1 air-CMC-2 air-CMC-3

0.5

0.4

0.4

0.5

0.6

0.7

εg-Experimental

Figure 20. Comparison of the experimental values of gas-phase holdup with those calculated from Eq.(18).

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Table 6. Comparative results of gas holdup of present works with other authors

Authors

Type of flow

Godbole et al. (1982)

Column Diameter(m)

batch with gas upflow

0.10

Schumpe & Deckwer (1982)

cocurrent upflow

0.10-0.14

Khatib & Richardson (1984)

cocurrent upflow

0.039

Liquid used

Liquid flow Gas flow range Gas holdup Range,Vl (m/s) Vg(m/s) εg

CMC CMC (0-1.8 wt%)

-

0.05- 0.3

0.1- 0.28

-

0.003- 0.025

0.03- 0.20

Kaolin suspension

0.305 - 0.61

0.30-3.5

0.15- 0.55

Ohkawa et al. (1986)

cocurrent downflow

0.02-0.026

water

0.05 - 0.20

0.05-0.20

0.01- 0.40

Bando et al. (1988)

cocurrent downflow

0.07

water

0.10 - 0.20

0.01-0.10

0.01- 0.32

Yamagiya et al. (1990)

cocurrent downflow

0.034-0.070

water

0.4 - 0.912

0.1- 0.50

0.15 - 0.40

Briens et al. (1992)

downflow venturi column

0.095

water

0.2 - 0.5

0.01 - 0.10

0.15 - 0.40

Briens et al. (1992)

upflow venturi column

0.095

water

0.2 - 0.5

0.01- 0.10

0.08 - 0.12

Das et al. (1992)

cocurrent horizontal

0.019

CMC (0.5-1.0 kg/m3)

0.141- 1.00

0.067-1.55

0.10 - 0.40

0.296 -1.00

0.17-1.60

0.12- 0.45

0.008 - 0.029

0.004-0.076

0.05 - 0.24

0.08 - 0.144

0.004 -0.058

0.38- 0.50

0.007 - 0.099

0.45 - 0.61

3

CMC (0.5-1.0 kg/m )

Das et al. (1992)

cocurrent upflow

0.019

Zahradnik et al. (1997)

cocurrent upflow

0.29

water

cocurrent downflow

0.0516

air-water

Present work Present work

cocurrent downflow

0.0516

3

CMC (1.0-2.5 kg/m ) 0.05-0.117

Hydrodynamic Studies on Two-Phase Gas-Liquid Flow…

319

Table 7. Parameters of Slip Velocity model (defined in Eq. 16) Fluid Water CMC-1 CMC-2

c1 0.902 0.718 0.610

ub (m/s) -0.254 -0.257 -0.250

Table 8. Values of Zuber-Findlay parameters (defined in Eq. 17)

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Investigator

System

Distribution Parameter, Co

Average drift velocity (Vd), m/s

Present work

Air-Water Air-CMC-1 Air-CMC-2 Air-CMC-3

1.56 1.23 1.24 1.28

-0.17 -0.084 -0.061 -0.092

Rosehart et al. (1975)

Air-CMC Air-Polyhall 295 Air-Carbopol 941

1.57 1.54 1.35

0.25 0.56 0.35

Kelkar et al. (1983)

Air-n-butanol Air-n-propanol Air-ethanol

1.24 1.30 1.66

0.060 0.066 0.074

Chhabra et al. (1984)

Air-CMC 1.25 % Air-CMC 1.00 % Air-CMC 0.75 %

1.24 1.16 1.23

0.17 0.12 0.33

Clark et al. (1985)

Air-water upflow Air-water downflow

1.07 1.17

0.25 -0.25

Das et al. (1992)

Air-CMC 0.50 kg/m3 Air-CMC 0.67 kg/m3 Air-CMC 0.83 kg/m3 Air-CMC 1.00 kg/m3

1.40 1.47 1.50 1.67

0.53 052 0.50 0.62

3.3. Pressure Drop Pressure drop characteristic is an important design parameter for scaling up a gas-liquid reactor. Literature review reveals that significant works have been done on pressure drop characteristics in two-phase cocurrent gas-Newtonian liquid flow in vertical or horizontal columns. Many books have been published and some of these are by Brodkey (1967), Wallis (1969), Brauer (1971), Govier and Aziz (1972), Butterworth and Hewitt (1977), Azbel (1981), Chisholm (1983), Deckwer (1992) etc. Scott (1963) has also contributed a chapter on

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properties of cocurrent gas-liquid flow in Advances in Chemical Engineering. However, most of these studies have been made on frictional pressure drop in ordinary pipeline contactors. Further, very few studies have been reported on two-phase gas- non-Newtonian liquid systems. In the present system of gas-non-Newtonian liquid two-phase flow, the experiments were carried out in the free suction regime, i.e. air was sucked through the secondary entrance of the ejector by the high velocity liquid jet. Unlike in horizontal two-phase flow, the frictional pressure drop in vertical flow systems cannot be directly obtained because the measured pressure drop is the sum of the frictional pressure drop (ΔPf), the column hydrostatic head (ΔPh) and pressure drop due to accelerative effects (ΔPa). Thus, the total pressure drop of bubbly flow zone (Δz) in the vertical column may be expressed as, ΔPT = ΔPf + ΔPh + ΔPa

(19)

According to Friedel (1980), the accelerative component is of significance only in evaporating or condensing flows and may be neglected in case of adiabatic systems and hence Eq. (19) reduce to the following equation, ΔPf = ΔPT − ΔPh

(20)

Hydrostatic head has been calculated by considering the actual insitu density of the twophase system as given by Eq. (21) ΔPh = g Δz (ρlεl + ρgεg)

(21)

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From Eq. (20) and Eq. (21), neglecting the density of gas compared to liquid, one can express the two-phase frictional pressure drop as,

ΔPf =ΔPT - gΔz ρlεl

Therefore,

ΔPf

ρ l gΔz

=

ΔPT -εl ρ l gΔz

(22)

(23)

Thus, two-phase frictional drop has been obtained from Eq. (23) by measuring the total pressure drop and the corresponding holdup fraction. Variation of the Two-Phase Frictional Pressure Drop The effect of gas flowrate on gas-non-Newtonian liquid two-phase frictional pressure drop, ΔPf /ρLgΔz, at constant liquid flowrate for typical air-CMC system has been illustrated in Figs. 21. It may be observed from this plot that the frictional pressure drop increases with increasing gas flowrate, Qg at constant liquid flowrate. This increase in pressure drop can be explained by the fact that an increase of Qg causes higher population of gas bubbles, which in turn increases the true liquid velocity. It is also seen that, lower pressure drops are observed at higher liquid flowrate for same gas flowrate. The reason for this can be elucidated by considering the increasing drag experienced by the bubbles and coalescence of gas bubbles.

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At higher liquid flowrate, comparatively bigger bubbles are formed due to coalescence (Mahalingam and Valle, 1972 and Godbole et al., 1982), which causes a decrease in the true liquid velocity because of increase in liquid flow area. Moreover, due to bigger bubbles size some of the liquid get entrapped in the wake of bubble (Collins, 1965; Narayanan et al., 1974 and Pal et al., 1980) and as a result the bulk liquid velocity reduces which causes decreasing frictional pressure drop at increasing liquid fowrate. From these plots it may be further seen that in the region of lower gas flowrate, the initial rate of increase frictional pressure drop is much higher and the rate gradually decreases with increasing gas flowrate. This may be explained from the fact that at lower gas flowrate, gas is dispersed as fine bubbles and as the gas flowrate is increased the bubble size increases by coalescence, thereby decreasing velocity of liquid in the column. Fig 22 depicts the variation of frictional pressure drop with Qg at same liquid flowrate for different CMC solutions, from which it is clear that the twophase frictional pressure drop increases with increase in concentration of CMC solution due to increased viscosity.

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Effect of Gas Holdup on Frictional Pressure Drop Fig 23 shows the variation of two-phase frictional pressure drop with gas-holdup,εg. As expected, frictional pressure drop increases with εg due to decreasing flow area of the liquid. A comparative pictures of gas-liquid two-phase frictional pressure drop and the corresponding gas holdup for the present system and that of gas-liquid contactors reported by several authors are given in Table 9. In the present system a moderate pressure drop was observed with a relatively higher gas holdup at very low gas flowrate indicating an efficient dispersion of gases into liquid phase. Analysis of Experimental Data The variations of frictional pressure drop with gas and liquid flowrate, nozzle diameter and properties of liquid have been investigated for the present system. Experimental data of frictional pressure drop were analyzed by two different approaches viz. (i) pressure drop ratio method and (ii) two-phase friction factor method. In the first approach, the experimental data of frictional pressure drop have been analyzed by various established correlations viz., Lockhart-Martinelli (1949), Davis (1963) correlation for vertical gas-liquid two-phase flow, and Aoki correlation (1965) for vertical two-phase gas-Newtonian liquid flow. Lockhart-Martinelli (1949) proposed correlations for the analysis of frictional pressure drop in the horizontal two-phase gas-Newtonian liquid systems for four different combinations i.e., laminar-laminar, laminar-turbulent, turbulent-laminar and turbulentturbulent of gas-liquid flow. They presented the frictional pressure drop data in the form of plots of pressure drop ratios,

φ l and φ g versus the parameter X, where, ⎛ Δp ⎞

⎛ Δp ⎞

f ⎟⎟ / ⎜⎜ f ⎟⎟ φ l2 = ⎜⎜ ⎝ Δz ⎠ T ⎝ Δz ⎠ l

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Ajay Mandal

⎛ Δp ⎞

⎛ Δp ⎞

f ⎟⎟ / ⎜⎜ f ⎟⎟ φ g2 = ⎜⎜ ⎝ Δz ⎠ T ⎝ Δz ⎠ g

X 2 = ⎛⎜ Δp f ⎜ Δz ⎝

⎞ ⎛ Δp f ⎟⎟ / ⎜⎜ ⎠ l ⎝ Δz

⎞ ⎟⎟ ⎠g

(25)

(26)

The frictional pressure drop, (Δpf/Δz)l and (Δpf/Δz)g for the flow of liquid and gas alone over a distance, Δz of the column have been evaluated by Fanning’s equation. Davis (1963) extended the applicability of the Lockhart-Martinelli approach for vertical flow through the modification of the parameter, X by incorporating the Froude number of the gas-liquid mixture, Frtp to allow the effects of gravity and velocity. Thus the modified parameter, X ′ is defined earlier in Eq (11). It has been observed that in the present case the data have deviated remarkably from the Lockhart-Martinelli correlation based on Davis modification (Mandal et el., 2004a). The experimental value of the frictional pressured drop for a particular value of X′ is higher than that given by this approach. This may be attributed to the better dispersion of gas in the liquid in the present system. The improved dispersion reduces the actual flow area available for liquid flow in the column thereby increasing the actual liquid velocity and thus increasing the pressure drop. Considering the enhanced gas-liquid mixing and hence higher ( Δp f / Δz ) T , the frictional multipliers, φ l of Lockhart-Martinelli correlation has been modified by multiplying a factor, obtained by regression analysis of the present experimental data.

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φ l/ =0.15 φ l

(27)

Fig 24 shows that the experimental results matched well with the Lockhart-Martinelli correlation by modification in frictional multiplier. Aoki (1965) developed a correlation for the prediction of Lockhart-Martinelli parameter,

φ l for the vertical two-phase air water flow as, ⎛ Q φ = 1 + 250⎜⎜ r ⎝ 1 − Qr 2 lA

The values of

⎞ ⎟⎟ ⎠

0.8

(28)

φ lA and X calculated from the experimental data have been plotted as

shown in Fig 25 along with the original Lockhart-Martinelli curve. It may be observed Aoki correlation qualitatively explain the experimental data of the present system and a little deviation implies an enhanced mixing in the present system. The analysis of the experimental data has also been done by a second method, i.e. friction factor method. For single phase liquid flow, Fanning’s friction factor, fl is related to the frictional pressure drop by the following equation:

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ΔPfl

ρ l gΔz

=

2 f lVl gd c

323

2

(29)

In analogous way a two-phase friction factor, ƒtpl based on the liquid superficial velocity in the column may be defined as,

(

ΔPf

ρ l gΔz

)l =

2 f tplVl

2

gd c

(30)

Substitution of Eq. (30) in Eq. (23) gives

f tpl =

⎤ gd c ⎡ ΔPT − εl ⎥ 2 ⎢ 2Vl ⎣ ρ l gΔz ⎦

(31)

From the experimental pressure drop data the two-phase friction factor, ftpl may be calculated by Eq. (31). The two-phase friction factor, ftpl has been correlated with the physical, dynamic and geometric variables of the system. The multiple regression analysis of the experimental data yielded the following correlation with correlation coefficient and standard deviation 0.96 and 4.75 % respectively;

f tpl = 0.356 × 10 4 Re l−1.702 Mo −0.384 Ar0.201 H r−1.069

(32)

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The friction factor values calculated from Eq. (32) are compared with excremental values and are shown in Fig. 26. Gas-holdup Distribution from Pressure Drop Data Estimation of gas-holdup in two-phase gas-liquid system by measuring the column pressure is a well-known method (Marchese et al., 1992). To obtain a cumulative gas-holdup throughout the column, pressures at different positions of the column were measured by manometers as shown in Fig. 27. An assumption has been made to incorporate this method that, the dynamic pressure component is negligible compared to hydrostatic pressure. Further for a particular gas and liquid flowrate frictional loss is assumed to be constant throughout the column. Under steady operation the overall gas-holdup can be obtained by measuring the total gas-liquid mixing height (hm) and the corresponding clear liquid height (hl) in the arm L (Fig. 27) neglecting the frictional and accelerating pressure terms, and it can be expressed as,

ε g = ε g1 =

hm − hl hm

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A pressure balance equation between the bottom part of the column and at any position, i, can be written as, (34)

where Pi is the pressure at point i (1,2,3…7) (Fig. 27) and εgi is the overall gas-holdup from point i to the top of the gas-liquid mixture (Mandal et al., 2004b). By measuring the pressures at different positions of the column and hmi, the height of a gas-liquid mixing zone from position i to the upper point of the gas-liquid mixture in the column, the gas-holdup values for that particular zone can be obtained. Fig. 28 show a typical characteristic picture of variation of εgi with hm. Analysis of the plots gives an average distribution of gas holdup in different zones of the column, as shown in Fig. 29. The whole column may be roughly divided into three different sections viz. A, B and C. From the Fig. 29, it is clear that Section-B contributes maximum gas-holdup to the overall. Whereas, top and bottom sections show decreasing trend of gas-holdup towards the end. Experimental observations also show that there are three distinct zones in the column at steady state. At the top, there is an intense gasliquid mixing zone where the jet penetrates the liquid, releases its energy and disperses the gas. A significant back mixing and hence, regeneration of gas bubbles occur in this section. This is followed by a uniform bubble zone with substantial bubble population. The bubble sizes in this section are relatively higher than the previous section. The bottom section consists of bubbles whose sizes permit them to move downward by the imparted momentum of liquid flow which prevail over the upward buoyant force. Bubble population in this section is relatively lower than the previous section and consequently lower gas holdup was observed. An increase in bubble population always leads to higher gas-holdup. An increase in bubble size and hence, slip velocity also leads to higher gas-holdup for cocurrent downflow, since the bubbles then spend more time in the column. In the intense mixing zone, there is significant bubble population but bubble sizes are much smaller compared to the following section and thus a lower gas-holdup results. 0.5 0.4 0.3

0.2

ΔPf /ρlgΔz

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P1 – Pi = hm(1-εg1) ρl g – hmi(1-εgi) ρl g

4

3

Symbol Ql x10 (m /s) 2.0 2.14 2.47 2.64

3

System: Air-CMC-2(1.5kg/m ) dn: 6 mm

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 -5

2.0x10

-5

4.0x10

-5

6.0x10

-5

8.0x10

3

Qg [m /s]

Figure 21. Variation of frictional pressure drop with Qg at constant Ql.

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0.5 4

3

Symbol Ql x10 (m /s) 2.99 3.40 3.53

0.4 0.3

3

System: Air-CMC-4 (2.5kg/m ) dn: 7 mm

ΔPf /ρlgΔz

0.2

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.0

2.0x10

-5

-5

4.0x10

6.0x10

-5

3

Qg [m /s]

Figure 22. Variation of frictional pressure drop with Qg at constant Ql. 0.20

System: Air-CMC-3 dn : 6 mm 4

ΔPf /ρlgΔz

0.15

3

Symbol Ql x10 (m /s) 2.10 2.62 2.79

0.10

0.05

0.40

0.42

0.44

0.46

0.48

0.50

0.52

0.54

εg

100 4

3

Symbol Solution Ql x10 (m /s) CMC-2 2.00-2.64 CMC-3 2.10-2.95 CMC-4 3.13-3.53 dn : 6 mm 10

/

LOCKHART-MARTINELLI CURVE

φl

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Figure 23. Variation of gas holdup with frictional pressure drop.

1

0.1 1

X'

10

Figure 24. Typical comparison of two-phase pressure drop with Davis parameter and modified Eq.(27).

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φ l/ by

326

Ajay Mandal 100

φlA

10

1

4

3

Symbol Solution Ql x10 (m /s) CMC-2 2.00-2.64 CMC-3 2.10-2.95 CMC-4 3.13-3.53 dn: 6 mm

0.1 20

30

40

50

60

70

80

90 100

X

Figure 25. Typical comparison of two-phase pressure drop with Aoki correlation.

ftpl-Calculated

10

1

Symbol

Solution CMC-1 CMC-2 CMC-3 CMC-4

0.1 0.1

1

10

ftpl-Experimental

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Figure 26. Comparison of ftpl calculated from Eq. (32) and the experimental values.

Figure 27. Distribution of gas bubbles and manometer positions.

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327

0.64

0.60

0.56

εgi

0.52 3

Symbol Ql(m /s) dn(mm)

0.48

-3

6.0

-3

7.0

-3

6.0

-3

7.0

0.264x10 0.200x10 0.247x10

0.44

0.214x10

0.40 0

20

40

60

80

100

120

140

hm [cm] Figure 28. Variation of εGi with gas-liquid mixing height (hm) Air-CMC-2 solution system.

0.8

A

C

B

0.7 -3

3

Ql:0.264x10 m /s dn : 6.0 mm

0.6

εg

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0.5 0.4 0.3 0.2 0.1 0.0 0

20

40

60

80

100

120

140

160

hm [cm] Figure 29. Distribution of gas holdup with gas-liquid mixing height for air-CMC-2 solution system.

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Table 9. Comparison of Gas-holdup, Pressure drop and Energy Dissipation for different Gas-Liquid Contacting systems Author

Type

Vl(m/s)

Vg(m/s)

εg

ΔPf /Δz (kPa/m)

E(kw/m3)

Zahardnik et al.(1982)

ejector bubble bolumn

0.008-0.028

0.004-0.067

0.05-0.25

⎯

Chandraker et al.(1985)

vertically upward

0.023-0.10

0.02-1.0

0.15-0.40

0.55-5.5

0.05-9.2

Akagawa et al.(1989)

vertically upflow

0.01-0.10

7.7-12.8

0.005-0.01

0 .30-1.5

2.3-13

Das et al. (1989)

horizontal tube

0.14-1.0

0.25-1.5

0.1-0.40

2.0-6.5

0.9-20

Iliuta and Thyrion(1997) downflow packed bed

0.005-0.012

0.04-0.40

0.15-0.30

10.0-35.0

0.7-20

Badie et al. (2000)

horizontal pipe

0-0.03

15-25

0.01-0.10

0.05-0.40

0.80-10.5

Meikap (2000)

multi-stage bubble column 0.003-0.005

0.11-0.20

0.21-0.65

3.1-6.2

0.25-1.0

Present work

downflow bubble column

0.004-0.043

0.40-0.57

0.30-1.5

0.1-0.50

0.095-0.16

1-12

Hydrodynamic Studies on Two-Phase Gas-Liquid Flow…

329

3.4. Energy Analysis in Ejector-Contactor System The phenomenon of gas dispersion in non-Newtonian liquid in an ejector induced downflow vertical contactor is extremely complex in nature. For designing such gas-liquid downflow contactor, a complete knowledge regarding the total energy requirement and the energy dissipation in the different sections of both the ejector and contactor are essential. Further the total energy loss in the system is also a measure of the interfacial area and mass transfer coefficient obtainable in such systems. In view of the overall importance, a detailed energy analysis on gas dispersion in a vertical downflow liquid-gas ejector system is of utmost important. Energy analysis has been carried out to estimate the energy loss in ejector, which includes the loss in the parallel throat and the divergent diffuser. Also, analysis has been carried out to predict mixing energy loss in the intense mixing zone inside the column. The analysis is based on one dimensional homogeneous flow model. The detail energy balance equations are discussed in our earlier paper (Mandal et al., 2005b). Total energy loss per unit mass flow in the ejector is given by:

ρ l eejt = ( K th + K di )( ρ lVln2 / 2)

(35)

and the rate of total energy dissipation will be given by:

E ejt = ρ l (Ql + Q g )eejt

(36)

E ejt = K ejt Ql H (1 + Qr )

(37)

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or,

where K ejt = K th + K di , is the coefficient of total frictional loss in the ejector and can be obtained from the knowledge of throat and diffuser loss coefficients (Mandal et al., 2005b). As there is no mixing in the ejector so the energy loss accounts for frictional loss only. Total energy loss in the ejector may be calculated from Eq. (37) with the knowledge of liquid flowrate, jet velocity head and Kejt. Similar to the total energy loss in the ejector, the total energy loss in the mixing zone and the total frictional energy loss in the bubbly flow zone can be calculated from Eq. (38) and (39);

E m = K m Ql H (1 + Qr )

(38)

Ef =ΔPf × Ql (1+Qr)

(39)

Fig. 30 shows the variations of the different energy losses and the total energy supplied, H, with motive liquid flowrate. The variations of different energy terms with liquid flowrate is different due to different mode of frictional losses as discussed earlier. A comparison of total energy loss (Et = Eejt + Em + Ef) with the supplied energy, H, have also been presented in the figure. It may be observed that the total energy loss, Et is almost equal to the supplied

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energy, H at a particular liquid flowrate. Hence, it may be concluded for the present system that the net energy of the liquid jet, H, is fully utilized in different sections of the ejector contactor system and subsequent entrainment of air. The total energy losses in the different sections of the ejector contactor system have been compared with the energy supplied by the liquid jet for different CMC solutions having different operating conditions with 410 experimental data in Fig.31. It has been found that the total energy loss, Et is as close to the supplied energy within standard deviation of 0.59. Thus conservation of energy for the present system signifies the validity of the different energy terms.

12

Symbol energy terms H Emix Eejt Ef Em+Eejt+Ef

11 10 9

Energy (Nm/s)

8 7 6 5 4 3 2 1 0

-4

-4

2.0x10

-4

2.2x10

-4

2.4x10

2.6x10

3

Ql (m /s)

32

Symbol CMC Conc. 3

1.0 kg/m

3

1.5 kg/m

24

3

2.0 kg/m

3

2.5 kg/m

Et (Nm/s)

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Figure 30. Variations of energy losses and energy supplied with Ql .

16

8

0 0

8

16

24

32

H (Nm/s)

Figure 31. Comparison of energy supplied by liquid jet with the total energy losses in different sections of ejector contactor system.

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3.4. Mass Transfer Characteristics In gas liquid two-phase system, interfacial area and volumetric mass transfer coefficient are the most important parameters for scale up and designing a gas-liquid reactor. In general, gasliquid interfacial area is a function of the geometric and operating parameters and the physical and chemical properties of specific material system. Interfacial area is the most dominant factor in determining reactor output, particularly in the region where absorption is accompanied by fast chemical reaction. On the other hand, in case of absorption accompanied by slow or instantaneous reaction, the mass transfer rate is determined by the volumetric mass transfer coefficient. A review of literature shows that considerable work has been carried out on the measurement of interfacial area in different types of equipments. In most of these studies one of the following techniques has been used.

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1. 2. 3. 4.

Light attenuation High speed photography Dynamic gas disengagement process Chemical method

In the light attenuation technique proposed by Calderbank (1958), a collimated beam of light is passed through the gas-liquid dispersion. The intensity of the emerging beam is measured by a photocell, photo-multiplier tube or other photosensitive devices. This intensity will depend on the number of inter-phases the beam has to cross, and hence is a measure of the interfacial area in the system. Vermeulen et al. (1955), Rodger et al. (1956) and Calderbank et al. (1958) have used this method for the measurement of interfacial area in dispersions. Though intrinsically simple and easy to use, the direct applicability of this method is limited to situations where the effect of multiple and forward light scattering are negligible. Because of this limitation, its use in dispersions of high interfacial area may be subject to serious error. In the photographic method a high-speed photograph of the system is analyzed for determining the mean bubble size of the dispersed phase. For a quantitative estimate of the bubble size distribution, it is necessary to ensure that only the bubbles in the focal plane are measured. This requirement is difficult to satisfy especially where the bubble size is large. However the error due to this may be reduced by using the comparative techniques discussed by York and Stubbs (1952). Cooper et al. (1964) have used the photographic method to measure interfacial area in liquid and gas dispersed systems. In case of two-phase flow where bubble concentrations are high, the image received by the camera is the resultant of a series of complex optical interactions, which are difficult to analyze. Due to this reason it has been demonstrated that the results obtained by this technique are subject to serious uncertainties (Landau et al., 1977). The dynamic gas disengagement technique was first introduced by Sriram and Mann (1977). It requires an accurate measurement of the rate at which the level of the gas-liquid dispersion drops once gas flow to the bubble column is shut off. The measured disengagement profile is then used to estimate the holdup structure that existed in the dispersion prior to flow interruption. The bubble size distribution, and therefore the gas-liquid interfacial area, can also be determined using appropriate correlations that relate bubble sizes to bubble rise velocities. Patel et al. (1989) followed this technique to measure holdup and interfacial area. In the analysis, they assumed that the dispersion is

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332

Ajay Mandal

axially homogeneous at the instant when gas flow is interrupted and there are no bubblebubble interactions. The main advantage of the technique lies in its simplicity and its ability to provide a wide range of information. However, there are some limitations resulting from the assumptions made, deficiencies in the data collection and from its dependence on correlations relating terminal bubble rise velocity to bubble size. The chemical method of interfacial area determination is based on the theory of gas absorption followed by chemical reaction. When certain specified conditions of the reaction are maintained the rate of gas absorption per unit area is constant, and it can be calculated from the physico-chemical rate of the reaction. The total rate of absorption is measured experimentally, and from the knowledge of the specific rate of absorption, the interfacial area is calculated. Among all chemical methods the sulfite oxidation and CO2 absorption into caustic solutions have found most widespread application (Schumpe and Deckwer, 1980). Further, physical measuring processes give the local values of interfacial area, whereas the whole reactor volume can be assessed by using chemical methods. Since in the present system, the intense mixing between the phases generates very high interfacial area and turbulence, the first three methods were considered unsuitable and, therefore, the chemical method was selected. Many researchers (Danckwerts, 1968; Sharma and Danckwerts, 1970; Biswas, 1975, Radhakrishan, 1978) have used the chemical method for determining the interfacial area in different types of equipment. Schumpe and Deckwer (1980) measured the interfacial area by two different chemical methods, viz. sulfite oxidation and CO2 absorption into alkali solution in the same contactor. They observed a lower effective interfacial area by carbon dioxide absorption method compared to sulfite oxidation. Bandyopadhyay et al. (1980) determined the mass transfer coefficient and interfacial area with arsenite at high ionic strength. Interfacial area in horizontal tubes have been measured by Biswas (1975), Wales (1966) and Gregory and Scott (1968) and vertical column by Kasturi and Stephanek (1972, 1974), and in helical coils by Eben and Pigford (1965) and Banerjee et al. (1968). Richards et al. (1964) and Degaleesan and Ladha (1966) measured interfacial area in packed column and extraction column by chemical method. Studies on ejector or ventury type bubble column are increasing these days, as they provide a high gas-liquid interfacial transfer area. Cramers et al. (1992) determined mass transfer characteristics in a downflow liquid jet ejector by cobalt catalyzed sulfite oxidation method. Dutta and Raghavan (1987) measured interfacial area and mass transfer coefficient in an ejector loop reactor by absorbing carbon dioxide in aqueous sodium hydroxide solution and in an aqueous buffer solution of sodium carbonate-bicarbonate respectively. Raghuram et al. (1992) developed a model for a fast pseudo-first order reaction with an ejector-contactor system and assume that the contactor comprises a series of stirred vessels followed by a plugflow reactor. They adopted the process of absorption of carbon dioxide in sodium hydroxide solution to test the model and proved the acceptability of the model with the experimental results. Evans et al. (2001) discussed the performance of confined plunging liquid jet bubble column by measuring volumetric mass transfer coefficient in chemical method in presence of hypochlorite catalyst. Van de Sande (1975), Evans and Machniewski (1999) and Ide et al. (2001) described the characterization of volumetric mass transfer coefficient of different plunging liquid jet systems.

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Hydrodynamic Studies on Two-Phase Gas-Liquid Flow…

333

Chemical Method of Measuring Interfacial Area The chemical method of interfacial area determination is based on gas absorption followed by chemical reaction. In general, when physical absorption is followed by a chemical reaction, the overall rate will be governed by both the physical rate of absorption and the kinetics of the reaction. Detailed review of the theory of mass transfer with chemical reaction has been published by Sherwood and Pigford (1952), Danckwerts (1970), Astarita (1967) and Ladha and Degaleesan (1976). The application of the chemical method for determination of interfacial area and mass transfer co-efficient has been reviewed by Danckwerts and Sharma (1966). Brian (1964) has discussed the theory of absorption followed by a chemical reaction of the general order. For the irreversible reaction, A + JB → Products

(40)

which is pth order in A and qth in B, the local rate of reaction is given by Eq. (41). If the o

concentration of B is same everywhere and is equal to the bulk concentration, C B , then

− rA = k pq [C Bo ] q [C A ] p

(41)

o q

since k pq [C B ] is a constant, the rate depends only on the concentration, CA. Such a reaction

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is called a pseudo pth order reaction. The condition to be satisfied for the reaction to be pseudo pth order is,

⎡ ⎢ M =⎢ ⎢ ⎢⎣

1

2 ⎤2 D A k pq (C A ) p −1 (C Bo ) q ⎥ C Bo p +1 ⎥ 0, vk,j+1/2 (ρk αk )nj n vk,j+1/2 (ρk αk )j+1 otherwise

(87)

=



if vk,j+1/2 > 0, vk,j+1/2 (ρk αk vk )nj n vk,j+1/2 (ρk αk vk )j+1 otherwise

(88)

for phase k ∈ {g, ℓ}. ˜I For the convective implicit part G j+1/2 we use the p-XF formulation of the fluxes (30) described in Section 3.1.5. We then obtain hybrid convective fluxes through (83), using (78) and (79). Note that ˜I the convective fluxes G j+1/2 are now calculated with the coupling (77) to the upwind flux U ˜ G , as described in more detail in Section 5.3.1. j+1/2

5.2.4. The Hybrid Pressure Flux We write the pressure fluxes as  0 ˜U =  0  H p˜U 

and

 0 ˜I =  0 . H p˜I 

(89)

By (78) and (79), we see that the hybrid pressure flux (84) becomes simply ˜ j+1/2 = H ˜I H j+1/2 ,

(90)

where p˜I is given by a fully implicit calculation in the form (31). Hence no definition of upwind pressure fluxes p˜U is required, the WIMF flux hybridization only affects the convective fluxes.

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414

5.3.

Evje, Fl˚atten and Munkejord

Implementation Details

Before extending the above results to Φ 6= 0, it may be instructive to focus in more detail on how this WIMF scheme is implemented in practice. As for the p-XF scheme, the computation consists of two steps: 1. Flux linearization: We calculate numerical fluxes through the implicit pressuremomentum coupling. 2. Conservative update: We use these numerical fluxes to update the conservative variables according to (35). Note that both these steps incorporate the flux hybridizations (83). We will address them in turn. 5.3.1. Implicit Step As for the p-XF scheme derived in Section 3, the pressure-momentum coupling yields 3 equations for each computational cell to be implicitly solved over the computational domain. However, an added complication arises from the implicit calculation also involving the explicit part of the system, as given by (77). In the following exposition, we will find it convenient to use the symbol M = Mg + Mℓ (91) to denote the total convective momentum flux. Herein Mg = ρg αg vg2 ,

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Mℓ =

(92)

ρℓ αℓ vℓ2 .

(93)

For convenience of notation, we also use the shorthand [v] = vj+1/2 .

(94)

˜U ˜I ˜U Applying (77)–(79) and the splitting (82), where we use G j+1/2 , Gj+1/2 , and Hj+1/2 , ˜I H as described in Section 5.2.3 and 5.2.4, we see that the WIMF mass fluxes j+1/2

(ρk αk vk )WIMF j+1/2 can be written in the form n U n U (ρg αg vg )WIMF j+1/2 = [αℓ ]j+1/2 (ρg αg vg )j+1/2 − [αg ρg /ρℓ ]j+1/2 (ρℓ αℓ vℓ )j+1/2

(95)

n U n U (ρℓ αℓ vℓ )WIMF j+1/2 = −[αℓ ρℓ /ρg ]j+1/2 (ρg αg vg )j+1/2 + [αg ]j+1/2 (ρℓ αℓ vℓ )j+1/2

(96)

n +[αg ]nj+1/2 (ρ^ g αg vg )j+1/2 + [αg ρg /ρℓ ]j+1/2 (ρ^ ℓ αℓ vℓ )j+1/2 ,

+[αℓ ρℓ /ρg ]nj+1/2 (ρ^ g αg vg )j+1/2

+

[αℓ ]nj+1/2 (ρ^ ℓ αℓ vℓ )j+1/2 .

Similarly, we calculate that   U n ^2 MWIMF g αg vg )j+1/2 + (ρg αg vg )j+1/2 j+1/2 = [αℓ v(1 − ρℓ /ρg )]j+1/2 (ρg αg vg )j+1/2 − (ρ^   ^2 +[αg v(1 − ρg /ρℓ )]nj+1/2 (ρℓ αℓ vℓ )U ℓ αℓ vℓ )j+1/2 + (ρℓ αℓ vℓ )j+1/2 , j+1/2 − (ρ^

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(97)

A WIMF Scheme for the Drift-Flux Model

415

where WIMF (ρg αg vg2 + ρℓ αℓ vℓ2 + p)WIMF ˜Ij+1/2 . j+1/2 = Mj+1/2 + p

Comparing the WIMF convective momentum flux (97) to the sum of (95) and (96) we see WIMF that MWIMF j+1/2 now can be written in terms of the WIMF mass fluxes (ρk αk vk )j+1/2 as follows:  fj+1/2 + [v] (ρg αg vg )WIMF − (ρ^ MWIMF = M g αg vg )j+1/2 j+1/2 j+1/2

 + (ρℓ αℓ vℓ )WIMF ℓ αℓ vℓ )j+1/2 , (98) j+1/2 − (ρ^

where ^2 2 fj+1/2 = (ρ^ M g αg vg )j+1/2 + (ρℓ αℓ vℓ )j+1/2 .

(99)

This suggests a natural splitting of MWIMF j+1/2 into

WIMF WIMF MWIMF j+1/2 = Mg,j+1/2 + Mℓ,j+1/2 ,

(100)

WIMF f MWIMF g αg vg )j+1/2 + [v](ρg αg vg )j+1/2 g,j+1/2 = Mg,j+1/2 − [v](ρ^

(101)

where

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MWIMF ℓ,j+1/2

fℓ,j+1/2 − [v](ρ^ = M ℓ αℓ vℓ )j+1/2 +

[v](ρℓ αℓ vℓ )WIMF j+1/2 .

(102)

As v is not properly defined when vg 6= vℓ , we propose to use the following natural modification of (101)–(102) for general slip relations: WIMF f MWIMF g αg vg )j+1/2 + [vg ](ρg αg vg )j+1/2 g,j+1/2 = Mg,j+1/2 − [vg ](ρ^ WIMF f MWIMF ℓ αℓ vℓ )j+1/2 + [vℓ ](ρℓ αℓ vℓ )j+1/2 . ℓ,j+1/2 = Mℓ,j+1/2 − [vℓ ](ρ^

(103) (104)

^2 2 We now want to represent ρ^ g αg vg in (26) by (103) and ρℓ αℓ vℓ in (27) by (104). Thus, the implicit pressure-momentum coupling corresponding to (26), (27), and (31), but now with mixture momentum fluxes MWIMF k,j+1/2 , take the following form: • Pressure equation: 1 n n pn+1 j+1/2 − 2 (pj + pj+1 )

∆t

(ρ^ g αg vg )j+1 − (ρ^ g αg vg )j ∆x (ρ^ ℓ αℓ vℓ )j+1 − (ρ^ ℓ αℓ vℓ )j = 0. ∆x

+ [κρℓ ]j+1/2

+ [κρg ]j+1/2

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(105)

416

Evje, Fl˚atten and Munkejord

• Gas momentum equation:



(ρg αg vg )nj

+

−

  2 − ρ^ α v g g g

[vg αg ρg /ρℓ ]j+1/2 (ρℓ αℓ vℓ )U j+1/2 

∆x − [vg αg ρg /ρℓ ]j−1/2 (ρℓ αℓ vℓ )U j−1/2

∆x  n+1   n+1 mg n pj+1/2 − pj−1/2 m ] g = Q . ρ j ∆x ρ j

• Liquid momentum equation:



2 ρ^ ℓ αℓ vℓ



  ^ 2 − ρℓ αℓ vℓ

n (ρ^ ℓ αℓ vℓ )j − (ρℓ αℓ vℓ )j j−1/2 j+1/2 + ∆t ∆x

[vℓ αg ]j+1/2 ρ^ ℓ αℓ vℓ j−1/2 ℓ αℓ vℓ j+1/2 − [vℓ αg ]j−1/2 ρ^ − ∆x

[vℓ αℓ ρℓ /ρg ]j+1/2 ρ^ g αg vg j−1/2 g αg vg j+1/2 − [vℓ αℓ ρℓ /ρg ]j−1/2 ρ^

+ −

(106)

∆x U [vg αℓ ]j+1/2 (ρg αg vg )U j+1/2 − [vg αℓ ]j−1/2 (ρg αg vg )j−1/2

+

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(ρ^ g αg vg )j − j+1/2 j−1/2 + ∆t ∆x

[vg αℓ ]j+1/2 ρ^ g αg vg j−1/2 g αg vg j+1/2 − [vg αℓ ]j−1/2 ρ^ − ∆x

[vg αg ρg /ρℓ ]j+1/2 ρ^ ℓ αℓ vℓ j+1/2 − [vg αg ρg /ρℓ ]j−1/2 ρ^ ℓ αℓ vℓ j−1/2 +

+

2 ρ^ g αg vg

(107)

∆x U [vℓ αg ]j+1/2 (ρℓ αℓ vℓ )U j+1/2 − [vℓ αg ]j−1/2 (ρℓ αℓ vℓ )j−1/2

[vℓ αℓ ρℓ /ρg ]j+1/2 (ρg αg vg )U j+1/2

+



∆x − [vℓ αℓ ρℓ /ρg ]j−1/2 (ρg αg vg )U j−1/2

∆x  n+1 n+1 mℓ n pj+1/2 − pj−1/2 ρ

j

∆x

=



m ] ℓ Q ρ



.

j

Here the linearized fluxes are given (as before) by (21) and (28)–(29) as:
1 ∆x
1 (ρ^ (ρ^ (ρk αk )nj − (ρk αk )nj+1 (108) k αk vk )j+1/2 = k αk vk )j + (ρ^ k αk vk )j+1 + 2 4 ∆t and 1 n 1 ∆x 1 n 2 ((ρk αk vk )j − (ρk αk vk )j+1 )n . (ρ^ k αk vk )j + (vk ·ρ^ k αk vk )j+1 + k αk vk )j+1/2 = (vk ·ρ^ 2 2 4 ∆t (109) n+1 In conclusion, we solve (105)–(107) to obtain the variables pj+1/2 , (ρ^ g αg vg )j and (ρ^ ℓ αℓ vℓ )j to be used in the following. As for the p-XF scheme, this step requires the inversion of a sparse linear system with a bandwidth of five (pentadiagonal linear system) – where the coefficients become slightly more complicated due to the hybridization (77).

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417

5.3.2. Conservative Update By use of ˜U G j+1/2 and ˜I G j+1/2

 (ρg αg vg )U j+1/2   (ρℓ αℓ vℓ )U = , j+1/2 2 )U + (ρ α v (ρg αg vg2 )U ℓ ℓ ℓ j+1/2 j+1/2 

 (ρ^ g αg vg )j+1/2   (ρ^ ℓ αℓ vℓ )j+1/2 = , ^ 2 ^ 2 (ρg αg vg )j+1/2 + (ρℓ αℓ vℓ )j+1/2 



˜I =  H

(110)

0 0 pn+1 j+1/2



,

(111)

˜ U as explained as defined in Section 5.2.3 and 5.2.4 (note that there is no need to specify H in Section 5.2.4), where the required quantities are obtained through the equations (105)– (107), the numerical scheme can be written in the conservative form Un+1 − Unj j ∆t

+

Fj+1/2 − Fj−1/2 e j, =Q ∆x

(112)

where Fj+1/2 is obtained from (77) and (82). Finally, the physical variables are obtained by the procedure described in Section 3.1.8. from Un+1 j

5.4.

Resolution of Contact Wave

We now provide some attractive theoretical results for the special case of Φ = 0. We consider the linear wave arising from the initial conditions

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pj = p Yj = Y (j)

∀j

∀j

(vg )j = (vℓ )j = v

(113) ∀j.

In particular, ν is constant across the computational domain as stated by (43). The pressure gradient now vanishes from the model (2)–(4), and the solution to the initial value problem (113) is that the distribution of Y will propagate with the uniform velocity v. That is, we have ∂µ ∂µ +v = 0, (114) ∂t ∂x in accordance with (44). For the corresponding linear wave associated with the two-fluid model, we proved in [11, 12] that the WIMF scheme possessed the following properties: (i) WIMF reduces to the explicit upwind flux for the linear wave (113); (ii) WIMF preserves uniformity of the pressure and velocity field for this linear wave; (iii) WIMF captures the wave exactly on uniform meshes if the time step corresponds to a convective CFL number 1, i.e. ∆x = v. (115) ∆t

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Evje, Fl˚atten and Munkejord

Here (ii) and (iii) are direct consequences of (i). An equivalent result holds for the current WIMF scheme for the drift-flux model. In particular, we have the following proposition: Proposition 3. The WIMF scheme described in Section 5.3, when applied to the linear wave (113), has a solution that satisfies pn+1 = p j vjn+1

= v

n+1 n αk,j = αk,j n+1 n αk,j = αk,j

∀j, n;

(116)

∀j, n;
∆t n n αk,j − αk,j−1 −v ∆x
∆t n n −v αk,j+1 − αk,j ∆x

(117) ∀j, n

for v ≥ 0;

(118)

∀j, n

for v < 0.

(119)

Herein n+1 (ρ^ k αk vk )j = ρk αk,j v,

(120)

ρk ≡ ρk (p) = const.

(121)

where

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Proof. Substitute (116)–(120) into the equations of Section 5.3. Through a rather lengthy calculation, this will reduce the discrete equations of the WIMF scheme to trivial identities.

In particular, this means that (i)–(iii) are satisfied also in the current context. In Section 6.1, these results will be illustrated numerically.

6.

Numerical Simulations

In this section, we present some selected numerical examples. We first numerically verify Proposition 3 by studying a simple contact discontinuity for the Φ = 0 model. We then investigate how this behaviour carries over to more general cases, by considering a couple of shock tube problems known from the literature. Reference results will be provided by the explicit Roe scheme described by Fl˚atten and Munkejord [17]. Finally, we investigate the performance of the scheme on a case more representative of industrial problems; a large-scale mass transport problem given a non-linear slip law. For the simulations, a convective CFL number is defined as follows C=

∆t max |(vg )nj |, ∆x j,n

(122)

as this corresponds to the expected velocity of the mass transport wave associated with the Zuber-Findlay slip law (see Proposition 2). Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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6.1.

419

No-Slip Contact Discontinuity

For our first test, we consider a linear wave where the slip law is given by Φ = 0.

(123)

We assume an isolated contact discontinuity separating the states   5 p 10 Pa  αℓ   0.75   WL =   vg  =  10 m/s vℓ 10 m/s



(124)

  5 p 10 Pa  αℓ   0.25   WR =   vg  =  10 m/s vℓ 10 m/s



(125)



  

and 

 . 

We assume a 100 m long pipe where the discontinuity is initially located at x = 0. We use a computational grid of 100 cells and simulate a time of t = 5.0 s. The discontinuity will then have moved to the centre of the pipe, being located at x = 50 m.

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6.1.1. Sensitivity of WIMF to the Convective CFL Number In Figure 1, the results of WIMF are plotted for various values of the convective CFL number. We observe that WIMF captures the contact exactly for C = 1, as stated by Proposition 3. The numerical dissipation increases as C decreases. For C > 1, the scheme becomes unstable. 6.1.2. Sensitivity of p-XF to the Convective CFL Number In Figure 2, the results of p-XF are plotted for various values of the convective CFL number. The scheme obtains maximal accuracy for C = 1, and the numerical dissipation increases for both smaller and larger values of C. The dissipation is always larger than for the WIMF scheme, in particular this is the case for C = 1. However, the p-XF scheme is observed to be unconditionally stable for this test case. For both the WIMF and p-XF schemes, we observe that the pressure and velocities remain constant to floating point precision, as is dictated by Proposition 3.

6.2.

Dispersed Law Contact Discontinuity

In this section, we consider a more general contact discontinuity where the slip law is given as Φ = −δ/αℓ . (126) This test case is similar to Experiment 4 of Baudin et al. [1].

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Evje, Fl˚atten and Munkejord reference C=1 C=0.9 C=2/3 C=0.01

0.8

0.7

Liquid fraction

0.6

0.5

0.4

0.3

0.2 0

20

40

60

80

100

Distance (m)

Figure 1. No-slip contact discontinuity, WIMF scheme, 100 cells. Various values of the convective CFL number.

reference C=0.025 C=0.05 C=0.1 C=0.2 C=1

0.8

0.7

0.7

0.6 Liquid fraction

Liquid fraction

0.6

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reference C=20 C=10 C=5 C=2 C=1

0.8

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2 0

20

40

60

80

100

0

Distance (m)

20

40

60

80

100

Distance (m)

Figure 2. No-slip contact discontinuity, implicit p-XF scheme, 500 cells. Various values of the convective CFL number. Left: Approaching C = 1 from below. Right: Approaching C = 1 from above.

According to Baudin et al. [1], the slip law (126) describes inclined pipe flows where small gas bubbles are dispersed in the liquid. We follow in their footsteps and use the following value for δ: δ = 0.045 m/s.

(127)

In the framework of the Zuber-Findlay slip relation (10), the slip relation (126) corresponds Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

A WIMF Scheme for the Drift-Flux Model 1

0.8

reference 20 000 cells 2 000 cells 200 cells 50 cells

0.9

Liquid volume fraction

Liquid volume fraction

1

reference 20 000 cells 2 000 cells 200 cells 50 cells

0.9

421

0.7 0.6 0.5 0.4 0.3 0.2

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.1

0.1 50

55

60

65

70

75

80

85

90

95

100

50

55

60

65

70

Distance (m)

75

80

85

90

95

100

Distance (m)

Figure 3. Dispersed law contact discontinuity. Grid refinement for the implicit p-XF and WIMF schemes. Left: p-XF scheme. Right: WIMF scheme.

to K = 1,

(128)

S = −δ.

(129)

The initial states are given by   p (105 + 7.8) Pa  αℓ   0.9   WL =   vg  =  1 m/s vℓ 1.050 m/s Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.



and

  p 105 Pa  αℓ   0.2   WR =   vg  =  1 m/s 1.224 m/s vℓ 



 . 

   

(130)

(131)

This discontinuity will now propagate, without change of shape, with the gas velocity vg = 1 m/s, as stated by Proposition 2 (The effect of the liquid compressibility is negligible). We assume a pipe of length 100 m where the contact is initially located at x = 50 m. The simulation runs for 25 s. 6.2.1. Convergence Test for p-XF and WIMF In Figure 3, we investigate the convergence of the WIMF and p-XF schemes as the grid is refined. For the p-XF scheme, we used a convective CFL number C = 1, with respect to the gas velocity vg = 1 m/s. For the WIMF scheme, where the condition Φ = 0 (under which the flux hybridizations were derived) no longer applies, instabilities occurred for C > 0.9. In addition, for 0.75 < C < 0.9, a persistent overshoot was produced in the contact wave. Hence the WIMF results presented here are produced with a convective CFL number of C = 0.75.

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Table 1. Dispersed law contact discontinuity. Convergence rates for the p-XF scheme. n 1 2 3 4

cells 50 200 2000 20000

||E||n 4.818 2.405 0.762 0.241

sn 0.5012 0.4991 0.4999

Table 2. Dispersed law contact discontinuity. Convergence rates for the WIMF scheme. n 1 2 3 4

cells 50 200 2000 20000

||E||n 2.260 1.094 0.341 0.107

sn 0.5234 0.5066 0.5014

However, with this reduction of the CFL number we observe that the WIMF scheme is in fact able to provide an accurate resolution of the contact – the desired upwind-type accuracy is retained, while the sonic CFL criterion is still violated. Convergence rates for the volume fraction variable are given in Tables 1 and 2, where the error is measured in the 1-norm X ref ∆x|αg,j − αg,j ||E|| = |, (132) j

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and the order of convergence s is obtained through sn =

ln(||E||n /||E||n−1 ) . ln(∆xn /∆xn−1 )

(133)

Both schemes uniformly approach the expected analytical solution, at similar convergence rates. 6.2.2. Start-up Errors Due to the particular choice of slip relation, there exists a persistent pressure jump across the contact - whereas the numerical schemes are obtained from considerations of a contact where the pressure is constant. As a consequence of this, no result analogous to Proposition 3 holds, and start-up errors in the form of pressure oscillations occur for the first steps of the simulation. We now define the pressure variation at each time step as ∆˜ p = max(pnj ) − min(pnj ). j

j

(134)

With a grid of 20 000 cells and a convective CFL number of C = 0.75, a plot of ∆˜ p against time is given in Figure 4. The behaviour is rather similar for both the p-XF and WIMF schemes, so these oscillations are not primarily associated with the flux hybridization. Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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55 reference WIMF p-XF

50 45

Pressure variation (Pa)

40 35 30 25 20 15 10 5 0

0.05

0.1

0.15

0.2

0.25

Time (s)

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Figure 4. Dispersed law contact discontinuity, start-up errors. Initial pressure oscillations produced by WIMF and p-XF schemes.

This seems to be a price to pay for the simplicity achieved by keeping the schemes independent of the structure of the slip relation Φ. However, we note that the pressure oscillations are rather small and decrease with time, indicating that such start-up errors may be of minor importance for practical calculations. This will be supported by our further numerical examples.

6.3.

Zuber-Findlay Shock 1

Using the Zuber-Findlay slip relation with K = 1.07

(135)

S = 0.216 m/s,

(136)

we consider a shock tube problem also investigated by Evje and Fjelde [8]. The initial states are given by     p 80450 Pa  αℓ    0.45    WL =  (137)  vg  =  12.659 m/s  vℓ 10.370 m/s and



  p 24282 Pa  αℓ   0.45   WR =   vg  =  1.181 m/s vℓ 0.561 m/s

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

 . 

(138)

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Figure 5. Zuber-Findlay shock tube 1. Grid refinement for the WIMF scheme. Top left: Gas volume fraction. Top right: Pressure. Bottom left: Gas velocity. Bottom right: Liquid velocity.

The initial discontinuity is located at x = 50 m in a pipe of length 100 m, and results are reported at the time t = 1.0 s. Reference solutions are calculated by the flux-limited Roe scheme of [17], using a grid of 20 000 cells. 6.3.1. Convergence Test for the WIMF Scheme We use a convective CFL number of C = 1, or more precisely ∆x = 13 m/s ≈ max |(vg )nj |. j,n ∆t

(139)

The results of the WIMF scheme are plotted in Figure 5 for various grid sizes. We observe an overshoot in the volume fraction for the coarsest grids. Apart from this, the WIMF scheme convergences smoothly to the reference solution. Convergence rates for the gas volume fraction are given in Table 3. 6.3.2. Comparison between the Various Schemes In Figure 6, the results of WIMF and p-XF are compared with the first-order Roe scheme, for a grid of 100 cells. For the WIMF and p-XF schemes we used a convective CFL number

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of C = 1 as given by (139). For the Roe scheme, we used ∆x = 32.6 m/s, ∆t

(140)

corresponding to the CFL criterion for the sonic waves, C = 0.4 with respect to convection. 0.6

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20

40

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20

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Distance (m)

40

60

80

100

Distance (m)

Figure 6. Zuber-Findlay shock tube 1. Roe, p-XF and WIMF schemes, 100 cells. Top left: Gas volume fraction. Top right: Pressure. Bottom left: Gas velocity. Bottom right: Liquid velocity.

We observe that the p-XF and WIMF schemes provide a similar resolution of the sonic waves, whereas they are both inferior to the Roe scheme in this respect. We further observe

Table 3. Zuber-Findlay shock 1. Convergence rates for the WIMF scheme. n 1 2 3 4 5 6 7

cells 50 100 200 400 800 3200 10000

||E||n 2.181 1.352 0.746 0.338 0.256 0.0812 0.0352

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sn 0.6897 0.8578 1.1417 0.4041 0.8269 0.7325

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Evje, Fl˚atten and Munkejord

that WIMF gives a sharper resolution of the contact wave than Roe, but as previously noted, also introduces an overshoot.

6.4.

Zuber-Findlay Shock 2

We now consider a second shock tube problem using the same Zuber-Findlay slip law (135)–(136) as in the previous example. This problem was investigated as Example 3 by Baudin et al. [1]. We here follow in their footsteps and modify the gas pressure law; in the context of (8), we use ag = 300 m/s (141) instead of ag =

√

105 m/s

(142)

which is used for all other numerical examples of this paper. However, as for our other simulations, the liquid remains compressible as described by (7). We also follow Baudin et al. [1] in transforming to the variables (see also Section 5.1): ρ Y v

-

mixture density, gas mass fraction, mixture velocity.

Herein, v is expressed as

mg vg + mℓ vℓ . ρ In this formulation, the initial states are given by [1]     453.197 kg/m3 ρ  0.00705 WL =  Y  =  24.8074 m/s v

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v=

and

   ρ 454.915 kg/m3 . 0.0108 WR =  Y  =  v 1.7461 m/s 

(143)

(144)

(145)

The initial discontinuity is located at x = 50 m in a pipe of length 100 m, and results are reported at the time t = 0.5 s. The flux-limited Roe scheme on a grid of 20 000 cells was used to compute the reference solutions. 6.4.1. Convergence Test for the WIMF Scheme We use a convective CFL number of 1, or more precisely ∆x = 30 m/s, (146) ∆t corresponding to the maximum gas velocity occurring during the simulation. The results of the WIMF scheme are plotted in Figure 7 for various grid sizes. Convergence rates, with respect to the gas mass fraction Y , are given in Table 4. We observe that the WIMF scheme converges uniformly to the reference solution, and for this case no overshoots are visible.

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A WIMF Scheme for the Drift-Flux Model 620

427

1.1e+06 reference 10 000 cells 3 200 cells 800 cells 400 cells 200 cells 100 cells

600

580

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0.011

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0.01 Gas mass fraction

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0.009

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0.007

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0 0

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Figure 7. Zuber-Findlay shock 2. Grid refinement for the WIMF scheme. Top left: Mixture density. Top right: Pressure. Bottom left: Gas mass fraction. Bottom right: Densityaveraged velocity.

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6.4.2. Convergence Test for the p-XF Scheme We now use a time step 4 times larger than for the WIMF scheme, i.e. in the context of (122) we use C = 4. Hence the CFL condition (1) is violated with respect to all waves of the system. The results of the p-XF scheme are plotted in Figure 8 for various grid sizes. We observe that also the p-XF scheme converges to the reference solution in a fully non-oscillatory manner. Due to the increased time step, there is a significant amount of numerical diffusion, enforcing the use of fine grids. However, as can be seen by Table 5, the convergence rate – with respect to gas mass fraction – is comparable to that of WIMF. Remark 3. This example illustrates that p-XF qualifies as a strongly implicit scheme whereas WIMF is weakly implicit by the terminology of [12].

6.5.

A More Complex Slip Relation

The purpose of this final test is to investigate the performance of the WIMF scheme for more realistic slip relations which do not have a simple linear form such as (10). In addition, this case features transitions between genuine two-phase and pure liquid regions. These are both challenges that are relevant for industrial applications of the drift-flux model.

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Evje, Fl˚atten and Munkejord 620

1.1e+06 reference 20 000 cells 10 000 cells 4 000 cells 2 000 cells 1 000 cells 500 cells

600

580

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0.011

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0.01 Gas mass fraction

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0 0

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Figure 8. Zuber-Findlay shock 2. Grid refinement for the implicit p-XF scheme. Top left: Mixture density. Top right: Pressure. Bottom left: Gas mass fraction. Bottom right: Density-averaged velocity.

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6.5.1. The Test Case This case was introduced as Example 4 by Evje and Fjelde [9], and has been further investigated by Munkejord et al. [17, 26]. We consider a pipe of total length L = 1000 m which is initially filled with almost-pure liquid (αg = 10−7 ). During the first 10 seconds of the simulation, the inlet liquid and gas mass flowrates are increased from zero to 12.0 kg/s and 0.08 kg/s respectively. The liquid flow rate is then kept constant for the rest of the simulation. At the time t = 50 s, the inlet gas mass flow rate is linearly decreased to

Table 4. Zuber-Findlay shock 2. Convergence rates for the WIMF scheme. n 1 2 3 4 5 6

cells 100 200 400 800 3200 10000

||E||n 7.204 · 10−3 4.864 · 10−3 3.208 · 10−3 2.220 · 10−3 9.501 · 10−4 4.819 · 10−4

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sn 0.5669 0.6005 0.5420 0.6067 0.5958

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429

zero in 20 s, and for the rest of the simulation only liquid flows into the pipe. Throughout the simulation, the outlet pressure is kept constant at 105 Pa. The results are reported at t = 175 s. 6.5.2. The Slip Relation We use the same nonlinear slip law as the previous works [9, 17, 26]. Writing the law on the standard form (10), we take K to be constant, whereas S is allowed to depend on αℓ in a non-linear way. In particular, we use the parameters √ K = 1.0 S = S(αℓ ) = αℓ × 0.5 m/s, (147) which may be viewed as a more complicated form of the dispersed slip law (126). 6.5.3. Friction Terms For this test case, we follow Evje and Fjelde [9] and include a simple friction model. More precisely, in the context of (4) we choose 32vmix µmix . (148) d2 Here d = 0.1 m is the diameter of the pipe. Furthermore, vmix is the mixture velocity Q=−

vmix = αg vg + αℓ vℓ

(149)

µmix = αg µg + αℓ µℓ .

(150)

and µmix is the mixture viscosity

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Here µg = 5 × 10−6 Pa · s

and

µℓ = 5 × 10−2 Pa · s.

(151)

6.5.4. Discretization of the Friction Terms For the Roe scheme, we used an explicit forward Euler discretization of the source terms. For the WIMF scheme, we have discretized (148) as e j = − 32 (µmix )n (vg Q mix )j , j d2 Table 5. Zuber-Findlay shock 2. Convergence rates for the p-XF scheme. n 1 2 3 4 5 6

cells 500 1000 2000 4000 10000 20000

||E||n 4.783 · 10−3 3.343 · 10−3 2.285 · 10−3 1.546 · 10−3 9.068 · 10−4 5.955 · 10−4

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sn 0.5168 0.5488 0.5637 0.5824 0.6067

(152)

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where (vg mix )j is calculated in a linearly implicit manner as (vg mix )j =

(ρ^ (ρ^ g αg vg )j ℓ αℓ vℓ )j + . n (ρg )j (ρℓ )nj

Using this, we discretize the right hand sides of (106) and (107) as     mk mk n e ^ Qj . Q = ρ ρ j j

(153)

(154)

In this manner, the scheme retains its linearity in the implicit terms. 6.5.5. Performance of the Roe and WIMF Schemes For the WIMF scheme, we used the time step ∆x = 3.8 m/s, (155) ∆t corresponding to a convective CFL number C = 1 as given by (122). For the Roe scheme, we used a CFL number C = 0.9 with respect to sonic propagation, which for this case is approximately al = 1000 m/s (156) due to the single-phase liquid regions. It is worth emphasizing that implicit methods are particularly useful on cases involving such single-phase liquid regions, due to the strict CFL requirements imposed by the rapid sonic propagation. Here ∆tWIMF /∆tRoe ≈ 300, (157)

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and the efficiency differences between the Roe and WIMF schemes are significant. 6.5.6. Comparison between the Roe and WIMF Schemes

Results for the first-order Roe and WIMF schemes are given in Figure 9, with a grid of 200 cells. The reference solution was computed by the flux-limited Roe scheme, using a grid of 10 000 cells and CFL number C = 0.5. Note that the highly improved efficiency of the WIMF scheme is accompanied by a similar improvement in the resolution of the slow dynamics, as was also seen in Sections 6.1 and 6.3. This attractive behaviour was also observed in [11, 12] for the two-fluid version of WIMF. Table 6. Mass transport problem, WIMF scheme. Convergence rates with respect to volume fraction. n 1 2 3 4

cells 200 400 800 4000

||E||n 16.442 9.642 5.982 1.557

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sn 0.7699 0.6889 0.8363

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350000 reference Roe WIMF

reference Roe WIMF

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Figure 9. Mass transport problem. WIMF vs Roe scheme, 200 cells. Top left: Gas volume fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

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6.5.7. Convergence As seen by Figure 10 and Table 6, the WIMF scheme converges to the same solution as the Roe scheme as the grid is refined. This is reassuring in light of the large disparity of the time steps, as well as the inclusion of boundary conditions and source terms. It should however be noted that for grids of less than 200 cells, the WIMF scheme requires a somewhat lower CFL number for stability.

7.

Conclusion

We have presented an implicit pressure-based central type scheme for a drift-flux two-phase model, denoted as p-XF. Generalizing a technique introduced in [11], denoted as WIMF, we have incorporated explicit upwind-type fluxes allowing for an accurate resolution of the mass transport waves of the system. The WIMF scheme improves on the accuracy of p-XF with little loss of stability, and is the scheme we propose for practical applications. A difficulty with the drift-flux model is that its formulation is sensitive to the specification of the closure law Φ, which may vary depending on the flow conditions of the application. In this paper, the numerical schemes have been derived by basing the implicit approximation of the fluxes on a linearization around the slip Φ = 0. By this, we ensure certain

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350000 reference 4 000 cells 800 cells 400 cells 200 cells

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Figure 10. Mass transport problem. Convergence of the WIMF scheme, convective CFL number C = 1. Top left: Gas volume fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

accuracy and robustness properties for this particular case. The numerical examples demonstrate that the desirable properties of the schemes essentially carry over to more general choices of Φ. The schemes are conservative in all numerical fluxes and consistent with a given slip relation, and numerical evidence confirms that convergence to correct solutions are obtained. Numerical overshoots and oscillations in some cases occur for the mass transport wave. We observe that such oscillations may to a large extent be tamed by reducing the CFL number. The WIMF scheme outperforms the explicit Roe scheme in terms of efficiency and accuracy on slow dynamics, and results compare well to existing semi-implicit methods presented in the literature [2, 15]. This demonstrates that the WIMF strategy introduced in [11] has applicability beyond the two-fluid model originally considered. With this paper, we have presented a general setting for the construction of WIMF type schemes and by that hope to pave the way for further application to additional models. In particular, the WIMF approach seems useful for models where the eigenstructure is too complicated for an efficient construction of approximate Riemann solvers.

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Acknowledgements The authors thank the Research Council of Norway for financial support.

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[7] W. Dai and P. R. Woodward, A high-order iterative implicit-explicit hybrid scheme for magnetohydrodynamics, SIAM J. Sci. Comput 19, 1827–1846, 1998. [8] S. Evje and K. K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys. 175, 674–701, 2002. [9] S. Evje and K. K. Fjelde, On a rough AUSM scheme for a one-dimensional two-phase model, Comput. Fluids 32, 1497–1530, 2003. [10] S. Evje and T. Fl˚atten, Hybrid central-upwind schemes for numerical resolution of two-phase flows, ESAIM-Math. Model. Num. 39, 253–274, 2005. [11] S. Evje and T. Fl˚atten, Weakly implicit numerical schemes for a two-fluid model, SIAM J. Sci. Comput. 26, 1449–1484, 2005. [12] S. Evje and T. Fl˚atten, CFL-violating numerical schemes for a two-fluid model, J. Sci. Comput. 29, 83–114, 2006. [13] S. Evje and T. Fl˚atten, On the wave structure of two-phase flow models, SIAM J. Appl. Math. 67, 487–511, 2007. [14] S. Evje, T. Fl˚atten and H. A. Friis, On a relation between pressure-based schemes and central schemes for hyperbolic conservation laws, Numer. Meth. Part. Diff. Eq. 24, 605–645, 2008. Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[15] I. Faille and E. Heintz´e, A rough finite volume scheme for modeling two-phase flow in a pipeline, Comput. Fluids 28, 213–241, 1999. [16] K. Falk, Pressure pulse propagation in gas-liquid pipe flow, Dr. Ing. thesis, Department of Petroleum Engineering, NTNU, Norway, 1999. [17] T. Fl˚atten and S. T. Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model, ESAIM-Math. Model. Num. 40, 735–764, 2006. [18] F. Franc¸a and R. T. Lahey, Jr, The use of drift-flux techniques for the analysis of horizontal two-phase flows, Int. J. Multiphase Flow 18, 787–801, 1992. [19] S. L. Gavrilyuk and J. Fabre, Lagrangian coordinates for a drift-flux model of a gas– liquid mixture, Int. J. Multiphase Flow 22, 453–460, 1996. [20] A. Harten and P. D. Lax, On a class of high resolution total-variation-stable finitedifference schemes, SIAM J. Numer. Anal. 21, 1–23, 1984. [21] T. Hibiki and M. Ishii, Distribution parameter and drift velocity of drift-flux model in bubbly flow, Int. J. Heat Mass Tran. 45, 707–721, 2002. [22] T. Y. Hou and P. G. Le Floch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comput. 62, 497–530, 1994.

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[23] M. Larsen, E. Hustvedt, P. Hedne, and T. Straume, Petra: A novel computer code for simulation of slug flow, in SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, 1997. [24] R. J. Lorentzen and K. K. Fjelde, Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells, Int. J. Numer. Meth. Fluids 48, 723–745, 2005. [25] J. M. Masella, Q. H. Tran, D. Ferre and C. Pauchon, Transient simulation of two-phase flows in pipes, Int. J. Multiphase Flow 24, 739–755, 1998. [26] S. T. Munkejord, S. Evje and T. Fl˚atten, The multi-stage centred-scheme approach applied to a drift-flux two-phase flow model, Int. J. Numer. Meth. Fluids 52, 679–705, 2006. [27] S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, Philadelphia, 1980. [28] C. Pauchon, H. Dhulesia, D. Lopez and J. Fabre, TACITE: A Comprehensive Mechanistic Model for Two-Phase Flow, BHRG Conference on Multiphase Production, Cannes, June 1993. [29] A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow, Cambridge University Press, New York, 2007. [30] V. H. Ransom and D. L. Hicks. Hyperbolic two-pressure models for two-phase flow, J. Comput. Phys. 53, 124–151, 1984. Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

A WIMF Scheme for the Drift-Flux Model

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[31] J. E. Romate, An approximate Riemann solver for a two-phase flow model with numerically given slip relation, Comput. Fluids 27, 455–477, 1998. [32] H. B. Stewart and B. Wendroff, Review article; two-phase flow : Models and methods, J. Comput. Phys. 56, 363–409, 1984. [33] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comput. 43, 369–381, 1984. [34] E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2nd Ed., Springer-Verlag, Berlin, 1999. [35] I. Toumi and D. Caruge, An Implicit Second-Order Numerical Method for ThreeDimensional Two-Phase Flow Calculations, Nucl. Sci. Eng. 130, 213–225, 1998.

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[36] N. Zuber and J. A. Findlay, Average volumetric concentration in two-phase flow systems, J. Heat Transfer 87, 453–468, 1965.

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In: Multiphase Flow Research Editors: S. Martin and J.R. Williams, pp. 437-495

ISBN: 978-1-60692-448-8 © 2009 Nova Science Publishers, Inc.

Chapter 8

MODELLING OF THE INTERACTION OF THE DISPERSED AND THE CONTINUOUS PHASES IN A LIQUID-LIQUID EXTRACTION COLUMN A.-H. Meniai*, A. Hasseine, O. Saouli and A. Kabouche Laboratoire de l’Ingénierie des Procédés d’environnement Université de Constantine, Algeria

Abstract

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Liquid-liquid extraction columns are a quite good illustration of a two-phase flow system. They are used for contacting two immiscible liquid phases, flowing co or counter currently, and are simply vertical tubes having often, at their base, a distributor which can be a perforated plate or a series of nozzles, enabling the dispersion of one phase (the dispersed) into the other one (the continuous). The key parameter for this type of systems is the drop sizes distribution. This latter affects directly both the hydrodynamic and the mass transfer taking place, and depends greatly on various factors particularly the influence of the flow conditions of the continuous phase on the behaviour of the dispersed phase where the dynamic of the drops is often accompanied by the breakage and/or coalescence phenomena. In fact the study of the behaviour of the dispersed phase in a continuous flow field, generally turbulent is not easy. This is due to the nature of the macroscopic interactions generated by the breakage and coalescence phenomena resulting in a randomly distributed population of drops with respect to their sizes, concentrations and ages. Consequently, it is necessary to consider detailed mathematical models capable of describing the events generated by the interaction between the turbulent continuous phase and the dispersed one (drops), including both phenomena i.e. breakage and coalescence. Generally, the drop breakage term considers the interaction of a simple drop with the turbulent continuous phase continue turbulent, where it effectively undergoes a breakage if the transmitted turbulent kinetic energy exceeds its surface energy. Similarly, drop coalescence can take place due to the interaction between two colliding drops in the turbulent continuous phase, and it is effective if the interstitial liquid film disposes of sufficient time to drain out .

*

E-mail address: [email protected], Tel +213 662 57 14 26, Fax 213 31 81 88 80

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A.-H. Meniai, A. Hasseine, O. Saouli et al. As a first part, this chapter considers the modelling of the continuous phase flow, under turbulent conditions, using the two equations model (k-ε) which has proven its ability for this type of problems. The influence of the continuous phase hydrodynamic on drop breakage and coalescence is examined, relying on the link between these phenomena and the dissipation rate of turbulent energy which can be determined from the model resolution and used for the calculation of important parameters like the maximal stable diameter, the breakage and coalescence efficiencies, etc. via computer experiments. In the second part, this two-phase flow system is solved by the drop population balance approach where the development of many experimental research programs has greatly contributed to the elaboration of realistic models which, globally, include the drop breakage and coalescence phenomena as well as their transport. However at this level, mathematical complexities are induced and analytical solutions are not easily obtained for the drop population balance equation. From this study one can see how complex is the modelling of the behaviour of a dispersed phase in a continuous one. The main difficulty is due to the interaction of the involved various factors as well as the nature of the problem which is rather stochastic.

Keywords: Extraction; dispersed phase; Continuous Phase; Genetic Algorithm; Hold-up

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1. Introduction Spray columns are the oldest but simplest type of equipment used for contacting two immiscible liquid phases which flow co or counter currently, as is the case, for liquid-liquid extraction processes. They are simple vertical and cylindrical columns having often, at their base, a distributor which can be a perforated plate or a series of nozzles, enabling the dispersion of one phase (the dispersed) into the other one (the continuous). However for a reliable design of such an equipment, the knowledge of the spread and hence the distribution of the drop sizes is required. This is a key parameter because it does affect directly both the hydrodynamic and the mass transfer taking place in the system. It is greatly influenced by various factors particularly the flow conditions of the continuous phase and the behaviour of the dispersed phase where the dynamic of the drops is often accompanied by the breakage and/or coalescence phenomena. A priori this chapter considers the modelling of the continuous phase flow, under turbulent conditions, using the two equations model (k-ε) [1] which has widely proven its ability to handle this kind of problems, as will be shown in the following section.

2. Hydrodynamic of the Continuous Phase The continuous phase is forced into the column from the top, under turbulent flow conditions and it is essential to know the local velocities of the liquid at any time and position inside it. Considering the geometry of this flow system, the most adequate choice of representation is the cylindrical coordinates as shown in Figure 1, in which the parameters to be determined are function of the corresponding axial and radial directions, excluding any variation in the angular direction. Therefore the flow is assumed as bi-dimensional and the physical properties of the fluid are assumed to remain constant.

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439

θ r

x

Figure 1. The adopted coordinates system for the column.

2.1. The Governing Differential Equations The governing equations for the considered problem are those of continuity and momentum. In a two dimensions and under steady state conditions, they take the following forms: Continuity equation:

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∂(ρU) 1 ∂(ρrV) + =0 ∂x r ∂r

(1)

with x and r denoting the axial and radial directions, U and V the corresponding fluid velocity components, respectively and ρ is its density. Axial momentum:

⎡ 1 ∂ ⎛ ∂U ⎞ ∂ 2U ⎤ ∂ P ∂ρUU 1 ∂rρUV + = μ⎢ ⎜r ⎟+ 2 ⎥− ∂x r ∂r ⎣ r ∂r ⎝ ∂r ⎠ ∂x ⎦ ∂x

(2a)

with P denoting the mean pressure over a cross section area. Radial momentum:

⎡⎛ 2 ⎞ ⎤ ∂ ( ρUV ) + 1 ∂ ( ρrVV ) = μ ⎢⎜⎜ ∂ V2 ⎟⎟ + 1 ∂ ⎛⎜ r ∂V ⎞⎟⎥ − μ 2V2 − ∂P ∂x r ∂r r ∂r ⎣⎝ ∂x ⎠ r ∂r ⎝ ∂r ⎠⎦ with μ the fluid viscosity.

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(2b)

440

A.-H. Meniai, A. Hasseine, O. Saouli et al. The equations 2 have been obtained by the use of three assumptions stated as follows:

a) The gravitational contribution is neglected because of the forced convection; b) The flow is assumed to be of the boundary-layer type, implying that momentum is not transferred by molecular mixing in the axial direction; c) A space average pressure is assumed to prevail at each cross section in the axial momentum equation while the radial momentum equation is influenced by the pressure variations. As reported in [2] this pressure ‘uncoupling’ is necessary for parabolic flows, particularly if a marching integration procedure is adopted, for the numerical resolution of the equations.

2.2. Turbulence Modelling In a turbulent flow regime, the momentum conservation equations are not applicable directly. A generally adopted approach for the representation of such conditions was proposed by Reynolds [3]. It consists of decomposing an instantaneous characteristic property into macroscopic and turbulent parts, in a random manner as shown in the following figure:

φ

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φ

Time (s) Figure 2. Reynolds propriety decomposition.

Therefore a property φ it can be written:

φ = φ+Φ

(3)

with Φ the fluctuation of the variable φ and φ its mean value between time t = 0 and t = T, defined as follows :

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φ=

1 T

441

t+T

∫ φdt

(4)

t

Hence the Reynolds decomposition for the instantaneous velocity U and the pressure P are given as follows:

U = U+u

(5)

P = P+p

The fluctuations are new random variables introduced in the initial equations. Clearly for −

the general variable φ, the mean of the fluctuations Φ is itself zero. From this result, important properties can be written as follows: a)

(

)(

)

UV = U + u V + v = UV + uv + Uv + Vu

hence:

UV = UV + uv + Uv + Vu = UV + uv = UV + uv 2

U=V, U = U + u 2

For the particular case of

2

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b) With r being the independent variables, one can write:

⎡ 1 t+T ⎤ ∂ ⎢ ∫ Udt ⎥ T ⎛ ∂U ⎞ 1 t + T ∂U ⎦ = ∂U dt = ⎣ t ⎜ ⎟= ∫ ∂r ∂r ⎝ ∂r ⎠ T t ∂r −

−

∂U ∂u c) It can easily be seen that = ∂t ∂t Finally the equations for the turbulent flow can be expressed in a general form as follows:

( )=0

∂ ρUi ∂x i

⎤ ⎡ ⎛ ∂U ∂U j ⎞ ⎟⎟ − ρu i u j ⎥ ∂ ⎢μ⎜⎜ i + ∂ ρUi U j ⎥⎦ ⎢⎣ ⎝ ∂x j ∂x i ⎠ ∂P =− + ∂x j ∂x i ∂x j

(

)

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(6)

(7)

442

A.-H. Meniai, A. Hasseine, O. Saouli et al.

⎛ μ ∂φ ⎞ − ρu jφ⎟⎟ ∂⎜⎜ ∂ ρU j φ ⎝ σ φ ∂x j ⎠ + Sφ = ∂x j ∂x j

(

)

(8)

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New terms of the form −ρu i u j representing the turbulence effect on the mean flow are introduced and are known as Reynolds stresses. However the number of unknowns is greater than the number of equations and hence the system is not closed. Different methods were proposed for the system closure leading to different turbulence models, where the mostly used ones are briefly described in the following section. The main difficulty encountered when treating the turbulence lies in the formulation of the Reynolds stresses in terms of the mean flow characteristics and all the developed models are different with respect to the manner they express them. However the fundamental idea is based on the fact that the turbulence is dissipative and that all the kinetic energy resulting from the mean flow by means of the Reynolds stresses action on the mean deformation rate, is dissipated by small structures known as eddies characterised by spectra of dimensions, velocities and fluctuation frequencies. Many turbulence models are proposed in the literature and enable the calculation of velocity, temperature and mean concentration distributions under turbulent flow conditions. The first proposed theories are mainly empirical, simply formulated and based on the use of a turbulent viscosity coefficient as well as on the mixing length model. These types of models and their various improvements have shown certain success in reproducing experimental data, despite their theoretical limitations, particularly in the turbulent layers. More realistic hypotheses on the turbulent fluid behaviour have been introduced, leading to the elaboration of more complex models which have enabled the calculation of the turbulent mean flow field as well as its characteristics. These are the transport equation models where the most famous one is the energy-dissipation known as the ‘k-ε’ model [1]. However, the current research work aims to develop universal models with a higher number of transport equations and capable of solving a wide number of different practical flow problems in different duct geometries, without having to readjust the numerical constants. Few turbulence models based on the apparent viscosity concept are described in the following section.

2.2.1. Zero Equation Models These models do not involve any differential equation for the calculation of the turbulent quantities. They are based on the Boussinesq’s concept [4], which, in a thin layer, is expressed as follows:

− uv = ν t

∂U ∂y

with y the transversal distance.

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(9)

Modelling of the Interaction of the Dispersed and the Continuous Phases…

443

In the region other than that of the boundary layer, the apparent viscosity concept can be written as follows:

− ui u j +

⎛ ∂ Ui ∂ U j ⎞ 2 ⎟⎟ + kδ ij = ν t ⎜⎜ 3 ⎝ ∂x j ∂x i ⎠

(10)

δij is the Kronecker operator. The mixing length model developed by Prandtl [5] has proven to be among the most important and reliable ones for the bi dimensional calculation of the boundary, by proposing the following expression:

ν t = l 2m

∂U ∂y

(11)

with lm the mixing length. This mixing length hypothesis is similar to the mean free path of the molecules according to the kinetic gas theory. This length is proportional to the wall distance. Although this model is simple, it remains the best for the flow predictions in the boundary layer.

2.2.2. One Equation Models

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This model class uses the turbulent kinetic energy k as the main variable in the additional transport differential equation, obtained from Navier-Stokes equations. Therefore, a priori, the variable k is obtained from the resolution of the corresponding transport equation using the turbulent viscosity concept according to the widely proposed following expression [1]:

ν t = c μ k 1/ 2 l

(12)

with l the length scale of the large eddies, which is specified by an algebraic and empirical expression [1].

2.2.3. Two Equations Models For these models, the length scale distribution is calculated by the introduction of a new transport differential equation. To date the most attractive model is the k-ε one [6]. Its popularity is mainly due to the following two points: It does not require additional terms in the near wall treatments; - The Prandtl number for the kinetic energy (ε) dissipation rate equation, contrarily to other models, takes a reasonable value of 1.3, excluding any corrections near the wall. To use the above equations (Equations 2) for a turbulent flow, the fluid viscosity μ has to be replaced by an effective viscosity μeff which is expressed as follows:

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A.-H. Meniai, A. Hasseine, O. Saouli et al.

μ eff = μl + μt

(13)

with μl and μt the laminar and the turbulent parts of the viscosity, respectively. The effective viscosity μeff can be calculated using the two-equation model proposed and described in details by Launder & Spalding [1] and which has proven its ability to handle this kind of situations. According to this model, one can write:

μt = Cμ ρk 2 / ε

(14)

with k and ε the turbulent kinetic energy and its dissipation rate, respectively and Cμ a constant equal to 0.09. Also the governing differential equations for turbulent kinetic energy k and its dissipation rate ε can be written as follows: Kinetic energy of turbulence:

∂ 1 ∂ 1 ∂ ⎡ ∂k ⎤ rμt / σ k ⎥ + G − ρε ( ρrVk ) + ( ρUk ) = ⎢ r ∂r r ∂r ⎣ ∂x ∂r ⎦

(15)

Dissipation rate:

1 ∂ 1 ∂ ⎡ ∂ε ⎤ ρε 2 Gε ∂ ( ρrVε ) + ( ρUε ) = − C2 rμt / σ ε ⎥ + C1 ∂r ⎦ k k ∂x r ∂r r ∂r ⎢⎣

(16)

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with G expressed as follows:

⎡ ⎧⎪⎛ V ⎞ 2 ⎛ ∂V ⎞ 2 ⎧ ∂U ⎫2 ⎫⎪ ⎧ ∂U ∂V ⎫2 ⎤ G ≡ μt ⎢2⎨⎜ ⎟ + ⎜ + ⎟ +⎨ ⎬ ⎥ ⎬ ⎬+⎨ ∂x ⎭ ⎦⎥ ⎣⎢ ⎪⎩⎝ r ⎠ ⎝ ∂r ⎠ ⎩ ∂r ⎭ ⎪⎭ ⎩ ∂r

(17)

C1, C2, σk and σε are constants of the model taking the numerical values 1.44, 1.92, 1.0 and 1.3, respectively [1]. It is interesting to note that all the above equations (Eqns15-16) can be put into the following general form:

1⎡ ∂ 1⎡ ∂ ∂φ ∂ ∂φ ⎤ ∂ (ρrUφ) + (ρrVφ)⎤⎥ = ⎢ ⎛⎜⎝ rΓφ ⎞⎟⎠ + ⎛⎜⎝ rΓφ ⎞⎟⎠ ⎥ + Sφ ⎢ r ⎣ ∂x ∂r ∂x ∂r ∂r ⎦ ⎦ r ⎣ ∂x

(18)

where x and r are the axial and radial directions, respectively, U and V are the axial and radial velocities, respectively, ρ the density of the liquid, φ is the dependent variable which can be a velocity component, temperature, concentration, etc. and it can be transported by convection

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Modelling of the Interaction of the Dispersed and the Continuous Phases…

445

or diffusion , Γφ is the effective diffusion coefficient of the transported variable φ and it can be the viscosity, the thermal conductivity or the mass diffusion coefficient, depending on the type of transfer phenomenon considered, and Sφ is the source term which can be a production or a destruction of the transported variable φ. For the present problem φ represents the axial or the radial velocity and for Γφ = μ and S taking a suitable form, the momentum equations (Equations 2) appear, while for ϕ = 1, Γ = 0, and S = 0, the continuity equation (Equation 1) is obtained.

2.3. Treatment Near the Wall In the near-wall regions, the k-ε model can only be applied for high Reynolds numbers [4] and the variations of the flow properties are much steeper than in the bulk of the fluid. Consequently, a special but conventional treatment is needed, as previously adopted by launder & Spalding [4] who have systematically neglected the presence of any laminar sublayer near the wall and where a uniform shear stress distribution has been assumed. This has brought great simplifications in the turbulent kinetic energy balance, since a sort of local equilibrium is reached where the production rate of turbulent kinetic energy is equal to its dissipation rate.

N

W

τP

E

P

rP

Paroi solide

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U

S Figure 3. Control volume near the wall.

Generally a logarithmic law is well suited for this region between the solid wall and a certain small distance yp,, where the axial velocity is given by the following expression as:

Up Uτ +

with y =

=

1

κ

ln( Ey + )

(19)

ρUτ y p expressing the local Reynolds number, E is set to 9.0 for smooth wall μ

surfaces, κ is the Von Karman constant and equal to 0.4 and U τ is the frictional velocity which is given as follows: Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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A.-H. Meniai, A. Hasseine, O. Saouli et al.

Uτ = (

τp ρ

)1/ 2

(20)

where τp is the shear stress at the wall. Near by the wall the convective and diffusive contributions for the k are negligible and hence one can write: 2

⎛ ∂U ⎞ ⎟⎟ = ρε G = μt ⎜⎜ ⎝ ∂y ⎠

(21)

This will lead to the following expression:

τ = ρCμ1/ 2 k

(22)

Combining this relation with the logarithmic law, one obtains:

τp =

+

U wa κμywa ywa ln Eywa +

(

)

(23)

To obtain the turbulent kinetic energy near the wall, modifications should be brought to the source term, taking into account the shear stress, and one can write: 2

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U ⎛ ∂U ∂V ⎞ G = ∫∫ μt ⎜ + ⎟ dv = τ wa wa vol ywa ⎝ ∂r ∂x ⎠ V

(24)

where vol denotes the volume of the control volume corresponding to a near-wall grid point. In the same manner, ε is integrated over the control volume to obtain:

∫∫ εdv = Cμ

3/ 4

k wa

3/ 2

U+

V

where

and

U + = ywa

U+ =

(

+

ln Eywa

κ

vol ywa

(25)

+

if ywa ≤ 11.63 +

)

+

if ywa ≥ 11.63

The dissipation rate is obtained by means of the following relation:

ε = k3/ 2 / L where L is the mixing length which is obtained from the usual relationship as follows:

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(26)

Modelling of the Interaction of the Dispersed and the Continuous Phases…

447

L = C μ−3 / 4κy wa

(27)

Finally this leads to the following expression:

ε p = Cμ 3 / 4 k wa 3 / 2 / (κywa )

(28)

2.4. The Solution Procedure The finite volume concept is selected and the conservation equations are integrated over a general control volume in the physical space. Figure 4 shows how the points are arrayed in the r-x plane and used for storing the variables U, V, k and ε.

Location

N

P, k, ε

n W

Variable

U

E

w

e V

U V

s

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S Figure 4. Variables locations in the r-z plane.

Non uniform grid spacing is adopted as is often the case for the great effectiveness of computing power obtained [7]. In the neighbourhood of the wall, fine grid spacing is adopted because of the steep variations of the properties. Before the differential equations of the preceding section can be expressed in a finite difference form, the following two assumptions are made: - The variation of a property ϕ is assumed to be linear between two main grid points and in both directions; - The convective velocities and the effective diffusivities are uniform through the corresponding faces. This enables the integration of Equation 18 to be written in the following form:

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(ρUea e )φe − (ρU w a w )φ w + (ρVna n )φ n − (ρVsa s )φs = (Γea e ) +( Γn a n )

∂φ ∂φ − ( Γw a w ) ∂x e ∂x

∂φ ∂φ − ( Γsa s ) + Sφ vol ∂r n ∂r s

w

(29)

with ae, aw, an and as the coefficients representing the areas of the ‘east’, ‘west’, ‘north’ and ‘south’ faces of the control volume, respectively, Sφ is the mean value of the source term in the control volume (r dr dx) and Γe , Γw , Γn , Γs are the diffusivities corresponding to control volume interfaces and are determined, for the case of a scalar variable by a simple linear interpolation, as follows:

1 (ΓP + ΓE / N ) 2 1 = ( ΓP + ΓW / S ) 2

Γe / n =

(30a)

Γw / s

(30b)

where the subscripts E/N or W/S denote ‘east’ or ‘north’ and ‘west’ or ‘south’ Finally Equation 18 is reduced to the following expression:

Ce φ e − C w φ w + C n φ n − Csφ s = D e Δφ e − D w Δφ w + D n Δφ n − D s Δφ s + Sφ vol

(31)

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with C and D the convective and diffusive terms, respectively, expressed as follows:

C e / w = ρUe / w a e / w

(32a)

C n / s = ρUn / sa n / s

(32b)

De / w =

Γe / w a e / w Δx e / w

(32c)

Dn / s =

Γn / sa n / s Δx n / s

(32d)

The gradient of the general variable φ is approximated by centred differences as follows:

Δφe / n = (φ E / N − φ P )

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(33a)

Modelling of the Interaction of the Dispersed and the Continuous Phases…

Δφe / n = (φ P − φW / S )

449 (33b)

According to the numerical method adopted for the resolution of the algebraic equations, the source term should be linearised in order to speed up convergence, then:

S φ = bφ P + c

(34)

with b representing the coefficient of φ P and c is the constant part of Sφ , and accordingly Equation 20 becomes:

C e φ e − C w φ w + C n φ n − C sφ s = D e ( φ E − φ P ) − D w ( φ P − φ W ) + D n ( φ N − φ P ) − D s ( φ P − φ S ) + b ′φ P + c ′

(35)

with b’=b.vol and c’=c.vol.

⎛ ∂φ ⎞ ⎛ ∂φ ⎞ ⎟ and ⎜ ⎟ are approximated using centred ⎝ ∂x ⎠ ⎝ ∂r ⎠

Since the partial derivatives ⎜

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differences, the problem of φ at the interfaces is solved by using the hybrid scheme which is solely a combination of the centred differences and another scheme known as the ‘upwind’. Different schemes are reported and described in the literature [7] and it is generally established that the choice of a particular one for a given problem depends more on the Peclet number. Finally for the variable φ one can write:

φe = f eφ P + (1 − f e )φ E

(36a)

φ w = f wφW + (1 − f w )φ P

(36b)

φ n = f nφ P + (1 − f n )φ N

(36c)

φ s = f sφ S + (1 − f s )φ P

(36d)

with fe, fw, fn and fs are coefficients between 0 and 1. After substitution and rearrangement, Equation 24 can be written as:

[Ce fe − Cw (1 − f w ) + Cn f n − Cs (1 − f s ) + De + Dw + Dn + Ds ]φ P = [De − Ce (1 − fe )]φ E + [Dw + Cw f w ]φW + [Ds + Cs f s ]φS + [Dn − Cn (1 − f n )]φ N + b′φ P + c′ (37) By letting:

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[Ce f e − Cw (1 − f w ) + Cn f n − Cs (1 − f s ) + De + Dw + Dn + Ds ] = AP

(38a)

[De − Ce (1 − f e )] = AE

(38b)

[Dw + Cw f w ] = AW

(38c)

[Ds + Cs f s ] = AS

(38d)

The discretisation of Equation 26 can be written in the final form as:

A P φ P = A E φ E + A W φ W + A S φS + A N φ N + Sφ

(39)

The continuity equation is simply obtained by substituting the values φ = 1 , Γeff = 0 and Sφ = 0 in Equation 18, followed by an integration to obtain the following expression:

Ce − C w + C n − Cs = 0

(40)

2.4.1. Discretisation of Source Terms a) Momentum equation in the x direction The source term Sφ with φ = U, represents simply the variation of the pressure in the x

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direction. The integration of this equation over the control volume with respect to U for the node (i,j) gives :

Sφ =

⎛ ∂P ⎞

∫∫ ⎜⎝ − ∂x ⎟⎠ rdrdx = − V

( Pe − Pw ) vol Δx ew

b’=0

c′ = −

( Pe − Pw ) vol Δx ew

(41a)

(41b) (41c)

The momentum equation can then be written:

A W UW =

∑A

V

U V + a ew ( PW − PP )

(42)

V

It should be noted that before the integration of the U equation, the effective viscosity must be calculated at the ‘north’ and ‘south’ interfaces due to the staggering of the grid. An arithmetic mean value is taken between the interfaces of the control volumes containing the grid points (i-1,j), (i-1, j+1), and (i, j), (i, j+1), respectively, to obtain:

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⎤ ⎡ ⎥ ⎢ dr ( j) dr ( j) ⎥ μn = ⎢ + Δr ( j) Δr ( j + 1) ⎥ ⎢ Δr ( j) + Δr ( j + 1) + ⎢⎣ μ ( i − 1, j) μ ( i − 1, j + 1) μ ( i , j) μ ( i , j + 1) ⎥⎦

(43a)

⎤ ⎡ ⎥ ⎢ dr ( j − 1) dr ( j − 1) ⎥ + μs = ⎢ Δr ( j − 1) Δr ( j) ⎥ ⎢ Δr ( j − 1) + Δr ( j) + ⎢⎣ μ (i − 1, j − 1) μ (i − 1, j) μ (i, j − 1) μ (i, j) ⎥⎦

(43b)

and

b) Momentum equation in the r direction Similarly to the preceding case the source term integration over the corresponding control volume leads to: e n

b′ =

μ eff (μ ) rdrdx = − 2 eff2 P vol 2 r rP

(44a)

( PP − PS ) vol ⎛ ∂P ⎞ rdrdx − = − ⎜ ⎟ ∫ ∫ ⎝ ∂r ⎠ ΔrPS

(44b)

∫ ∫ −2 w s

c′ =

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and the momentum equation can be written :

A S VS =

∑A

V

VV + a s ( PS − PP )

(45)

V

For the same reasons mentioned above for the case of U, the viscosity at the interfaces ‘east’ and ‘west’ are evaluated in the same manner as:

⎡ ⎤ ⎢ ⎥ dx ( i − 1) dx ( i − 1) ⎥ + μw = ⎢ Δx ( i − 1) Δx ( i ) ⎥ ⎢ Δx ( i − 1) + Δx ( i ) + ⎢⎣ μ ( i − 1, j − 1) μ ( i , j − 1) μ ( i − 1, j) μ ( i , j) ⎥⎦

(46a)

⎡ ⎤ ⎢ ⎥ dx(i) dx(i) ⎥ + μe = ⎢ Δx(i) Δx(i + 1) ⎥ ⎢ Δx(i) + Δx(i + 1) + ⎢⎣ μ (i, j − 1) μ (i + 1, j − 1) μ (i, j) μ (i + 1, j) ⎥⎦

(46b)

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A.-H. Meniai, A. Hasseine, O. Saouli et al. c) Source term of k For this case one can write: e n

b′ =

ε

εP

∫ ∫ −ρ K rdrdx = −ρ K w s

vol

(47a)

P

e n

c′ =

∫ ∫ Grdrdx = [ G]vol

(47b)

w s

with:

⎧⎪ ⎡⎛ U − U ⎞ ⎛ V − V ⎞ ⎛ V ⎞ 2 ⎤ ⎡ U − U V − V ⎤ 2 ⎫⎪ w s s w [ G ] = μ t ⎨2 ⎢⎜ e ⎟ +⎜ P⎟ ⎥+ ⎢ n ⎟ +⎜ n + e ⎥ ⎬ (48) Δ Δ Δ Δ x r r r x ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎥⎦ ⎣ ew ns P ns ew ⎦ ⎪ ⎪⎩ ⎢⎣ ⎭

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Therefore:

VP = (Vn + Vs ) / 2

(49a)

U n = (U e + U w + U Ne + U Nw ) / 4

(49b)

U s = (U e + U w + U Se + U Sw ) / 4

(49c)

Ve = (Ve + Vw + VnE + VsE ) / 4

(49d)

Vw = (Vs + Vn + VnW + VsW ) / 4

(49e)

d) Source term of ε Similarly one can write: e n

b′ =

∫∫−C w s

e n

c′ =

∫∫C w s

ε ε ρ rdrdx = − C ε 2ρ P vol k kP

(50a)

ε ε Grdrdx = C ε1 [ G ]vol k k

(50b)

ε2

ε1

where [G] is given by Equation 32.

2.4.2. Numerical Solution The above equation 39 can be rewritten in following form:

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′ A P φ P = A N φ N + A Sφ S + c′′

(51)

with

′ A P = A P − b′ c′′ = A E φ E + A Wφ W + c′

The coefficients in this equation depend on the variable values and therefore the system is not linear and the use of the TDMA (Tri-Diagonal Matrix Algorithm), particularly devised for this type of systems and is described in the following section.

2.4.2.a. The TDMA Algorithm Once the boundary conditions are introduced, a set of non linear algebraic equations is obtained. Due to the difficulty in resolving a such system, it is frequent to use a semi-iterative method consisting in the determination of the φ values on the ith grid line, after assuming those at the nearest grid lines (i+1)th and (i-1)th . Therefore for each grid point of line i, the equation is reduced in order to contain only the three unknowns φ P , φ N , φ S , according to Equation 51 as follows:

− β 2 φ1 + D 2 φ 2 − α 2 φ 3

= c′′2

− β 3φ 2 + D3φ 3 − α 3φ 4

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.

.

= c′′3 .

.

−β jφ j−1 + D jφ j − α jφ j+1 .

.

.

. = c′′j

.

−β NJ −1φ NJ − 2 + DNJ −1 φ NJ −1 − α NJ −1φ NJ = c′′NJ −1 with D = A P , α = A N ,β = A S and c′′ = A Wφ W + A E φ E + c′ , as assumed. The TDMA algorithm is then used to solve the obtained tri diagonal system by rearranging each grid point equation as follows:

φj =

By letting Q j =

αj Dj

; Rj =

βj Dj

βj Dj

φ j−1 +

αj Dj

and Z j =

φ j +1 +

c′′j Dj

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c′′j Dj

, the line equations become:

(52)

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φ 2 = Q2 φ 3 + R 2φ1 + Z2

(i)

φ3 = Q3φ 4 + R3 φ 2 + Z3

(ii)

. . .

.

.

.

.

.

. .

.

.

φ NJ −1 = QNJ −1φ NJ + R NJ −1φ NJ − 2 + ZNJ −1

.

;

.

.

(NJ-2)

where NJ is the number of lines. Since φ1 is known, φ 2 is eliminated from (ii) then φ 3 from (iii) and so on to obtain for

φ j , the following general relationship: φ j = A jφ j+1 + B j

(53)

with

(

A j = α j / D j − β j A j −1

(

)

)(

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B j = β jB j−1 + c′′j / D j − β jA j−1

)

and H1 =0, T1 =φ1 (2≤j≤NJ-1) To get the values of φ on the line i, the coefficients Aj and Bj (j=2,NJ-1) are first calculated, then φNJ-1 from φNJ which is known from the boundary conditions, φNJ-2 from φNJ1 and so on up to φ2 , φ1 is also known from the boundary conditions. Once φ is known on the ith column its value is calculated at the next (i+1) th column and so on up to the last NIth column (NI is the number of columns). The resolution process is then based on a sweeping in the direction west-east, alternating with a sweeping north-south to complete an iteration.

2.4.2.b. Under Relaxation and Convergence Under relaxation is used to accelerate the convergence of the equations which are not linear. For a variable φ stored in a point P, the explicit expression for the under relaxation is given as follows:

φ P = ωφ P + (1 − ω )φ P R

N

0

(54)

with ω the coefficient of under relaxation lying between 0 and 1 and φP0 is the value at the previous iteration, φPN the new value and φPR the value after under relaxation. Substituting the new value in the general equation gives the following expression:

(A P − b) R φP = ω

∑A

V

φV + c +

(1 − ω ) 0 φP ω

The diagonal dominance is reinforced, insuring the stability of the solution.

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2.4.2.c. Convergence Criteria To optimise the number of iterations, the convergence is checked at the end of each iteration by means of a test expressed in terms of momentum and mass residues as follows:

Rφ =

∑A

V

φ V + c − ( A P − b )φ P

(56)

V

Finally a forward-marching procedure is applied to solve the algebraic equations obtained from the discretisation work, using the iterative method based on the Tri- Diagonal Matrix Algorithm (TDMA) [7]. The equations are solved by marching from an upstream station, where the flow conditions are known, to successive stations downstream. It is also important to note that the SIMPLE (Semi-Implicit Method for Pressure Linked Equation) algorithm has been used. This algorithm has been widely described in the literature like in [7, 8] and basically it starts by guessing the downstream pressure field so as to lead to a preliminary velocity field which may satisfy the momentum equations but not the continuity equation, then corrections are made in a systematic manner to the pressure field until all the equations are satisfied.

2.5. Boundary Conditions The resolution of the above equations requires boundary conditions which, generally, are obtained by considering different regions of the system like the entry, the axis of symmetry, near the wall and the exit. For the present problem these conditions are stated as follows:

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a) At the entry of the column : Uen =Uunif with Unif the fluid velocity supposed to be uniform; Ven =0; 2

Ken =I U en ; 3/ 2

εen = k en /lm . with I the intensity of turbulence taken equal to 0.012 from [1], lm is the length scale of eddies and is equal to (λ D/2), with D the column diameter and λ a constant equal to 0.12, still from [1]; b) At the column axis: at this level the radial gradient of any property is equal to zero; c) At the column wall: in this region all the velocity components are zero and as mentioned previously a particular treatment is then necessary; d) At the exit of the column: at this stage, it is assumed that the flow is fully developed and hence the axial gradient of any property is also zero, and the velocity is obtained so that to satisfy the continuity. Figures 2a, b, c & d give the evolution of the axial velocity in the radial direction and one can see qualitatively that the curves representing these velocity profiles have the typical and

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representative shape of a turbulent flow, where near the wall the velocity is varying very rapidly and is practically uniform further near the axis, comparatively to a laminar flow where the profile is simply parabolic. To test the reliability of the results, the profiles obtained numerically are compared with those obtained by the use of the 1/7 power law, for the same Reynolds numbers considered above, as shown in the same Figures 5a, b, c & d. For low values of Reynolds numbers, the curves corresponding to 1/7 power law are above those obtained numerically by the use of the k-ε model, whereas when the Reynolds number increases progressively, the two curves tend towards one another to become practically the same. This is in a perfect agreement with the fact that in the approximation represented by the 1/7 power law, the thickness of boundary sublayer has been neglected which is effectively the case for high Reynolds numbers where this thickness is almost zero.

1,0

0,8

r/R

0,6

Numerical profile éi Theoretical profile thé i Re=5000

0,4

0,2

0,0 0,0

0,2

0,4

0,6

0,8

1,0

0,8

1,0

U/Umax

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(a)

1,0

0,8

r/R

0,6

Numerical profile éi Theoretical profile thé i Re=20000

0,4

0,2

0,0 0,0

0,2

0,4

0,6

U/Umax (b) Figure 5. Continued on next page.

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1,0

0,8

r/R

0,6

Numerical profile éi Theoretical profile thé i Re=100000

0,4

0,2

0,0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0,7

0,8

0,9

1,0

1,1

U/Umax (c)

1,0

0,8

r/R

0,6

Numerical profile éi Theoretical profile thé i Re=200000

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0,4

0,2

0,0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

1,0

1,1

U/Umax (d) Figure 5. Axial velocity profiles for different Reynolds Numbers.

The profile of the turbulent energy dissipation rate in function of the radial position in a given station of the column and for a fixed Reynolds number was also explored, in this part. For instance, Figure 6 shows such a variation for a Reynolds number equal to 5000. This can also provide a mean to test the reliability of the values obtained for the velocity profiles. Since as it can be seen that at the axis the value of the energy dissipation rate is zero and it increases rapidly and then slowly when approaching the walls and this is consistent with the velocity profiles obtained. Finally to be confident about the results and their convergence the evolution of the residue of the energy k as well as those of the mass and the axial velocity component were considered in terms of the number of iterations where it can be seen from

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Figure 4 that the three residues tend towards very low values of the order of 10-3 indicating that the results obtained are accurate enough. 1,0

0,8

r/R

0,6

Re=5000

0,4

0,2

0,0 0,0

0,1

0,2 2

0,3

0,4

3

ε(m /s ) Figure 6. Radial variation of energy dissipation rate.

Rφ

100

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10

x

Rk : K energy res idue Rm : Mass res id ue Ru : U veloc ity res id ue

1 0,1 0,01 1E-3 1E-4 0

50

100

150

200

250

300

Niter Figure 7. Variable residue variations with the iterations number.

Since the values of the velocity components (radial and axial) are available at any point of the column, Figures 8, for the sake of illustration show a 3-D representation of the continuous phase flow.

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459

V

Height

Radius

(a)

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U

Height

Radius

(b) Figure 8. Three-dimensional representation of velocity profile along the column height.

3. Modelling of the Influence of the Hydrodynamic of the Continuous Phase on Drop Breakage and Coalescence The drop dynamics is accompanied by drop breakage and coalescence processes which have a direct influence on the drop sizes and hence their distribution as well as interfacial area between the two phases which commands the mass transfer and the column efficiency. In this work, the treatment of the breakage and coalescence dynamic is essentially based on the continuous phase hydrodynamic. At this stage all the hydrodynamic field of the continuous phase is known and available at any station of the column. In this part its influence on the drop behaviour once injected into the column are presented and discussed.

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The processes of drop breakage and coalescence are also described where models representing the two phenomena reported in the literature have been discussed. This part is rather concerned with the effect of the hydrodynamic of the continuous phase surrounding the drop, and which has inevitably an influence on its evolution and behaviour.

3.1. Hydrodynamic Influence on Drop Breakage The drop deformation in a turbulent flow is mainly caused by the pressure fluctuation [8], the eddy–drop interactions [9] as well the fluctuations of the velocity [10]. The most widely used model considers the collision or the interaction of the drop with eddies of the same size, as the main factors causing this deformation. However the breakage takes place according to a certain probability which will be discussed further. Eddies which can generate the drop breakage are generally of the same size of the drop itself. Whereas the eddies which are much larger serve more for the transport of the drops. Also from other works [11], it has been shown experimentally the existence of a maximum stable diameter of the drops dmax, which is defined as the highest value below which the breakage probability is zero. In other words all the drops having a diameter less than dmax, are generally stable enough. This diameter depends on physical properties of the liquid forming the drop as shown by the following equation obtained by Shinnar [12]: 3

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d max

⎛ σ ⎞ 5 − 25 = k⎜ ⎟ ε ⎝ ρc ⎠

(57)

with ρc the density of the continuous phase, σ the superficial tension, ε the turbulent energy dissipation rate and k a numerical constant equal to 0.084 from the experimental data reported in [11]. The drops of sizes greater than dmax have a breakage probability ηbreak given by the following expression:

ηbreak = exp(−

ES ) e

(58)

with e the energy of an eddy transmitted to the drop and is expressed according to [11], as follows: 11 3

e = 0.43πρd ε

2 3

(59)

and ES is the energy associated to the surface generated by the drop breakage and can be considered as being the minimum required for this process. It can also be expressed according to [11], as:

E S = πσd 2

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Figure 9. Radial variation of energy dissipation rate at different Reynolds numbers.

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Therefore this phenomenon is directly related to the nature of the liquid flow around the drop. This process is stochastic in nature and therefore it is necessary to introduce the breakage probability which can be calculated from the relation given in Equation 58. However the use of this equation requires the knowledge of the quantity ε representing the dissipation rate of the turbulent energy but the measure of this latter is very difficult and almost impossible experimentally. Consequently, a tool was developed which enables to carry out computer experiments leading to the values of the turbulent energy dissipation rate ε, to end up finally with the value of the breakage probability for a drop of a given diameter. A priori to put into evidence the effect of ε, Figure 9 shows for an arbitrarily chosen station that an increase of the Reynolds number causes also an increase in the energy dissipation rate and this has a direct effect on the probability of drop breakage. 0,7

ηBre

0,6 0,5 0,4 0,3

Re=3000 Re=4000 Re=5000 Re=6000 Re=7000 Re=8000

0,2 0,1 0,0 -0,1 0,000

0,001

0,002

0,003

d(m) Figure 10. Drop size effect on breakage efficiency.

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0,004

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Figure 10 shows that the more Re increases the higher is the probability of breakage. This can be explained by the fact that an increase of the turbulence generates a great number of eddies of small sizes, which, as mentioned above, can cause drop deformations and ultimately breakage (eddies of greater sizes rather serve to transport the drops).

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Figure 11. Exemple de détermination de dmax à Re=5000.

Figure 12. Effect of the dissipation rate on dmax.

It is noted that all the curves for different Reynolds numbers present the same shape, starting with an horizontal line where the probability of breakage is zero to increase

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afterwards when the drop diameter reaches a certain critical value. Consequently these numerical experiments make it possible to have the different values of the turbulent energy dissipation rates to enable the determination of the maximum stable value of the diameter, corresponding to the size below which the drop is able to resist to the intensity of turbulence, because as suggested by Hinze5, a drop deforms and breaks in a turbulent medium, indicating that the interfacial and viscosity forces inside the drop are unable to counterbalance those due to the pressure fluctuations which cause the deformation. As an example, Figure 11 shows the value of dmax for Re =5000. Also these experiments demonstrate clearly that dmax, is much related to the value of Reynolds number and consequently to that of the energy dissipation rate. To see better this relation Figure 12 shows directly the influence of the dissipation rate on the value of dmax, for a given station in the column. It is clear that dmax increases when ε diminishes, reinforcing the preceding results. Another argument consolidating this is the fact that eddies of small size have very short lifetime and an energy less important. Also as mentioned and established previously the dissipation rate is relatively high near the walls of the column as illustrated by Figure 6 and hence it is expected a great number of drops which may undergo a breakage in this region, in agreement with the results reported in [11]. Figure 13 shows a 3-D representation of this dependence which is in a perfect agreement with the curves shown in Figures 5 & 10 where it can be affirmed that for different Reynolds numbers a drop can be influenced by the same rate of energy dissipation in different positions only. This result is important because in a great number of models dealing with the processes like drop breakage, it is a usual practice to neglect any radial variation of certain parameters like the mass transfer coefficient, concentrations, velocities (plug flow) etc., and as seen from this work, this is not always justified, particularly for large columns. Consequently the model proposed in this work is important and has the advantage to take into account the local variations of the above parameters, but the computing cost is generally the price to pay.

Figure 13. Three-dimensional representation of the variation of dmax with the radial position and the energy dissipation rate.

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3.2. Hydrodynamic Influence on Drop Coalescence Coalescence between two drops is due to the breakage of liquid film of the continuous phase which separates the two drops. For this case also only binary coalescence is considered. However in many research works [13], it has been shown that after colliding two drops, can remain in contact with one another a certain time without undergoing coalescence and therefore this process similarly to drop breakage is also conditioned by a certain probability which is defined in the next paragraph. Many research workers have proposed expressions for the probability of coalescence, and the one considered in this work is that reported in [9], as follows:

ηcoal

⎡ μ ρ ε ⎛ d d ⎞2⎛ 1 1 ⎞⎤ = exp ⎢− c 2c ⎜ 1 2 ⎟ ⎜ 2 − 2 ⎟ ⎥ h0 ⎠ ⎥ ⎢⎣ σ ⎝ d 1 + d 2 ⎠ ⎝ h ⎦

with h0 = k.a where k = 0.1 and a =

(61)

r1r2 , r1 and r2 being the radii of the two coalescing r1 + r2

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drops, respectively, h is the thickness of the liquid film between the two drops, μc and ρc are the viscosity and the density of continuous phase and σ is the superficial tension. Generally it is recommended that the breakage of the film occurs if h < 500Å [11]. The hydrodynamic of the continuous phase flowing under turbulent conditions, down a vertical and cylindrical column having the same geometrical parameters of the one used in [14] i.e. a diameter of 9.7 mm and a height of 500 mm, is first given. Then the influence of this hydrodynamic field of the continuous phase on the drop breakage and coalescence phenomena is presented for the chemical system used in [15], namely Water- Acetic acid – Isopropyl Ether.

Figure 15. Energy dissipation rate effect on coalescence efficiency.

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The coalescence phenomenon is related and depends directly on the resistance of the interstitial liquid film between two drops to breakage. Consequently the influence of the hydrodynamic field of the continuous phase on the liquid film thickness has been examined in this study as shown by Figure 14 where the variation of coalescence efficiency in function of the film thickness, is represented. It can be noticed that for low values of film thickness, the efficiency of coalescence increases rapidly to reach unity. Outside this very thin region the coalescence probability is practically zero excluding any possibility of coalescence between the considered drops. This figure shows also clearly the fact that even two drops are as very near as possible to one another, coalescence is not certain to occur and hence the notion of coalescence efficiency and the stochastic nature of the phenomenon. The effect of the energy dissipation rate on coalescence has also been examined and is less pronounced than the case of breakage as shown in Figure 15 where it can be seen that above a certain critical value of ε, the coalescence efficiency is practically constant and equal to unity, showing a first branch which increases rapidly to stabilise with ε having no effect. Probably this indicates that at energy dissipation rate values higher than the critical value the liquid film between the drops does not resist anymore to turbulence and film breakage is instantaneous to enable coalescence. From these results one can see the importance of the influence of local hydrodynamic field on the behaviour of the drop and hence on other processes like mass transfer.

4. Drop Motion in the Liquid-Liquid Extraction Column

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4.1. Drop Formation at the Distributor a) Drop formation: the orifices or the nozzles used to introduce the drops should have neither small diameters to minimise the clogging of the distributor nor large diameters to avoid the formation of excessively large drops which are not often desirable for systems such as liquid-liquid extraction column which involve mass transfer. Perforation diameters lying between 1 and 6 mm [16] are generally recommended, minimising the accumulation of the dispersed phase at the distributor surface. Quantitatively, the prediction of drop size is not always easy as reported in [17] whereas qualitatively, it is noticed that at low drop velocities through the perforations, the formed drops are of small and uniform sizes, which is very often desirable in practice. For more important velocities, a jet is formed the breakage of which gives rise to drops with a non uniform size distribution. b) Drop velocities: the rise or fall velocities of immersed drops in a static liquid, was initially, reported by Bond and Newton [18] who developed the following expression:

⎛ 2Δρgr ⎞ ⎛ μ d rσ ⎞ ⎟⎟ f ⎜⎜ ⎟⎟ u t = ⎜⎜ , ⎝ 9μ c ⎠ ⎝ μ c W ⎠

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(62)

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With utg the drop terminal velocity, μc and μd , the viscosities of the continuous and dispersed phases, respectively, r the drop radius, Δρ the density difference between the two liquids, g the gravity acceleration, W the apparent mass of the drop and σ the superficial

⎛ 2Δρgr ⎞

⎟⎟ is simply an expression of Stokes’s law for rigid spheres tension. The quantity ⎜⎜ ⎝ 9μ c ⎠ ⎛ μ d rσ ⎞ , ⎟ is a correction factor taking into consideration the drop size effect. ⎝ μc W ⎠

and f ⎜

An improvement of the expression of Equation 62, for the case of a drop displacement in a stagnant phase was proposed by Vignes [19] as follows: 2

1

d ⎡ gΔρ ⎤ 3 ⎡ ρ c ⎤ 3 ⎡ gΔρd 2 ⎤ ut = ⎢ ⎥ ⎢ ⎥ ⎢1 − ⎥ 6σ ⎦ 4.2 ⎣ ρ c ⎦ ⎣ μ c ⎦ ⎣

(63)

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with the variables as defined previously. It should be noted that both expressions given in Equations 62 and 63 are based on the equality of the Archimedean and the interfacial forces expressed by (VgΔρ) and (πdσ), respectively. However, for the present case, it is necessary to have a relationship which gives the drop velocities in a flowing liquid. For this purpose, many researchers have tried to generalise Equation 62 for the case of a flowing continuous phase, accompanied by the formation of drops at the orifices, after the jet disintegration [17]. The best expression for the calculation of these velocities is due to Kagan et al [20]. It has the advantage to be an explicit relationship in term of the drop volume, vg, which is expressed as follows:

⎤ ⎡ ⎥ ⎢ 1 πσdF ⎢ d 3 1 + 2.39 vg = We − 0.485We⎥ ⎥ Δρg ⎢ 8σ ⎥ ⎢ Δρg ⎥⎦ ⎢⎣

(64)

with We the Weber number expressed as:

We =

(ρ

c

+ ρ d )du 2inj 4σ

(65)

with uinj the velocity of drop injection and F a correction factor, due to Harkins and Brown [21], which, under static conditions, expresses the relative volume of the formed drop with respect to the liquid just above au the plate or the nozzle, at it detaches. In many relations F was fixed at 0.655 [17], but, more accurately, it can be calculated according to the following expression:

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467

⎛ Δρg ⎞ F = 0.6 + 0.4 exp⎜ −2 d 3 ⎟ πσd ⎠ ⎝

(66)

However, it should be noted that, generally, the calculation of the drop velocity in a liquid is very complex problem due to the instability in the shape of the drops, a fact which has encouraged the development of other relationships of empirical nature as the ones described in [22]. The above two relationships show that the large drops tend to be very slow as reported in [23] but have important terminal velocities, compared to the small drops.

4.3. Influence of the Drop Velocities Field At this stage, it is necessary to know how the drops will behave once they are injected into the column to undergo the influence of the fields of the radial and axial velocity components of the continuous phase, obtained by means of the integration of the Navier-Stokes equations, as described in the previous section, and of its proper velocity calculated by means of Equations 4.2 or 4.3. Figure 16 shows how a drop inside the control volume is under the influence of the different velocities involved in the system.

Z

v(i+1,j)

v(i+1,j+1

v2

Z(i+1)

drop (ij,8)

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u(i+1,j) drop (ij,10)

drop (ij,3) u1

drop (ij,9) v1

v(i,j) Z(i)

u(i,j)

u(i+1,j+1 ) u2 v(i,j+1)

u(i,j+1) r(j+1)

r(j) drop(ij,2)

r

Figure 16. Different velocity components acting on a drop inside the control volume.

As shown in the above figure, the drop is inside the control volume and its position does not necessarily coincide with a nodal point, as shown by the following figure:

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v1

v0

v2

A

0

B

Figure 17.Calculation Scheme of velocity components acting on a drop.

According to this figure, the point 0 represents the drop centre, A and B are two grid points where the different values of the axial and radial velocities are already available from the previous part. For example, the velocity (radial or axial) exerted on the drop is given according to a lever rule arm as follows:

vO =

v1. OA + v 2 . OB AB

(67)

vO can be either the radial or the axial velocity and v1 and v2 are the corresponding velocities at the adjacent grid points. In another words, the point representing the drop can be regarded as the barycentre of the two left and right surrounding grid points. On this basis, the resulting velocity is calculated from Figure 17 to obtain: v1= (1-Fr).v(i,j) +Fr.v(i,j+1)

(68a)

v2= (1-Fr).v(i+1,j) + Fr.v(i+1,j+1)

(68b)

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and

with Fr a factor lying between 0 and 1 which expresses the ratio of the distances between the point representing the projection of the drop centre on the north and south faces of the control volume. Therefore the radial component of the resultant velocity is given by: Vradial = (1-Fz).v1 + Fz.v2

Fr =

FZ =

rdrop − r ( j ) r ( j + 1) − r ( j )

Z drop − Z (i ) Z (i + 1) − Z (i )

(69) (70a)

(70b)

with Fz, similarly to Fr, expresses the ratio of the distances between the projection of the drop center on the east and west faces of the control volume. Analogously to the radial velocity component, the axial component of the resultant velocity can be obtained as follows: u1= (1-Fz).u(i+1,j) + Fz.u(i,j) and

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Modelling of the Interaction of the Dispersed and the Continuous Phases…

469

u2= (1-Fz).u(i+1,j+1) + Fz.u(i,j+1)

(71b)

Uaxial = (1-Fr).u1 + Fr.u2 + Vrise

(72)

The energy dissipation rate can also be calculated similarly to the velocity components which, once calculated with respect to each direction, enable to follow the trajectory of the drops in the column and hence the calculation of their number at every height. In this part all the different steps described above are linked in order to follow the motion of the drops in the continuous phase. Also, the model describes the evolution of the drop population which is under the influence of the hydrodynamic field of the continuous phase. This enabled to obtain the drop distribution curves along the column height and hence the hold up of the dispersed phase as shown in Figures 18. The shapes of the curves are similar to those reported in the literature [24, 25] for different types of columns. Therefore, qualitatively, the model results are in a good agreement with those reported in the literature. 0,014

Re=3000

0,012 0,010

β 0,008 0,006 0,004 0,002 0,000

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0

5

10

15

20

25

30

Ne (a) 0,014

Re=4000

0,013 0,012 0,011 0,010 0,009

β

0,008 0,007 0,006 0,005 0,004 0,003 0,002 0,001 0,000 0

5

10

15

20

Ne (b) Figure 18. Continued on next page.

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25

30

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Re=5000

0,016

0,014

β

0,012

0,010

0,008

0

5

10

15

20

25

30

Ne (c) 0,022

Re=6000

0,020 0,018

β

0,016 0,014 0,012 0,010 0,008 0,006 0,004 0,002 0,000 0

5

10

15

20

25

30

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Ne (d) 0,07

Re=7000

0,06 0,05

β

0,04 0,03 0,02 0,01 0,00 0

5

10

15

20

Ne (e) Figure 18. Continued on next page.

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25

30

Modelling of the Interaction of the Dispersed and the Continuous Phases… 0,30

471

Re=8000

0,25

β

0,20

0,15

0,10

0,05

0,00 0

5

10

15

20

25

30

Ne (f) Figures 18. Variation of dispersed phase retention with the column height.

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However, it should be noted that, a priori, a preliminary study was necessary to determine an adequate range for the value of Reynolds number, for the column operation. From Figures 18 it can be noticed that for low Reynolds numbers of the order of 3000, the drops are relatively free to move in the continuous phase, along the column. This explains the oscillatory character of the curves representing the variation of the dispersed phase retention with the column height. As Reynolds number is increased progressively, a drop ‘crowding’ takes place at the distributor. This can be explained by the fact that at high Reynolds numbers, the eddies are vigorous forcing the drops to remain near the entry at the distributor.

5. Modelling of the Dispersed Phase Hold-up The determination of the dispersed phase hold-up and the particle size distributions in two phases contactors is always necessary for the computation of chemical engineering processes, such as solvent extraction, absorption, reaction engineering, etc. The modelling of such systems can be based on the use of the population balance approach which enables the description of the variations in size distribution of the dispersed phase by averaging functions related the behaviour of individual particles like drops or bubbles, as well as their interactions. The development of the corresponding models relies on a parameter fitting to match the experimental size distributions. This defines the so-called inverse problem which is suitably considered when the mathematical solution to a population balance model is known, but phenomenological functions are not.

5.1. The Drop Population Balance Approach This chapter section introduces the drop population approach through the following brief literature review.

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Hulburt and Katz [26] and Valentas and Amundson [27] were among the first researchers to introduce the model of the population balance for the modelling of certain chemical processes involving a dispersed phase. In these models, a drop size distribution is generated along the spatial coordinate of the liquid-liquid contactor, contrarily to models where drop size distributions based on small mean diameter values, are assumed to be uniform as reported in [28, 29]. Therefore modelling based on drop population balance contributes to the determination of the dispersed phase hold up as well as any other integral propriety related to the drop mean size and the interfacial specific area for mass and heat transfer calculations [15, 30-31]. The drop population balance approach was applied for the modelling of the behaviour of liquid-liquid interactive dispersions in two different manners. The first is based on the notion of stage and the second one on the differential models. In the stage approach [30, 32-33], the column is assimilated to a series of agitated reactors. Practical examples can be cited such as the perforated columns, the pulsed columns, Scheible column, and cascades of mixer-decanters. For these systems, a population balance equation is written for each reactor with adequate boundary conditions. For the approach based on the differential model, the phases are in continuous contact and separate at the contactor exit, as is the case of the jet columns, the rotating disc contactor (RDC), pulsed plate columns, Kühni and Oldshue-Rushton columns. For these cases, the drop population balance model is usually formulated as a conservation law in terms of volumetric concentrations [15, 34-37]. The resulting differential model takes into account the transport, breakage and coalescence of the drops, along with the necessary boundary conditions. However these latter are not clearly expressed in the literature. A complete review of the mathematical modelling of liquid-liquid extraction columns as well as the advantages and disadvantages of the corresponding models is reported in [38]. Globally the application of the population balance equation provides relevant information for any modelling including breakage and coalescence as well as transport of drops. However the development of research programs aiming at finding experimentally the breakage and the coalescence kinetics and the drop transport, has largely contributed to the elaboration of more realistic models as reported by research workers in [39, 40]. The level reached by the model development has induced mathematical complexities which require a high computing cost, since no analytical solution is easily obtained for the drop population balance general equation. Consequently, a numerical solution is necessary for any reliable simulation of the dispersed phase process. A great number of research works concerning only the numerical solution were published during the last two decades. One can cite particularly [41-43]. However despite all this intensive research and up to date, no general numerical approach does exist and systematically applicable to the drop population balance equation, when the distributions depending on many variables such as the size, the concentration and the ge of the drops and known as the internal coordinates, do intervene [26]. Such multivariable distribution in the liquid-liquid dispersions is reported in [44, 45]. However the problem is even complicated when external coordinates of the continuous phase intervene, as in the case of differential models of drop population balance.

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5.2. The Model 5.2.1. The Model Equations The polydispersed character of one of the phases is represented by a volume distribution law denoted P(z,d,t) and considered as the basic variable of the model. It represents the volume fraction of drops with diameter d in an unit volume of the column at level z and time t [46]. Hence, the local hold-up of the dispersed phase can be calculated from P(z,d,t) as follows :

φ (z, t ) =

d max

∫ P(t , z, d ) ∂d 0

(73)

5.2.1.a. The Dispersed Phase Mass Balance The volume balance equation of unit volume of the column can be expressed as: Q ∂ ∂ ∂ ∂ P(t , z , d ) + (v d (t , z, d , ϕ ) P(t , z, d )) = ( Dax P(t , z, d )) + d Pin (d ) δ ( z − hd ) + PV (t , z , d ) A ∂z ∂z ∂z ∂t (74)

With transient and convective terms balanced against a back mixing term expressed by means of the dispersion coefficient Dax . The solvent feed is handled as a point source by Dirac’s δ − function . The drop break-up and coalescence processes are taken into account

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in the term PV, which is given by:

PV ( t,z,d ) = ∫

dmax

0

β ( d0,d )g ( t,z,d0 ) P ( t,z,d0 ) ∂d0-g ( t,z,d ) P ( t,z,d ) +

(75)

2 3 3 3 dmax−d3 P( t, z, d1) P( t, z, d2) ⎛ d ⎞ V(d) d / 2 P(t, z, d1) ∂d1 ω( d1, d2) ω( d1, d2) ⎜ ⎟ ∂d1 − P(t, z, d)∫0 ∫ 0 2 V ( d1) V ( d2) ⎝ d2 ⎠ V ( d1)

The first integral accounts for gain and loss due to break-up of the mother droplet d0 according to the daughter droplet distribution β . Similarly, the second integral holds for coalescence according to the coalescence frequency ω . 5.2.1.b. The Continuous Phase Mass Balance The flow of the continuous phase is opposite to that of the dispersed one, and for a volume between z+Δz/2 and z-Δz/2, a mass balance leads to the following differential equation:

∂ [(1 − φ (t , z ))] + ∂ [(1 − φ (t , z )) ⋅ v c (t , z )] = ∂ ⎡⎢ D c (z ) ⋅ ∂ [(1 − φ (t , z ))]⎤⎥ ∂t ∂z ∂z ⎣ ∂z ⎦ Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

(76)

474

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+

Qd p in (d ) δ ( z − h d ) + PV (t , z , d ) A

5.2.2. The Initial and Boundary Conditions The liquid-liquid extraction column can be divided into four different regions as shown in the following figure:

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Figure 19. A liquid-liquid extraction column at counter-current.

The obtained differential equations shown above enable a physical description of the contactor and their resolution requires boundary conditions for the four regions of the column shown above in Figure 19 and which are the feed stages for the continuous and dispersed phases and the top and the bottom stages (the exits). These boundary conditions are expressed as follows: For the dispersed phase:

∂ ⎛ ∂ ⎞ ∂ ∂ Ρ⎟ + S Ρ + (v d Ρ ) = ⎜ Dd ∂z ⎝ ∂z ⎠ ∂z ∂t

(77)

with S = S F + S B + S C , the source term. Boundary condition at the column bottom (z = 0):

z=0

v d+ Ρ − Dd

∂Ρ = 0 with v d+ = max (v d ,0 ) ∂z

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475

Boundary condition at the top of the column (z = H) :

z=H

Dd

∂Ρ =0 ∂z

(78b)

Initial condition:

Ρ(0, z , d ) = Ρ 0 ( z , d ) for z ∈ [0, H ], d ∈ [0, d max ]

At t = 0

(78c)

In this case, it is assumed initially that the column contains the continuous phase only and

hence Ρ ( z , d ) = 0 0

Condition at the dispersed phase feed stage : d max

Q S F ( z , d ) = d ΡF (d )δ ( z − z d ) with A

∫ Ρ (d )δd = 1 F

(78d)

0

For the following model equation:

∂ ∂ ∂ ⎛ ∂ Ρ ∂Φ ⎞ Ρ + (v d Ρ ) = ⎜ Dd Ρ + Dc ⎟ + S F + S B + SC ∂t ∂z ∂z ⎝ ∂z 1 − Φ ∂z ⎠

(79)

The boundary and initial conditions are as follows:

Ρ(0, z , d ) = Ρ 0 ( z , d )

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t=0 z=0

v d+ Ρ − Dd

∂Ρ =0 ∂z

z=H

Dd

with v d+ = max (v d ,0 )

∂Ρ =0 ∂z

(80a)

(80b)

(80c)

For the following equation:

φ (t , z ) =

d max

∫ Ρ(t , zd )δd

(81a)

0

v d (t , z , d , φ ) = v r (1 − φ ) − with vc ,sup =

Qc A

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vc ,sup

1 − φ (t , z )

(81b)

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Moreover to the initial formulation adopted by Casamatta in [47], these equations are written in term of a volumetric balance, completed by a solvent feed term represented as punctual source described by the delta (δ) function of Dirac. The dispersed phase feed is according to a drop size distribution which simulates the existence of the distributor. Each equation is based on two contributions: the first one is dispersive and represents the axial mixing, whereas the second one is convective. For the dispersed phase this enables to include explicitly the mixing due to the differences in velocities of different drops of various sizes, as well as the back mixing due to the agitation of the medium. The drop breakage and coalescence phenomena are incorporated in the model by means of the term PV known as the surface transfer area production term. In this second part of the chapter, the hydrodynamic study concerns mainly the dispersed phase. It should be noted that the presented model is in the continuation of previous works carried out in [46-48].

5.3. Modelling of the Hydrodynamic The modelling is based on different laws which are empirical in nature, in most cases. For the continuous phase, the representation is based on the evaluation of the axial dispersion. For the dispersed phase, there are many and complex related phenomena. The prediction of drop size distributions is an important step for the determination of the hold-up, the mean residence time, as well as the behaviour of the drops at flooding. The distribution is also important for the interphase mass transfer since it influences directly the transfer area. Inversely, the transfer intensity may affect considerably the drop population behaviour, leading to a decrease of the global column efficiency.

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5.3.1.a. The Hydrodynamic Parameters Drop velocity: in the model the relation between the absolute velocities of each phase leads to the slip velocity which is generally derived from the velocity limit of a unique drop in an infinite and stagnant medium. The slip velocity is function of drop sizes and the physicochemical proprieties of the system. However, in an agitated packed column the trajectory of a drop is not straight and can be affected by various external conditions such as turbulence, recirculation, interactions with the wall, other drops, etc.). The effective velocity Ut results from the unique drop model in a stagnant medium and it is corrected to take account of the presence of the other drops, the agitation and the column geometry. Figure 20 shows the evolution of the Ut in term of the drop diameter. As shown in the above figure, three different regimes with unclear boundaries can take place and the terminal velocity is expressed as follows: 1

1 ⎛4 Δρ ⎞ 2 ⋅ d⎟ ⋅ c w − 2 Ut = ⎜ ⋅ g ⋅ ρC ⎠ ⎝3

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Velocity Ut

I

II

477

III

Drop size Figure 20. Velocity of a single drop.

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This relation is the result of a force balance expressing and equilibrium between the gravity and the drag forces exerted on a drop. The use of this equation requires the knowledge of the drag coefficient which is function of the Reynolds number with respect to the drop, as shown in Figure 21. For the first regime, the drops are assimilated to solid spherical particles, corresponding to low Reynolds numbers. For higher values a surface motion and an internal recirculation start to develop, due to viscous constraints (regime II), inducing a certain resistance to the drop ascension. In regime III, the drop behaviour deviates suddenly from spherical drop model. The drop becomes unstable and flat particularly for low interfacial tensions and then starts to oscillate due to the fact that the interfacial tension no more compensates the external forces. The ascension velocity decreases down to a constant value even for increasing diameters up to drop breakage.

Figure 21. Drag coefficient of a single drop.

The determination of the drag coefficient cw for rigid drops is not difficult, comparatively to drops with internal recirculation.

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As mentioned previously, the different effect which perturb the rise of an isolated drop, are mainly induced by the presence of other drops, the agitation system, the packing of the column as well as its walls. For instance for disk columns, these phenomena are also observed with great intensity as reported in the study reported in [49]. This author showed that the decrease in the rise velocity for this geometry type is of the order of 40% to 50%, compared to the case of no packing. These latter prevent the drop having a straight trajectory as shown in Figure 22. Globally, the slowing of the drops increases with their sizes and decreases with the agitation intensity.

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Figure 22. An example of a simulated drop trajectory [49].

For the present model, all these effects are grouped into a coefficient kV which integrates these phenomena leading to the following simple relationship for the drop velocity in a column:

vr* = k V ⋅ vt

(83)

The influence of the presence of other drops is taken into account by means of correlations a list of which is given in [48]. In the present case the expression reported by Pratt et al [50] is adopted and showed to be satisfactory in various columns. It is expressed as follows:

vr = v0 (1 − φ )

(84)

where v0 is a characteristic velocity which depends on the operating conditions, the physicochemical proprieties of the system as well as the obstacles in the column. It was identified to the drop velocity rise defined in Equation 2.11, to end up with the following expression:

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vr = kV vT (1 − φ )

(85)

The term (1 − φ ) comes to reduce the effect of the Archimedean force exerted on the drop. The computation of kv(d) depends also on the column type as follows : For an RDC column, a correlation reported by Godfrey and Slater [51], is expressed as follows:

(

kv (d ) = 1 − 1.037 N D 3

)

5 0.12 R

⎛ ⎞ d ⎟⎟ − 0.62 ⎜⎜ ⎝ DS − DR ⎠

0.44

(86)

with N the rotor speed DS and DR, are the stator and rotor diameters, respectively. For a Kühni column this term is expressed as follows [52]:

⎛ 7.18.10−5 ReR / e ⎞ kv = 1 − (1 − e ) ⎜ ⎟ -5 ⎝ 1 + 7.18.10 ReR / e ⎠

(87)

The axial dispersion coefficient: this phenomenon is essentially due to the contactor mechanical agitation, and describes the back mixing. For the continuous phase it expressed as follows:

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⎡ ⎢ Dax ,c ⎛v ⎞ ⎛ NDR = 0.42 + 0.29 ⋅ ⎜⎜ d ⎟⎟ + ⎢c1 ⎜⎜ ⎢ vc H ⎝ vc ⎠ ⎢ ⎝ vc ⎢⎣

⎤ ⎥ ⎞ 13.38 ⎥⎛⎜ vc DR ρ c ⎟⎟ + ⎥⎜ ⎠ 3.18 + ⎛⎜ NDR ⎞⎟ ⎥⎝ η c ⎜ v ⎟ ⎝ c ⎠ ⎥⎦

⎞ ⎟⎟ ⎠

− 0.08

⎛D ⋅ ⎜⎜ K ⎝ DR

⎞ ⎟⎟ ⎠

0.16

⎛ DK ⎜⎜ ⎝ Hc

0.1

⎞ ⎟⎟ e (88) ⎠

The importance of the axial dispersion coefficient for the dispersed phase is more limited and negligible since for this case the forward mixing due to the drop rise velocity differences, is the dominant factor. It is expressed as follows:

⎛ NDR ⎞ = 0.7 + 0.02 ⋅ ⎜ ⎟ vd H ⎝ vT ⎠

Dax ,d

(89)

To calculate vd ( t,z,d,φ ) , it is necessary to find an expression for the continuous phase velocity. The column hydrodynamic is described by means of the plug-diffusion model [53]. The continuous phase flowrate through a right column section is expressed according to the following expression:

Qc ( z ) = A.vc ( z ).(1 − φ ( z )) − A ⋅ Dc ( z )

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∂ (1 − φ (z )) ∂z

(90)

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hence:

vc ( z , d ) =

1 ∂φ ( z ) Qc − Dc ( z ) (1 − φ (z )) ∂z A(1 − φ ( z ))

(91)

5.3.2. Drop Transport The velocities of both phases are expressed according to the two layers model [54] as follows:

vd ( t,z,d,φ ) = vr

( t,z,d,φ)

+

vc (t,z,φ)

(92)

with vc and vd the continuous and the dispersed phase velocities, respectively and vr the velocity of a drop of diameter d relative to the surrounding continuous phase. It can be obtained from the single drop terminal velocity vT by means of the following correlation:

vr ( d,φ ) = kV ( d ) vT

(1 − φ )

(93)

where kv(d) is the slowing coefficient taking by definition values between 0 and 1, depending on the type of the extraction column. For these systems, a correlation from the literature [55] has been proposed as follows:

(

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kv (d , N ) = 1 − 1.037 N D 3

)

5 0.12 R

⎞ ⎛ d ⎟⎟ − 0.62 ⎜⎜ D D − R ⎠ ⎝ S

0.44

(94)

with N the rotor speed, DS and DR, as the stator and rotor diameters, respectively.

5.4. Drop Breakage In the case of stirred extraction column, different break-up mechanisms exists and strongly depend on geometries. For instance, in a RDC column the drops are generated when passing near the rotor disc. The probability of drop break-up is supposed homogeneous in each compartment of the column and therefore a local approach is used, and the entire mechanism of break-up probability is correlated in terms of the geometries of the column, the system physical properties and the energy dissipation. For these systems the prediction of the drop break-up probability p (d ) and frequency

g ( z , d ) are calculated with the correlations reported in [39, 56-57] recommended by Modes [58]:

[

(

) ]

p(d ) 1.8 = 1.2.10 −6 ρ c0.8 μ c0.8 dDR1.6 (2π ) N 1.8 − N c1.8 / σ 1 − p(d )

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2.88

(95)

Modelling of the Interaction of the Dispersed and the Continuous Phases…

481

The breakage frequency depends on the residence time of the drops and are given by:

g ( z, d ) =

p (d )vd ( z , d ) HC

(96)

For the toluene/water N is expressed as follows:

⎛ ρ D3 N = 1.148⎜⎜ c R ⎝ σ

⎞ ⎟ ⎟ ⎠

−0.5

⎛ d ⎜⎜ ⎝ DR

⎞ ⎟⎟ ⎠

−0.667

(97)

Whereas for the n-butyl acetate/water, it is given as follows:

N crit = 0,016

DR− 2 / 3 μ d d − 4 / 3

(ρ c ρ d )1 / 2

⎡ ⎤ DR− 2 / 3 μ d d − 4 / 3 2 σ + ⎢(0,008 + ) 0 . 127 ⎥ ρ c DR1 / 3 d 5 / 3 ⎦⎥ (ρ c ρ d )1 / 2 ⎣⎢

0.5

(98)

The daughter drop-size distribution is assumed to follow the beta distribution, based on the mother droplet diameter d0, expressed as follows:

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β ( d 0 ,d ) = 3xm ( xm − 1 ) (1 -

d 3 ( xm - 2 ) d 5 ) d 03 d 06

(99)

Where xm is the mean number of daughter drops produced upon breakage of mother droplet of diameter d0. It is experimentally correlated and found dependent on the energy dissipation and having a value greater or equal to 2 for these two systems.

5.5. Drop Coalescence For this process, the hydrodynamics, the system properties at the interfaces, the intensity of the collision and the contacting time between the colliding drops are key parameters. The coalescence efficiency between two approaching, deformable drops according to the film drainage theory is due to drainage of the flow of the film into the bulk. It results in a resistance that hinders the drops to approach one another. The coalescence efficiency due to this resistance depends on interfacial mobility of the drops. Assuming that the drops approach along the centreline of the two drops, the relationship between the force and the approaching velocity of the drops has been calculated using a lubrication approximation [13]. It was assumed that coalescence occurs when the gap between two drops reaches a critical value, hc at which the film between drops ruptures resulting in coalescence. theoretically that [13]:

hc = (A k d eq / 16πσ)

1/ 3

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It has been shown

(100)

482

A.-H. Meniai, A. Hasseine, O. Saouli et al. In general, the coalescence process is described by the drop coalescence rate which is the

product of the collision rate h( d1 , d 2 ) , and the coalescence efficiency,

λ (d1 , d 2 ) , between

two drops of diameter d1 and d2. However, the determination of this (drop pair) coalescence rate remains an extremely complex phenomenon and its modelling is still not fully developed. From the literature [59- 60] the expressions for h( d1 , d 2 ) can be derived for these two systems as follows: - Collision rate:

h (d1 , d 2 ) = C1

ε 1/ 3

(1 + CV φ )

(d1 + d 2 )7 / 3

(101)

This equation is derived from the relative velocity of two mass points in the liquid phase on a distance of d1 + d 2 . The turbulence damping factor was introduced to account for the effects of two phase flow on the intensity of turbulence as defined by [59, 61]. The determination of the coalescence efficiency depends mainly on the nature of the two drops involved in the coalescence process as well as the properties of their interfaces. Some of the relevant models reported above are described as follows: -Coalescence efficiency: In the work of Chesters [13], the collisions are divided as viscous and inertial and for any detail how derived the following formula for the coalescence efficiency see [62]:

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⎛ μ d ρ c1 / 2 ε 2 / 3 d eq11 / 6 (d 1 + d 2 )1 / 3 ⎞⎟ ⎜ λ (d1 , d 2 ) = exp − C 2 ⎜ ⎟ σ 3 / 2 (1 + C vφ )2 hcrit ⎝ ⎠

(102)

- Determination of coalescence parameters Coalescence is a stochastic process where the probability of coalescence of the drops depends on their sizes, their physico-chemical proprieties, the hold up, etc. [63, 64]. The coalescence in the agitated liquid-liquid extraction columns mainly takes place in the low turbulence regions which correspond to plates below the stator [32, 63]. Due to the fact that the breakage and the coalescence processes take place simultaneously, it is not possible to study and estimate them independently. Hence the coalescence parameters are calculated numerically, by solving the inverse problem of the drop population balance. Similarly to the work reported in [65], the genetic algorithm which is described below, is used to minimise the following objective function:

Fobj =

NDC

∑ ⎡⎣Q k =1

k ,out 3

(exp) − Q3k ,out ( sim) ⎤⎦

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2

(103)

Modelling of the Interaction of the Dispersed and the Continuous Phases…

483

5.6. The Numerical Method: Genetic Algorithm Generally, a mathematical programming procedure has to be applied to find the best fit of the experimental data. The application of an inverse problem leads to an explicit analytical or approximate solution which is highly sensitive to errors in the experimental data [66]. The parameter-fitting problem is solved by a genetic algorithm (GA), where a finite volume and generalized fixed-pivot [67] techniques are used for solving the population balance problem and the calculation of the parametric derivatives of the solution. This approach is based on a global search method which has proven to be more robust than many traditional search techniques [68] with the following features:

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- The GA makes no assumption about the function to be optimized and thus it can also be used for nonconvex objective functions; - The GA makes a trade off between exploring new points in the search space and exploiting the current information; - The GA is a randomized algorithm whose results are governed by probabilistic transition rules rather than deterministic rules; - The GA operates on several solutions simultaneously, gathering information from current search points and using it to direct subsequent searches which makes it less susceptible to the problems of local optima; - The GA uses an objective function or fitness information only, instead of using derivatives or other auxiliary knowledge, as required by traditional optimization methods.

Figure 23. Flow diagram of the general genetic algorithm.

A genetic algorithm (GA) is an optimization search algorithm based on evolution principles and natural genetics [68]. The basic idea of GA is to place the parameters of the real problem to be optimised within what is referred to as a chromosome (or individual) which consists of genes. Each parameter is mapped to a gene in the chromosome. Parameters may be real numbers, integers, or even complex data structures. A general GA creates an initial generation (a population or a discrete set of decision variables), G(t=0), and for each

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generation, G(t), it generates a new one, G(t+1). The general genetic algorithm is described in the following A chromosome, also called a solution string of length n, is a vector of the form {x1, x2, …, xi} where each xi is an allele or a gene representing a set of decision variable values. The domain of values from which the xi value is chosen is called the alphabets of the problem. The initial population G(0) can be chosen heuristically or randomly. The populations of the generation G(t+1) are chosen from G(t) by a randomized selection procedure which is composed of four operators:

a. Selection

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Selection is a process in which individual strings are copied according to their objective function or fitness (f). In the process of selection, only solution strings with high fitness values are reproduced in the next generation. This means that the solution strings which are fitter and have shown better performance will have higher chances of contributing to the next generation. There are many selection methods available, such as tournament selection, roulette wheel selection, etc. These methods use the fitness value of each chromosome to decide whether it will survive or not.

Figure 24. Crossover and mutation operators.

b. Crossover The crossover operator, randomly, exchanges parts of genes of two parent solution strings of generation G(t) to generate two child solution strings of generation G(t+1). There are three variants of crossover: one-point, two-point, and one-gene crossover (see Figure 24). In a

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simple one-point crossover, a random cut is made and genes are switched across this point. A two-point crossover operator randomly selects two crossover points and then exchanges genes in between. However, in a one-gene crossover, a single gene is exchanged between chromosomes of two parents at a random position. The main purpose of the crossover operator is to search the parameter space. Other aspect is that the search needs to be performed in a way to preserve the information stored in the parent strings maximally, because these parent strings are instances of good strings selected using the Selection operator.

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c. Mutation Mutation operates on an individual (chromosome) producing offspring that is very different from its parent. A GA randomly selects a gene of the chromosome or solution string and then changes the value of this gene within its permissible range with a small mutation probability (Pm), as shown in Figure 24. Pm controls the rate at which new genes are introduced into the population. A low rate will prevent the introduction of potential genes and a high rate will give too much random perturbation. Mutations encourage a population that is converging onto some optimum to jump into a different part of the solution space, thus increasing the probability of detecting a different point leading to the global optimum solution. The need for mutation is to maintain diversity in the population. After selection, crossover, and mutation are applied to the whole population, one generation of a GA is completed. These three operators are simple and straightforward. Selection operator selects good strings and crossover operator recombines good substrings from two good strings together to hopefully form a better substring. Mutation operator alters a string locally to hopefully create a better string. Even though none of these claims are guaranteed and/or tested while creating a new population of strings, it is expected that if bad strings are created they will be eliminated by the reproduction operator in the next generation and if good strings are created, they will be emphasized.

d. General Remarks GAs have some unique characteristics which make them a more robust global search method than many traditional search techniques [68]: a GA makes no assumption about the function to be optimized and thus can also be used for nonconvex objective functions; a GA optimizes the tradeoff between exploring new points in the search space and exploiting the information discovered thus far; a GA is implicitly parallel; a GA is a randomized algorithm whose results are governed by probabilistic transition rules rather than deterministic rules; a GA operates on several solutions simultaneously, gathering information from current search points and using it to direct subsequent searches which makes a GA less susceptible to the problems of local optima and noise; a GA only uses objective function or fitness information, instead of using derivatives or other auxiliary knowledge, as are needed by traditional optimization methods. The efficiency of a GA depends on the customization of the parameters to the specific problems. Among the ideas that have been introduced are fitness scaling, generation gap, rank selection, and replacement criterions. The key GA parameters, which are common to all strategies explained in later sections, are the population size in each generation (NPOP), the

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percentage of the population undergoing reproduction (R), crossover (C), and mutation (M), and the number of generations (NGEN).

5.7. Description of the Considered Extraction System In the present study we consider a liquid–liquid flow in a pilot-scale rotating disc contactor (RDC), operating with two different chemical systems: Toluene/Water and N-Butyl Acetate/Water. These are usually recommended by the European Federation of Chemicals Engineering (EFCE) as test systems for liquid extraction studies [69, 70]. The physical properties of water, toluene and n-butyl acetate are reported in Table 1, with water as the continuous phase and toluene and n-Butyl Acetate forming the dispersed phase, respectively. Table 1. Chemical system properties System H2O/n-butyl acetate H2O/toluene

µx (mPa) 0.73 0.59

µy (mPa) 1.34 1.00

ρx (kg m-3) 881.5 865.6

ρy (kg m-3) 998.9 997.7

σ mN m-1 14.0 36.1

The operating conditions and the dimensions of each of the columns are presented in Table 2, as follows:

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Table 2. RDC column dimensions Column diameter (m) Compartment height (m) Active part (m) Stator diameter (m) Rotor diameter (m)

0.15 0.03 1.65 0.105 1.09

In this section the experimental and simulated results are presented and discussed as follows. We attempt to present the possible flow situations concerning the coalescence process for non-deformable drops [13]. The coalescence parameters were estimated in a small laboratory scale device, which consists of five compartments of an RDC column. Several rotational speeds and flow rates for the saturated toluene/water and n-butyl acetate /water systems were used to estimate the constants of the coalescence model. As mentioned above, they were estimated using the GA method [68] and the generalized fixed-pivot [67] technique to solve the inverse problem of the droplet population model, which consists of minimizing the objective function, S [69], given by equation 103. For the investigated systems (toluene/water, n-butylacetate/water) a set of parameters is obtained. They are not influenced by the operational conditions (rotor speed, volumetric flow rates) but are specific for each chemical system. For the system toluene/water the best parameters are c1 =0.001 and c2 = 0.041 and for the system n-butylacetate/water, they are c1 = 0.010 and c2 = 0.011 for two populations considered for each system. The parameters should be, in principle, of the same order of magnitude. However, further work is required, considering different chemical systems to improve the coalescence model.

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Within the range of the experimental accuracy the mean values for these constants describe satisfactory well these systems and are used further on.

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Figure 25. Influence of rupture and coalescence on the hold-up and Sauter diameters for the toluene/water system. N=350 r.p.m. (Qc=100 l/h, Qd=112 l/h). (-O- Sim. -◄- Exp. breakup and coalescence), (-*- Sim. breakup).

Figure 26.Comparison of a simulated hold-up and Sauter diameters with experimental data for nbutylacetate system. Influence of agitation effect on hold-up and Sauter diameters.(Qc=100 l/h, Qd=112 1/h), (200 r.pm.-O- Sim. -►- Exp.), (250 r.p.m. –*- Sim.-◄- Exp.).

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In Figures 25 and 26, the simulation results and the experimental data for these two systems in RDC column are presented. The simulation includes axial dispersion, break-up and coalescence. The axial dispersion coefficient after Modes [58] for the dispersed phase was with a correlation of Rod [71] and for the continuous phase after Kumar and Hartland [72]. In Figure 25, the hold-up and the Sauter diameter profiles are represented versus the dimensionless column height for the toluene/water system. Different situations such as breakup solely and break-up and coalescence are considered. The results further can be regarded as a sensitivity analysis of the model based on the concept of a population balance with the parameters derived and the validation by experimental data. The first case with no coalescence and only break-up leads to smaller drops which have a smaller slip-velocity and greater residence time. Therefore the hold-up in the column increases and the Sauter diameter decreases. For the inverse case with presence of coalescence larger drops are present. This leads to a lower hold-up and a higher Sauter diameter. Figure 26 shows the simulation and experimental data for n-butylacetate/water system for different rotating speed; clearly any increase in agitation speed encourages drop break-up and greatly reduces their coalescence. This results in drops with small average sizes and hence an improved hold-up and small Sauter diameter values.

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6. Conclusion From this chapter one can see how complex is the modelling of the behaviour of a dispersed phase in a continuous one. The main difficulty is due to the interaction of the various factors intervening and also to the nature of the problem which is rather stochastic. However this work aims more to examine the link between the hydrodynamic of the continuous phase under turbulent conditions and the drop breakage and coalescence phenomena, relying on certain relationships reported in the literature. For instance it has been shown how the turbulent energy dissipation rate is obtained and used to calculate important parameters like the maximal stable diameter, the breakage and coalescence efficiencies, etc. enabling computer experiments to be carried out. Finally and at this stage, this study should only be regarded as a preliminary step for a more complete model which would include the mass transfer between the continuous and the dispersed phases and where experimental results are available for known, enabling a better assessment of the obtained results. This chapter also explored the use of GA to estimate the coalescences parameters for liquid – liquid extraction. The systematic approach of GA to solve the inverse problem was proposed and implemented. The results showed that the GA method eliminated complex calculations and is proved to be an effective tool in the calculation of the coalescences parameters. An integrated optimisation-simulation algorithm was developed to inversely estimate the coalescence parameters and was applied to hydrodynamics simulation based on droplets population balance model in two-phase liquid–liquid flow for two systems. The algorithm allows for the calculation of the derivatives of the solution with respect to the empirical coefficients of the model. As soon as the parametric derivatives are known, an efficient optimization genetic algorithm method can be used to minimize the difference between the observed and the numerical results, that is, the unknown empirical parameters of

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the model can be extracted from the available experimental data. One hundred iterations of the genetic algorithm are required to identify the unknown coefficients. The coefficients identified on the basis of one set of experimental data can be used to predict the behaviour of these two systems under different operation conditions. The proposed algorithm also provides information about sensitivity of the solution to the parameters of the model; this information can be used to decide which parameters have to be identified with best solution. At present, this simulation tool allows an interpretation and a comprehension of the various phenomena implied in the operation of liquid-liquid extraction columns essentially with the dimensioning of these apparatus.

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References [1] Launder B.E. and Spalding D.B., in ‘Mathematical Models of Turbulence’, Academic Press London and New York, 1972; [2] Abdelmaguid A.M. and Spalding D.B., in ‘Turbulent Flows and Heat Transfer in Pipes with Buoyancy Effects’, J. Fluid Mech, 94, 383, 1979; [3] Davidov, B. I., in ‘On the Statistical Dynamics of an Incompressible Fluid’, Dokh. (English version), AMSSSR, 136, p47, 1961; [4] Launder B.E. and Spalding D.B., in ‘The Numerical Computational of Turbulent Flows’, Computer Methods in Applied Mechanics and Engineering, 3, 269, 1974; [5] Escudier, M. P., in ‘The Distribution of Mixing Length in Turbulent Flows Near Walls’, Imperial College, Heat transfer Section, London, 1966; [6] Gosman, A. D., Idriah, F. J. K., in ‚Ageneral ComputerProgram for Two Dimensional Turbulent Recirculation Flow’, Imperial College, London, 1976; [7] Patankar S.V., in ‘Numerical Heat Transfer and Fluid Flow’, Series in Computational Methods in Mechanics and Thermal Sciences, 1980; [8] Hinze J.O., in ‘Fundamentals of the Hydrodynamic Mechanism of Splitting up in the Dispersion Processes’, AIChE J. 1, 189, 1955. [9] Coulaloglou C.A. and Tavlarides L.L., in ‘Description of Interaction Processes in Agitated Liquid-Liquid Dispersions’, Chem. Eng. Sci., 32, 213, 1977. [10] Narsimhan G.D., Nejfelt G., Ramkrishna D., in ‘Breakage Functions of Droplets in Agitated Liquid-Liquid Dispersions’, AIChE. J., 30, 457, 1984. [11] Tsouris C. and Tavlarides L. L., in ‘Breakage and Coalescence Models for Drops in Turbulent Dispersions’, AIChE. J., 40, 395, 1994. [12] Shinnar R., in ‘On the Behaviour of Liquid Dispersions in Mixing Vessels’, J. Fluid Mech., 10, 259, 1961. [13] Chesters A. K., in ‘The Modelling of Coalescence Processes in Fluid-Fluid Dispersions‘, Trans IchemE, vol. 69, Part A, p259, July 1991. [14] Sawant S. B. and Sikdar S. K., in ‘Hydrodynamics and Mass Transfer in Two-Phase Aqueous Extraction Using Spray Columns’, Biotechnology and Bioengineering, vol. 36, pp 109-115, 1990. [15] Al Khani S. D., Gourdon C., Casamatta G., in ‘Simulation of Hydrodynamics and Mass Transfer of a Disks and a Pulsed Column’, Ind. Eng. Chem. Res., 27, 329-333, 1988.

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[16] Perry, R. H. and Chilton C. H., Chemical Engineers’ Handbook, 5-47, Fifth Edition, Mc Graw Hill, Kogakusha Ltd, 1971; [17] Hayworth C. B. and Treybal R. E., Ind. Eng. Chem., 42, p1174, 1950; [18] Bond W. N. and Newton D.A., Phil Mag., 5, Series 7, 794, 1928; [19] Vignes, A., in ‘Hydrodynamique des dispersions’, Génie Chimique, 93, 129, 1965; [20] Kagan, S. Z., Kovalev, Y. N., Zakharychev, A. P., Theor. Osn. Chem. Eng., Moscow, 7, p 514, 1973 ; [21] Harkins, W. D. and Browns F. E., J. Am. Chem. Soc., 41, p 499, 1919; [22] Slater, M. J., Chem. Eng. Tech. , 46, MS 116/74, 1974; [23] Steiner, L. and Hartland S., in ‘Hydrodynamics of Liquid-liquid Spray Columns’, Handbook of Fluid Motion, Butterworths, Ltd, Great Britain, 1983; [24] Meniai A. H. and Kaabouche A., in ‘Simulation of the Interaction of the Dispersed and Continuous Phases in Liquid-liquid Extraction Processes by the Monte Carlo Method’, Proceedings of the 1st Algerian Congress of Process Engineering, Algiers, December, 1996; [25] Collins S. B. and Knudsen J. G., in ‘Drop size distributions produced by turbulent pipe flow of immiscible liquids’, AiChE J, 1970; [26] Hulburt, H., & Katz, S. (1964). Some problems in particle technology. A statistical mechanical formulation. Chem. Eng. Sci., 19, 555-574. [27] Valentas, K. J. , & Amundson, A. R. (1966). Breakage and coalescence in dispersed phase systems. Ind.Eng. Chem. Fundam., 5, 533-542. [28] Alatiqi, I., Aly, G., Mjalli, F., & Mumford, C. J. (1995). Mathematical modeling and steady-state analysis of a Scheibel extraction column. Can. J. Chem. Eng., 73, 523-533. [29] Weinstein, O., Semiat, R., & Lewin, D. R. (1998). Modeling, simulation and control of liquid-liquid extraction columns. Chem. Eng. Sci., 53, 325-339. [30] Tsouris, C., Kirou, V. I., & Tavlarides, L. L. (1994). Drop size distribution and holdup profiles in amultistage extraction column. AIChE J., 40, 407-418. [31] Alopaeus, V., Koskinen, J., Keskinen, K. I., & Majander, J. (2002). Simulation of the population balancesfor liquid-liquid systems in a nonideal stirred tank: Part 2- parameter fitting and the use of multiblockmodel for dense dispersions. Chem. Eng. Sci., 57, 18151825. [32] Kentish, S. E., Stevens, G. W., & Pratt, H. R. C. (1998). Estimation of coalescence and breakage rate constants within a Kühni column. Ind. Eng. Chem. Res., 37, 1099-1106. [33] Steiner, L., Bamelli, M., & Hartland, S. (1999). Simulation of hydrodynamic performance of stirredextraction column. AIChE J., 45, 257-267. [34] Casamatta, G. , & Vogelpohl, A. (1985). Modelling of fluid dynamics and mass transfer in extraction columns. Ger. Chem. Eng., 8, 96-103. [35] Al Khani, S. D., Gourdon, C., & Casamatta, G. (1989). Dynamic and steady-state simulation of hydrodynamics and mass transfer in liquid-liquid extraction column. Chem. Eng. Sci., 44, 1295-1305. [36] Cabassud, M., Gourdon, C., & Casamatta, G. (1990). Single drop break-up in a Kühni column. Chem.Eng. J., 44, 27-41. [37] Modes, G., Bart, H.-J., Rodrigue-Perancho, D., & Broder, D. (1999). Simulation of the fluid dynamics ofsolvent extraction columns from single droplet parameters. Chem. Eng. Tech., 22, 231-236.

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[38] Mohanty, S. (2000). Modeling of liquid-liquid extraction column: A review. Rev. Chem. Eng., 16, 199-248. [39] Cauwenberg, V., Degreve, J., & Slater, M. J. (1997). The interaction of solute transfer, contaminants and drop break-up in rotating disc contactors: Part I. Correlation of drop breakage probabilities. Can. J. Chem. Eng., 75, 1046-1055. [40] Colella, D., Vinci, D., Bagatin, R., & Masi, M. (1999). A study on coalescence and breakage mechanisms in three different bubble columns. Chem. Eng. Sci., 54, 47674777. [41] Gelbard, F. , & Seinfeld, J. H. (1978). Numerical solution of the dynamic equation for particulate systems. J. Comput. Phys., 28, 357-375; [42] Hounslow, M. J. (1990). A discretized population balance for continuous systems at a steady state. AIChE J., 36, 106-116; [43] Vanni, M. (1999). Discretization procedure for the breakage equation. AIChE J., 45, 916-919; [44] Ribeiro, L. M., Regueiras, P. F. R., Guimaraes, M. M. L., Madureira, C. M. C., & CruzPintu, J. J. C.(1995). The dynamic behavior of liquid-liquid agitated dispersions-I. The hydrodynamics. Comput.Chem. Eng., 19, 333-343. [45] Gerstlauer, A.,: Herleitung und Reduktion populationsdynamischer Modelle am Beispiel der Fluessig-Fluessig-Extraktion. Fortschritt-Berichte VDI Reihe, 3, 612, 1999; [46] T. Kronberger, A. Ortner, W. Zulehner, H.J. Bart, Computers Chem. Eng. 1995, 19, 639; [47] Casamatta G., « Comportement de la population de gouttes dans une colonne d’extraction : Transport, rupture, coalescence et transfert de matière », Thèse de Docteur ès Sciences, INP Toulouse, (1981) ; [48] Gourdon C., « Les colonnes d’extraction par solvant : Modèles et comportement », Thèse de Docteur ès Sciences, INP Toulouse, (1989) [49] Bardin N., «Simulations et expériences Lagrangiennes d’écoulements diphasiques dans les colonnes pulsées à garnissage disques-couronnes », Thèse de Doctorat, INP Toulouse, (1998) [50] Pratt H.R.C., Glayer R., Roberts N.W., « Liquid-liquid extraction. Part IV : a further study of hold-up in packed columns », Trans.Inst. Chem.Eng., Vol. 31, (1953); [51] Godfrey, J.C. and Slater, M.J., Trans. IchemE, 1991, 69, 130; [52] J. Fang, J.C. Godfrey, Z.Q. Mao, M.J. Slater and C. Gourdon, Chem. Eng. Technol. 1995, 18, 41.; [53] Miyauchi T., Vermeulen T., « Longitudinal dispersion in two-phase continuous-flow operations », Ind. Eng. Chem. Fund., Vol.2 ; pp.113, (1963); [54] R. Gayler, N.W. Roberts, H.R.C. Pratt, Trans. Am. Inst. Chem. Eng. 1953, 31, 57;. [55] H.J. Bart, T. Misek, M.J. Slater, J. Schröter, B. Wachter, Recommended Systems for Liquid Extraction Studies, in Liquid-Liquid Extraction Equipment, J.C. Godfrey and M.J. Slater, (Eds), John Wiley & Sons, London 1994, [56] M. Simon, H.-J. Bart, Chem. Eng.Tech. 2003, 75; [57] Schmidt, S., Simon, M., & Bart, H-J. (2003). TropfenpopulationsmodellierungEinfluss von Stoffsystem und technischen Geometrien. Chemie Ingenieur Technik, 75, 62-67;

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[58] Modes, G., (2000): Grundsatzliche Studie zur Populationsdynamik einer Extraktionskolonne auf Basis von Einzeltropfenuntersuchungen, Dissertation, Shaker Verlag. [59] Coulaloglou, C. A. , & Tavlarides, L. L. (1977). Description of interaction processes in agitated liquidliquid dispersions. Chem. Eng. Sci., 32, 1289-1297. [60] A.M. Kamp, A.K. Chesters, C. Colin, J. Fabre, Int. J. Multiphase Flow 2001, 27, 1363; [61] C. Tsouris, L.L. Tavlarides, AIChE J. 1994, 40, 395 ; [62] Hasseine A., Meniai A-H, Bencheikh Lehocine M., H J Bart, Chem. Eng. Technol., 2005, 28 (5), 552 ; [63] Simon, M., Bart, H.-J., (2001). Experimentelle Untersuchung zur Koaleszenz in Flüssig/Flüssig-Systemen, Chem. Ing. Tech., 73, 988-992. [64] M. Simon, H.-J. Bart, Chem. Eng.Tech. 2003, 75. [65] Attarakih, M. M., Simon, M., Lagar, Lagar G., Schmidt, S. A., Bart, H.-J., (2004). Internal report, Technical University Kaiserslautern; [66] Mahoney, A. W. , & Ramkrishna, D. (2002). Efficient solution of population balances equations with discontinuities by finite elements. Chem. Eng. Sci., 57, 1107-1119; [67] Attarakih, M. M., Bart, H. J., Faqir, N. M., in Proc. European Symp. On Computeraided process Engineering (Eds. A. Kraslawski, I. Turunen), Elsevier, Amsterdam, 2003; [68] Goldberg, D. E., Genetic algorithms in searching, optimization, and machine learning, Addison-Wesley, Reading, MA, 1989; [69] Misek, R. Berger, J. Schröter, Standard Test Systems for Liquid Extraction, The Inst. Chem. Engineers, Rugby, 1978 (http://www.icheme.org/learning/:Feb 2002). [70] H.J. Bart, T. Misek, M.J. Slater, J. Schröter, B. Wachter, Recommended Systems for Liquid Extraction Studies, in Liquid-Liquid Extraction Equipment, J.C. Godfrey and M.J. Slater, (eds), John Wiley & Sons, London 1994. [71] V. Rod, Coll. Czech. Chem. Commun. 1968, 33, 2855. [72] A. Kumar, S. Hartland, Ind. End. Chem. Res. 1996, 35, 2682.

Nomenclature A

DC

Convection/Diffusion coefficient in the discretisized equations Hamaker constant (J) Area of control volume Convective term in discretisized equations Collision frequency constant Coalescence efficiency constant Turbulence (damping) constant Diffusive term in discretisized equations Axial dispersion coefficient (m2/s) Column diameter (m)

DR

Rotor diameter (m)

DS

Stator diameter (m)

Ak

a C C1 C2 CV D Dax

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d eq

Equivalent diameter (m)

d0

Mother drop diameter (m)

d 32

d E e F f G

(,d) z HC

Sauter mean drop diameter (m) Drop diameter (m) Energy, Wall constant in Eqn. 19 (equal to 9 for smooth pipe) Energy of an eddy Frequency of breakage/coalescence (m3/s) Weighing coefficient between 0 and 1 Production rate of energy of turbulence g Break-up frequency (s-1) Compartment height (m)

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h Film thickness (m) h(d1,d2) Collision frequency in unit volume (m3/s)

I k L lm N NI NJ P P(d) Q Re R r S U V Vol V(d) v vr vT x xm y Z z

Intensity of turbulence Kinetic energy (m2/s2) Scale length Mixing length (m) Rotor speed (s-1) Number of lines in the grid space Number of rows in the grid space Pressure (Pa), Distribution density function (m-1) Break-up probability for a drop of diameter d Phase flux (m3/s) Reynolds number Residue of a variable Radius (m) Source term Axial velocity (m/s) Radial velocity (m/s) Volume (m3) Drop volume with size d (m3) Phase velocity (m/s) Relative velocity (m/s) Terminal velocity (m/s) Axial co-ordinate, direction Mean number of daughter droplets distance from wall (m) Column length (m) Height (m)

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494

A.-H. Meniai, A. Hasseine, O. Saouli et al.

Greek Letters

β (d0, d)

Daughter drop size distribution

ε

Dissipation rate of kinetic energy or Mechanical power Dissipation per unit mass Constant equal to 0.12

λ ρ σ η μ κ φ Γ τ

φ

Fluid density [kg/m3] Interfacial tension Breakage/coalescence efficiency Fluid dynamic viscosity [Pa.s] Von Karman constant (0.418) Generalised property Property Diffusivity Shear Stress Volume fraction holdup of the dispersed phase

λ(d1, d2)

Coalescence probability

ω (d1, d2 )

Coalescence rate

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Subscripts Bre Coa c d E, e l t W, w S, s N, n eff P unif en wa rup

Breakage Coalescence Continuous Dispersed East Laminar Turbulent West South, surface North Effective Principal node Uniform Entry Wall rupture

Superscripts * ′

Estimated Fluctuating Mean, averaged

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Modelling of the Interaction of the Dispersed and the Continuous Phases…

Underscripts continuous phase critical Dispersed phase Maximal dispersed phase continuous phase

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c crit d max x y,

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In: Multiphase Flow Research Editors: S. Martin and J.R. Williams, pp. 497-542

ISBN: 978-1-60692-448-8 © 2009 Nova Science Publishers, Inc.

Chapter 9

TWO-PHASE GAS-LIQUID FLOW PROPERTIES IN THE HYDRAULIC JUMP: REVIEW AND PERSPECTIVES Frédéric Murzyn1 and Hubert Chanson2 1

ESTACA Campus Ouest, Parc Universitaire de Laval Changé, Rue Georges Charpak, 53061 Laval Cedex 9, France 2 Department of Civil Engineering, The University of Queensland, QLD 4072, Brisbane Australia

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Abstract Research on multiphase flows has been strongly improved over the last decades. Because of their large fields of interests and applications for chemical, hydraulic, coastal and environmental engineers and researchers, these flows have been strongly investigated. Although they are some promising and powerful numerical models and new computing tools, computations can not always solve all actual practical problems (weather forecast, wave breaking on sandy beach…). The recent and significant developments of experimental techniques such as Particle Imagery Velocimetry (PIV) and conductivity or optical probes have particularly led scientists to physical modeling that provide series of data used to calibrate numerical models. Flows with time and length scales that were not achievable in the past are now studied leading to a better description of physical mechanisms involved in mixing, diffusion and turbulence. Nevertheless, turbulence is still not well understood, particularly in two-phase flows. In the present chapter, we focus on a classical multiphase flow, the hydraulic jump. It occurs in bedrock rivers, downstream of spillways, weirs and dams, and in industrial plants. It characterizes the transition from a supercritical open-channel flow (low-depth and high velocity) to a subcritical motion (deep flow and low velocities). Experimentally, this twophase flow can be easily studied. Furthermore, it involves fundamental physical processes such as air/water mixing and the interaction between turbulence and free surface. This flow contributes to some dissipation of the flow kinetic energy downstream of the impingement point, in a relatively short distance making it useful to minimize flood damages. It is also associated with an increase of turbulence levels and the development of large eddies with implications in terms of scour, erosion and sediment transport. These are some of the reasons that make studies on this flow particularly relevant. Although numerical and analytical studies

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Frédéric Murzyn and Hubert Chanson exist, experimental investigations are still considered as the best way to improve our knowledge. After a brief description of the hydraulic jumps, the first part of this chapter aims to review some historical developments with special regards to the experimental techniques and physical modeling (similitude). In the second part, we describe and discuss the basic properties of the two-phase flow including void fraction, bubble frequency, bubble velocity and bubble size. The free surface and turbulence properties are presented as well. In the last part, we develop some conclusions, perspectives and further measurements that should be undertaken in the future.

Keywords: Hydraulic jump, Two phase flows, Turbulence, Turbulence length and time scales, Froude number, Reynolds number, Void fraction, Free-surface, Bubble frequency

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Notations C Cmax Dt

Void fraction defined as the volume of air per unit volume of mixture Maximum void fraction in the air bubble diffusion layer Turbulent diffusivity (m2/s) of air bubbles in air-water flow

D*

Dimensionless turbulent diffusivity: D =

dmbcl d1 d2 F Fmax Fr Fscan g hc Lc Lf Lg Lr Lxx Lxz lc Nab Q

Mean bubble chord length (m) Upstream flow depth (m) Downstream flow depth (m) Bubble count rate (Hz) or bubble frequency Maximum bubble count rate (Hz) at a given cross-section Upstream Froude number Sampling rate (Hz) Acceleration of gravity: g=9.80m/s2 Channel height (m) Channel length (m) Longitudinal length scale of turbulence (m) for free surface [42] Transversal length scale of turbulence (m) for free surface [42] Length of the roller (m) Auto-correlation length scale (m) in the bubbly flow [12, 13] Transverse air-water integral length scale (m) in the bubbly flow [12, 13] Channel width (m) Number of air bubbles per record Water discharge (m3/s)

Re

Reynolds number: Re =

Rxx Rxz

Normalized auto-correlation function (reference probe) Normalised cross-correlation function between two probe output signals

St

Strouhal number: St =

*

Dt U 1 d1

ρU 1 d1 U 1 d1 = μ υ

Ftoe d 1 U1

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Average air-water interfacial travel time between the two probe sensors (s) Sampling duration (s) Free surface integral time scale of turbulence measured by wire gages (s) Measure of the turbulence level in the air-water flow

Txx

Auto-correlationintegral time scale (s): Txx =

τ = τ ( R xx = 0 )

∫R

xx

499

dτ

τ=0

T0.5 U1 x x1 y y* yCmax yFmax z

Characteristic time lag τ for which Rxx = 0.5 (s) Depth-averaged flow velocity upstream of the hydraulic jump (m/s) Longitudinal distance from the upstream gate (m) Longitudinal distance from the gate to the jump toe (m) Distance measured normal to the bed channel (m) Distance measured normal to the channel bed corresponding to the boundary between the turbulent shear layer and the mixing layer (m) Distance normal to the bed corresponding to C=Cmax (m) Distance normal to the bed corresponding to F=Fmax (m) Transverse distance from the channel centreline (m)

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Greek Symbols δ Δx ΔH η η’ ν μ ρ σ τ τ0.5

Boundary layer thickness (m) Longitudinal distance between probe sensors for dual-tip conductivity probe (m) Head loss (m) Free surface mean level of the jump above the channel bottom (m) Root mean square of the free surface level fluctuation (m) Kinematic viscosity of water (m2/s) Dynamic viscosity of water (Pas) Density of water (kg/m3) Surface tension between air and water (N/m) Time lag (s) Characteristic time lag for which Rxz = 0.5 (Rxz)max

Subscripts 1 2

Upstream flow conditions Downstream flow conditions

Abbreviations PD FD

Partially-Developed inflow conditions Fully-Developed inflow conditions

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1. Introduction 1.1. Multiphase Flows and the Hydraulic Jump

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Multiphase flows are everywhere around us. According to Andrea Prosperetti and Gretar Tryggvason [43], “it is estimated that over half of anything which is produced in a modern industrial society depends to some extent on a multiphase flow process”. By multiphase flows, we consider a mix of at least two phases among liquid, gas and solid. In the environment, mixing of air, water and solid particles (sediment) is particularly encountered in coastal and hydraulic engineering. For instance, when a wave breaks on a beach due to a progressive or abrupt modification of the bottom slope, air is entrapped and sediment may also be swept into the water column leading to a complex mix. Figure 1 shows a wave breaking (plunging breaking) on a beach at North Stradbroke Island (Queensland, Australia). The white upper part of the wave is explained by the large amount of air entrapped by the breaking. Scars and turbulent structures are also clearly shown at the free surface on the lower part of this figure that enhance beach erosion or accretion.

Figure 1. Wave breaking at North Stradbroke Island, Queensland, Australia, June 2007. Photo by Frédéric Murzyn.

In chemical engineering, multiphase flows are involved in flocculation processes with applications to water treatment. The study of chemical mixing is also of primary interest regarding reactor efficiency and diffusion processes. Pollutant transport and dispersion in river streams are among the environmental problems that are linked to multiphase flows as well. In automotive engineering, the mixing of air and fuel (internal combustion engine) is

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also of primary importance to improve engine power and reduce pollution with ecological issues. Foam is a substance that is formed by trapping gas bubbles in a liquid. Their low density makes them particularly suitable for thermal insulation, for flotation devices or packing materials. They can also be used as fire retardant when liquid. Multiphase flows are also implied in clouds and rain formation or in pipelines flows. These examples demonstrate our need to improve knowledge on the topic. Furthermore, these flows imply a wide range of spatial and temporal scales from the millimetric (foams) to geophysical world (cyclone and tropical storms) that require several experimental techniques from the Pitot tube to optical sensors. In the present chapter, our interest is focused on a classical two-phase flow encountered by coastal and hydraulic engineers: the hydraulic jump. When a transition occurs from a supercritical flow (shallow water, high velocity) to a subcritical motion (deep flow, low velocity), a hydraulic jump takes place. Figure 2 presents the typical sketch of a hydraulic jump. Figure 3 shows a hydraulic jump during a laboratory experiments while Figure 4 shows a hydraulic jumps in nature. In most laboratory experiments, the hydraulic jump is formed a short distance downstream of the sluice gate (x1 < 0.5 m, Figure 2) and far upstream from a weir located at the end of the channel. Good controls on the flow rate, weir height and aperture beneath the sluice gate ensure the stability of the toe (limited horizontal oscillations, low Strouhal number) and govern the flow regime. Several minutes are generally allowed to elapse before any measurements to ensure good conditions and avoid unexpected effects. The experimental arrangements may sometimes differ.

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Undershoot gate

Recirculation region

Outer edge of boundary layer

Impingement point y d2

y

δ

d1

Air-water shear layer

V

Boundary layer flow x1

x L air

Figure 2. Sketch of a hydraulic jump with relevant notations.

Indeed, an obstacle can be set on the bottom of the channel instead of a sluice gate. The flow section is reduced and the hydraulic jump appears a short distance downstream of the top of obstacle.

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Figure 3. Hydraulic jump during laboratory experiments at the University of Southampton (School of Engineering and the Environment, Southampton, UK), Flow from left to right, Fr = 3.65, d1 = 0.032 m. Photo by Frédéric Murzyn.

Figure 4. Hydraulic jumps in natural streams in Québec, Canada (Vallée de la Gatineau, Le grand remous, viewed from the right bank, july 2002). Courtesy of Mr and Mrs Chanson.

The hydraulic jump is characterized by a highly turbulent flow with an air-water shear layer and a recirculating area (Figure 2). Macro-scale vortices develop inside and interact with the free surface leading to splashes and droplets formation in the two-phase flow region.

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Beyond the region of turbulence production (downstream of the impingement point), some significant kinetic energy dissipation takes place. In terms of environmental aspects, this property is used, for instance, with low impact structures for river restoration and to minimize flood damages. The main parameter which characterizes the hydraulic jumps is its Froude number defined as:

Fr =

U1

(1)

gd1

where U1 is the inflow velocity (m/s), d1 is the inflow water depth (m) and g is the acceleration of gravity (m/s2). The Froude number is always greater than 1 for hydraulic jumps. Depending on that dimensionless number, different kinds of jumps are referenced including [7, 10, 15, 16]: -

The undular hydraulic jump for 1 < Fr < 1.5 to 4; The breaking jump for larger Froude numbers encompassing the oscillating jump studied by Mossa [35] and the steady jump for large Froude numbers.

The study of Gualtieri and Chanson [28] showed that the air-water flow properties were similar in both steady and strong jump flows. For a horizontal rectangular channel and neglecting boundary friction, the continuity equation leads to Bélanger equation [2, 6, 7, 15]:

)

(

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d2 1 = 1 + 8Fr 2 − 1 d1 2

(2)

where d1 is the inflow water depth (m) at the impingement point and d2 is the downstream flow depth (m) far from the impingement point. The hydraulic jump is mainly used as an energy dissipator [6, 7, 10, 29]. The stronger the jump is, the highest the energy dissipation is. The dimensionless head loss is given by:

(

)

3

ΔH 1 + 8Fr 2 − 3 = d 1 16 1 + 8Fr 2 − 1

(

)

(3)

For steady hydraulic jump, the energy dissipation goes from 45% to 70%. It reaches up to 85% for strongly turbulent jump. Although hydraulic jumps dissipate a large part of the incoming flow energy, the high levels of turbulence in these flows tend to increase scour and erosion at bridge piers or in rivers (Figures 5 and 6). For physical modeling and similitude purposes, the Reynold number (Re) is often used as well to characterize the flow:

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Re =

U1d 1 υ

(4)

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Figure 5. A large flood flow of the Todd river in Alice Springs (NT, Australia), January 2007. Flow from left to right. Courtesy of Sue McMinn.

Figure 6. Hydraulic jump stilling basin and spillway on Chain Lakes dam (Southern Alberta, Canada), June 2005. Flow from top right to bottom left. Courtesy of John Remi.

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Figure 7. River kayaking at Flage (Norway). Courtesy of Øyvind Thiem.

On Figure 5, the current strength is clearly seen by looking at the foot of the road panel. Here, the formation of the hydraulic jump is caused by the sudden and abrupt change on the road profile. Note the brown color of the water which carries solid particles. On Figure 6, the upcoming supercritical flow (right top of the picture) becomes subcritical (lower part). At the bottom of the spillway, a transition occurs leading to a turbulent shear flow with macro-vortices associated with large and rapid fluctuations of the free surface, strong mixing and scars that both enhance mixing and energy dissipation. Note that droplets and splashes can be observed on this figure. When the weather conditions are “good”, hydraulic jumps are also much appreciated by kayakers and surfers. They become funny spots as illustrated by Figures 7 and 8. After strong rains, the flow rate of some rivers strongly increases. Particularly in mountain areas, the river beds are generally made of gravels with strong and sudden flow depth variations. Hydraulic jumps are not rare providing some wonderful spots for funny moments. On Figure 8, a spilling breaker is ridden by a surfer while a second surfer is waiting for the next wave. This picture illustrates once again the strong interest of researchers and engineers for the hydraulic jump: the hydraulic jump may be considered as a steady spilling breaker. As a consequence, the broad spectrum of situations where hydraulic jumps take place has reinforced our need to explore this exciting two-phase flow. Air/water mixing, bubble dynamics, sediment transport, erosion and scour, bubble coalescence, free surface and turbulence interactions and wave breaking processes are among research topics that can be studied through hydraulic jumps.

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Figure 8. Surfers and spilling breaker on Plage du Minou. Viewed from Pointe du Grand Minou (Hameau de Toulbroc'h), March 2004 (Mid flood tide). Photo by Hubert Chanson.

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1.2. Reviews of Historical Works and Experimental Techniques In this part, we aim to review some of the most significant contributions on hydraulic jumps. In the last 50 years, many studies have been undertaken through numerical, analytical and experimental (physical) modeling requiring a wide range of instrumentation. Table 1 summarizes the main experimental studies that have been undertaken on hydraulic jumps in the last 50 years. This review is not exhaustive but encompasses the some significant contributions obtained with different experimental techniques over the last 50 years. Here, we are particularly interested in experimental contributions. Indeed, although some recent developments, numerical and analytical approaches are not yet enough accurate or developed to give exact solutions. The complexity of the flow and the high number of equations to solve make such computations too difficult. The first significant experimental contributions have been made by Rajaratnam [44, 45]. In 1965, he gave a description of the velocity fields in hydraulic jumps using a Pitot-Prandtl tube [45]. He particularly showed the analogy between the wall-jet flow and non-aerated hydraulic jumps. Indeed, he observed the development of a boundary layer next to the bed characterized by a rapid increase of the velocity over a thin layer above the channel bottom. Then, a gradual decrease was measured. These results followed a first extensive study made by Rajaratnam in 1962 where conductivity probes were used to present some basics results on bubbly flow properties such as the void fraction [44]. Ten years later, Resch and Leutheusser [46, 47] have probably brought one of the most important contribution (to date) on the topic. Their results were obtained with conical hot-film

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probes in the bubbly flow region. Resch and Leutheusser have mainly showed that the air entrainment, momentum transfer and energy dissipation processes were strongly affected by the incoming flow conditions. This point is fundamental and means that the quality of the results depends on the experimental conditions. As a consequence, the relative boundary layer thickness (δ/d1) developing upstream of the impingement point is a key-point that affect the bubbly flow. Depending on this relative boundary layer thickness, the inflow conditions are either partially developed (PD) or fully developed (FD) (Table 1). Murzyn et al. [41] gave an estimation of their boundary layer thickness developing over a flat bed made of PVC. They found that 0.18d1 < δ < 0.36d1 which corresponds to partially-developed (PD) inflow conditions. In 1981, Babb and Aus [1] used conical hot film probes which sizes were smaller than those of Resch and Leutheusser. This point is of interest. Indeed, intrusive probes may disturb the flow. In hydraulic jumps, this is particularly true in the upper part of the flow where negative horizontal velocities are observed. Thus, the disturbances caused by the probes propagate upstream leading to some possible significant flow modifications. Thus, intrusive probes must be as small as possible to minimize such disturbances and optimize the accuracy of the results. Babb and Aus investigated the movement of air bubbles in and out of the jump for one experimental condition (Fr = 6). They found that large bubbles were mostly located near the impingement point. Their lifetime was very short because of the turbulence, shear and buoyancy. In the last years, the improvement of these experimental techniques in terms of response times, spatial and temporal resolution or data analysis procedures has led to an extraordinary number of studies with either conductivity and optical probes, wire gages or flow visualizations [5, 11, 12, 13, 14, 15, 16, 19, 23, 28, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 48]. Since the beginning of the 90’s, the most important contributions are probably those of Chanson [5, 11, 12, 13, 14, 15, 16], Chanson and Brattberg [19], Gualtieri and Chanson [28], Mossa [35, 36], Kucukali and Chanson [31], Mouazé et al. [37] or Murzyn et al. [41, 42]. Other contributions include [23, 31, 38, 39, 40]. They have brought new developments leading to a better understanding of the physical mechanisms involved in hydraulic jumps. Nevertheless, this is not enough. Chanson [5], Chanson and Brattberg [19], Chanson and Toombes [24] have preferentially used conductivity probes (single or dual tip conductivity probes) which have been manufactured, tested and used at the University of Queensland with a range of sensor size from 0.025 mm to 0.35 mm including 0.1 mm and 0.25 mm. These phase-detection probes are designed to pierce bubbles. Based upon the difference of electrical resistivity between air and water, they are well-adapted for bubbly flows such as hydraulic jumps. Figure 9 presents a sketch of a single-tip probe. For instance, the study of Chanson and Brattberg [18] used a 0.025 mm sensor size. When the sensitive part of the probe is in water, current flows between the tip and the supporting metal. Then, an output voltage is collected. It becomes nearly null when the tip is in air. Because the output voltage is subject to some fluctuations caused by dust flowing in water, a single threshold technique is applied on the output signal to define time lags corresponding to air and water [31, 38].

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air bubble

probe sensor piercing a bubble

external electrode (stainless steel) inner electrode (Platinum)

Flow direction

insulant (Araldite epoxy)

Figure 9. Sketch of the single-tip conductivity probe (Left: bubble piercing, Right: sketch of the tip).

Data analysis on this voltage provides basic information on the bubbly flow such as the void fraction (C), the bubble frequency (F) and bubble size (dmbcl). The main strength of these intrusive sensors concerns their size (0.35 mm or 0.25 mm). Thus, they are enough accurate to detect very small bubbles (down to the size of the sensor). Their dynamic response time is high (less than 10μs). This makes them accurate and suitable for measurements in highly turbulent and fluctuating flows such as hydraulic jumps. Chanson [5, 12] first detailed the air-water mixing processes providing pertinent information on void fraction, bubble frequency, bubble size and bubble velocity downstream of the impingement point. His main conclusions concern:

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-

The void fraction (C): in the turbulent shear layer, C satisfies a diffusion equation. Then, the vertical profiles of C fit a gaussian distribution given by:

⎛ ⎛ y − yC ⎞2 ⎞ max ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜⎝ d 1 ⎠ ⎟ C = C max exp⎜ − ⎜ * ⎛ x − x1 ⎞ ⎟ ⎜⎜ 4D ⎜⎜ d ⎟⎟ ⎟⎟ ⎝ 1 ⎠⎠ ⎝ -

-

-

(5)

The bubble frequency (F): in the bubbly flow region, the vertical profiles of F exhibit two distinctive peaks. The major peak is found to be in the turbulent shear layer while the second minor peak is mostly found in the mixing layer ; The vertical positions of the maximum of void fraction (yCmax/d1) and bubble frequency (yFmax/d1): they do not coincide showing two competitive diffusion processes ; The importance of scale effects: the dynamic similarity (Froude similitude) is important and some scale effects may arise for largest Reynolds numbers (Re > 105).

When a dual-tip conductivity probe is used (Figure 10), further information can be obtained such as the bubble velocity (V), turbulence levels (Tu) which correspond to the fluctuations of the air-water interfacial velocity and some time (Txx) and length (Lxx, Lxz) scales of turbulence [12, 13] representing the turbulent structures of the flow. New results have been discussed by Kucukali and Chanson [31] and Murzyn and Chanson [38, 39, 40]. The turbulence length and time scales bring new information on the physical mechanisms involved in such flows that may be useful to calibrate numerical models (mesh size, spatial and temporal resolutions…). They will be discussed in the present chapter.

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Figure 10. Dual-tip conductivity probe at the University of Queensland. The inner electrode diameter is 0.25 mm and the longitudinal distance between both tips is Δx = 7 mm. Flow from right to left, d1 = 0.018 m, x1 = 0.75 m, Fr = 8.45, U1 = 3.6 m/s.

Table 1. Most significant experimental investigations of hydraulic jumps

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References

Fr

Re

Measurement techniques Conductivity probes

Rajaratnam 1962 [44]

2.68 to 8.72 34000 to 110500

Rajaratnam 1965 [45]

2.68 to 9.78 52500 to 128000

Resch and Leutheusser 1972 [46, 47]

2.98 to 8.04 33360 to 71760 Conical hot-film probe

Babb and Aus 1981 [1]

6.00

122850

Prandtl type Pitot static tube

Conical hot-film probe

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Comments 1/8 inch diameter copper tube leaving a length of 1/64 inch bare at the bottom and fitted inside a brass tube leaving a conducting ring of 1/64 inch in length at the bottom 3 mm external diameter, hemispherical head 0.6 mm sensor size Partially and Fullydeveloped inflow conditions 0.4 mm sensor size Partially developed inflow conditions

510

Frédéric Murzyn and Hubert Chanson Table 1. Continued

References

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Chanson 1995 [5]

Measurement techniques 5.10 to 8.60 39500 to 63800 Pitot tube + single and dual-tip conductivity probe Fr

Re

Mossa and Tolve 1998 [36] Chanson and Brattberg 2000 [19]

6.42 to 7.30 57300 to 58000 Video camera CCD

Waniewski et al 2001 [48] Liu et al 2004 [33]

11.5 to 19.3 10940 to 16680 Phase Doppler Anemometry

Murzyn et al 2005 [41], 2007 [42]

2.00 to 4.80 45990 to 88500 Optical fibre probes (double and single tips) Wire gages

Lennon and Hill 2006 [32] Chanson 2006 [5]

1.40 to 3.00 23930 to 27940 Single camera Particle Image Velocimetry 5.00 to 8.10 31500 to 51000 Single-tip (Estimation) conductivity probes Pitot tube (3.3 mm external diameter) 4.60 to 8.60 25000 to 98000 Single-tip conductivity probes

Chanson 2007 [12, 13]

Gualtieri and Chanson 2008 [28]

6.33 to 8.48 36120 to 48580 Pitot tube + dual-tip conductivity probe

2.00 to 3.32 86100 to 147680

Comments Pitot tube: 3.3 mm external diameter Conductivity probe: single tip (0.35 mm inner electrode) Partially-developed inflow conditions CCD 200,000 pixels Partially-developed inflow conditions Dual-tip: 25 μm inner electrode Partially-developed inflow conditions Resolution: 1μm

Micro Acoustic 1 mm diameter Doppler Velocimeter conical sapphire tip

5.20 to 14.3 24680 to 58000 Single-tip conductivity probe

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0.010 mm sensor size, 1 mm tip spacing Sampling rate up to 1 MHz Two thin wire: 1 mm apart Partially-developed inflow conditions

Inner electrode: 0.35 mm in diameter Partially-developed inflow conditions Inner electrode: 0.35 mm in diameter Partially-developed inflow conditions Inner electrode: 0.35 mm in diameter Partially-developed inflow conditions

Two-Phase Gas-Liquid Flow Properties in the Hydraulic Jump

511

Table 1. Continued References

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Murzyn and Chanson 2007 [38], 2008 [39, 40]

Measurement Comments techniques 5.10 to 8.30 38550 to 64100 Dual-tip conductivity 0.25 mm inner probes electrode, 7 mm tip spacing Partially-developed inflow conditions Fr

Re

Figure 11. Optical fibre probe in bubbly flow. The presence of air or water is detected at the ends of each two parallel 10 μm diameter optical fibre probes, 1 mm apart and situated in the circle (although non visible). The fibres extend 5 mm beyond the end of 25 mm long, 0.8 mm diameter cylindrical supports. Photo by Frédéric Murzyn.

In 2005, Murzyn et al. [41] published their results on hydraulic jumps obtained with optical fibre probes. This is an intrusive technique in which tips are also designed to pierce bubbles. The difference compared to conductivity probes is that, in this case, the detection of air or water is based upon the difference of the refractive index between both media. This is a robust and highly accurate technique as the sensor tip size is only 0.01 mm (Figure 11). Bubbles size down to 10 μm can thus be detected. Furthermore, the response time of this probe is less than 1μs making the optical probe well-adapted to scan rapidly-changing and highly turbulent two-phase flows. Their results showed that the data were consistent down to C = 0.001 or better. This is certainly better than for conductivity probes. In the mixing layer, they also found that all vertical void fraction profile fit an error function whatever the Froude number is. Coupled with the results of Chanson [11], a complete description of the vertical

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void fraction profiles is available in the turbulent shear layer as well as on the mixing layer for a wide range of Froude numbers (2 < Fr < 8.5). Murzyn et al. [42] have also investigated the free surface motion using two home-made thin wire gages (diameter = 0.05 mm, 1 mm apart). New description of the free surface dynamics has been made in order to investigate the free surface interaction with turbulence. This part may be considered as independent of the bubbly flow. Nevertheless, the free surface behavior strongly influences the flow dynamics. Thus, it will be discussed in this chapter as well. Concerning the free surface levels, their results pointed out a peak of turbulent fluctuations in the first half of the roller where turbulence production is the most intense. Downstream of this maximum, turbulence levels regularly decreased to reach smallest levels. Furthermore, using a correlation technique, typical length scales have also been estimated to be smaller than 5d1 depicting the turbulent structures developing at the free surface and indicating that the upstream conditions have some influence on the predominant scales of the flow. Other contributions were obtained in different ways with different goals. They refer either to non intrusive techniques such as PIV or LDV or to other intrusive techniques that were not fully satisfying. A relevant contribution has been made by Mossa and Tolve [36]. Using a video camera, they proposed to analyze the air concentration in hydraulic jumps through image processing. Their non invasive technique was able not only able to evaluate air concentration successfully but also to visualize coherent structures of turbulence. Nevertheless, they were limited in terms of experimental conditions. Indeed, only three Froude numbers (in the same range) were studied. Anyway, their technique seemed to be very promising and would need further investigations.

Figure 12. LDV measurements in hydraulic jumps at the University of Caen (France) in 2002. Flow from right to left. The hydraulic jump is formed by a round-shaped obstacle situated on the bottom of the channel. The four Laser beams (two green and two blue) converge at the measurement point. With this 2D system, both horizontal and vertical components of the velocity are simultaneously recorded.

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Following the development of new experimental techniques in fluid mechanics such as Laser Induced Fluorescence (LIF) or Particle Tracking Velocimetry (PTV), Laser Doppler Velocimetry (LDV) and Particle Imagery Velocimetry (PIV) have really been important over the ten past years. For instance, figure 12 shows LDV measurements in hydraulic jumps (upstream part of the jump). PIV systems have been for all kind of applications including in-situ measurements. Theoretically, this is a non intrusive technique although it is now used in natural environments such as seas or oceans. It offers many advantages such as high spatial and temporal resolution and 2D/3D instantaneous measurements of velocity fields. As a non invasive optical technique (in laboratory experiments), it is particularly wellsuited for monophase flows. Indeed, the measurement system consists on a thin Laser sheet lighting a region of the flow. Positions of particles situated in this Laser sheet are recorded. Correlation analysis of these particle positions between different images leads to the corresponding velocity vectors. Actually, this is probably the most powerful technique for flow velocity, vorticity and turbulence measurements in liquid or gas flows. Nevertheless, recent studies have started on two-phase flows and particularly on hydraulic jumps (but limited to low Froude numbers). Lennon and Hill [32] tended to scan the flow in hydraulic jumps using Particle Imagery Velocimetry (PIV) system. The technique is necessarily quite limited in hydraulic jumps with high Froude number because of the large amount of bubbles that disturb the optical path of the Laser sheet. Thus, their experimental works were limited to Fr = 1.37, 1.65 and 3.0. In the two first cases, the jumps were undular while the last one has a low rate of air entrainment. Nevertheless, they obtained some plots of mean velocities and vorticity. This first approach is interesting but seems to be limited to low-aerated flows. Indeed, they argued that “the problem with very bubbly flows, however, is that the bubbles distort the optical rays between the image and the acquisition plane. Despite the relatively short optical path in the present experiments, a high bubble fraction will blur the images of the illuminated seed particles resulting in poor image correlations. As such, optical and methods such as PIV and LDV will have the same difficulties as acoustic and thermal methods when it comes to turbulence measurements in a highly aerated roller”. Liu et al. [33] used a microADV to measure the flow velocity in free hydraulic jumps with Froude numbers of 2.0, 2.5 and 3.32. They found that this acoustic method was not accurate enough due to boundary effects that strongly increase the relative error on velocity measurements. Compared to Prandtl tube, the error on mean velocity measured by the ADV increased linearly with the air concentration at very low void fraction. Matos et al. [34] found that accuracy of ADV is limited in two-phase flows when void fraction exceeds 8 %. Thus, it is reasonable to think that ADV are not well-suited for studies in hydraulic jumps with large Froude numbers. This review of the most significant experimental contributions on hydraulic jumps leads to different conclusions. Although some new powerful techniques are available for researchers and scientists such as PIV, LDV or ADV, it is still believed that the most accurate techniques are conductivity and optical probes. These intrusive methods have proven their robustness in terms of response time and space-time resolutions. Particularly, they were used by Rajaratnam [44], Chanson [5, 11, 12, 13] and Murzyn et al. [41]. Furthermore, the small size of tips limits weak effects. The Reynolds number associated with the flow past the probe is low. For instance, it was less than 30 in the measurements of Murzyn et al. [41]. This

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contributes to minimize uncertainty on measurements. Optical probes can measure bubble sizes down to 10 μm and void fraction down to 0.001 or better. Concerning the data acquisition, attention must be focused on duration and sampling rate to ensure their quality. For conductivity probes, Chanson [12, 14] performed a sensitivity analysis on the effects of sampling duration Tscan and sampling rate Fscan on void fraction and bubble count rate in hydraulic jumps. The sampling duration was selected within the range 0.7 s < Tscan < 300 s and the sampling rate was between 600 Hz < Fscan < 80 kHz. First, the data showed that the sampling rate had almost no effect on the void fraction for a given sampling duration. However, the bubble count rate was underestimated for sampling rates below 5 to 8 kHz. Second, the sampling duration had little effect on both void fraction and bubble count rate for scan periods longer than 30 to 40 s. Then, he recommended a sampling duration of 45 s and a sampling rate of 20 kHz. For optical probes, Murzyn et al. [41] acquired their data during a maximum sampling duration of 120 seconds. Nevertheless, as soon as 10000 bubbles were recorded, data acquisition stopped. To date, these experimental conditions are supposed to be large enough.

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1.3. Physical Modeling of the Hydraulic Jump Hydraulic jumps are commonly encountered in natural streams. This complex two-phase flow always requires more and more investigations. Despite numerous studies over the last decades, there is still a lack of knowledge on the physical mechanisms involved in diffusion, turbulence and mixing processes occurring in such flows. Thus, developments of numerical codes, analytical methods and experimental works are fundamental to improve our knowledge on the flow dynamics. Nevertheless, analytical and numerical studies of multiphase flows are quite difficult due to the large amount of relevant equations to solve. Furthermore, many interactions take place between bubbles, droplets, particles, free surface, turbulence that make analytical methods quite limited. The same complexity often imposes reduced descriptions (averaged equations, basic hypothesis) for numerical modeling which can then not be fully successful and satisfying. Experimental investigations are thus required to help numerical modelers to calibrate their codes with empirical data. This is the strong interaction between numerical and experimental studies that will lead research to a better knowledge of the physical mechanisms involved in these flows. Experimental investigations are numerous and many sets of data are available in the literature regarding the two-phase flow properties (void fraction, bubble frequency, bubble velocity…). These were mainly obtained using phase-detection probes (optical fibre probes or conductivity probes). The accuracy of these measurements, the results and their meanings for natural flows are not only linked to the data acquisition parameters but also to the experimental conditions (similitude). Data interpretation (extrapolation) to natural streams requires some similitude criterion that must be achieved (geometric, kinematic or dynamic similitude). Generally-speaking, two-phase flow laboratory experiments are based on a geometric similitude. This means that model and prototype scales are geometrically similar. Note that dynamic similitude is sometimes used as well. In fluid mechanics, the governing equations for fluid flow motion are known as the Navier-Stokes equations. In a free surface open-channel flow, gravity effect can not be neglected. The corresponding Navier-Stokes equations are then given by:

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du 1 ∂p + υΔu =− dt ρ ∂x dv 1 ∂p =− + υΔv ρ ∂y dt dw 1 ∂p =− − g + υΔw ρ ∂z dt

(6)

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Where (u, v, w) are the velocity components (m/s), ρ the density of water (kg/m3), p the pressure (Pa), g the acceleration of gravity (m/s2), (x, y, z) the coordinate axis (z positive upward), ν the kinematic viscosity of water (m2/s). To date, these nonlinear differential equations remain still unsolved. Indeed, turbulent flows are characterized by random processes that can not be exactly modeled unless some restrictive hypothesis. To date, only approximated solutions can be proposed. That is one of reason which explains why weather forecasts become meaningless after more than 1 week. This is partially due to random and unpredictable phenomena that may occur between forecasts and reality. It is not unbelievable to think that the Navier-Stokes equation will remain unsolved for a long time. This partially explains the limited number of numerical studies on hydraulic jumps compared to experimental investigations: modeling turbulence is difficult but modeling turbulence of two-phase flows becomes nearly impossible. Equations (6) are not dimensionless. To make them so, let us introduce new dimensionless variables:

x+ =

x y z ; y+ = ; z+ = d1 d1 d1

u+ =

u v w ; v+ = ; w+ = U1 U1 U1

t+ =

(7)

U1 p t ; p+ = 2 d1 ρU 1

Then, let us introduce them in equations (6). We obtain new dimensionless Navier-Stokes equations given by:

∂p + du + 1 =− + + Δu + + Re ∂x dt + + dv 1 ∂p =− + + Δv + + Re dt ∂y

(8)

dw + 1 ∂p + 1 = − − + Δw + + + Fr Re dt ∂z To ensure similarity between model (laboratory) and prototype (at scale), one must ensure that Navier-Stokes equations are similar in both cases. Indeed, same equations will

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have same solutions. For a free-surface open-channel flow (such as the hydraulic jump), this is achieved when Froude and Reynolds numbers for model and prototype are equal (Eq. 8). Theoretically, this means that Froude and Reynolds similitude must be simultaneously achieved. Unless rare cases, they can not be simultaneously achieved. Nevertheless, for a free surface flow, the Reynolds condition may be eliminated when diffusion process can be neglected (Re >> 1). The dimensional analysis proposed by Chanson [12] leads to a similar conclusion. Considering a hydraulic jump in a horizontal, rectangular channel, it is supposed that all flow properties (C, F, U, dmbcl,…) depend on the fluid and experimental set-up properties. That is:

C, F, U, d mbcl ,... = F1 (x , y, z, d 1 , δ, υ air , υ water , ρ air , ρ water ,...)

(9)

where C is the void fraction, F the bubble frequency, V the velocity, dmbcl the mean bubble chord length, x, y and z the coordinates, νair and νwater the kinematic viscosity of air and water respectively, ρair and ρwater the density of air and water respectively. δ is the boundary layer thickness of the inflow (defining partially or fully-developed inflow conditions). Neglecting compressibility effects, one can ignore ρair and νair such as equation (9) may be expressed in dimensionless terms:

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C,

⎛ x − x 1 y z U 1 U 1d 1 ⎞ Fd1 U d mbcl , , ,... = F2 ⎜ , , , , ,... ⎟ ⎜ ⎟ V1 d d d d υ gd 1 gd 1 1 1 1 ⎝ 1 ⎠

(10)

Figure 13. Hydraulic jump at the University of Queensland, flow from left to right, d1 = 0.018 m, x1 = 0.75 m, Fr = 5, Re = 37800, U1 = 2.1 m/s, PD inflow conditions. Photo by Hubert Chanson.

In the right hand side of equation (10), Froude and Reynolds numbers appear in fourth and fifth positions respectively. Chanson [12] found that significant scale effects in terms of

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air-water flows properties (void fraction, bubble count rate and bubble chord time distributions) may arise if only a Froude similitude is used. Thus, attention must be drawn on Reynolds number as well. Recently, Gualtieri and Chanson [28] analyzed Froude number effect on air entrainment in the hydraulic jumps with Froude numbers up to 14.3. They showed that aeration properties are enhanced by higher Froude numbers. Figures 13 and 14 show some laboratory experiments on hydraulic jumps at the University of Queensland. The channel was 3.2 m in length, 0.50 m in width and 0.45 m in height.

Figure 14. Hydraulic jump at the University of Queensland, flow from left to right, d1 = 0.018 m, x1 = 0.75 m, Fr = 7.9, Re = 59400, U1 = 3.3 m/s, PD inflow conditions. Photo by Hubert Chanson.

Figure 13 corresponds to a Froude number of 5 whereas Figure 14 deals with a Froude number of 7.9. The free surface motion is more turbulent in the second case and flow aeration is much more important as well as found by Gualtieri and Chanson [28]. In the second part of this chapter, we aim to describe the most significant properties of the two-phase flow. Our interest is focused on void fraction, bubble count rate (bubble frequency), bubble velocity and mean bubble chord length. The present results are presented and compared with other experimental studies (mostly Gualtieri and Chanson [28], Kucukali and Chanson [31], Chanson [5, 12], Chanson and Brattberg [19] and Murzyn et al. [41, 42]).

2. Bubbly Flow Properties The first basic property concerns the void fraction. Figure 15 presents three vertical dimensionless distribution of void fraction in hydraulic jumps with Froude numbers of 5.1, 7.6 and 8.3 with partially-developed inflow conditions. They correspond to the same relative distance downstream of the impingement point, (x-x1)/d1 = 12.5. Theoretical results are

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plotted for the highest Froude number according to equation (5). Note that they are limited to the turbulent shear layer. 10 Fr=5.1 Fr=7.6 Fr=8.3 Fr=8.3 (Theory)

8

6

y/d1 4

2

0 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

C

Figure 15. Dimensionless distribution of void fraction at (x-x1)/d1 = 12.5 for 3 different Froude numbers. Data from Murzyn and Chanson [38]. Comparison with theoretical solution of the diffusion

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equation (Eq. 5)

⎛ ⎛ y − yC ⎞2 ⎞ max ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜⎝ d 1 ⎠ ⎟ ⎜ C = C max − ⎜ * ⎛ x − x1 ⎞ ⎟ ⎜⎜ 4D ⎜⎜ d ⎟⎟ ⎟⎟ ⎝ 1 ⎠⎠ ⎝

for Fr = 8.3.

For all experimental conditions, this figure shows a rapid increase of the void fraction from the bottom up to a well-defined position called yCmax/d1 where C = Cmax. The corresponding peak of void fraction is clearly marked. Then, a slight decrease is observed. In the upper part of the flow (mixing layer or recirculation region), the void fraction increases again up to 100 % which corresponds to air. First, we note that theoretical solution of the diffusion equation is in agreement with the experimental data in the turbulent shear layer. At a given position (x-x1)/d1 downstream of the toe, it is evident that Cmax changes with the Froude number. Whereas it is nearly 30% for Fr = 8.3, it is only 6.3% for Fr = 5.1. The highest values of Cmax are found for the largest Froude numbers. This is confirmed by Figure 16 where Cmax is plotted against (x-x1)/d1. At a fixed position downstream of the impingement point, it is obvious that the lower aeration correspond to the smallest Froude number. For a given Froude number, this Figure 16 also indicates that void fraction decreases when the distance to the toe increases. Figure 17 describes the position of the maximum void fraction as a function of the distance to the toe. The vertical position yCmax/d1 linearly increases with increasing the distance from the toe. Smaller shear stresses in the flow and larger buoyancy effects explain this trend. The best fit is given by equation (11):

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0,6 Fr=6.51 (2007-3) Fr=8.37 (2007-3) Fr=10.76 (2007-3) Fr=14.27 (2007-3) Fr=5.1 (2007-1) Fr=7.6 (2007-1) Fr=8.3 (2007-1) Fr=4.7 (2007-2) Fr=5.8 (2007-2) Fr=6.9 (2007-2)

0,5

0,4

Cmax 0,3 0,2

0,1

0 0

10

20

30

40

50

60

(x-x1)/d1 Figure 16. Dimensionless longitudinal distribution of maximum void fraction in the shear layer for different Froude numbers. 2007-1 refers to the experimental works of Murzyn and Chanson [38], 20072 to those of Kucukali and Chanson [31] and 2007-3 to those of Gualtieri and Chanson [28]. 6

5

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4

yCmax/d1 3 2

Fr=6.51 (2007-3) Fr=8.37 (2007-3) Fr=10.76 (2007-3) Fr=14.27 (2007-3) Fr=5.1 (2007-1) Fr=7.6 (2007-1)

1

Fr=8.3 (2007-1) Fr=4.7 (2007-2) Fr=5.8 (2007-2) Fr=6.9 (2007-2) Linear fit Bes t fit (2000)

0 0

10

20

30

40

50

60

(x-x1)/d1 Figure 17. Dimensionless longitudinal distribution of yCmax/d1 for different Froude numbers.2007-1 refers to the experimental works of Murzyn and Chanson [38], 2007-2 to those of Kucukali and Chanson [31], 2007-3 to those of Gualtieri and Chanson [28] and 2000 to those of Chanson and Brattberg [19].

y C max d1

= 0.0881

x − x1 + 1.134 d1

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(11)

520

Frédéric Murzyn and Hubert Chanson

A comparison is also shown with the best linear fit obtained by Chanson and Brattberg [19]. For this study, measurements were performed with a smaller sensor. The close agreement between all data set would show that the main findings are nearly independent of the instrumentation characteristics. 10 Fr=5.1 Fr=7.6 Fr=8.3 8

6

y/d1 4

2

0 0

20

40

60

80

100

120

140

Frequency (Hz)

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Figure 18. Vertical distribution of bubble count rate at (x-x1)/d1 = 12.5 for 3 different Froude numbers. Data from Murzyn and Chanson [38].

Figure 18 presents vertical distribution of bubble count rate at (x-x1)/d1 = 12.5 for different Froude numbers. It points out that the two distinctive peaks are measured. The first one (the most important) is found in the turbulent shear layer while the second one (minor peak) is at a higher elevation in the recirculating area. At a given position downstream of the toe, these maximum bubble count rates are related to the Froude number. Fmax is only 25.4 Hz for Fr = 5.1 whereas it reaches 123.9 Hz for Fr = 8.3. Dimensionless longitudinal distribution of yFmax/d1 for different Froude numbers is shown on Figure 19. We observe the same linear trend as for yCmax/d1. Nevertheless, the equation of the linear fit differs. Here, it is given by equation (12):

y Fmax d1

= 0.073

x − x1 + 0.6441 d1

(12)

It indicates that the relative position of the maximum bubble count rate above the channel bottom increases with the distance to the toe. The data analysis of several data sets reveals that, for a given Froude number, the dimensionless maximum bubble count rate (Fmaxd1/U1) is a function of the distance to the toe (Figure 20). Furthermore, at a given distance (x-x1)/d1, Fmaxd1/U1 is also dependent on the Froude number. The highest bubble count rate is found for the largest Froude number.

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6 Fr=5.18 (2007-3) Fr=8.37 (2007-3) Fr=10.76 (2007-3) Fr=14.27 (2007-3) Fr=5.1 (2007-1) Fr=7.6 (2007-1)

5

Fr=8.3 (2007-1) Fr=4.7 (2007-2) Fr=5.8 (2007-2) Fr=6.9 (2007-2) Linear fit Bes t fit (2000)

4

yFmax/d1 3 2

1

0 0

10

20

30

40

50

60

(x-x1)/d1 Figure 19. Dimensionless longitudinal distribution of yFmax/d1 for different Froude numbers.

2007-1 refers to the experimental works of Murzyn and Chanson [38], 2007-2 to those of Kucukali and Chanson [31], 2007-3 to those of Gualtieri and Chanson [28] and 2000 to those of Chanson and Brattberg [19].

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0,8 Fr=5.18 (2007-3) Fr=8.37 (2007-3) Fr=10.76 (2007-3) Fr=14.27 (2007-3) Fr=5.1 (2007-1) Fr=7.6 (2007-1) Fr=8.3 (2007-1) Fr=4.7 (2007-2) Fr=5.8 (2007-2)

0,6

Fmaxd1/U1 0,4

0,2

0 0

10

20

30

40

50

60

(x-x1)/d1

Figure 20. Longitudinal distribution of the dimensionless maximum bubble count rate Fmaxd1/U1 in the hydraulic jumps for different Froude numbers.2007-1 refers to experimental works of Murzyn and Chanson [38], 2007-2 to those of Kucukali and Chanson [31] and 2007-3 to those of Gualtieri and Chanson [28].

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A close comparison of Figures 17 and 19 shows that the position of maximum bubble count rate is always below the maximum of air concentration whatever the Froude number is. Figure 21 clearly exhibits this trend. 8 Fr=5.18 (2007-3) Fr= 6.51 (2007-3) Fr=8.37 (2007-3) Fr=10.76 (2007-3) Fr=14.27 (2007-3) Fr=5.1 (2007-1) Fr=7.6 (2007-1) Fr=8.3 (2007-1) yCm ax/d1=yFm ax/d1 Bes t fit (2000)

7 6 5

yFmax/d1 4 3 2 1 0 0

1

2

3

4

5

6

7

8

yCmax/d1 Figure 21. Dimensionless relationship between yCmax/d1 and yFmax/d1 in hydraulic jumps for different Froude numbers.2007-1 refers to the experimental works of Murzyn and Chanson [38], 2007-3 to those of Gualtieri and Chanson [28] and 2000 to those of Chanson and Brattberg [19].

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4 Data (2005) Fit (y/d1y*/d1) 3

y/d1 2

1

0 1E-5

0,0001

0,001

0,01

0,1

1

C Figure 22. Typical void fraction distribution for Fr = 2.4 at (x-x1)/d1 = 8.7. The lower part of the data corresponds to the turbulent shear layer while the upper part is related to the mixing layer.Data 2005 refers to experimental works of Murzyn et al. [41].

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This finding was first observed by Chanson and Brattberg [19]. In 2006, Chanson [12] argued that it could be due to a “double diffusion process where vorticity and air bubbles diffuse at a different rate and in a different manner downstream of the impingement point”. This situation would mean some dissymmetric turbulent shear stress. Indeed, at a given void fraction, it is well-known the maximum bubble count rate is generally found in regions of high shear and large velocity. The distinction between the turbulent shear layer and the recirculating region is obvious when plotting dimensionless void fraction vertical profile Figure 22 presents one classical flow condition obtained by Murzyn et al. [41]. A marked separation is observed at a given elevation y*/d1. In the lower part, C follows equation (5) whereas in the upper part it is best fitted by an error function [41]. Figure 23 shows the position y*/d1 for different experimental conditions investigated by Gualtieri and Chanson [28]. A comparison with data fit of Murzyn et al. [41] is plotted as well showing strong agreements between both studies. Figure 24 describes dimensionless distributions of bubble velocity V/U1 for different Froude numbers and for (x-x1)/d1 < 15. Data of Kucukali and Chanson [31] are plotted and compared with those of Murzyn and Chanson [38]. Considering the boundary condition on the channel bottom (no slip condition), these profiles point out the development of a boundary layer next to the bottom. This thin layer is characterized by a rapid increase of the velocity over a short distance. Then, a maximum is reached which is followed by a decay of the bubble velocity. Although some data scattering (due to the intrusive technique), the thickness of the boundary layer increases gradually with the distance to the toe. These data, compared to the experimental work of Rajaratnam [45] confirmed the similarity of the velocity profiles with those obtained for a wall-jet flow.

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10

8

6

y*/d1 4 Fr=5.18 (2007-3) Fr=6.51 (2007-3) Fr=8.37 (2007-3) Fr=10.76 (2007-3) Fr=14.27 (2007-3) Linear Fit (2005)

2

0 0

10

20

30

40

50

60

(x-x1)/d1 Figure 23. Dimensionless position of the boundary between the turbulent shear layer and mixing layer.2007-3 refers to experimental works of Gualtieri and Chanson [28] and 2005 to those of Murzyn et al. [41].

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Frédéric Murzyn and Hubert Chanson 4 Fr=5.1, (x-x1)/d1=4,17 Fr=5.1, (x-x1)/d1=8,33 Fr=5.1, (x-x1)/d1=12,5 Fr=7,6, (x-x1)/d1=12,5

(2007-1) (2007-1) (2007-1) (2007-1)

Fr=8,3, (x-x1)/d1=12,5 Fr=6,9, (x-x1)/d1=4,17 Fr=6,9, (x-x1)/d1=8,33 Fr=6,9, (x-x1)/d1=12,5

(2007-1) (2007-2) (2007-2) (2007-2)

3

y/d1 2

1

0 0

0,2

0,4

0,6

0,8

1

V/U1 Figure 24. Dimensionless distributions of interfacial velocity V/U1 for (x-x1)/d1 < 15 for different Froude numbers.2007-1 refers to experimental works of Murzyn and Chanson [38] and 2007-2 to those of Kucukali and Chanson [31]. 7 Fr=5,1, (x-x1)/d1=4,17 (2007-1) Fr=5,1, (x-x1)/d1=8,33 (2007-1) Fr=5,1, (x-x1)/d1=12,5 (2007-1) Fr=7,6, (x-x1)/d1=12,5 (2007-1) Fr=7,6, (x-x1)/d1=16,67 (2007-1)

6

Fr=7,6, (x-x1)/d1=25 (2007-1) Fr=8,3, (x-x1)/d1=12,5 (2007-1) Fr=8,3, (x-x1)/d1=16,67 (2007-1) Fr=8,3, (x-x1)/d1=25 (2007-1)

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5 4

y/d1 3 2 1 0 0

2

4

6

8

10

dmbc l (mm) Figure 25. Distribution of mean bubble chord length in hydraulic jumps for different Froude numbers and (x-x1)/d1 < 25.2007-1 refers to experimental works of Murzyn and Chanson [38].

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Figure 25 presents the mean bubble chord length for different Froude numbers and (x-x1)/d1 < 25. Data are those of Murzyn and Chanson [38]. The data indicate that maximum mean bubble chord length do not exceed 10 mm in the turbulent shear layer. This is in agreement with photos, visual observations and videos made during the experiments. Furthermore, the smallest bubbles are found to be closed to the bottom. At a given position (x-x1)/d1, the smallest bubbles were found in the region of higher shear stress. The order of magnitude is in agreement with those observed in previous studies [31, 41].

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3. Free Surface, Sprays and Splashing The free surface has been investigated using different experimental techniques. The most significant contributions regarding its dynamics with acoustic displacement meters are due to Kucukali and Chanson [31] and Murzyn and Chanson [38]. Other studies include Mouazé et al. [37] and Murzyn et al. [42] with wire gages and Chanson [6, 11, 12], Mossa and Tolve [36], Kucukali and Chanson [31] and Murzyn and Chanson [38] for flow visualizations. While acoustic displacement meters and flow visualizations are non invasive, wire gages are intrusive. Nevertheless, their size (wire diameter = 0.05 mm) ensures minimized disturbances. Photographic techniques and conductivity probes were also used to study the free surface [6, 31, 38]. In this part, we aim to present some results that bring information on the air/water interface. This is an important point because its large amplitude (spatial) motions and rapid (temporal) variations affect the mixing. Figure 26 shows the harbour of Fécamp (Seine Maritime, France) during a storm in winter. On the right part of the bottom, the white color of the water demonstrates a large mixing of air and water due to the wave breaking associated with strong turbulence. This spilling breaking may be considered as a moving hydraulic jump. On the picture, other spilling breakers are seen showing important air/sea gas exchanges. While in-situ measurements are quite difficult in such weather conditions, experimental studies are more practical and needed to investigate the physical mechanisms involved in turbulent processes. Figures 27 to 34 are flow visualizations made at the University of Southampton. The flume was 0.3 m in width. The Froude number ranges from 1.98 to 4.82 with 0.021 m < d1 < 0.059 m with upstream velocity between 1.14 m/s and 2.19 m/s. On the left part, zooms on the toe are presented while on the right part, the entire width of the flow is seen. During experiments on hydraulic jumps, oscillations of the front of the jump are often observed with frequency less than 1 hertz. At the University of Southampton, a 10 mm square bar was placed across the floor of the channel 1.2 m downstream of the foot. This improved the stability of the hydraulic jump. According to Mossa and Tolve [36], Chanson [12] and Murzyn and Chanson [38], based on the oscillation frequency of the toe, the Strouhal number is between 0.0038 and 0.013 for Froude numbers between 4.6 and 8.6.

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Frédéric Murzyn and Hubert Chanson

Figure 26. Harbour of Fécamp (Seine Maritime, France) during a storm. Courtesy of Frédéric Malandain.

(27)

(28)

Figures 27-28. Front toe of the hydraulic jump for Fr = 1.98. Flow from top to bottom. Left: zoom on the toe (27), Right: Front toe of the jump (28).Experimental conditions: Fr = 1.98, d1 = 0.059 m, U1 = 1.5 m/s On the left, the width corresponds to approximately 22 cm. On the right, the vertical walls of the flume can be seen: the horizontal image csize orresponds to about 30 cm.

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(29)

527

(30)

Figures 29-30. Front toe of the hydraulic jump for Fr = 2.13. Flow from top to bottom. Left: zoom on the toe (29), Right: Front toe of the jump (30). Experimental conditions: Fr = 2.13, d1 = 0.029 m, U1 = 1.14 m/s On the left, the width corresponds to approximately 22 cm. On the right, the vertical walls of the flume can be seen: the horizontal image size corresponds to about 30 cm.

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(31)

(32)

Figures 31-32. Front toe of the hydraulic jump for Fr = 3.65. Flow from top to bottom. Left: zoom on the toe (31), Right: Front toe of the jump (32).Experimental conditions: Fr = 3.65, d1 = 0.032 m, U1 = 2.05 m/sOn the left, the width corresponds to approximately 22 cm. On the right, the vertical walls of the flume can be seen: the horizontal image size corresponds to about 30 cm.

(33)

(34)

Figures 33-34. Front toe of the hydraulic jump for Fr = 4.82. Flow from top to bottom. Left: zoom on the toe (33), Right: Front toe of the jump (34).Experimental conditions: Fr = 4.82, d1 = 0.021 m, U1 = 2.19 m/sOn the left, the width corresponds to approximately 22 cm. On the right, the vertical walls of the flume can be seen: the horizontal image size corresponds to about 30 cm.

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From figures 27 to 34, Froude number increases. Much more bubbles are observed and the front of the toe becomes more and more turbulent. The aeration increases with Froude number and different length and time scales are measured depending on the inflow conditions. The gate aperture seems to be of primary importance regarding length scales developing at the free surface. Side views tend to confirm this (Figures 13 and 14). 14 12

Fr=3.1 Fr=4.2 Fr=5.3 Fr=6.4 Fr=7.6 Fr=8.5

(2007-1) (2007-1) (2007-1) (2007-1) (2007-1) (2007-1)

Fr=4.7 Fr=5.0 Fr=5.8 Fr=6.9 Fr=8.5

(2007-2) (2007-2) (2007-2) (2007-2) (2007-2)

10 8

η/d1 6 4 2 0 -15

0

15

30

45

60

(x-x1)/d1

Figure 35. Dimensionless free surface levels in hydraulic jumps for different Froude numbers. 2007-1 refers to experimental works of Murzyn and Chanson [38] and 2007-2 to those of Kucukali and Chanson [31].

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1,6 Fr=3.1 Fr=4.2 Fr=5.3 Fr=6.4 Fr=7.6 Fr=8.5 Fr=4.7 Fr=5.0 Fr=5.8 Fr=6.9 Fr=8.5

1,2

η'/d1 0,8

(2007-1) (2007-1) (2007-1) (2007-1) (2007-1) (2007-1) (2007-2) (2007-2) (2007-2) (2007-2) (2007-2)

0,4

0 -15

0

15

30

45

60

(x-x1)/d1

Figure 36. Dimensionless turbulent fluctuations of the free surface in hydraulic jumps for different Froude numbers. 2007-1 refers to experimental works of Murzyn and Chanson [38] and 2007-2 to those of Kucukali and Chanson [31].

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Figure 35 presents the longitudinal dimensionless free surface profiles for Froude numbers up to 8.5. These experimental results correspond to measurements made at the University of Queensland at Brisbane by Kucukali and Chanson [31] and Murzyn and Chanson [38] using ultrasonic displacement meters. For (x-x1)/d1 > 0, a regular increase of the free surface is observed. Then, a constant level is reached (which corresponds to the dissipation area). For the highest Froude numbers, a constant level is not always observed. Indeed, only six acoustic displacement meters were used which was not sufficient enough. Nevertheless, the profiles (shapes) are in agreement with flow observations made during experiments and previous studies [31, 37, 42]. On Figure 36, the turbulent fluctuations of the free surface are plotted for the same experimental conditions. The same shapes are depicted for all Froude numbers. A rapid increase is noticed and a peak of turbulent fluctuations is reached in the first half of the roller. This corresponds to a turbulence production region. Then, a regular decrease is measured indicating a dissipative area. The roller length can be deduced from the mean free surface levels. It corresponds to the length over which the free surface increases. Results are plotted on figure 37 and compared to previous studies made at the University of Southampton by Murzyn et al. [42] and presented by Chanson [10] showing strong agreements. The best linear fit is given showing that the roller length is proportional to the Froude number according to relation (13):

Lr = 6.26(Fr − 1.2) d1

(13)

This relation is obtained considering that air entrainment starts at Fr = 1.2. This hypothesis is meaningful regarding some laboratory experiments [15, 16, 42].

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40 2007-1 2007-2 2007-4

Linear fit Hager et al. [30]

30

Lr/d1 20

10

0 0

2

4

6

8

Fr

Figure 37. Dimensionless length of the roller in hydraulic jumps for different Froude number. 2007-1 refers to experimental works of Murzyn and Chanson [38], 2007-2 to those of Kucukali and Chanson [31], 2007-4 to those of Murzyn et al. [42] and Hager et al. [30] to the correlation of Hager and coworkers as presented by [10].

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Based on a correlation technique, output signals of the wire gages can be analyzed to study the integral length and time scales of the structures developing at the free surface. Figure 38 presents the results regarding the dimensionless longitudinal length scales of turbulence for different Froude numbers. These results were obtained by Murzyn et al. [42] in the context of a research project which aimed to study the strong turbulence developing at the free surface. These length scales correspond to the size of the largest structures developing at the free surface. It is also a measure of the longitudinal length scale of vortical structures advecting air bubbles [12]. The figure shows that longitudinal turbulence length scale grows with the distance to the impingement point. It ranges from less than 0.5d1 to 3d1. It also demonstrates that the upstream aperture has a strong influence on the turbulent structures developing in the jump. The best fit is given by:

Lf x − x1 = 0.06 + 0.35 d1 d1

(14)

Figure 39 describes the dimensionless transverse length scales of turbulence. They are related to the structures developing at the free surface perpendicular to the mean flow direction. They can be seen in Figure 30. The same trend is observed as for Lf/d1. Transverse length scales of turbulence grow with the distance to the toe. They do not exceed more than 2d1 for most of the experimental conditions. They are smaller than Lf/d1 and the best fit is given by:

Lg d1

= 0.11

x − x1 + 0.14 d1

(15)

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3 Fr=1.98 (2007-4) Fr=2.43 (2007-4) Fr=3.65 (2007-4) Fr=4.82 (2007-4) Linear fit

2

Lf/d1 1

0 0

10

20

30

40

(x-x1)/d1 Figure 38. Dimensionless distributions of the streamwise turbulence length scales of the free surface measured by wire gages in hydraulic jumps for different Froude number. 2007-4 refers to experimental works of Murzyn et al. [42].

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5 Fr=1.98 (2007-4) Fr=2.43 (2007-4) Fr=3.65 (2007-4) Fr=4.82 (2007-4) Linear fit

4

3

Lg/d1 2

1

0 0

10

20

30

40

(x-x1)/d1 Figure 39. Dimensionless distributions of the transverse turbulence length scales of the free surface measured by wire gages in hydraulic jumps for different Froude number. 2007-4 refers to experimental works of Murzyn et al. [42]. 6

5

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4

TtU1/d1 3 2 Fr=1.98 Fr=2.13 Fr=3.65 Fr=4.82

1

(2007-4) (2007-4) (2007-4) (2007-4)

0 0

10

20

30

40

(x-x1)/d1 Figure 40. Dimensionless distributions of the turbulence time scales of the free surface measured by wire gages in hydraulic jumps for different Froude number. 2007-4 refers to experimental works of Murzyn et al. [42].

From auto-correlation measurements, integral time scales can be deduced. They represent the “lifetime” of the largest structures developing at the free surface. Figure 40 presents dimensionless integral time scales of turbulence as a function of the distance to the impingement point for different Froude numbers. Contrary to the longitudinal and transverse

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Frédéric Murzyn and Hubert Chanson

length scales of turbulence, a dependence on the Froude number is depicted in Figure 40. The highest time scales are found for the highest Froude numbers. At a given position downstream of the toe, TtU1/d1 increases with Fr. Lastly, TtU1/d1 increases with the distance to the impingement point until a maximum is reached. Then, a slight decrease is observed.

4. Turbulence Information on turbulence properties of the air-water flow can be obtained through optical or conductivity probes. With a dual-tip conductivity probe, correlation analysis can be undertaken to get details on the turbulent structure of the flow. The aim of this part is to present some results deriving from several experiments [12, 13, 23, 31, 38, 39, 40]. We are particularly interested in turbulence intensity and turbulence length and time scales. The turbulence intensity is derived from a cross-correlation analysis between the two tips of the dual-tip conductivity probes. These were aligned either in the streamwise direction or perpendicular to the mean flow. Murzyn and Chanson [38] used dual-tip probes aligned in the streamwise direction and separated by a distance Δx = 7 mm. Data analysis was led on the output signals of the sensors to get some turbulence intensities and integral time scales. From a theoretical point of view, this approach is based on the relative width of the auto and cross correlation functions. The procedure was first presented and detailed by Chanson and Toombes [24, 25]. Further relevant references include Chanson [8, 12, 13], Chanson and Carosi [20, 21, 22] and Carosi and Chanson [4]. Thus, the turbulence intensity was deduced from the shapes of the cross-correlation Rxz and auto-correlation Rxx functions:

τ 0.5 − T0.5 2

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Tu = 0.851

T

2

(16)

where τ0.5 is the time scale for which the normalized cross-correlation function is half of its maximum value such as Rxz(T+τ0.5) = (Rxz)max/2, (Rxz)max is the maximum cross-correlation coefficient for τ=T, T0.5 is the time for which the normalized auto-correlation function equals 0.5 (Figure 41). More details can be found in Murzyn and Chanson [38]. The turbulence intensity characterizes the fluctuations of the interfacial velocity. Typical results are presented on Figure 42 for different experimental conditions. These results show very high levels of turbulence intensity up to 500 %. The most important levels are found close to the impingement point for the highest Froude numbers. For a given Froude number, the turbulence intensity decays with the distance to the toe. Furthermore, the turbulence intensity rapidly increases with the distance to the bottom. Far from the toe, the dimensionless distributions show a homogeneous distribution over the whole water depth. An analysis of the auto-correlation function provides some information on the turbulent time scales (Txx) of the flow. This treatment was applied to the leading tip in order to get Txx. This integral time scale characterizes the streamwise coherence of the two-phase flow. This is also an estimation of the longest longitudinal connection in the air-water flow structure.

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Figure 41. Definition sketch of the auto and cross-correlation functions for a dual-tip phase detection probe [12]. 5 Fr=5,1, (x-x1)/d1=4,17 (2007-1) Fr=5,1, (x-x1)/d1=8,33 (2007-1) Fr=5,1, (x-x1)/d1=12,5 (2007-1) Fr=7,6, (x-x1)/d1=12,5 (2007-1) Fr=7,6, (x-x1)/d1=16,67 (2007-1) Fr=8,3, (x-x1)/d1=12,5 (2007-1)

4

Fr=8,3, (x-x1)/d1=16,67 (2007-1) Fr=6,9, (x-x1)/d1=4,17 (2007-2) Fr=6,9, (x-x1)/d1=8,33 (2007-2) Fr=6,9, (x-x1)/d1=12,5 (2007-2) Fr=6,9, (x-x1)/d1=16,67 (2007-2)

3

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y/d1 2

1

0 0

1

2

3

4

5

Tu Figure 42. Dimensionless distributions of streamwise turbulence intensity Tu in hydraulic jumps for (xx1)/d1 < 20. 2007-1 refers to experimental works of Murzyn and Chanson [38] and 2007-2 to those of Kucukali and Chanson [31].

Figure 43 presents results obtained for different experimental conditions corresponding to different Froude numbers. Results show that integral time scales range from 0 (bottom of the channel) to less than 12 ms. Similar trends are found whatever the experimental conditions are. The integral time scales grow linearly from the bottom of the channel up to the free surface. This suggests that

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the largest structures develop in the turbulent shear layer up to the air-water interface. Far downstream, the vertical profiles tend to be homogeneous with constant integral time scales. 10 Fr=5.1, (x-x1)/d1=4.17 (2007-1) Fr=5.1, (x-x1)/d1=8.33 (2007-1) Fr=5.1, (x-x1)/d1=12.5 (2007-1) Fr=7.6, (x-x1)/d1=12.5 (2007-1) Fr=7.6, (x-x1)/d1=16.67 (2007-1) Fr=7.6, (x-x1)/d1=25 (2007-1)

8

Fr=8.3, (x-x1)/d1=12.5 (2007-1) Fr=8.3, (x-x1)/d1=16.67 (2007-1) Fr=8.3, (x-x1)/d1=25 (2007-1) Fr=6.9, (x-x1)/d1=4.08 (2006) Fr=6.9, (x-x1)/d1=8.16 (2006)

6

y/d1 4

2

0 0

0,003

0,006

0,009

0,012

0,015

Txx (msec) Figure 43. Distributions of auto-correlation integral time scales (Txx) in hydraulic jumps for (x-x1)/d1 < 25. 2007-1 refers to experimental works of Murzyn and Chanson [38] and 2006 to those of Chanson [12].

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6

5

4

y/d1 3 2 Fr=4.7, (x-x1)/d1=4.17 Fr=4.7, (x-x1)/d1=8.33 Fr=4.7, (x-x1)/d1=12.5 Fr=5.8, (x-x1)/d1=8.33 Fr=5.8, (x-x1)/d1=12.5

1

(2007-2) (2007-2) (2007-2) (2007-2) (2007-2)

0 0

2

4

6

8

Lxx/d1 Figure 44. Dimensionless distributions of longitudinal integral length scales (Lxx/d1) in hydraulic jumps for (x-x1)/d1 < 20. 2007-2 refers to experimental works of Kucukali and Chanson [31].

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Associated with integral time scales, integral length scales can also be defined. They provide information on the largest structures developing in the air-water flow. Chanson [12, 13] gave description of the behavior. Some results are presented on Figure 44 for the longitudinal integral length scales and on Figure 45 for the transverse integral length scale. Their results indicate that both longitudinal and transverse integral length scales increase with relative flow depth. Whereas Lxz/d1 ranges from 0 to 1, Lxx/d1 grows from 0 to more than 6 showing a preferential stretching in the streamwise direction. The largest structures are also found close to the free surface. It is also evident that length scales become larger as the Froude number increases. 6

5

4

y/d1 3 2 Fr=4.7, (x-x1)/d1=4.17 (2007-2) Fr=4.7, (x-x1)/d1=8.33 (2007-2) Fr=4.7, (x-x1)/d1=12.5 (2007-2) Fr=5.8, (x-x1)/d1=8.33 (2007-2) Fr=5.8, (x-x1)/d1=12.5 (2007-2) Fr=5.8, (x-x1)/d1=16.67 (2007-2)

1

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0 0

0,2

0,4

0,6

0,8

1

Lxz/d1 Figure 45. Dimensionless distributions of transverse air-water integral length scales (Lxz/d1) in hydraulic jumps for (x-x1)/d1 < 25. 2007-2 refers to experimental works of Kucukali and Chanson [31].

All these results provide important information regarding the air-water flow properties and turbulence. They bring useful information on the structure of the hydraulic jump flow. They may be helpful for many reasons. For instance: -

-

They improve our knowledge on the flow dynamics by providing information on diffusion processes, bubble size, bubble frequency, void fraction…; They may provide interesting data to calibrate numerical codes. Indeed, the turbulence length and time scales give an idea for sizes and “lifetime” of the turbulent structure developing in the flow; They allow the development of new experimental techniques such as the optical probes used by Murzyn et al. [41] that were not used before in such flows. They allow new investigations in flows with length and time scales that are smaller and

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-

faster than before. Furthermore, the sensor dimensions decrease increasing thus the accuracy of the results; They may contribute to improve and develop new methods for data analysis that could be used for other studies; They may give new insights on other problems such as spilling breaker on a beach, mixing and diffusion processes…

Nevertheless, there is still an important lack of knowledge on the field that requires further investigations. In the next part, we develop some examples of further research topics to study in the future.

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5. Conclusions and Perspectives The two-phase gas-liquid flow properties have been investigated by many researchers using different experimental techniques. Basic results were clearly found in terms of void fraction, bubble frequency and bubble velocity vertical profiles. A similitude with wall-jet flows was demonstrated. Some information on a double diffusion process were suggested as well. The turbulent flow structures were discussed in terms of both the free surface and the two-phase flow. Yet the huge amount of available data still needs further investigations and analyses. The present knowledge should be improved to assist the numericians to develop and calibrate their codes. Furthermore, new research topics may be developed with a focus on fundamental issues. Some developments have already started. The most promising research areas are probably those on bubble clustering, assessment of scale effects, effect of water quality on the flow properties and the influence of bubbles on sediment transport (three-phase flow). In terms of bubble clustering, significant contributions were recently published for stepped cascades flows [25], dropshaft flows [9] and hydraulic jumps [12, 13]. By cluster, one means a groupment of two or more bubbles separated from other bubbles by a “significant distance”. In the cluster, bubbles are close together and surrounded by a “sizeable volume of water”. In the context of this chapter, the basic relevant references are [12, 13]. In these experimental works, two complementary approaches were presented [12, 13]. First, a clustering analysis is based upon the study of water chord time between adjacent bubbles. Second, a study is led on the interparticle arrival time. The main conclusions were obtained by analyzing the longitudinal flow structure [13]. These showed comparatively little bubble clustering in the air-water shear layer region. However, an interparticle arrival time analysis suggested some preferential bubble clustering for bubble chord times below 3 ms within the investigated flow conditions. Altogether both approaches are complementary but the interparticle arrival time analyses give a greater insight into the range of particle classes affected by non-random clustering. A further study was based on the analysis of the water chord time statistics [26, 27]. A cluster was recorded when a water chord time was less than 10 % of the median water chord time. The results showed that the number of bubbles associated with cluster in hydraulic jumps was around 21 % and this percentage decreases with increasing Froude number. They also showed that the average number of bubbles per cluster was 2.3 with mostly 2 bubbles per detected cluster: 70 % to 95 % of the clusters contain 2 bubbles [27]. These results may be helpful to assess vorticity production rate and bubble-turbulence interactions as well as new typical turbulent scales of the flow. Note that

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these works are unfinished and further experiments are then required and new developments needed to assess these characteristics. Recently, the effects of water quality on bubble entrainment and dispersion were investigated in circular plunging jet flows [17, 18]. This kind of flow has several applications including waste-water treatment or river re-oxygenation (Figure 46). Based upon laboratory experiments under controlled flow conditions using conductivity probes, the results demonstrated that lesser air entrainement takes place in seawater for identical experimental conditions compared to freshwater [17, 18]. Furthermore, they showed more fine bubbles in seawater than in freshwater or salty freshwater. Two main reasons may explain these findings. Living organisms may inhibite bubble entrainment. Furthermore, the surfactant and biochemicals are thought to harden the induction trumpet at plunge point, implying that surface tension σ could play a role in the air-water entrapment process. Nevertheless, its influence on the turbulent length scales developing at the free surface were investigated independently [42]. The results suggested that the longitudinal length scales of the free surface (Lf) do not depend upon the Weber number ( We = U 1

ρd 1 ) but mostly by the σ

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upstream water depth d1.

Figure 46. Strong mixing of air and water at Vallée de la Gatineau, Québec, Canada. Le Grand Remous viewed from le Pont Couvert Savoyard, July 2002. Photo by Hubert Chanson.

Scale effects may be very significant. Although it is easier to undertake laboratory experiments instead of in-situ measurements, one must be sure that results obtained from

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these experiments would be also true in nature. A study assessed the scale effects affecting two-phase flow properties in hydraulic jumps [39]. Based upon hydraulic jumps with identical Froude number (Fr = 5 and 8.5) but with different Reynolds numbers ranging from 24000 to 98000, the results demonstrated some drastic scale effects in the smaller hydraulic jumps in terms of void fractions, bubble count rate and bubble size. Simply dynamic similarity can not be achieved with a Froude similitude only and viscous effects may be significant even with Reynolds numbers larger than 105 as illustrated in [23, 39]. One challenging research topic deals with the interaction between air, water and solid particles. This is particularly encountered in coastal environments where sediments are entrapped by wave breaking or in river engineering during flood periods (Figure 47). Figure 47 illustrates a hydraulic jump in a muddy creek. The brownish colour of the waters is evidences of the large sediment suspension load. To date, these three-phase flows are too complex to be fully understood, with too many equations and parameters to be taken into account. For numerical modeling, fully non linear equations strongly increase the difficulties [3] and available experimental data are very limited. Even with the most powerful computers, it is not conceivable that the flow dynamics could be fully understood in the close future. Furthermore, exact similitude can not be achieved because of the numerous paramaters such as sediment size, density, shapes, flow scales… In situ measurements are also difficult because of the invasive technique used which disturb the flow and the dangers associated with the field measurements in a river in flood (Figure 47). Drastic progresses are needed in this field to ensure a better comprehension of the phenomena.

Figure 47. Todd river in Alice Springs NT (Australia), January 2007. A complex mixing of water, air and sediment during floods (1 in 5 or 1 in 10 years flood). Courtesy of Sue McMINN.

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Acknowledgements The first author thanks François Stephan, head of department at ESTACA Paris (Ecole Supérieure des Techniques Aéronautiques et de la Construction Automobile, http://www.estaca.fr) for his support and Frédéric Malandain (Cany-Barville, France, http://www.frederic-malandain.fr/ ) for providing nice pictures of Normandy’s coastlines.

References [1] [2]

[3] [4]

[5]

[6]

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[7] [8]

[9]

[10] [11] [12]

[13] [14]

Babb A.F., Aus H.C. Measurements of air in flowing water, Journal of Hydraulic Division, ASCE, 107 (HY12), 1981, pp 1615-1630. Bélanger J.B. Essai sur la solution numérique de quelques problèmes relatifs au mouvement permanent des eaux courantes, Carialian-Goeury, Paris, France, 1828 (in french). Brennen C.E. Fundamentals of multiphase flows, Cambridge University Press, 2005, 345 pages. Carosi G. Chanson H. Air-water time and length scales in skimming flows on a stepped spillway. Application to the spray characterization, Research Report CH59/06, Department of Civil Engineering, The University of Queensland, Brisbane, Australia, July 2006, 142 pages. Chanson H. Air entrainment in two dimensional turbulent shear flows with partiallydeveloped inflow conditions, International Journal of Multiphase Flow, 31, 6, 1995, pp 1107-1121. Chanson H. Air bubble entrainment in free surface turbulent shear flows, Academic Press, London, UK, 1997, 401 pages. Chanson H. The hydraulics of open-channel flow : an introduction, Edward Arnold, London, UK, 1999, 512 pages. Chanson H. Air-water flow measurements with intrusive phase-detection probes. Can we improve their interpretation? Journal of Hydraulic Engineering, ASCE, 128, 3, 2002, pp 252-255. Chanson H. An experimental study of Roman dropshaft operation : hydraulics, twophase flow, acoustics, Report CH 50/02, Department of Civil Engineering, The University of Queensland, Brisbane, Australia, 2002, 99 pages. Chanson H. The hydraulic of open-channel flow : an introduction, ButterworthHeinemann, Oxford, UK, 2nd edition, 2004, 630 pages. Chanson H. Bubble entrainment, spray and splashing at hydraulic jumps, Journal of Zhejiang University Science A, 7, 8, 2006, pp 1396-1405. Chanson H. Air bubble entrainment in hydraulic jumps. Similitude and scale effects, Research Report CH57/05, Department of Civil Engineering, The University of Queensland, Brisbane, Australia, January, 2006, 119 pages. Chanson H. Bubbly flow structure in hydraulic jump, European Journal of Mechanics, 26, 3, 2007, pp 367-384. Chanson H. Dynamic similarity and scale effects affecting air bublle entrainment in hydraulic jumps, Proceedings of the 6th International Conference on Multiphase Flow

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[15]

[16] [17]

[18]

[19] [20]

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[21]

[22] [23] [24]

[25]

[26]

[27]

Frédéric Murzyn and Hubert Chanson ICMF 2007, Leipzig, Germany, July 9-13, M. Sommerfeld Editor, Session 7, 2007, 11 pages. Chanson H. Hydraulic jumps: bubbles and bores, 16th Australasian Fluid Mechanics Conference AFMC, Crown Plaza, Gold Coast, Australia, December 2-7, P. Jacobs, T. McIntyre, M. Cleary, D. Buttsworth, D. Mee, R. Clemens, R. Morgan, C. Lemckert Editors, 2007, Plenary Address, pp 39-53. Chanson H. Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results, European Journal of Mechanics B/Fluids, in press, 2008. Chanson H., Aoki S.I., Hoque A. Scaling bubble entrainment and dispersion in vertical circular plunging jet flows : freshwater versus seawater, Proceedings of the 5th International Conference on Hydrodynamics, Tainan, Taïwan, 2002. Chanson H., Aoki S.I., Hoque A. Bubble entrainment and dispersion in plunging jet flows : freshwater versus seawater, Journal of Coastal Research, 22, 3, 2006, pp 664677. Chanson H., Brattberg T. Experimental study of the air-water shear flow in a hydraulic jump, International Journal of Multiphase Flow, 26, 4, 2000, pp 583-607. Chanson H., Carosi G. Advanced post-processing and correlation analyses in highvelocity air-water flows. 1- Macroscopic properties, Proceedings of the International Junior Researcher and Engineer Workshop on hydraulic structures (IJREWHS’06), Montemor-o-Novo, Jorge Matos and Hubert Chanson Eds, Report CH61/06, Division of Civil Egineering, The University of Queensland, Brisbane, Australia, December 2006, pp 139-148. Chanson H., Carosi G. Advanced post-processing and correlation analyses in highvelocity air-water flows. 1- Microscopic properties, Proceedings of the International Junior Researcher and Engineer Workshop on hydraulic structures (IJREWHS’06), Montemor-o-Novo, Jorge Matos and Hubert Chanson Eds, Report CH61/06, Division of Civil Egineering, The University of Queensland, Brisbane, Australia, December 2006, pp 149-158. Chanson H., Carosi G. Turbulent time and length scale measurements in high-velocity open-channel flows. Experiments in Fluids, 42, 3, 2007, pp 385-401. Chanson H., Gualtieri C. Similitude and scale effects of air entrainment in hydraulic jumps, Journal of Hydraulic Research, 46, 1, 2008, pp 35-44. Chanson H., Toombes L. Experimental investigations of air entrainment in transition and skimming flows down a stepped chute. Application to embankment overflow stepped spillways, Research Report CE158, Department of Civil Engineering, The University of Queensland, Brisbane, Australia, July, 2001, 74 pages. Chanson H., Toombes L. Air-water flows down stepped chutes: turbulence and flow structure observations, International Journal of Multiphase Flow, 28, 11, 2002, pp 1737-1761. Gualtieri C., Chanson H. Clustering process and interfaccial area analysis in a large-size dropshaft, Advances in Fluid Mechanics V, A. Mendes, M. Rahman, C.A. Brebbia Editors, WIT Press, 2004. Gualtieri C., Chanson H. Clustering process analysis in a large-size dropshaft and in a hydraulic jump, 32nd IAHR Biennial Congress, Venice, July 2007, 11 pages.

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[28] Gualtieri C., Chanson H. Experimental analysis of Froude number effect on air entrainment in the hydraulic jump, Environmental Fluid Mechanics, 7, 2007, pp 217238. [29] Hager W.H. Energy dissipators and hydraulic jump, Kluwer Academic Publisher, Water Science and Technology Library, Volume 8, Dordrecht, The Netherlands, 288 pages. [30] Hager W.H., Bremen R., Kawagoshi N. Classical hydraulic jump: length of roller, Journal of Hydraulic Research, IAHR, 28, 5, 1990, pp 591-608. [31] Kucukali S., Chanson H. Turbulence in hydraulic jumps: experimental measurements, Report CH62/07, Department of Civil Engineering, The University of Queensland, Brisbane, Australia, July 2007, 96 pages. [32] Lennon J.M., Hill D.F. Particle Image Velocity measurements of undular and hydraulic jumps, Journal of Hydraulic Engineering, 132, 12, 2006, pp 1283-1294. [33] Liu M., Zhu D.Z., Rajaratnam N. Evaluation of ADV measurements in bubbly twophase flows, Proceedings of the conference on Hydraulic Measurements and Experimental Methods, EWRI, Tony L. Wahl, Clifford A. Pugh, Kevin A. Oberg, Tracy B. Vermeyen (Editors), Estes Park, USA, 2002. [34] Matos J., Frizell K.H., André S., Frizell K.W. On the performance of velocity measurement techniques in air-water flows, Proceedings of speciality conference “Hyraulic Measurements and Experimental Methods”, EWRI, ASCE, IAHR, Estes Park, USA, August 2002, 11 pages. [35] Mossa M. On the oscillating characteristics of hydraulic jumps, Journal of Hydraulic Research, IAHR, 37, 4, 1999, pp 541-558. [36] Mossa M. Tolve U. Flow visualization in bubbly two-phase hydraulic jump, Journal of Fluids Engineering, 120, 1998, pp 160-165. [37] Mouazé D., Murzyn F., Chaplin J.R. Free surface length scale estimation in hydraulic jumps, Journal of Fluids Engineering, Transaction of ASME, 127, 2005, pp 1191-1193. [38] Murzyn F., Chanson H. Free surface, bubbly flow and turbulence measurements in hydraulic jumps, Report CH63/07, Division of Civil Engineering, The University of Queensland, Brisbane, Australia, July, 2007, 113 pages. [39] Murzyn F., Chanson H. Experimental assessment of scale effects affecting two-phase flow properties in hydraulic jumps, Experiments in Fluids, in press (corrected proofs), 2008, 9 pages. [40] Murzyn F., Chanson H. Experimental investigation of bubbly flow and turbulence in hydraulic jumps, Environmental Fluid Mechanics, in press (corrected proofs), 2008, 17 pages. [41] Murzyn F., Mouazé D., Chaplin J.R. Optical fibre probe measurements of bubbly flow in hydraulic jumps, International Journal of Multiphase Flow, 31, 1, 2005, pp 141-154. [42] Murzyn F., Mouazé D., Chaplin J.R. Air-water interface dynamic and free surface features in hydraulic jumps, Journal of Hydraulic Research 45, 5, 2007, pp 679-685. [43] Prosperetti A., Tryggvason G. Computational methods for multiphase flow, Cambridge University Press, 2007, 470 pages. [44] Rajaratnam N. An experimental study of air entrainment characteristics of the hydraulic jump, Journal of Instrumental Engineering, The Institution of Enginners (India), 42, 7, 1962, pp 247-273.

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[45] Rajaratnam N. The hydraulic jump as a wall jet, Journal of Hydraulics Division, HY 5, 1965, pp 107-132 (+ Discussion HY 3, pp 110-123, + Closure HY 1, pp 74-76). [46] Resch F.J., Leutheusser H.J. Le ressaut hydraulique : mesure de turbulence dans la région diphasique, La Houille Blanche, 4, 1972a, pp 279-293. [47] Resch F.J. Leutheusser H.J. Reynolds stress measurements in hydraulic jumps, Journal of Hydraulic Research, 10, 4, 1972b, pp 409-429. [48] Waniewski T.A., Hunter C., Brennen C.E. Bubble measurements downstream of hydraulic jumps, International Journal of Multiphase Flow, 27, 2001, pp 1271-1284.

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Reviewed by Dominique Mouazé Lecturer Morphodynamique Continentale et Côtière University of Caen - Basse Normandie, 24 rue des Tilleuls, 14000 Caen (France) E-mail address: [email protected] Ph: +33.(0)2.31.56.57.11

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Chapter 10

TWO PHASE FLOW AND HEAT TRANSFER OF SUB- AND SUPER-CRITICAL CO2 IN MACRO- AND MICRO-CHANNELS Lixin Cheng1,*and John R. Thome2 1

School of Engineering, University of Aberdeen, King’s College, Aberdeen, AB24 3FX, Scotland, the UK 2 Laboratory of Heat and Mass Transfer (LTCM), Faculty of Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Station 9, Lausanne, CH-1015, Switzerland

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Abstract This chapter addresses various issues related to sub- and super-critical CO2 two-phase flow and heat transfer in macro- and micro-channels such as flow boiling, two-phase flow patterns and pressure drops without and with lubricating oil effect, supercritical gaslubricating oil two-phase flow, two-phase flow and supercritical flow distribution in heat exchanger headers. Emphasis is given to our newly developed two-phase flow pattern map for CO2 evaporation, flow pattern based flow boiling heat transfer and pressure drop models. Furthermore, some simulation results for electronic chips cooling using CO2 evaporation are presented and discussed. So far, little information on CO2 condensation is available in the literature. Therefore, this chapter does not include CO2 condensation but it is recommended that future research be conducted in this aspect. The future research needs in CO2 two-phase flow and heat transfer have been identified.

Keywords: CO2, natural refrigerant, two-phase flow, evaporation, flow boiling, flow patterns, flow pattern map, pressure drop, model, oil, electronic chips cooling, sub-critical, supercritical, gas cooler, two-phase flow distribution, supercritical flow distribution.

*

E-mail address: [email protected]

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1. Introduction Over the past years, natural refrigerants such as such as CO2 (R744), ammonia (R717) and hydrocarbons have been studied to replace CFCs, HCFCs and HFCs in refrigeration, airconditioning and heat pump systems due to environmental issues such as ozone depleting and global warming [1-10]. The ozone depletion potentials (ODP) of these natural refrigerants are zero and most of them have close to zero global warming potentials (GWP) as well. CO2 was widely used in industry early in the last century and had been gradually phased out as it lost the competition with artificial refrigerants. A few years ago, a revival of CO2 as a working fluid has commenced [3]. CO2 has no ozone depletion potential (ODP = 0) and a negligible direct global warming potential (GWP = 1) when used as a refrigerant. Therefore, CO2 has been receiving renewed interest as an efficient and environmentally safe refrigerant in a number of applications, including automotive air conditioning, residential heat pumps and as a primary and secondary refrigerant in refrigeration systems at low temperatures. Due to its low critical temperature (Tcr = 31.1°C) and high critical pressure (pcr =7.38 MPa), CO2 is utilized at much higher operating pressures in air-conditioning and heat pump systems compared to other conventional refrigerants. Figure 1 shows a schematic of CO2 automobile air-conditioning system. For usual ambient air temperatures, the heat transfer process on the high pressure side of a CO2 cycle is not a condensation process as in conventional systems but a supercritical gas cooling process [9, 10]. Hence, the conventional condenser is replaced by a gas cooler that rejects heat under variable temperature at high pressure. To improve the performance of the CO2 system, an internal heat exchanger is used to subcool the refrigerant between gas cooler and the expansion device and to superheat the refrigerant in the suction line of the compressor. Figure 2 shows a comparison of pressureenthalpy diagrams of CO2 and R134a in automobile air-conditioning systems. For the evaporation side at low pressures, CO2 evaporates at much higher pressure than R134a. In this case, the physical and transport properties of CO2 are quite different from those of conventional refrigerants at the same saturation temperatures. CO2 has higher liquid and vapor thermal conductivities, a lower vapor-liquid density ratio (lower liquid and higher vapor densities), a very low surface tension, and a lower liquid-vapor viscosity ratio (lower liquid and higher vapor viscosities) than conventional refrigerants (comparisons of the CO2 and R134a physical properties are shown in Figs. 5-13 in section 2.1), thus flow boiling heat transfer, two-phase flow pattern and pressure drop characteristics are quite different from those of conventional low pressure refrigerants. Previous experimental studies have shown that CO2 has higher flow boiling heat transfer coefficients and lower pressure drops than those of conventional refrigerants at the same saturation temperatures. The available flow boiling heat transfer correlations developed for conventional low pressure refrigerants generally significantly underpredict the experimental data of CO2. In addition, dryout may occur much earlier (at moderate vapor quality) in CO2 flow boiling, particularly at high mass flux and high temperature conditions. Significant deviations for the flow patterns of CO2 compared to the flow pattern maps that were developed for other fluids at lower pressures have been observed as well. Two-phase pressure drops of CO2 are also much lower than conventional low pressure refrigerants. Furthermore, lubricant oil has great effect on heat transfer and pressure drop, which should be clarified for both flow boiling, supercritical gas heat transfer and pressure drops. Therefore, it is very important to understand and to predict

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the flow patterns, pressure drop and heat transfer in flow boiling without and with the oil effect in evaporators and oil-gas two-phase flow in the gas coolers.

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Figure.1. Schematic of CO2 automobile air-conditioning system.

Figure 2. Pressure-enthalpy diagrams of CO2 and R134a in automobile air-conditioning systems.

Furthermore, CO2 has very positive attributes as a secondary refrigerant at low temperatures in commercial refrigeration used in supermarkets, shops, large kitchens etc., in indirect and low temperature cascade systems and as a primary refrigerant in all CO2 centralized refrigeration systems. Due to its excellent safety characteristics (nonflammable, non-explosive, inexpensive and relatively nontoxic), CO2 is an ideal refrigerant to be used in the refrigerated spaces. Owing to the high working pressure, CO2 as a phase change secondary refrigerant has a high volumetric refrigeration capacity, which equates to approximately 5 times or more that of R22 and NH3. In addition, due to the good thermophysical properties, flow boiling heat transfer and two-phase flow characteristics of

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CO2, smaller pipe dimensions can be used in its refrigeration systems. Figure 3 shows a schematic of an ammonia-CO2 secondary loop of an indirect refrigeration system which operates at low temperatures. In this refrigeration system, CO2 evaporation and condensation processes are nearly at the same pressures. Also this system does not involve any oil effect for the CO2 secondary loop. The main benefits for using CO2 in indirect system arrangements include the simplicity of the system and the possibility of using components for other refrigerants to build the CO2 circuit. In recent years, other arrangements, such as cascade and multistage systems have been used commercially [10, 11]. Advantages of CO2 cascade systems include greatly reduced low-temperature compressor size, the absence of a liquid pump and fewer stages of heat transfer. Furthermore, two-stage and multistage CO2 centralized refrigerant systems are also used in supermarket refrigeration. These systems are most suitable for cold climates or where heat sinks are available [11]. The operation of such refrigeration systems is mostly in the sub-critical region. If the ambient temperature is higher than the critical temperature, the system will be supercritical and this generally should be connected to another thermal system such as hot water or space heating heat pumps for efficient energy use. The use of CO2 in these refrigeration systems requires the understanding and prediction of convective boiling and condensation heat transfer, two-phase flow patterns and pressure drops at low temperatures for achieving more accurate designs and more energyefficient cycles using CO2.

Figure 3. Schematic of ammonia-CO2 in secondary loop of an indirect refrigeration system.

Both macro- and micro-scale channels are used in the CO2 refrigeration, air-conditioning and heat pump systems. In automobile air-conditioning systems, micro-scale channels with diameters of 0.6 to 1 mm are generally used in evaporators, internal heat exchangers and gas coolers while macro- and micro-channels are used for CO2 refrigeration and heat pump systems. Due to the significant differences of two-phase flow and heat transfer phenomena in micro-scale channels as compared to conventional size channels or macro-scale channels,

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Figure 4. Comparison of the definitions of macro- and micro-scale channels for CO2 according to Kandlikar [12] and Confinement number Co [17].

emphasis has been put on the characteristics of two-phase flow and heat transfer in small and micro-scale flow passages due to the rapid development of micro-scale devices in recent years [12-19]. A number of studies on CO2 flow boiling and two-phase flow characteristics have been conducted in both macro- and microchannels. At first, one very important issue should be the clarification of the distinction between micro-scale channels and macro-scale channels. However, a universal agreement is not clearly established in the literature. Instead, there are various definitions on this issue [16, 19]. Here, just to show two examples, based on engineering practice and application areas such as refrigeration industry in the small tonnage units, compact evaporators employed in automotive, aerospace, air separation and cryogenic industries, cooling elements in the field of microelectronics and micro-electro-mechanicalsystems (MEMS), Kandlikar [12] defined the following ranges of hydraulic diameters Dh which are attributed to different classifications: • • •

Conventional channels: Dh > 3 mm. Minichannels: Dh = 200 μ m – 3 mm. Microchannels: Dh = 10 μ m – 200 μ m.

According to this definition, the distinction between small and conventional size channels is 3 mm and the distinction between mini and micro channels is 200 μm. Kew and Cornwell [17] earlier proposed the Confinement number Co for the distinction of macro- and microscale channels, as

Co =

1 Dh

4σ g ( ρ L − ρG )

(1)

which is actually based on the definition of the Laplace constant [16, 19]. Other different definitions are also proposed [16, 19] but not elaborated here. Thus, the definition of a micro-

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scale channel is quite confusing. Figure 4 shows the comparable definitions macro- and micro-scale channels for CO2 according to Kandlikar [12] and the Confinement number Co (Equation 1), which shows the big difference among these criteria. In this chapter, the distinction between macro- and micro-scale channels by the threshold diameter of 3 mm is adopted due to the lack of a well-established theory, but is in line with those recommended by Kandlikar [12] and also for the practical use in the CO2 air-conditioning, heat pump and refrigeration systems. From a predictive standpoint, many features of the existing heat transfer and two-phase pressure drop correlations require refinement to attain the desired level of accuracy for refrigerant heat exchanger (evaporators, internal heat exchangers, gas coolers and condensers) design as pointed out by Thome [6]. Therefore, this chapter addresses all the issues related to CO2 two-phase flow and heat transfer such as flow boiling, two-phase flow patterns and pressure drops without and with oil effect, supercritical gas-oil two-phase flow and two-phase flow distribution in headers, including experimental work. Emphasis is given to newly developed CO2 flow boiling methods, two-phase flow pattern maps and pressure drop models. Furthermore, some simulation results for electronic chips cooling using CO2 evaporation are presented and discussed. So far, little information on CO2 condensation is available in the literature. Therefore, this chapter does not include CO2 condensation but it is recommended that future research should be conducted in this aspect. The future research needs in the CO2 two-phase flow and heat transfer are also identified.

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2. Flow Boiling and Two-Phase Flow of CO2 under Subcritical Conditions A number of experiments on pool boiling, flow boiling, two-phase pressure drops and flow patterns have been undertaken over the past years. Gorenflo and Kotthoff [20] did an overall review on pool boiling heat transfer of CO2 which is an interesting topic but beyond the scope of this chapter. For two-phase flow and flow boiling of CO2 in macro- and micro-channels, Thome and Ribatski [7] did a comprehensive review. Here we do not do another comprehensive review but present a systematic knowledge on this topic. In this section, first, the physical properties of CO2 are discussed and compared to those of R134a. Then, behaviors of flow boiling heat transfer, two-phase flow patterns and two-phase pressure drops without oil effect are briefly summarized according to the available studies. Finally, our newly developed CO2 flow map, flow pattern based flow boiling heat transfer model and phenomenological two-phase frictional pressure drop model are mainly presented.

2.1. Physical Properties of CO2 Physical properties have a significant effect on two-phase flow and heat transfer characteristics. The physical and transport properties of CO2 are quite different from those of conventional refrigerants when compared at the same saturation temperature. Comparisons of the physical properties of CO2 and R134a obtained using REFPROP.NIST Ver 7.0 [21] are shown in Figs. 5-13. CO2 has a much higher saturation pressure than R134a at the same saturation temperature as shown in Fig. 5. Furthermore, CO2 has a much lower vapor-liquid density ratio (lower liquid and higher vapor densities), higher liquid and vapor specific

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heats, a lower liquid-vapor viscosity ratio (lower liquid and higher vapor viscosities), a higher latent heat (only near the critical point, the CO2 latent heats are lower than R134a), much higher liquid and vapor thermal conductivities and much lower surface tensions than R134a and other low pressure refrigerants. The different physical properties result in quite different flow boiling heat transfer, two-phase flow pattern and pressure drop behaviors as compared to those of conventional low pressure refrigerants. In the next section, these behaviors and their mechanisms are described and explained according to the physical properties.

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Figure 5. Comparison of saturation pressures of CO2 and R134a [21].

Figure 6. Comparison of liquid and vapor densities of CO2 and R134a [21].

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Figure 7. Comparison of liquid-vapor density ratios of CO2 and R134a [21].

Figure 8. Comparison of liquid and vapor specific heats of CO2 and R134a [21].

Figure 9. Comparison of liquid and vapor dynamic viscosities of CO2 and R134a [21].

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Figure 10. Comparison of liquid and vapor dynamic viscosity ratios of CO2 and R134a [21].

Figure 11. Comparison of latent heats of CO2 and R134a [21].

Figure 12. Comparison of liquid and vapor thermal conductivities of CO2 and R134a [21].

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The physical properties of CO2 may be obtained from several software packages [21-23]. However, it should be pointed out that there are some differences among these software packages. Just to show two examples for thermal conductivity and dynamic viscosity here, Figures 14 and 15 show the differences for CO2 liquid and vapor thermal conductivities from three different software packages: REFPROP.NIST version 6.01 [22], REFPROP.NIST version 7.0 [21] and Engineering Equation Solver (EES) [23]. Figures 16 and 17 show the differences for CO2 liquid and vapor dynamic viscosities from these software packages. Other physical property differences are also observed but not shown here. Utilization of different software packages for physical properties has some effect on reducing experimental results and implementation of prediction methods. For example, the flow boiling heat transfer and two-phase pressure drop models described in section 2.3 were developed using the physical properties from REFPROP.NIST version 6.01 [22]. When using other software packages, the results differ but not significantly.

Figure 13. Comparison of surface tensions of CO2 and R134a [21].

Figure 14. Comparison of liquid thermal conductivities of CO2 from three different software packages.

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Figure 15. Comparison of vapor thermal conductivities of CO2 from three different software packages.

Figure 16. Comparison of liquid dynamic viscosity of CO2 from 3 software packages.

Figure 17. Comparison of vapor dynamic viscosity of CO2 from 3 software packages.

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2.2. Behavior of CO2 Flow Boiling, Two-Phase Flow Patterns and Pressure Drops

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Different flow boiling heat transfer and two-phase flow behaviors of CO2 have been shown for high and low reduced pressures in the available studies. The boiling heat transfer and twophase flow of CO2 at saturation temperatures ranging from 0 to 25°C show different characteristics from those of conventional refrigerants due to the significant differences in physical properties. Generally, CO2 has much higher flow boiling heat transfer and much lower pressure drops than other low-pressure refrigerants. One feature is the dominance of the nucleate boiling at low/moderate vapor qualities prior to dryout [24-28]. Another feature is that the dryout in CO2 flow boiling occurs much earlier (at relatively lower vapor qualities) than conventional refrigerants. Furthermore, the effect of the saturation temperature on heat transfer coefficient is more noticeable. At high saturation temperatures, nucleate boiling is more pronounced and plays an important role at low vapor quality.

Figure 18. The experimental heat transfer coefficients in two different studies showing two opposite trends with the increase of saturation temperature. Arrow 1 showing the trend of the experimental flow boiling heat transfer coefficients (solid symbols) of Pettersen [24]: Dh = 0.8 mm, G = 280 kg/m2s and q = 10 W/m2 at 0, 20 and 25 °C. Arrow 2 showing the trend of the experimental flow boiling heat transfer coefficients (hollow symbols) of Yoon et al. [28]: Dh = 7.53 mm, G = 318 kg/m2s and q =16.4 W/m2 at 5, 15 and 20 °C.

It should be mentioned that the experimental data from the different independent studies show somewhat different heat transfer trends at similar test conditions. Just to show several examples here, Figure 18 depicts two opposite heat transfer behaviors with saturation temperature in the studies of Pettersen [24] and Yoon et al. [28]. Heat transfer coefficients increase with increasing saturation temperature in the study of Pettersen while they decrease in the study of Yoon et al. The only big difference between the two studies is the diameter of the test channels as indicated in Fig. 18. Figure 19 shows comparison of the experimental data of Yun et al. [29] for two diameters of 1.53 and 1.54 at the same test conditions. According to their results, heat transfer coefficients can be higher up to 80% with a very little change of

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hydraulic diameter from 1.53 mm to 1.54 mm at the same test conditions. No explanation why there is such a big difference even was offered in their paper. Figure 20 shows the comparison of the heat transfer coefficients of Pettersen [24] with those of Koyama et al. [30]. The biggest difference between them is that in Koyama et al. the heat flux is 32.06 kW/m2 while in Pettersen it is 10 kW/m2. The heat transfer coefficients fall off at a vapor

2

Heat transfer coefficient [W/m K]

2 x 10

4

1 2

1.8 1.6 1.4 1.2 1 0.8 0

0.1

0.2

0.3

0.4 0.5 0.6 Vapor quality

0.7

0.8

0.9

1

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Figure 19. The experimental flow boiling heat transfer coefficients in the same study showing different results with a very little change of hydraulic diameters from 1.53 mm to 1.54 mm. Solid symbols showing the experimental flow boiling heat transfer coefficients of Yun et al. [29]: Dh = 1.53 mm, G = 300 kg/m2s, Tsat = 5 °C and q = 20 W/m2. Hollow symbols showing the experimental flow boiling heat transfer coefficients of Yun et al. al. [29]: Dh = 1.54 mm, G = 300 kg/m2s, Tsat = 5 °C and q =20 W/m2.

Figure 20. The experimental flow boiling heat transfer coefficients in two different studies showing opposite flow boiling heat transfer coefficient trends. Solid symbols showing the experimental flow boiling heat transfer coefficients of Pettersen [24]: Dh = 0.8 mm, G = 190 kg/m2s, Tsat = 0 °C and q = 10 W/m2. Hollow symbols showing the experimental flow boiling heat transfer coefficients of Koyama et al. al. [30]: Dh = 1.8 mm, G = 260 kg/m2s, Tsat = 0.26 °C and q =32.06 W/m2.

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2

Heat transfer coefficient [kW/m K]

14 12 10 8 6 4 D = 4 mm D = 6 mm

2 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Vapor quality

0.7

0.8

0.9

1

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Figure 21. Experimental heat transfer data of Hihara [31] (G = 360 kg/m2s, Tsat = 15 °C and q = 18 kW/m2).

Figure 22. Comparison of different experimental data of CO2 at experimental conditions: (1) Bredesen et al. [32]: Tsat = 5°C, D = 7 mm, G = 200 kg/m2s, q = 6 kW/m2; (2) Park and Hrnjak [35]: Tsat = 30°C, D = 6.1 mm, G = 200 kg/m2s, q = 15 kW/m2; (3) Bredesen et al. [32]: Tsat = -30°C, D = 7 mm, G = 200 kg/m2s, q = 6 kW/m2;; (4) Zhao and Bansal [34]: Tsat = -28.7°C, D = 4.57 mm, G = 196.8 kg/m2s, q = 17 kW/m2; and (5) Knudsen and Jensen [33]: Tsat = -28°C, D = 10.08 mm, G = 80 kg/m2s, q = 8 kW/m2.

quality of about 0.7 in the study of Pettersen while the heat transfer coefficients increase even at qualities larger than 0.7 in the study of Koyama et al. It is difficult to explain why the heat transfer coefficients fall off at the lower heat flux in one study while they still increase at the higher heat flux in the other study. This could be an effect of the heating methods or because

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of multi-channel vs. single channel test setups. Figure 21 shows the heat transfer data of Hihara [31] at a mass velocity of 360 kg/m2s, a saturation temperature of 15 °C and a heat flux of 18 kW/m2 with two different tube diameters, 4 and 6 mm. It shows that the heat transfer coefficients of the 4 mm tube are twice those of the 6 mm tube. In addition, the trends of the heat transfer coefficients are totally different. As both diameters are in the range of macro scale, it is surprising that the diameter has such a big effect on the heat transfer values and trends. Hence, in summary, there is still not a clear view of why CO2 data do not conform to conventional trends and also differ widely from one study to another. However, the available studies have shown different heat transfer behaviors at lower saturation temperatures from those at higher saturation temperatures. In fact, at low evaporation temperatures down to -40°C, the CO2 reduced pressures (e.g. the reduced pressure pr = 0.136 at -40°C) are still much higher than those of conventional refrigerants such as R134a (e.g. the reduced pressure pr = 0.0126 at -40°C). The physical properties at the lower temperatures are much different from those of R134a as shown in Figs. 5 to 13, showing the similar trend as those of CO2 at higher temperatures as indicated. It is difficult to explain the experimental results in some studies at low temperatures. So far, there are several studies of CO2 at low temperatures in the literature but still very limited information available. Bredensen et al. [32] performed the boiling heat transfer experiments with CO2 at temperatures of -10°C and -25°C. The experimental results show the heat transfer coefficient increases with vapor quality until dryout, which is opposite to the trend of their data at 0°C. Knudsen and Jensen [33] measured flow boiling heat transfer coefficients of CO2 in a horizontal tube of diameter 10.06 mm at the saturation temperatures of -28°C and -30°C. Their boiling heat transfer coefficients are much lower than others’ data. Zhao and Bansal [34] presented experimental heat transfer data at -30°C. Park and Hrnjak [35] showed the heat transfer coefficients in a 6.1 mm inner diameter tube at -30 and -15 °C for various mass fluxes and heat fluxes. Figure 22 shows the comparison of the experimental heat transfer data in these studies. Quite big differences among these data are found. It is difficult to explain why there are such big differences although the test conditions are similar. Zhao and Bansal also found that the Liu and Winterton [49] correlation predicted their data rather well while it does not predict other data. In fact, there are only few data points in their study. Considering the big differences among the available data, it is recommended that more and accurate experimental data at low temperatures are needed by careful and well designed experiments. Empirical heat transfer methods do not capture the parametric trends in dryout and mist flow regimes and cannot explain the physical mechanisms although they predict some data well in some cases. Therefore, an improved heat transfer model based on flow regimes for flow boiling is needed, but first accurate experimental data under wide test conditions are needed. Furthermore, no two-phase pressure drop data at low temperatures are available so far. Regarding the flow boiling heat transfer mechanisms, high reduced pressures and low surface tensions for CO2 compared to conventional refrigerants have major effects on nucleate boiling heat transfer characteristics. Previous studies have suggested a clear dominance of nucleate boiling heat transfer even at very high mass flux. Therefore, CO2 has much higher heat transfer coefficients than those of conventional refrigerants at the same saturation temperature and the available heat transfer correlations generally underpredict the experimental data of CO2. In addition, previous experimental studies have demonstrated that dryout trends occur earlier at moderate vapor qualities in CO2, particularly at high mass flux and high temperature

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conditions. However, it is difficult to explain the available boiling data at low temperatures according to these mechanisms although nearly all these studies pointed to nucleate boiling dominant mechanism with respect to their data. From the physical properties at low temperatures, it seems that these heat transfer behaviors should be similar to those at high saturation temperatures but they are not indeed. Thus, understanding of the two-phase flow and heat transfer characteristics of CO2 at low temperatures is essential. Furthermore, for flow boiling in enhanced tubes, Koyama et al. [36] conducted experiments on flow boiling in a smooth copper tube and in a micro-fin copper tube at 5.3°C. From their results, the heat transfer coefficients are only slightly higher than in the microfin tubes with a slight pressure drop increase as well. In this case, microfin tubes are not appropriate for CO2. Cho and Kim [37] conducted experimental studies of CO2 for microchannels and their data show that the average evaporation heat transfer coefficients ware 150 to 210 % higher than those of smooth tubes. The increase of pressure drop was much lower than the heat transfer increase. So far, only limited studies of CO2 flow boiling in micro-fin tubes are available. Whether they significantly enhance CO2 flow boiling heat transfer or not is still unclear due to the lack of such information. Furthermore, Koyama et al. [92, 93] conducted experimental studies of flow boiling of CO2-oil mixture in micro-fin tubes. Similar conclusions to those in smooth tube were obtained. Analysis of the oil effect on flow boiling and two-phase flow is presented in section 3. An overall review of flow boiling heat transfer and two-phase flow of CO2 in the literature has been conducted by Thome and Ribatski [7]. The review addressed the extensive experimental studies on flow boiling heat transfer and two-phase flow in macro and micro channels, two-phase flow patterns and flow pattern maps, macro- and microchanel flow boiling heat transfer and two-phase pressure drop prediction methods for CO2. They found that not one of the available prediction methods was able to predict the experimental data of CO2 well. Therefore, they suggested that a new flow boiling heat transfer prediction method should be developed and the flow boiling heat transfer model should include CO2 effects on the annular to dryout and dryout to mist flow transitions in order to more accurately predict heat transfer coefficients at moderate/high vapor qualities. In response, Cheng et al. [38, 39] proposed a new flow boiling heat transfer model and flow pattern map for carbon dioxide evaporating inside horizontal tubes. They were developed by modifying the model of Wojtan et al. [40, 41], which is an updated version of the Kattan-Thome-Favrat [42-44] flow pattern map and flow boiling heat transfer model. The Cheng-Ribatski-Wojtan-Thome flow boiling heat transfer model is a flow pattern based flow boiling heat transfer model which relates the flow patterns to the corresponding heat transfer mechanisms, thus, different from the numerous empirical models, such as the correlations of Chen [45], Shah [46], Gungor and Winterton [47], Kandlikar [48], Liu and Winterton [49], etc., which do not include flow pattern information. In fact, some of these correlations predicted the data well to some extent but fail to capture the parametric trends, or ignore the dryout and mist flow regimes which are typical working conditions in automobile air-conditioning systems. The Cheng-RibatskiWojtan-Thome flow boiling heat transfer model is applicable to a wide range of conditions: tube diameters (equivalent diameters for non-circular channels) from 0.8 to 10 mm, mass fluxes from 80 to 570 kg/m2s, heat fluxes from 5 to 32 kW/m2, saturation temperatures from – 28 to 25oC (the corresponding reduced pressures are from 0.21 to 0.87). The model reasonably predicts the database and it covers channel diameters found in most CO2 flow boiling applications. However, their model is limited by its parameter ranges from being

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applicable to some important applications, for example, the mass velocity ranges from 50 to 1500 kg/m2s in CO2 automobile air conditioning systems. In addition, the heat fluxes in some applications go beyond the maximum value in the Cheng-Ribatski-Wojtan-Thome flow boiling heat transfer model. Furthermore, the model does not extrapolate well to these conditions. In addition, the heat transfer model does not include heat transfer methods for CO2 in mist flow and bubbly flow regimes due to the lack of the experimental data in these regimes, which were not available at that time. Therefore, it is necessary to update the heat transfer model for CO2 to cover a wider range of conditions and these flow regimes. Furthermore, a flow pattern based two-phase pressure drop model is also needed for CO2.

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2.3. CO2 Flow Pattern Map, Flow Boiling Heat Transfer and Two-phase Pressure Drop Models Flow patterns are very important in understanding the very complex two-phase flow phenomena and heat transfer trends in flow boiling. To predict the local flow patterns in a channel, a flow pattern map is used. In fact, successful flow pattern based flow boiling heat transfer and two-phase flow pressure drop models [40-44, 50-53] have been proposed in recent years. Over the past decades, many flow pattern maps have been developed to predict two-phase flow patterns in horizontal tubes, such as those by Baker [54], Taitel and Dukler [55], Hashizume [56], Steiner [57] and so on, just to name a few. Most were developed for adiabatic conditions and then extrapolated by users to diabatic conditions, thereby creating big discrepancies. For this reason, a number of diabatic flow pattern maps related to the corresponding heat transfer mechanisms have been developed [40-44]. However, none of these is applicable to CO2 evaporation in horizontal tubes because the two-phase flow characteristics of CO2 evaporation are greatly affected by the very high reduced pressures and low surface tensions of CO2. In addition, the very low viscosities of CO2 at high reduced pressures may affect the two-phase pressure drop greatly. Recently, Cheng et al. [58, 59] proposed a new flow pattern map and a new general flow boiling heat transfer model, flow map and two-phase pressure drop model for CO2 in macro- and micro-scale channels to meet the wide range of parameters used. The details of the flow pattern map, flow pattern based flow boiling heat transfer model and pressure drop model are presented in the following sections.

2.3.1. A General Flow Pattern Map for CO2 Evaporation An updated CO2 flow pattern map was developed based on the Cheng-Ribatski-WojtanThome CO2 flow pattern map [38, 39] according to sharp changes of trends in flow boiling data that indicate such things as onset of dryout and onset of mist flow. The physical properties of CO2 have been obtained from REFPROP version 6.01 of NIST [22]. The flow map is intrinsically related to the flow boiling heat transfer model in section 2.3.2. For non-circular channels, equivalent diameters rather than hydraulic diameters were used in the flow pattern map, as Deq =

4A

π

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(2)

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Lixin Cheng and John R. Thome

Using the equivalent diameter gives the same mass velocity as in the non-circular channel and thus correctly reflects the mean liquid and vapor velocities, something using hydraulic diameter in a two-phase flow does not. In the updated CO2 flow pattern map, several new features were developed as compared to the Cheng-Ribatski-Wojtan-Thome flow pattern map [38, 39]: (1) Combining with the updated flow boiling heat transfer model for CO2 in section 2.3.2, the annular flow to dryout region (A-D) transition boundary was further modified so as to better fit the sharp changes in flow boiling heat transfer characteristics for higher mass velocities; (2) Based on experimental heat transfer data, a new criterion for the dryout region to mist flow (D-M) transition was proposed; (3) Bubbly flow occurs at very high mass velocities and very low vapor qualities and a bubbly flow pattern boundary was integrated into the map to make it more complete.

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With these modifications, the updated flow pattern map for CO2 is now applicable to much higher mass velocities. Complete flow pattern transition criteria of the updated flow pattern map for CO2 are described below.

Figure 23. Schematic diagram of stratified two-phase flow in a horizontal channel.

As shown in Fig. 23, the six dimensionless geometrical parameters used in the flow pattern map are defined as:

hLD =

hL Deq

(3)

PLD =

PL Deq

(4)

PVD =

PV Deq

(5)

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PiD =

Pi Deq

(6)

ALD =

AL Deq 2

(7)

AVD =

AV Deq 2

(8)

where Deq is the internal tube equivalent diameter (for non-circular channels, equivalent diameter Deq is used. So for circular channels, equivalent diameter Deq equals hydraulic diameter Dh), PL is the wetted perimeter, PV is the dry perimeter in contact with vapor, AL and AV are the corresponding cross-sectional areas of the liquid and vapor phases, Pi is the length of the phase interface and hL is the height of the liquid phase from the bottom of the tube. As a practical option and for consistency between the flow pattern map and the flow boiling heat transfer model, an easier to implement version of the flow map was proposed by Thome and El Hajal [60]. The void fraction ε which is determined with the Rouhani-Axelsson drift flux model [61] by Thome and El Hajal is kept the same in the present new flow map for CO2 as: 1/ 4 ⎛ x 1 − x ⎞ 1.18 (1 − x ) ⎡⎣ gσ ( ρ L − ρV ) ⎤⎦ ⎤ x ⎡⎢ ⎥ ε= (1 + 0.12 (1 − x ) ) ⎜ ρ + ρ ⎟ + ρV ⎢ G ρ L1/ 2 ⎥ V L ⎠ ⎝ ⎣ ⎦

−1

(9)

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Then, the dimensionless parameters are determined as follows: ALD =

A (1 − ε )

AVD =

Deq 2 Aε Deq 2

(10)

(11)

⎛ ⎛ 2π − θ strat ⎞ ⎞ hLD = 0.5 ⎜1 − cos ⎜ ⎟⎟ 2 ⎝ ⎠⎠ ⎝

(12)

⎛ 2π − θ strat ⎞ PiD = sin ⎜ ⎟ 2 ⎝ ⎠

(13)

where the stratified angle θstrat (which is the same as θdry shown in Fig. 23) is calculated with the equation proposed by Biberg [62]:

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θ strat

1/ 3 ⎧ ⎫ 1/ 3 ⎛ 3π ⎞ 1/ 3 ⎪π (1 − ε ) + ⎜ ⎟ ⎡⎣1 − 2 (1 − ε ) + (1 − ε ) − ε ⎤⎦ ⎪ ⎪ ⎪ ⎝ 2 ⎠ = 2π − 2 ⎨ ⎬ 2 ⎪ 1 2 − (1 − ε ) ε ⎡⎣1 − 2 (1 − ε )⎤⎦ ⎡⎣1 + 4 (1 − ε ) + ε ⎤⎦ ⎪⎪ ⎩⎪ 200 ⎭

(14)

Taking into account the modifications in the annular flow to dryout (A-D), dryout to mist flow (D-M) and intermittent flow to bubbly flow (I-B) transition curves which were newly developed in this study, the implementation procedure of the updated flow pattern map for CO2 is as follows: The void fraction ε and dimensionless geometrical parameters ALD, AVD, hLD and PiD are calculated with Eqs. (9) to (13). The stratified-wavy to intermittent and annular flow (SWI/A) transition boundary is calculated with the Kattan-Thome-Favrat criterion [42-44]:

Gwavy

⎧ 3 ⎪ 16 AVD gDeq ρ L ρV =⎨ 2 1/ 2 ⎪ x 2π 2 ⎡1 − ( 2hLD − 1) ⎤ ⎣ ⎦ ⎩

1/ 2

⎡ π2 ⎢ 2 ⎢⎣ 25hLD

⎛ FrL ⎜ ⎝ WeL

⎫ ⎞ ⎤⎪ 1 + ⎟ ⎥⎬ ⎠ ⎥⎦ ⎪ ⎭

+ 50

(15)

where the liquid Froude number FrL and the liquid Weber number WeL are defined as FrL =

G2 ρ L 2 gDeq

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WeL =

(16)

G 2 Deq

(17)

ρ Lσ

Then, the stratified-wavy flow region is subdivided into three zones according the criteria by Wojtan et al. [40, 41]: • • •

G > Gwavy(xIA) gives the slug zone; Gstrat < G < Gwavy(xIA) and x < xIA give the slug/stratified-wavy zone; x ≥ xIA gives the stratified-wavy zone.

The stratified to stratified-wavy flow (S-SW) transition boundary is calculated with the Kattan-Thome-Favrat criterion [42-44]: ⎡ 226.32 ALD AVD 2 ρV ( ρ L − ρV ) μ L g ⎤ =⎢ ⎥ x 2 (1 − x ) π 3 ⎢⎣ ⎥⎦

1/ 3

Gstrat

(18)

For the new flow pattern map: Gstrat = Gstrat(xIA) at x < xIA. The intermittent to annular flow (I-A) transition boundary is calculated with the ChengRibatski-Wojtan-Thome criterion [38, 39]:

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Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2… −1/ 7 −1/1.75 ⎡ ⎤ ⎛ μL ⎞ 1/ 0.875 ⎛ ρV ⎞ + 1⎥ xIA = ⎢1.8 ⎜ ⎟ ⎜ ⎟ ⎢⎣ ⎥⎦ ⎝ ρL ⎠ ⎝ μV ⎠

563

−1

(19)

Then, the transition boundary is extended down to its intersection with Gstrat. The annular flow to dryout region (A-D) transition boundary is calculated with the new modified criterion of Wojtan et al. [40] based of the dryout data of CO2 in this study: 1.471

Gdryout

−0.17 −0.17 −0.27 −0.25 ⎧ 1 ⎡ 0.58 ⎤ ⎛ ρV ⎞ ⎛ q ⎞ ⎫⎪ ⎤ ⎛ Deq ⎞ ⎡ 1 ⎪ ⎛ ⎞ =⎨ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ ⎟ ⎬ ⎟ + 0.52⎥ ⎜ ⎢ln ⎜ ⎦ ⎝ ρVσ ⎠ ⎣⎢ gD eq ρV ( ρL − ρV ) ⎦⎥ ⎝ ρL ⎠ ⎝ qcrit ⎠ ⎪⎭ ⎪⎩ 0.236 ⎣ ⎝ x ⎠

(20)

which is extracted from the new dryout inception equation in this study: 0.27 ⎤ ⎡ 0.17 Fr 0.17 ρ / ρ 0.25 q / q ⎢0.52 − 0.236WeV ⎥ V , Mori V L crit ⎦

(

xdi = 0.58e ⎣

)

(

)

(21)

This equation remains the same as in the Wojtan et al. [40] flow map for low pressure refrigerants, except that new empirical parameters were obtained based on the CO2 data since the previous expression did not extrapolate well to reduced pressures far higher than its underlying database. The vapor Weber number WeV and the vapor Froude number FrV,Mori defined by Mori et al. [63] are calculated as WeV =

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FrV , Mori =

G 2 Deq

(22)

ρV σ

G2 ρV ( ρ L − ρ V ) gDeq

(23)

and the critical heat flux qcrit is calculated with the Kutateladze [64] correlation as qcrit = 0.131ρV 0.5 hLV ⎡⎣ gσ ( ρ L − ρV ) ⎤⎦

0.25

(24)

The dryout region to mist flow (D-M) transition boundary is calculated with the new criterion developed in this study based on the dryout completion data for CO2: 1.613

−0.15 −0.16 −0.72 0.09 ⎧ ⎡ ⎤ ⎛ ρV ⎞ ⎛ q ⎞ ⎪⎫ 1 ⎤ ⎛ Deq ⎞ ⎪ 1 ⎡ ⎛ 0.61 ⎞ ln 0.57 + GM = ⎨ ⎢ ⎥ ⎟ ⎬ ⎜ ⎟ ⎜ ⎟ ⎢ ⎜ ⎥⎜ ρ σ ⎟ ⎦⎝ V ⎠ ⎢⎣ gDeq ρV ( ρ L − ρV ) ⎥⎦ ⎝ ρ L ⎠ ⎝ qcrit ⎠ ⎪⎭ ⎪⎩ 0.502 ⎣ ⎝ x ⎠

(25)

which is extracted from the dryout completion (which means the wall remains completely dry) equation developed in this study by solving for GM from: xde = 0.61e

0.72 ⎤ ⎡ 0.16 Fr 0.15 ρ / ρ −0.09 q / q ⎢0.57 − 0.502WeV ⎥ V , Mori V L crit ⎣ ⎦

(

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)

(

)

(26)

564

Lixin Cheng and John R. Thome

Again, this equation and its dimensionless groups remain the same as those used in the previous method [40] for conventional low reduced pressure refrigerants and only some empirical values were changed when correlating it to the CO2 data. The vapor Weber number WeV and the vapor Froude number FrV,Mori are calculated with Eqs. (22) and (23). The intermittent to bubbly flow (I-B) transition boundary is calculated with the criterion which arises at very high mass velocities and low qualities [42-44]: 2 1.25 ⎪⎧ 256 AVD ALD Deq ρ L ( ρ L − ρ V ) g ⎪⎫ GB = ⎨ ⎬ 1.75 2 0.25 ⎩⎪ 0.3164(1 − x) π PiD μL ⎭⎪

1/1.75

(27)

If G > GB and x < xIA, then the flow is bubbly flow (B). The following conditions are applied to the transitions in the high vapor quality range:

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• • •

If Gstrat(x) ≥ Gdryout(x), then Gdryout(x) = Gstrat(x) If Gwavy(x) ≥ Gdryout(x), then Gdryout(x) = Gwavy(x) If Gdryout(x) ≥ GM(x), then Gdryout(x) = GM(x)

Figure 24. Flow patterns observed by Gasche [65] at the experimental conditions: G=149 kg/m2s, Tsat = 23.3°C, Deq = 0.833 mm, q = 1.86 kW/m2 where (1), (2), (3) and (4) ⎯ plug flow; (5) ⎯ slug/annular flow; (6) ⎯ annular flow.

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Gashe [65] recently conducted an experimental study of CO2 evaporation inside a 0.8 mm diameter rectangular channel for various mass velocities and observed flow patterns by flow visualization as well. The updated CO2 flow pattern map was compared to his observations. It should be mentioned here that different names for the same flow patterns are used by different authors. Gasche in particular used the definition of plug flow, which is an intermittent flow in our flow pattern map. Just to show one example, Figure 24 shows the observed flow patterns of CO2 by Gashe for Deq = 0.833 mm (equivalent diameter is used here for the rectangular channel). Figure 25 shows the observations in Fig. 24 compared to the updated flow pattern map (in the flow pattern map, A is annular flow, D is dryout region, I is intermittent flow, M is mist flow, S is stratified flow and SW is stratified-wavy flow. The stratified to stratified-wavy flow transition is designated as S-SW, the stratified-wavy to intermittent/annular flow transition is designated as SW-I/A, the intermittent to annular flow transition is designated as I-A and so on.). It should be mentioned that the observed slug/annular flow of Gashe is counted as an annular flow in the updated flow pattern map. From the photographs in Fig. 24, it seems that the annular flow is the predominant flow in the slug/annular flow defined by Gashe. The observations (3) and (4) are near their correct regimes, especially by the typical flow pattern map standards.

Figure 25. The experimental data of the observed flow patterns by Gashe [65] in Fig. 2 shown in the updated CO2 flow pattern map where (1), (2), (3) and (4) ⎯ plug flow; (5) ⎯ slug/annular flow; (6) ⎯ annular flow.

Statistically, 82% of the total 28 flow pattern data of Gashe [65] are identified correctly by the updated flow map, or more specifically, 75% of the intermittent flows and 88% of the annular (slug/annular flow) flows [58]. The updated CO2 flow pattern map thus predicts the flow patterns observed by Gasche rather well. The lack of other new data in the literature should justify future experimental studies to obtain more. Furthermore, it is commonly understood that flow pattern transitions do not occur abruptly but over a range of conditions to complete the transition from one stable regime to the other, whereas transition lines on a map only represent the probable “centerline” of this transition range. With the limited data

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available for CO2 at this point, predicting the “width” of a transition zone around the transition line is not yet feasible, but it should be a good topic for future research.

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2.3.2. A General Flow Pattern Based Flow Boiling Heat Transfer Model for CO2 A new general flow boiling heat transfer model was developed by modifying the ChengRibatski-Wojtan-Thome [38, 39] flow boiling heat transfer model [58, 59]. By incorporating the updated new flow pattern map in the above section, the new heat transfer model is physically related to the flow regimes of CO2 evaporation, and thus correspondingly the new model has been extended to a wider range of conditions and to include new heat transfer methods in mist flow and bubbly flow regimes. The proposed new general flow boiling heat transfer model predicted reasonably well an extensive experimental database derived from the literature. To develop a general flow boiling heat transfer prediction method, it is important that the method is not only numerically accurate but that it also correctly captures the trends in the data to be useful for heat exchanger optimization. Most importantly, the flow boiling heat transfer mechanisms should be related to the corresponding flow patterns and be physically explained according to flow pattern transitions. Besides significantly extending the range of the heat transfer database here, several new modifications were implemented in the updated general flow boiling heat transfer model and will be presented below. Changes to the flow pattern map also have an effect on the heat transfer model: the new dryout inception vapor quality correlation (Eq. (21)) and a new dryout completion vapor quality correlation (Eq. (26)) are used to better segregate the data into these regimes, which have sharply different heat transfer performances. Accordingly, the flow boiling heat transfer correlation in the dryout region was updated. In addition, a new mist flow heat transfer correlation for CO2 was developed based on the CO2 data and a heat transfer method for bubbly flow was adopted for completeness sake. With these modifications, a new general flow boiling heat transfer model for CO2 was developed to meet a wider range of conditions and to cover all flow regimes [58, 59]. The Kattan-Thome-Favrat [42-44] general equation for the local flow boiling heat transfer coefficients htp in a horizontal tube is used as the basic flow boiling expression: htp =

θ dry hV + ( 2π − θ dry ) hwet 2π

(28)

where θdry is the dry angle defined in Figs. 23 and 26. The dry angle θdry defines the flow structures and the ratio of the tube perimeter in contact with liquid and vapor. In stratified flow, θdry equals the stratified angle θstrat which is calculated with Eq. (14). In annular (A), intermittent (I) and bubbly (B) flows, θdry = 0. For stratified-wavy flow, θdry varies from zero up to its maximum value θstrat. Stratified-wavy flow has been subdivided into three subzones (slug, slug/stratified-wavy and stratified-wavy) to determine θdry.

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Figure 26. Schematic diagram of film thickness.

For slug zone (Slug), the high frequency slugs are assumed to maintain a continuous thin liquid layer on the upper tube perimeter. Thus, similar to the intermittent and annular flow regimes, one has:

θ dry = 0

(29)

For stratified-wavy zone (SW), the following equation is proposed:

θ dry

⎛ Gwavy − G = θ strat ⎜ ⎜ Gwavy − Gstrat ⎝

⎞ ⎟⎟ ⎠

0.61

(30)

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For slug-stratified wavy zone (Slug+SW), the following interpolation between the other two regimes is proposed for x < xIA:

θ dry = θ strat

x ⎛ Gwavy − G ⎜ xIA ⎜⎝ Gwavy − Gstrat

⎞ ⎟⎟ ⎠

0.61

(31)

The vapor phase heat transfer coefficient on the dry perimeter hV is calculated with the Dittus-Boelter [66] correlation assuming tubular flow in the tube: hV = 0.023ReV 0.8 PrV 0.4

kV Deq

(32)

where the vapor phase Reynolds number ReV is defined as follows: ReV =

GxDeq

μV ε

(33)

The heat transfer coefficient on the wet perimeter hwet is calculated with an asymptotic model that combines the nucleate boiling and convective boiling heat transfer contributions to flow boiling heat transfer by the third power:

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Lixin Cheng and John R. Thome hwet = ⎡( Shnb ) + hcb3 ⎤ ⎣ ⎦ 3

1/ 3

(34)

where hnb, S and hcb are respectively nucleate boiling heat transfer coefficient, nucleate boiling heat transfer suppression factor and convective boiling heat transfer coefficient and are determined in the following equations The nucleate boiling heat transfer coefficient hnb is calculated with the Cheng-RibatskiWojtan-Thome [38, 39] nucleate boiling correlation for CO2 which is a modification of the Cooper [67] correlation: hnb = 131 pr −0.0063 ( − log10 pr )

−0.55

M −0.5 q 0.58

(35)

The Cheng-Ribatski-Wojtan-Thome [38, 39] nucleate boiling heat transfer suppression factor S for CO2 is applied to reduce the nucleate boiling heat transfer contribution due to the thinning of the annular liquid film: If x < xIA, S = 1

(36)

⎛ Deq ⎞ If x ≥ xIA, S = 1 − 1.14 ⎜ ⎟ ⎝ 0.00753 ⎠

2

⎛ δ ⎞ ⎜1 − ⎟ δ IA ⎠ ⎝

2.2

(37)

Furthermore, if Deq > 7.53 mm, then set Deq = 7.53 mm. The liquid film thickness δ shown in Fig. 1 is calculated with the expression proposed by El Hajal et al. [68]:

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2

⎛ Deq ⎞ 2 AL δ= − ⎜ ⎟ − 2 ⎝ 2 ⎠ 2π − θ dry Deq

(38)

where AL, based on the equivalent diameter, is cross-sectional area occupied by liquid-phase shown in Fig. 23. When the liquid occupies more than one-half of the cross-section of the tube at low vapor quality, this expression would yield a value of δ > Deq/2, which is not geometrically realistic. Hence, whenever Eq. (38) gives δ > Deq/2, δ is set equal to Deq/2 (occurs when ε < 0.5). The liquid film δIA is calculated with Eq. (38) at the intermittent (I) to annular flow (A) transition. The convective boiling heat transfer coefficient hcb is calculated with the following correlation assuming an annular liquid film flow from the original model [44]: hcb = 0.0133Reδ 0.69 PrL 0.4

kL

δ

(39)

where the liquid film Reynolds number Reδ is defined as [43]: Reδ =

4G (1 − x ) δ

μ L (1 − ε )

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Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2…

569

The void fraction ε is calculated with Eq. (9) and δ is calculated with Eq. (38). The heat transfer coefficient in mist flow is calculated by a new correlation developed in this study, which is a modification of the correlation by Groeneveld [69], with a new lead constant and a new exponent on ReH according to CO2 experimental data: hM = 2 × 10−8 Re H 1.97 PrV 1.06 Y −1.83

kV Deq

(41)

where the homogeneous Reynolds number ReH and the correction factor Y are calculated as follows: ReH =

GDeq ⎡ ⎤ ρV (1 − x )⎥ ⎢x + μV ⎣ ρL ⎦

⎡⎛ ρ ⎤ ⎞ Y = 1 − 0.1 ⎢⎜ L − 1⎟ (1 − x ) ⎥ ⎠ ⎣⎢⎝ ρV ⎦⎥

(42)

0.4

(43)

The heat transfer coefficient in the dryout region is calculated by a linear interpolation proposed by Wojtan et al. [41]:

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hdryout = htp ( xdi ) −

x − xdi ⎡ htp ( xdi ) − hM ( xde ) ⎤⎦ xde − xdi ⎣

(44)

where htp(xdi) is the two-phase heat transfer coefficient calculated with Eq. (28) at the dryout inception quality xdi and hM(xde) is the mist flow heat transfer coefficient calculated with Eq. (41) at the dryout completion quality xde. Dryout inception quality xdi and dryout completion quality xde are respectively calculated with Eqs. (21) and (26). If xde is not defined at the mass velocity being considered, it is assumed that xde = 0.999. A heat transfer model for bubbly flow was added to the model for completeness sake. In the absence of any data, the heat transfer coefficients in bubbly flow regime are calculated by the same method as that in the intermittent flow. Eq. (28) is used to calculate the local flow boiling heat transfer coefficients. In bubbly (B) flow, the dryout angle θdry = 0. The updated general flow boiling heat transfer model was compared to an extensive database [59]. Just to show one example here, Figure 27(a) shows the comparison of the predicted flow boiling heat transfer coefficients to the experimental data of Yun et al. [26] and Figure 27(b) shows the corresponding flow map. The updated general flow boiling heat transfer model not only captures the heat transfer trends well but also predicts the experimental heat transfer data well. As it is harder to predict (and harder to accurately measure) heat transfer data in the dryout and mist flow regimes, the updated general heat transfer model does not always predict the experimental data in these two flow regimes satisfactorily. Some examples of such comparisons can be found in [59]. Figure 28 shows simulation of the updated flow pattern map and flow boiling model for CO2 at the indicated conditions, superimposed on the same graphs by Cheng et al. [70]. The

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process path for the vapor quality variation from x = 0.01 to x = 0.99 is shown as the horizontal broken line (dash-dot line) while the variation in the heat transfer coefficient as it changes vapor quality and flow pattern is depicted by the dashed line. The flow pattern boundaries are in solid lines. The line (dash line with arrows) indicates the calculated heat transfer coefficient at the indicated mass velocity and vapor quality. Notice the various changes in trends in the heat transfer coefficient as this occurs. For example, when the flow regime passes from annular flow into the dryout regime, there is a sharp inflection in the heat transfer coefficient as the top perimeter of the tube becomes dry.

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(a)

(b) Figure 27. (a) Comparison of the predicted flow boiling heat transfer coefficients to the experimental data of Yun et al. [26]; (b) the corresponding flow pattern map (at the test conditions: Deq = 2 mm, q = 30 kW/m2, Tsat = 5°C and G = 1500 kg/m2s )s.

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571

Figure 28. Simulation of flow boiling heat transfer model and flow pattern map for 3 mm channel at the conditions: q = 20 kW/m2, Tsat = 10°C and G = 390 kg/m2s with indicated value at x = 0.70 [70].

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2

Predicted heat transfer coefficient [kW/m K]

30

25 +30% 20

15 -30% 10

5

0 0

5 10 15 20 25 2 Experimental heat transfer coefficient [kW/m K]

1 2 3 4 5 6 7 8 9 10 11 12 13

30

Figure 29. Comparison of the predicted flow boiling heat transfer coefficients to all heat transfer data without the dryout and mist flow data points in the entire database: 1 ⎯ Knudsen and Jensen [33], 2⎯ Yun at al. [25], 3 ⎯ Yoon et al. [28], 4 ⎯ Koyama et al. [30], 5 ⎯ Pettersen [24], 6 ⎯ Yun et al. [29], 7 ⎯ Gao and Honda [90, 91], 8 ⎯ Tanaka et al. [72], 9 ⎯ Hihara [27], 10 ⎯ Shinmura et al. [73], 11 ⎯ Zhao et al. [75, 76], 12 ⎯ Yun et al. [26, 71] and 13 ⎯ Jeong et al. [74] (Note: 1⎯6 were used in our previous study [38, 39]).

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2

Predicted heat transfer coefficient [kW/m K]

30

25 +30% 20

15

-30%

10

5

0 0

1 2 3 4 5 6 7 8 9 10 11

5 10 15 20 25 2 Experimenal heat transfer coefficient [kW/m K]

30

Figure 30. Comparison of the predicted flow boiling heat transfer coefficients to all dryout heat transfer data points in the entire database: 1 ⎯ Yun at al. [25], 2 ⎯ Koyama et al. [30], 3 ⎯ Pettersen [24], 4 ⎯ Yun et al. [29] 5 ⎯ Gao and Honda [90, 91], 6 ⎯ Tanaka et al. [72], 7 ⎯ Hihara [27], 8 ⎯ Shinmura et al. [73], 9 ⎯ Zhao et al. [75, 76], 10 ⎯ Yun et al. [26, 71] and 11 ⎯ Jeong et al. [74] (Note: 1 ⎯ 4 were used in our previous study [38, 39]).

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2

Predicted heat transfer coefficient [kW/m K]

30

25

20

+30%

15 -30% 10

5

0 0

5 10 15 20 25 2 Experimenal heat transfer coefficient [kW/m K]

1 2 3 4 5 6 7 8 9 30

Figure 31. Comparison of the predicted flow boiling heat transfer coefficients to all mist flow heat transfer data points in the entire database: 1 ⎯ Yun at al. [25], 2 ⎯ Koyama et al. [30], 3 ⎯ Yun et al. [29] 4 ⎯ Gao and Honda [90, 91], 5 ⎯ Tanaka et al. [72], 6 ⎯ Hihara [27], 7 ⎯ Shinmura et al. [73], 8 ⎯ Yun et al. [26, 71] and 9 ⎯ Jeong et al. [74] (Note: 1 ⎯ 3 were used in our previous study [38, 39]).

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573

In further analysis, comparisons have also been made by classes of flows, i.e. the predictions versus all the heat transfer data excluding dryout and mist flow data (essentially the all wet perimeter data), versus all dryout heat transfer data (the partially wet perimeter data) and versus the mist flow data (all dry perimeter data) [59]. Figure 29 shows the comparison to the first group, Figure 30 the second and Figure 31 the third. The statistical analysis has shown the following fraction of the database are predicted within ±30%: 71.4% of the entire database (1124 points), 83.2% of the all wet wall data points (773 points), 47.6% of the partially wet wall data points (191 points) and 48.2% of the all dry wall data points (160 points). Overall, the updated general flow boiling heat transfer model predicts the overall database quite well. However, for the dryout and mist flow regimes with partially or all dry perimeters, the heat transfer model is only partially satisfactory. For these last two regimes, many of the experimental data sets have a level of scatter ranging up 40% themselves. In part, the larger errors are due to the very sharp change in trend in these data with vapor quality, where an error of a 2-3% in vapor quality in the energy balance of the experiments or in the prediction of xdi and/or xde immediately results in a heat transfer prediction error of 50%. Therefore, more careful experiments are needed in these two regimes to provide more accurate heat transfer data, with attention to also determine the transitions xdi and xde, because they are typical working conditions in the micro-scale channels of extruded multi-port aluminum tubes used for automobile air-conditioners that operate over a wide range of mass velocities up to as high as 1500 kg/m2s.

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2.3.3. A Phenomenological Two-Phase Pressure Drop Model Based on Flow Patterns for CO2 As the predictions of two-phase flow frictional pressure drops with the leading methods often cause errors of more than 50% [50-53], efforts are increasingly being made to improve on these methods. In addition, the leading pressure drop prediction methods do not usually contain any flow pattern information, which is intrinsically related to the two-phase frictional pressure drop. As for CO2, the leading prediction methods do not work well. The reason is that these methods do not usually cover the much lower liquid-to-vapor density ratios and very small surface tension characteristics of CO2 at high pressures. Due to these characteristics, normally the two-phase flow pressure drops of CO2 are much lower than those of other refrigerants [38, 39]. Significantly, there is no proven generally applicable two-phase pressure drop prediction method for CO2, although there are a number of studies on CO2 pressure drops in the literature [28, 74-79]. Some researchers proposed pressure drop correlations for CO2 based on their own experimental data but such methods do not work well when extrapolated to other conditions. For example, Yoon et al. [28] proposed a modified Chisholm method to fit their data in a macro-scale channel but it cannot be applied to other conditions because they tested only one diameter. As opposed to the completely empirical two-phase pressure drop models, a flow pattern based phenomenological model relating the flow patterns to the corresponding two-phase flow pressure drops is a promising approach in the two-phase pressure drop predictions. Ould Didi et al. [50] used local flow patterns to analyze two-phase flow pressure drops, which resulted in a significant improvement in accuracy. Based on that, a new flow pattern based phenomenological model of two-phase frictional pressure drops was recently developed by

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Two-phase frictional pressure gradient [Pa/m]

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Moreno Quibén and Thome [51-53]. The model physically respects the two-phase flow structure of the various flow patterns while maintaining a degree of simplicity as well. The model predicts their experimental data well but not the present CO2 experimental database. Cheng et al. [58] compiled a large database of CO2 two-phase flow pressure drop was set up and compared the database to the leading two-phase frictional pressure drop methods: the empirical two-phase frictional pressure drop methods by Chisholm [80], Friedel [81], Grönnerud [82] and Müller-Steinhagen and Heck [83], a modified Chisholm correlation by Yoon et al. [28] and the flow patterned based pressure drop model by Moreno Quibén and Thome [51-53]. The CO2 database includes the experimental data of Bredesen et al. [32], Pettersen [24], Pettersen and VestbØstad [77], Zhao et al. [75, 76] and Yun and Kim [78, 79]. The test channels include single circular channels and multi-channels with circular, triangular and rectangular cross-sections and electrical and fluid heated test sections. The data were taken from tables where available or by digitizing the pressure drops from graphs in these publications. All together 387 two-phase pressure drop data points were obtained. Figure 32 shows the comparison of these leading pressure drop methods to the experimental data of Bredesen et al. [32] at the indicated test conditions. There are big differences among these methods. Overall, not one of these models is able to predict the CO2 two-phase frictional pressure drop data well (note that all have been extrapolated beyond their original conditions to make this comparison for CO2). The Friedel method gave reasonably good predictions, but it failed to predict the pressure drop in smaller channels. Therefore, it is necessary to develop a new model for CO2 two-phase pressure drop in macro- and micro-channels.

9000 Experimental 1 2 3 4 5

8000 7000 6000

3

1

5000 2

4000 4

3000 2000

5

1000 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Vapor quality

0.7

0.8

0.9

1

Figure 32. Comparison of the several leading methods to the experimental data of Bredesen et al. [32] at the experimental conditions: G = 400 kg/m2s, Tsat = -10 °C, Deq = 7 mm and q =6 kW/m2; 1- The Moreno-Quibén and Thome model [51, 52]; 2- The Friedel method [81]; 3- The Grönnerud method [82]; 4- The Müller-Steighagen-Heck method [83]; 5- The Chisholm method [80].

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575

A new two-phase frictional pressure drop model for CO2 was made by modifying the model of Moreno Quibén and Thome developed for R-22, R-410a and R-134a and incorporating the updated CO2 flow pattern map in section 2.3.1, using the CO2 pressure drop database by Cheng et al. [58]. In developing this pressure drop model, two-phase frictional pressure drop data were used. The total pressure drop is the sum of the static pressure drop (gravity pressure drop), the momentum pressure drop (acceleration pressure drop) and the frictional pressure drop: Δptotal = Δpstatic + Δpm + Δp f

(45)

For horizontal channels, the static pressure drop equals zero. Furthermore, the momentum pressure drop can be calculated as ⎧ ⎫ 2 ⎡ (1 − x )2 x2 ⎤ x2 ⎤ ⎪ ⎪ ⎡ (1 − x) ⎥ ⎬ Δpm = G ⎨ ⎢ + + ⎥ −⎢ ρ (1 − ε ) ρV ε ⎥⎦ out ⎢ ρ L (1 − ε ) ρV ε ⎥ ⎪ ⎣ ⎦ in ⎭ ⎩⎪ ⎢⎣ L 2

(46)

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Thus, diabatic experimental tests that measure total pressure drops can be reduced using the above expressions to find the two-phase frictional pressure drops. The details of the updated two-phase flow frictional pressure drop model for CO2 are as follows (For non-circular channels, the equivalent diameter Deq is used in the pressure drop model to remain consistent with that in the flow pattern map. Using the equivalent diameter gives the same mass velocity as in the non-circular channel and thus correctly reflects the mean liquid and vapor velocities, something using hydraulic diameter in a two-phase flow does not. Thus, equivalent diameter Deq is used in the following method: 1) CO2 frictional pressure drop model for annular flow (A): The basic equation is the same as that of the Moreno- Quibén-Thome [51-53] pressure drop model: Δp A = 4 f A

L ρV uV 2 Deq 2

(47)

where the two-phase flow friction factor of annular flow fA was correlated according to CO2 experimental data here (considering the main parameters which affect the two-phase pressure drops for CO2) as: f A = 3.128ReV −0.454WeL −0.0308

(48)

This correlation is thus different from that of the Moreno Quibén-Thome [51-53] pressure drop model. The mean velocity of the vapor phase uV is calculated as uV =

Gx

ρV ε

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Lixin Cheng and John R. Thome

The void fraction ε is calculated using Eq. (9). The vapor phase Reynolds number ReV and the liquid phase Weber number WeL based on the mean liquid phase velocity uL are calculated as ReV =

GxDeq

μV ε

(50)

WeL =

ρ L uL 2 Deq σ

(51)

uL =

G (1 − x) ρ L (1 − ε )

(52)

2) CO2 frictional pressure drop model for slug and intermittent flow (Slug+I): A proration is proposed to avoid any jump in the pressure drops between these two flow patterns, so that the Moreno Quibén-Thome [51-53] pressure drop model is updated to become: ⎛ ⎛ ε ⎞ ε ⎞ ΔpSLUG + I = ΔpLO ⎜ 1 − ⎟ + Δp A ⎜ ⎟ ε IA ⎠ ⎝ ⎝ ε IA ⎠

(53)

where ΔpA is calculated with Eq. (47) and the single phase frictional pressure drop considering the total vapor-liquid two-phase flow as liquid flow ΔpLO is calculated as

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ΔpLO = 4 f LO

L G2 Deq 2 ρ L

(54)

The friction factor is calculated with the Blasius equation as f LO =

0.079 0.25 ReLO

(55)

where Reynolds number ReLO is calculated as ReLO =

GDeq

μL

(56)

3) CO2 frictional pressure drop model for stratified-wavy flow (SW): The equation is kept the same as that of the Moreno Quibén-Thome [51-53]pressure drop model: ΔpSW = 4 f SW

L ρV uV 2 Deq 2

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Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2…

577

where the two-phase friction factor of stratified-wavy flow fSW is calculated with the following interpolating expression (a modification of that used in the Moreno Quibén-Thome [51-53] pressure drop model) based on the CO2 database: * 0.02 * f SW = θ dry fV + (1 − θ dry )0.02 f A

(58)

* and the dimensionless dry angle θ dry is defined as

* θ dry =

θ dry 2π

(59)

where θdry is the dry angle as shown in Fig. 23. The dry angle θdry defines the flow structure and the ratio of the tube perimeter in contact with vapor. For the stratified-wavy regime (SW), θdry is calculated with Eq. (30). The single-phase friction factor of the vapor phase fV is calculated as fV =

0.079 ReV 0.25

(60)

where the vapor Reynolds number Rev is calculated with Eq. (50).

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4) CO2 frictional pressure drop model for slug-stratified wavy flow (Slug+SW): It is proposed to avoid any jump in the pressure drops between these two flow patterns and to updated the Moreno Quibén-Thome [51-53] pressure drop model as: ⎛ ⎛ ε ⎞ ε ⎞ ΔpSLUG + SW = ΔpLO ⎜ 1 − ⎟ + ΔpSW ⎜ ⎟ ⎝ ε IA ⎠ ⎝ ε IA ⎠

(61)

where ΔpLO and ΔpSW are calculated with Eqs. (54) and (57), respectively. 5) CO2 frictional pressure drop model for mist flow (M): The following expression is kept the same as that in the Moreno-Quibén-Thome [51-53] pressure drop model: ΔpM = 4 f M

L G2 Deq 2 ρ H

(62)

The homogenous density ρH is defined as

ρ H = ρ L (1 − ε H ) + ρV ε H where the homogenous void fraction εH is calculated as

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Lixin Cheng and John R. Thome ⎛

ε H = ⎜1 + ⎝

(1 − x) ρV ⎞ ⎟ x ρL ⎠

−1

(64)

and the friction factor of mist flow fM was correlated according to the CO2 experimental data, which is different from that in the Moreno Quibén-Thome [51-53] pressure drop model, as: fM =

91.2 Re M 0.832

(65)

The Reynolds number is defined as: ReM =

GDeq

(66)

μH

where the homogenous dynamic viscosity is calculated as proposed by Ciccitti et al. [84]:

μ H = μ L (1 − x) + μV x

(67)

The constants in Eq. (65) are quite different from those in the Blasius equation. The reason is possibly because there are limited experimental data in mist flow in the database and also perhaps a lower accuracy of these experimental data. Therefore, more accurate experimental data are needed in mist flow to further verify this correlation or modify it if necessary in the future.

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6) CO2 frictional pressure drop model for dryout region (D): The linear interpolating expression is kept the same as that in the Moreno Quibén-Thome pressure drop model as: Δpdryout = Δptp ( xdi ) −

x − xdi ⎡ Δptp ( xdi ) − ΔpM ( xde ) ⎤⎦ xde − xdi ⎣

(68)

where Δptp(xdi) is the frictional pressure drop at the dryout inception quality xdi and is calculated with Eq. (47) for annular flow or with Eq. (57) for stratified-wavy flow, and ΔpM(xde) is the frictional pressure drop at the completion quality xde and is calculated with Eq. (62). xdi and xde are respectively calculated with Eqs. (21) and (26). 7) CO2 frictional pressure drop model for stratified flow (S): No data fell into this flow regime but for completeness, the method is kept the same as that in the Moreno QuibénThome [51-53] pressure drop model as: For x ≥ xIA: Δpstrat ( x ≥ xIA ) = 4 f strat ( x ≥ xIA )

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L ρV uV 2 Deq 2

(69)

Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2…

579

where the mean velocity of the vapor phase uV is calculated with Eq. (49) and the two-phase friction factor of stratified flow f strat ( x ≥ x ) is calculated as IA

* * f strat ( x ≥ xIA ) = θ strat fV + (1 − θ strat ) fA

(70)

The single-phase friction factor of the vapor phase fV and the two-phase friction factor of annular flow fA are calculated with Eqs. (60) and (48), respectively, and the dimensionless * is defined as stratified angle θ strat * θ strat =

θ strat 2π

(71)

where the stratified angle θstrat is calculated with Eq. (14). For x < xIA: ⎛ ⎛ ε ⎞ ε ⎞ Δpstrat ( x < xIA ) = ΔpLO ⎜ 1 − ⎟ + Δpstrat ( x ≥ xIA ) ⎜ ⎟ ⎝ ε IA ⎠ ⎝ ε IA ⎠

(72)

where ΔpLO and Δpstrat ( x ≥ x ) are calculated with Eqs. (54) and (69), respectively. IA

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8) CO2 frictional pressure drop model for bubbly flow (B): No data are available for this regime but keeping consistent with the frictional pressure drops in the neighboring regimes and following the same format as the others without creating a jump at the transition (there is no such a model in the Moreno-Quibén-Thome [51-53] pressure drop model), the following expression is used: ⎛ ε ΔpB = ΔpLO ⎜ 1 − ε IA ⎝

⎞ ⎛ ε ⎞ ⎟ + Δp A ⎜ ⎟ ⎠ ⎝ ε IA ⎠

(73)

where ΔpLO and ΔpA are calculated with Eqs. (54) and (47), respectively. Further experimental data are needed to verify or modify this model. The new updated CO2 two-phase frictional pressure drop model was compared to the CO2 two-phase pressure drop database [58]. Figure 33 shows the comparison of the new CO2 pressure drop model to the experimental data of Bredesen et al. [32] at the indicated experimental conditions and the corresponding flow pattern map. The model predicts the data well and also captures the pressure drop trend. Figure 34 shows the comparative results of the predictions by the new CO2 pressure drop model to the entire two-phase pressure drop database. In addition, the statistical results of the predicted and experimental data for individual research according to the percent of data predicted within ±30% are presented in Fig. 34. In addition, the detailed breakdown of the statistical analysis for the new pressure drop model is summarized [58]. Most of the experimental data points (75.5%) are in annular flow and 75.7% of experimental data in annular flow are predicted within ±30%. However, the predictions in some regions such as S-Slug and SW are not satisfactory. Generally, the new

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Lixin Cheng and John R. Thome

Two-phase frictional pressure gradient [kPa/m]

pressure drop model reasonably predicts the database and importantly captures the trends in the data too. Nonetheless, there are not many experimental data available covering some flow patterns and future experimental work is recommended to address these conditions. The new CO2 two-phase flow pressure drop model predicts the CO2 pressure drop database better than the existing methods. Due to the very few and less accurate experimental data in micro-scale channels currently available, the new CO2 pressure drop model does not predict these data satisfactorily. It is suggested that additional, more accurate experimental CO2 pressure drop data be obtained to further test or improve the model in the future. 5

4

3

2 Predicted Experimental

1

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Vapor quality

1

(a) M

600 Mass Velocity [kg/m2s]

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700

500

I

D

A

400 300 200 SW+ Slug 100 0 0

SW

S 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Vapor quality

1

(b) Figure 33. (a) Comparison of the new CO2 pressure drop model to the experimental data of Bredesen et al. [32] at the experimental conditions: G = 400 kg/m2s, Tsat = -10 °C, Deq = 7 mm and q = 3 kW/m2; (b) The corresponding flow pattern map at the same experimental condition as that in (a) (I represents intermittent flow, A represents annular flow, D represents dyrout region, M represents mist flow, S represents stratified flow and SW represents stratified-wavy flow).

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Predicted frictional pressure gradient [kPa/m]

Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2…

581

2

10

1 2 3 4

1

10

+30%

-30% 0

10

-1

10

-1

10

0

1

2

10 10 10 Experimental frictional pressure gradient [kPa/m]

Figure 34. Comparison of the predicted frictional pressure gradients by the new model to the entire database (74.7% of the data are predicted within ±30%). By individual study, the following percent of data are captured with ±30%: 1−Bredesen et al. [32] 81.5%, 2−Yun and Kim [78. 79] 33.3%, 3− Pettersen [24] and Pettersen and VestbØstad [77] 43.2%, and 4−Zhao et al. [75, 76] 55.6% (Note that 2, 3 and 4 are the data of micro-scale channels).

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3. Oil Effect on CO2 Two-Phase Pressure Drops and Flow Boiling Heat Transfer Flow boiling heat transfer and two-phase flow of refrigerant-lubricating oil mixtures are very complex phenomena but very important topics for air-conditioning, heat pump and refrigeration systems. The presence of lubricating oil may considerably affect the transport properties of a refrigerant and have a significant effect on the flow boiling heat transfer, twophase pressure drop and flow pattern characteristics. Due to its large viscosity, surface tension and insulating effect, lubricating oil tends to decrease the flow boiling heat transfer coefficient of a refrigerant in most cases while it may increase the flow boiling heat transfer coefficient of the refrigerant at low lubricating oil concentrations. Therefore, it is of great importance to understand how the lubricating oil concentration affects the heat transfer, twophase pressure drop and flow regime characteristics and to predict the heat transfer coefficient and two-phase pressure drop of such mixtures. Refrigerants are classified as completely miscible, partially miscible or immiscible according to their mutual solubility relations with lubricating oils. The miscibility of lubricating oils and refrigerants often has a great effect on the flow boiling heat transfer and two-phase pressure drop characteristics. To select the most suitable lubricating oils for CO2 automotive air-conditioning and heat pump systems, there are several potential issues which should be considered as

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(i) Lubricant transport: To ensure good oil return to the compressor, the refrigerant oil needs to have either a higher density than liquid CO2 or good miscibility with it. (ii) Wear: (a) CO2 is an excellent solvent and this solvency can cause excessive lubricant dilution leading to potential wear and foaming problems; (b) CO2 requires the use of higher operating pressures (e.g. in automotive air-conditioning, the working pressure is about one order of magnitude higher compared to R134a). Therefore, high loads increase the stress on bearings which can lead to increased wear. (iii) Stability: CO2 can react with water to form carbonic acid that can then accelerate potential hydrolysis processes. A number of lubricants for CO2 have been studied over the past years. According to the studies of lubricants for CO2 by Kaneko et al. [85] and Ikeda [86], PAGs (Plyalkyleneglycols) are the most suitable lubricants used in CO2 automotive air-conditioning so far. Unlike R134a, PAGs are immiscible with CO2. A literature survey of flow boiling heat transfer and two-phase pressure drop of CO2 with lubricating oil has been conducted. Only several papers have been found for the effect of lubricating oil on CO2 flow boiling heat transfer and two-phase pressure drop. Of these studies, some are related to flow boiling heat transfer and two-phase pressure drop in macroscale channels and others are related to flow boiling heat transfer in micro-scale channels. Analysis of these studies on flow boiling and two-phase pressure drop of CO2-oil mixtures has been carefully conducted to present a state-of-the-art status of the study on these topics. As the lubricating oil (PAG) used in CO2 automotive air-conditioning and heat pump systems is immiscible with CO2, emphasis has been placed on flow boiling heat transfer and two-phase pressure drop of CO2 with immiscible oil in micro-scale channels which are of interest to CO2 air-conditioning and heat pump systems. Unfortunately, there is only one study of flow boiling heat transfer and two-phase pressure drop of CO2 with immiscible oil in a 3 mm ID tube. With the very limited experimental data available, it is impossible to propose a model for flow boiling heat transfer of CO2 with immiscible oil. In addition, there is no study of two-phase pressure drop of CO2 with lubricating oil in micro-scale channels. Katsuta et al. [87] conducted an experimental study of flow boiling heat transfer and twophase pressure drop of CO2 and lubricating oil (PAG) mixture in a horizontal tube with an inner diameter of 4.59 mm. The oil concentrations were 1% (wt) and 5% (wt), respectively. The oil concentration had little effect on the two-phase pressure drops. However, the twophase pressure drops increased with increasing the oil concentrations at vapor qualities larger than 0.6. The possible reason of this trend is that the local oil concentration plays an important role in increasing two-phase pressure drops for the low mass velocities. Figure 35 shows their pressure drop data at two test conditions. It is interesting to see that at a mass velocity of 800 kg/m2s, the pressure drops of 5% oil are slightly lower than those of 1%. No explanation for such a result was presented in their paper. Katsuta et al. [88, 89] presented the experimental heat transfer and two-phase pressure drop data of the lubricating oil concentrations with 0 to 1.1% (wt). For the very low oil concentrations of 0.05% to 0.08% (wt), the flow boiling heat transfer coefficients are decreased about 40% compared to those without oil. This is quite different from the experimental results of Gao and Honda [90, 91] for a 3 mm I.D. tube. According to the study of Gao and Honda, the oil concentration does not affect the flow boiling heat transfer

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coefficients if the oil concentration is less than 0.1% (wt). Furthermore, there is no obvious difference between the flow boiling heat transfer coefficients of the oil concentrations of 0.05% to 0.08 % (wt) and those of the oil concentrations with 0.3% to 1.1% (wt). However, when the oil concentration is larger than 1%, heat transfer coefficient decreases greatly.

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(a)

(b) Figure 35. The effect of the oil on two-phase pressure drop [87]: G = 400 kg/m2s, q = 10 kW/m2; (b) G = 800 kg/m2s, q = 10 kW/m2.

Koyama et al. [92, 93] studied flow boiling of CO2-PAG mixtures inside a horizontal smooth tube with an inner diameter of 4.42 mm and a micro-fin tube with an inner diameter of 4.76 mm. The authors have concluded that the heat transfer coefficient for the smooth tube decreases abruptly with the addition of a very small amount of oil, while the deterioration rate of heat transfer coefficient for the micro-fin tube is smaller than that of the smooth tube. The two-phase pressure drop in the smooth tube increases with increasing the oil concentration

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especially at high vapor qualities larger than around 0.6 while that of the micro-fin tube increases slightly with increasing the oil concentration.

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Figure 36. The oil effect on heat transfer coefficient [90, 91]: Tsat = 10°C, G = 390 kg/m2s, q = 20 kW/m2.

Figure 37. The effect of oil on heat transfer coefficient [95] (Tsat = 10°C, G = 300 kg/m2s, q = 11 kW/m2).

For all the above studies in macro-scale channels, the oil concentrations were measured by sampling. There is no information on the measurement of the local oil concentration which is of great importance in modeling flow boiling heat transfer and two-phase pressure drop. There are only a few studies of flow boiling heat transfer in micro-scale channels and there is no study of two-phase pressure drop in micro-scale channels with oil. Siegismund and Kauffeld [94] only presented a description of their test facility and the data reduction methods

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585

in their paper. No experimental data were presented in their paper although the title of their paper is the influence of lubricant oil on CO2 heat transfer in mini-channel tubes. Gao and Honda [90, 91] conducted an experimental study on flow boiling heat transfer of CO2 and oil (PAG) mixtures inside a horizontal tube with an inner diameter of 3 mm. Three oil concentrations were used in their study: less than 0.1% (wt), 1% (wt) and 2% (wt). When the concentration of the lubricating oil is more than 1% (wt), the local heat transfer coefficient was much lower than that without the lubricating oil. According to the authors’ analysis, the reason for this decrease is considered due to the nucleate boiling being suppressed by the oil film. Therefore, the flow boiling heat transfer is considered to change from the nucleate boiling dominated mechanism to the convective evaporation dominated mechanism. The dryout vapor quality decreases with increasing mass velocity, and is a little influenced by the saturation temperature and the lubricating oil concentration. The experimental heat transfer data of the oil concentration less than 0.1% (wt) were already used in the database in our previous study of the CO2 flow boiling heat transfer model [59]. As independent experimental heat transfer data, the flow pattern based flow boiling heat transfer model of Cheng et al. [58, 59] predicts well the heat transfer data of the oil concentration less than 0.1% (wt). Therefore, it can be concluded from this study that the oil does not affect the heat transfer when the oil concentration is less than 0.1% (wt) [59]. Figure 36 shows the effect of lubricating oil on the local heat transfer coefficient at the indicated test conditions [90, 91]. The local heat transfer coefficient falls considerably when the lubricating oil concentration is larger than 1% (wt). Furthermore, when the lubricating oil concentration increases, the change of heat transfer coefficient with vapor quality becomes small. Therefore, when the oil concentration is larger than 1% (wt), the heat transfer coefficient is not greatly affected by further increasing the oil concentration. Zhao et al. [95] conducted an experimental study of flow boiling heat transfer of CO2 with miscible oil in multi micro-channels with an inner diameter of 0.86 mm (an equivalent diameter of 1.15 mm). Figure 37 shows their results of the oil effect on heat transfer coefficient at the indicated test conditions. A small concentration of the lubricating oil (< 3%) increases the flow boiling heat transfer coefficients at low vapor qualities less than 0.4. The authors attributed this to a foaming effect that increases wall-wetting. In addition, they concluded that increasing mass flux also improved the heat transfer coefficient in the presence of oil since the average vapor quality decreased for constant heat flux and inlet vapor quality. On the other hand, they concluded that increasing the vapor quality and the saturation temperature decreased the heat transfer coefficient in the presence of oil. It must be pointed out here that the authors did not mention what kind of oil was used in their study. It was a miscible oil. Apparently, the effect of the miscible oil on the heat transfer characteristics of CO2 is quite different from that of an immiscible oil. According to the analysis, for the study of flow boiling heat transfer and two-phase pressure drop of CO2 with an immiscible oil (PAG) in micro-scale channels, there is no such information in the literature so far. In general, very small amounts of the lubricating oil (< 0.5% wt) seem to have little effect while larger concentrations (> 1% wt) tend to dramatically reduce flow boiling heat transfer coefficients. For two-phase pressure drops, the effect of lubricating oil only occurs at vapor qualities larger than around 0.6. Based on this analysis, it is recommended that further experimental studies on flow boiling and two-phase flow of CO2-immiscible oil mixtures in both macro-scale and micro-

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scale channels be conducted over a wide range of test conditions to provide more experimental data on these aspects. In order to understand the heat transfer mechanisms of flow boiling with CO2 and immiscible oil, flow visualization and local measurement of oil concentration are also suggested to be done.

-2

-1

Heat Transfer Coefficient [W m K ]

7000 1

1: 0% 2: 0.5% 3: 1% 4: 3% 5: 5%

6000 5000

2 3

4000

4 5

3000 2000 1000 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Vapor Quality

0.7

0.8

0.9

1

5000

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-1

Frictional Pressure Gradient [Pa m ]

Figure 38. The oil effect (mass concentration from 0 to 5%) on flow boiling heat transfer coefficient of R134a in a horizontal tube (Simulated results by the flow pattern based heat transfer model of Wojtan et al. [40, 41]): G = 300 kg/m2s, Tsat = 10°C, D = 13.84 mm and q = 7.5 kW/m2.

1: 0% 2: 0.5% 3: 1% 4: 3% 5: 5%

4000

3000

2 4 5

3 1

2000

1000

0 0

0.1

0.2

0.3

0.4 0.5 0.6 Vapor Quality

0.7

0.8

0.9

Figure 39. The oil effect (mass concentration from 0 to 5%) on two-phase frictional pressure gradient of R134a in a horizontal tube for flow regimes before dryout region (The two-phase frictional pressure gradients were calculated using the flow pattern based two-phase pressure drop model of MorenoQuiben and Thome [51, 52] for pure R134a with a correction factor of oil effect on two-phase pressure drop): G = 300 kg/m2s, Tsat = 10°C, D = 13.84 mm and q = 7.5 kW/m2.

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As for the prediction methods for flow boiling heat transfer and two-phase pressure drop with CO2-immsicible oil mixtures, the flow pattern based flow boiling heat transfer and twophase pressure drop models of Cheng et al. [58, 59] may be used by introducing the physical properties of CO2-immiscible lubricating oil mixtures. As the heat transfer [59] model predicts well the independent experimental heat transfer data of the lubricating oil concentration less than 0.1 % (wt) by Gao and Honda [90, 91], it is expected that the heat transfer model may be modified to handle flow boiling heat transfer of CO2-immiscible lubricating oil mixtures. For the prediction of the two-phase pressure drops of CO2immiscible lubricating oil mixtures, the flow pattern based two-phase pressure drop model of Cheng et al. [58] for pure CO2 may be a good basis to start from when accurate and sufficient experimental data become available. Figure 38 shows the predicted heat transfer results of R134a-oil using the heat transfer model of Wojtan et al. [40, 41] and Figure 39 shows the predicted pressure drop results using the Moreno- Quibén-Thome [51, 52] pressure drop model with a correct factor [96]. These simulations show the effect of oil concentrations on flow boiling heat transfer coefficients and two-phase pressure drops. Due to the immiscibility of lubricant oil and CO2, whether these methods for miscible oil can be applied to CO2-lubricant oil or not still need to be further verified. Therefore, extensive experimental data are needed to verify the available methods or to develop new methods for CO2.

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4. Simulation of Cooling Electric Chips Using CO2 Evaporation Advances in micro-electronics technology continue to develop with surprisingly rapidity and the energy density of electronic devices to be dissipated is becoming much higher. Thus, the thermal emission delivered from micro-electronic elements and components is still increasing considerably. Therefore, it is essential to develop new high heat flux cooling technology to meet the challenging heat dissipation requirements. With the rapid miniaturization of devices to nanoscale and microscale, the new technologies taking advantage of these advances are faced with very serious heat dissipation problems per unit volume (cooling CPUs and power electronics). The main issues in future trends for cooling of microprocessors are to dissipate footprint heat fluxes as high as 300 W/cm2 or more while maintaining the chip safely below its maximum working temperature (less than 85°C) and providing a nearly uniform chip base temperature, the whole with a minimal energy consumption. This means that the conventional cooling technology, i.e. air cooling, will no longer be able to satisfy these heat duties. New solutions must be found to solve this problem. One possible solution is to use forced vaporization in micro-channels (multi-channels made of silicon or copper which may be used in copper cooling elements attached to CPUs in stacked CPUs, or directly in the silicon chip itself) by making use of the high heat transfer performance of two-phase flows [97-104]. In this aspect, flow boiling of low pressure refrigerants in multi-channel evaporators is a promising technique. The recent experimental results of high heat flux flow boiling of two refrigerants R236fa and R245fa in silicon multi-microchannels by our laboratory have shown very good heat transfer behaviors [103, 104]. To further explore the use of environmentally friendly refrigerants for the cooling of micro-processors, Cheng and Thome [105] did simulations of cooling of microprocessors using flow boiling of CO2 in a silicon multi-microchannel evaporator is presented in this

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section. The flow pattern based flow boiling heat transfer and two-phase pressure drop models for CO2 of Cheng et al. [58, 59] were used to predict the thermal performance of CO2 in a silicon multi-microchannel evaporator (67 parallel channels with a width of 0.223 mm, a height of 0.68 mm and a length of 20 mm) used for cooling of micro-processors. First, some simulation results of CO2 flow boiling heat transfer and two-phase pressure drops in microscale channels are presented. The effects of channel diameter, mass flux, saturation temperature and heat flux on flow boiling heat transfer coefficients and two-phase pressure drops are addressed. Then, simulations of the base temperatures of the silicon multimicrochannel evaporator using R236fa and CO2 evaporation were performed for the following conditions: base heat fluxes qb = 20-100 W/cm2, mass flux G = 987.6 kg/m2s and saturation temperature Tsat = 25°C. Figure 40 shows the schematic of the silicon multi-microchannel heat sink assumed for the present simulation. It consists of 67 microchannels. The dimensions of the microchannels are the same as those of Agostini et al. [103, 104] and are given in Table 1, where the hydraulic diameter and equivalent diameter are defined as

Dh =

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Deq =

2WH W +H

4A

π

=

(74)

4WH

(75)

π

Figure 40. Schematic of a silicon multi-microchannel evaporator used for chips cooling [105].

Table 1. Channel dimensions. L (mm)

H (mm)

W (mm)

t (mm)

e (mm)

N

20

0.68

0.223

0.08

0.32

67

D (mm) 0.336∗ 0.44∗∗

∗ hydraulic diameter and ∗∗equivalent diameter.

The physical model is described as follows: (i) the bottom of the heat sink is uniformly heated by electric heating at a base heat flux qb, (ii) the top of the heat sink is adiabatic, (iii) the coolant (CO2 or R236fa) flows in the silicon multi-microchannel evaporator without inlet

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subcooling, ΔTsub = 0 K. With these conditions, the base temperature Tb was simulated using CO2 and R236fa evaporation in the silicon microchannel evaporator at the same operational conditions and the simulated results compared. The recent flow boiling heat transfer and two-phase pressure drop models for CO2 of Cheng et al. [58, 59] are used to simulate the heat transfer and pressure drop behaviors of CO2 in microchannels with diameters ranging from 0.44 to 1 mm. It should be pointed out that these models are thus extrapolated here below its minimum diameter limit of 0.6 mm. However, the predicted heat transfer and pressure drop results still show some interesting parametric trends which are helpful in selecting heat transfer coefficients for the simulation of base temperatures of the silicon multi-microchannel evaporator in Fig. 40. The experimental results for R236fa together with their test conditions and methods can be found in [103].

Figure 41. Simulation of flow boiling heat transfer of CO2: The effect of channel diameter [105].

Figure 42. Simulation of flow boiling heat transfer of CO2: The effect of saturation temperature [105].

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Figure 43. Simulation of flow boiling heat transfer of CO2: The effect of heat flux [105].

Figure 44. Simulation of flow boiling heat transfer of CO2: The effect of mass flux [105].

The simulations of flow boiling heat transfer are shown in Figs. 41-44. Figure 41 shows the effect of channel diameter on CO2 heat transfer. Heat transfer coefficient increases with decreasing channel diameter. However, this variation is not so notable in the selected diameter range of micro-channels. Figure 42 shows the effect of saturation temperature on CO2 heat transfer. The heat transfer coefficient increases with increasing saturation temperature from 0 to 15°C. Figure 43 shows the effect of heat flux on CO2 heat transfer. The heat transfer coefficient increases strongly with increasing heat flux. Figure 44 shows the effect of mass flux on CO2 heat transfer. The heat transfer coefficient increases with increasing mass flux, with the variation becoming larger at high vapor qualities. However, the increase of heat transfer coefficient is not so notable.

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Figure 45. Simulation results of two-phase frictional pressure drops of CO2: The effect of channel diameter [105].

Figure 46. Simulation results of two-phase frictional pressure drops of CO2: The effect of saturation temperature [105].

The simulations of two-phase pressure drops are shown in Figs. 45-47. Figure 45 shows the effect of channel diameter on CO2 two-phase frictional pressure drop for G = 1500 kg/m2s and Tsat = 0°C. The two-phase frictional pressure drop increases with decreasing tube diameter, where the diameter effect is significant. Figure 46 shows the effect of saturation temperature on CO2 two-phase pressure drop for G = 1500 kg/m2s, D = 0.44 mm (equivalent diameter of the microchannels in Fig. 1 is used here) and L = 20 mm. The two-phase frictional pressure drop decreases with increasing saturation temperature, where the effect of saturation temperature becomes more significant at high vapor qualities. Figure 47 shows the

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effect of mass flux on CO2 two-phase frictional pressure drop D = 0.44 mm, Tsat = 0°C and L = 20 mm, where the two-phase frictional pressure drop increases with mass flux.

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Figure 47. Simulation results of two-phase frictional pressure drops of CO2: The effect of mass flux [105].

From the simulated pressure drop results, it may be concluded that a high saturation temperature and a low mass flux should be selected to obtain low pressure drops, so as to reduce the pumping power for a fixed channel diameter. In the case of a base heat flux qb = 250 W/cm2 (2.5MW/m2) and a mass flux G =1500 kg/m2s indicated by the dashed line in Fig. 46, this results in an exit vapor quality of 0.5, where the frictional pressure gradient is 615 kPa/m. This corresponds to a frictional pressure drop less than 12.3 kPa for the microchannel length L = 20 mm. Considering the acceleration pressure drop Δpa evaluated by Eq. (46) (about one-third of the two-phase frictional pressure drop), the total two-phase pressure drop Δptotal evaluated by Eq. (45) in a single microchannel is less than 16.3 kPa, which is much less than that for R236fa (162 kPa evaluated by the homogeneous model [103]). Thus, using CO2 requires much less pumping power than R236fa for the same conditions. Simulations of base temperature of the silicon multi-microchannel evaporator in Fig. 40 were performed. First, wall heat fluxes qW were assumed to predict the flow boiling heat transfer coefficients of CO2 for certain conditions. Then, Eq. (76) was used to determine the wall temperature TW. Finally, Eqs. (77) to (81) were iteratively solved to obtain the base temperatures. The heat transfer coefficient was defined as

htp =

qW TW − Tsat

and the base heat flux was evaluated as [106]:

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(76)

Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2…

qb = qW

W + 2η H W +t

593

(77)

where the fin efficiency was calculated as follows [106]:

η=

tanh(mH ) mH

(78)

m=

h(W + L) k siWL

(79)

The thermal conductivity of silicon was calculated as in [103]: k si = 0.0018T 2 − 0.7646T + 143.25

(80)

The base temperature was calculated with a one dimensional heat conduct model as

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Tb = TW −

qb e ksi

(81)

Figure 48. Simulation of base temperatures of CO2 for different saturation temperatures at a fixed mass flux [105].

Figure 48 shows the simulated base temperatures of CO2 flow boiling in the silicon multimicrochannel evaporator at three different saturation temperatures and a mass flux of 1000 kg/m2s. It can be seen that a lower saturation temperature can contribute to lower base temperatures. However, in practical applications, the higher energy cost caused by using a lower temperature should be considered. Figure 49 shows the simulated base temperatures of

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CO2 flow boiling in the silicon multi-microchannel evaporator at mass fluxes of 500 kg/m2s and 1500 kg/m2s and at a saturation temperature of 15°C. With the higher mass flux, a higher base heat flux may be cooled. For the lower mass flux, a more limited heat flux may be dissipated by the evaporator because of the limitation of the exit vapor quality xe. For the lower mass flux at higher heat flux, the exit vapor quality may reach 1 and thus cooling is no longer effective. Figure 50 shows the comparison of the simulated base temperatures of CO2 and R236fa flow boiling in the silicon multi-microchannel evaporator at a mass flux of 987.6 kg/m2s and a saturation temperature of 25°C for base heat fluxes in the range from 20 to 100 W/cm2. It can be seen that CO2 can achieve lower base temperatures than R236fa by 4 to 6 K.

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Figure 49. Simulation of base temperature of CO2 for two different mass fluxes at a fixed saturation temperature [105].

Figure 50. Comparisons of simulation results of base temperature of CO2 and R236fa at the indicated conditions [105].

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It should however be pointed out that very high heat flux cooling is beyond the applicable conditions of the CO2 models. In the simulations, heat transfer coefficients at much lower heat fluxes (less than 5W/cm2) were used in the simulation. This makes the simulated results much more conservative. However, further experiments are needed for CO2 flow boiling in microchannels to extend the heat transfer model to extremely high heat fluxes and to understand the physical mechanisms which might be quite different from those in macro-scale channels. Apparently, no such information is available so far in this range. Although CO2 has much better thermal performance and much lower pumping power requirements than R236fa, it must be realized that the operational pressure of CO2 is much higher than R236fa. Therefore, CO2 also presents a great technological challenge just like any other new cooling technology.

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5. Analysis of Supercritical CO2–Oil Two-Phase Flow and Heat Transfer In a CO2 gas cooling heat transfer process, the pressure of the gas cooler during heat removal is maintained above the critical pressure, so the physical and transport property variations can be severe, as can be seen in Fig. 51. For a constant pressure larger than the critical pressure, an important characteristic is that the specific heat reaches a sharp maximum as shown in Fig. 51 (c). This point is called the pseudocritical point as indicated by the dash line in Fig. 51 (c) and the corresponding pressure and temperature are the pseudocritical pressure (ppc) and the pseudocritical temperature (Tpc). Near the critical pressure, the thermal conductivity also reaches a maximum value. During a supercritical heat transfer process, the physical and transport properties of CO2 change drastically with temperature around the critical point in an isobaric process, especially near the pseudocritical and critical points. In the vicinity of the pseudocritical points, with an increase in pressure, these changes become less pronounced. The density and dynamic viscosity undergo a significant drop near the critical point, which is almost vertical within a very narrow temperature range while the enthalpy undergoes a sharp increase. The specific heat, thermal conductivity and Prandtl number have peak values near the critical points. The magnitude of these peaks decreases very quickly with increasing pressure. All of these physical property effects strongly influence heat transfer and pressure drop under supercritical conditions. The heat transfer coefficients and pressure drops of supercritical CO2 are greatly dependent on both local mean temperature and local heat flux because of the strong physical property effects caused by the temperature gradients. Importantly, the ε-NTU and LMTD methods require that the specific heat and thermal conductivity be nearly constant over the design section or test section. Thus, when data are measured or thermal designs made, it should be carefully checked to see if their values are relatively constant or vary significantly. A comprehensive literature survey of supercritical heat transfer and pressure drop of CO2 with and withiout lubricating oil under cooling conditions has been recently done by Cheng et al. [9]. Here we only present a state-of-the-art of oil-gas two-phase flow and heat transfer.

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Lixin Cheng and John R. Thome 1000 1--- 7.5 MPa 2--- 8 MPa 3--- 9 MPa 4---10 MPa 5---12 MPa

-3

Density [kgm ]

800 600 4

1

5

400 2 200 0 0

3 20

40 60 80 o 100 Temperature [ C]

120

140

(a)

600 3

-1

Enthalpy [kJkg ]

500 2

400

5

1 300

4

200 100 20

40 60 80 100 Temperature [oC]

120

140

(b)

50 -1 -1

Specific heat [kJkg K ]

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0 0

1--- 7.5 MPa 2--- 8 MPa 3--- 9 MPa 4---10 MPa 5---12 MPa

1 1--- 7.5 MPa 2--- 8 MPa 3--- 9 MPa 4---10 MPa 5---12 MPa

40 2 30 20 3 10

4 5

0 0

20

40 60 80 o 100 Temperature [ C]

120

(c) Figure 51. Continued on next page.

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140

-1 -1

Thermal conductivity *1000 [Wm K ]

Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2…

597

120 1--- 7.5 MPa 2--- 8 MPa 3--- 9 MPa 4---10 MPa 5---12 MPa

100 80 60

1

4

5

40 2 20 0 0

3 20

40 60 80 o 100 Temperature [ C]

120

140

(d)

Dynamic viscosity *10 [Pas]

120 1--- 7.5 MPa 2--- 8 MPa 3--- 9 MPa 4---10 MPa 5---12 MPa

6

100 80 3 60

4 5

40

1

20

2

0 0

20

40 60 80 o 100 Temperature [ C]

120

140

50 1

1--- 7.5 MPa 2--- 8 MPa 3--- 9 MPa 4---10 MPa 5---12 MPa

40 Prandtl number [-]

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(e)

30 20 2 10 0

0

3 20

4 5

40 60 80 o 100 Temperature [ C]

120

140

(f) Figure 51 Physical properties of supercritical CO2 at five different pressures versus temperatures [21]: (a) Density; (b) Enthalpy; (c) Specific heat (the dash line indicates the pseudo-critical point at the pressure of 9 MPa); (d) Thermal conductivity; (e) Dynamic viscosity; (f) Prandtl number.

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In practical air-conditioning, heat pump and refrigeration systems, there are some problems arising from the partial miscibility of the lubricant oil in CO2. The oil has a great adverse effect on heat transfer coefficients and pressure drops. In determining the effect oil on the heat transfer and pressure drops of supercritical CO2 in a gas cooler, the following aspects should be considered:

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(i) The partial miscibility of CO2-lubricant oil mixtures at conditions of process operation; (ii) The densities of the continuous gas phase and the dispersed oil phase with dissolved gas; (iii) The viscosities of CO2 and oil with dissolved gas; (iv) The interfacial tension between CO2 and oil at operating conditions. Mori et al. [107] studied heat transfer characteristics of cooling CO2-oil mixtures in a macro-channel at supercritical pressures and found that oil resulted in significantly lower heat transfer coefficients. Figure 52 shows the comparison of heat transfer coefficients for pure CO2 and a CO2 oil mixture at the indicated conditions by Mori et al. [107]. The heat transfer coefficients of the mixture are much smaller than those of pure CO2, particularly in the temperature range of 40 to 55°C. Flow visualization revealed that the oil near the tube wall became an obstacle to heat transfer and the distribution of oil changed according to the fluid velocity and the density variation due to its temperature variation. Kuang et al. [108] studied the effect of oil on heat transfer and pressure drop in supercritical gas cooling in microchannels. Three different oils (two immiscible and one miscible) were tested at oil concentrations from 0 to 5% by mass. It was found that the oils had a significant negative effect on performance. Just one example is shown here in Figs. 53 and 54 for heat transfer and pressure drop, respectively. At higher oil concentrations, the heat transfer coefficients are as much as 65% lower and the pressure drops are up to 50% higher. As far as the type of oil is concerned, the immiscible oils demonstrated more negative influence than the miscible oil. Dang et al. [109] studied cooling of supercritical CO2 with PAG-type lubricating oil with concentrations from 0 to 5% by mass. They found that heat transfer coefficients decreased and the pressure drops increased with the addition of oil. The maximum degradation in heat transfer was about 75 % and occurred in the vicinity of the pseudiocritical temperature as in the above study. From their visual observations, it was confirmed that the degradation in heat transfer was due to the formation of an oil-rich layer along the tube wall. However, when the oil concentration exceeded 3% (mass), no further degradation in heat transfer was found while the effect of oil on pressure drops continued to rise. As noted in their study, for a large tube at lower mass velocities, no significant degradation in heat transfer coefficients was observed until the oil concentration reached 1% (mass). This is due to the transition of the flow pattern from an annular-dispersed flow to a wavy flow for the large tube, with CO2 flowing on the upper side and the oil-rich layer on the low side of the tube [109].

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Heat transfer coefficient [W/m2K]

3000 CO2 CO2 +oil

2000

p = 9.5 MPa D = 6 mm

1000 CO2 mass velocity: 400 kg/m2s Oil mass velocity: 25 kg/m2s 0 0

10

20 30 40 50 Temperature [oC]

60

70

Heat transfer coefficient [W/m2K]

Figure 52. The effect of oil (PAG/AN) on heat transfer by Mori et al. [107]. 2

x 10

without oil 1 wt.% oil 2 wt.% oil 3 wt.% oil 5 wt.% oil

1.5

1

0.5

0 0

10

20 30 40 Temperature [oC]

50

60

Figure 53. The effect of oil (PAG) on heat transfer by Kuang et al. [108]. 50

Pressure drop [kPa]

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4

40 30 20

Without oil 1 wt.% oil 2 wt.% oil 3 wt.% oil 5 wt.% oil

10 0 0

10

20 30 40 Temperature [oC]

50

60

Figure 54. The effect of oil (PAG) on pressure drops by Kuang et al. [108].

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Yun et al. [110] studied the effects of PAG oil on the convective gas cooling heat transfer and pressure drop characteristics of supercritical CO2 cooling in a minichannel. They also presented a brief review on supercritical CO2 gas cooling in their study, which is also of interest to the community. Their experimental results showed that the average gas cooling heat transfer coefficient was decreased by 20.4% and the average pressure drop was increased by 4.8 times when the oil concentration was increased from 0 to 4 wt.%. Furthermore, they compared their oil effect heat transfer data to the existing heat transfer models of Tichy et al. [111], Schlager et al. [112] and Bassi and Bansal [113], and their oil effect pressure drop data to a modified Darcy-Weisbach model [114]. From their compassions, the Tichy et al. [111] heat transfer model gave the lowest average mean deviation of 3.9% and the modified DarcyWeisbach pressure drop model gave a mean deviation of 23.3%. Apparently, these empirical heat transfer and pressure drop correlations with the oil effect were not developed for supercritical CO2 cooling, the predictions are not satisfactory so far. Further effort should be made to propose the heat transfer and pressure drop correlations for supercritical CO2 cooling considering the oil effect. Experiments should be done under a wide range of test conditions to build a database for this purpose in the future. So far, there are no correlations for the oil effect on heat transfer and pressure drop for cooling of supercritical CO2. In fact, only a few studies have been performed. More experimental studies are necessary for studying the influence of oil before a general correlation can be developed. The principal effects of oil are caused by

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1. Modification of the local physical properties in the near critical region by the oil (thus, CO2-oil properties need to be measured at these conditions); 2. Temperature gradient effects may be altered by the presence of oil; 3. Immiscible oils create a two-phase flow out of the single-phase flow and create many new flow aspects.

6. Two-Phase Flow and Supercritical CO2 Two-Phase Flow Distribution Issues The function of two-phase flow distributor (inlet header) is to evenly distribute the liquid and vapor phases of the two-phase flow to the channels of the boiling side of the heat exchanger and thereby give each parallel channel equal capacity to absorb heat. Therefore, developing distributors to uniformly distribute two-phase (liquid/vapor) flow is a difficult but a very important aspect in the design of the heat transfer components in air-conditioning, heat pump and refrigeration systems. One of the difficulties in the design of a two-phase distributor is to uniformly divide the two-phase fluid stream into several parallel streams of equal mass flow and vapor quality [115, 116]. Sometimes, it is possible to achieve good distribution under certain geometries for some conditions. However, in automotive air conditioning, heat pump and refrigeration systems, the working conditions change continuously over a wide range of refrigerant flow rates, qualities, saturation temperatures, etc. Consequently, the interaction between inertial and gravitational forces that might be in balance under some conditions can create an uneven supply under other operational conditions. Two-phase maldistribution reduces the

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effectiveness of evaporators and creates an uneven frosting of the evaporators. Therefore, flow maldistribution causes deterioration of both thermal and hydraulic performances. Generally, heat exchangers experience three main types of flow mal-distribution:

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(a) Port flow maldistribution: port flow maldistribution is caused by port pressure variation in a given pass that produces flow rate variations from channel to channel along the ports. (b) Flow maldistribution resulting from different plate groups arranged in a given pass: the plate group is defined as a group of plates which consists of channels having the same geometric and operating characteristics. This type of flow maldistribution occurs primarily when hydraulic flow resistances between two plate groups are different. Fluid passes more easily through the plate group with the lower hydraulic flow resistance. (c) Channel flow maldistribution: channel flow maldistribution, which occurs within any channel, results from asymmetrical parallel and diagonal channel flows. Channel flow maldistribution is more likely occur in parallel flow. Proper design of the inlet and outlet headers may minimize this problem. Two-phase flows in compact heat exchangers are very complex and only a few studies of two-phase flow distribution in such geometries have been conducted so far. For example, in the case of actual compact heat exchangers containing multiple channels which are assembled in parallel flow and have a common inlet connection, the shape and size of flow channels in the inlet region are expected to have a significant effect on the flow distribution among the multiple channels. However, most of the available studies address two-phase flow in single T junctions, the simplest case of the header geometry, while flow characteristics in the actual headers still remain largely unknown. Furthermore, most of the studies have been conducted using air-water as the working fluids. Only few studies are related to refrigerants, which are summarized in this section. Very limited studies on two-phase refrigerant distributors [121-129] have been conducted, only one involving CO2. As there are limited studies on this topic, there are no general conclusions that can be drawn on two-phase distributors so far. Some researchers proposed two-phase distribution models based on their own data. But these models lack generality. Because of the form of the published results on two-phase flow in T-junctions or distributors, it is not possible to compare the experimental data nor to compare experimental data to the prediction methods. For the study of two-phase distributors with micro-channels, there is only one study available so far. A common characteristic of two-phase flows through T-junctions is the maldistribution of the gas and liquid phases between the inlet and outlet branches. The occurrence of this maldistribution in the heat exchanger can constitute a major problem in the operation of the thermal system. Several researchers investigated the two-phase distribution in compact heat exchangers. For example, Samson et al. [117] tested a number of two-phase flow header concepts using air-water to determine which would equally distribute the fluid to parallel legs of a heat exchanger. None of these designs distributed the flow perfectly over a wide range of operating conditions. Some configurations worked well at a single inlet condition, but had poor to moderate performance at the other conditions tested. Based on these preliminary test results, they developed a new fan shaped header that could distribute the liquid of the two-phase flow almost equally at lower qualities and less acceptably at higher qualities. Because of the

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sensitivity of these headers to the inlet flow conditions, the heat exchanger operation may be restricted to a single design point in its operating range. They recommended that further research is needed to better understand the underlying problems of two-phase header design. Rong and Kawaji [118, 119] studied two-phase flow distribution in a single channel and multi-channels. The two-phase flow distributions were quite uneven over most of the test conditions. They concluded that the two-phase flow pattern in the header was an important factor affecting the flow distributions and also the flow distributions could be adjusted by changing the inlet geometry of the channel. They suggested that further studies on two-phase flow distribution phenomena in the evaporator be conducted. Bernoux et al. [120] did experiments on two-phase distribution at the inlet manifold of compact heat exchangers. Eight rectangular downward channels (2×50 mm) were connected in parallel to the bottom of a horizontal header (circular cylinder of 50 mm in diameter). A two-phase mixture of R-113 was introduced to the header through an orifice at one end and another end at the opposite side was plugged. The effects of mass flux and vapor quality were examined. Their results showed that the vapor distribution to the channels became more uniform with increasing of the vapor quality but the liquid distribution still remained unbalanced. Furthermore, the liquid distribution through the channels was not very sensitive to the mass flux. Apparently, these limited studies on two-phase distribution are mostly in the large size range of tubes, mostly using air-water as the working fluids. In addition, the only one study on refrigerant two-phase flow distribution was also conducted in large channels [120]. Each study only focused on the specific geometry and test conditions used in their study. It seems that the most important factors are the inlet geometry and inlet flow conditions. However, no general conclusions and models have been made so far. So far, there is apparently no study on supercritical flow or supercritical gas-oil twophase flow distributors. Also, there is no study involving lubricating oil in the refrigerant. Careful analysis of these studies has been presented here to provide a state-of-the-art review and also to identify the future research needs. As most of the available studies deal with the two-phase distribution in macro channels, here we do not show the details of their results but present an overview of these studies. Some results from the selected studies are also presented. In addition, due to nearly not study on two-phase separation models for distributors with micro-channels, no information on this aspect can be presented here. Only discussion of experimental studies is presented in the following. Asoh et al. [121] observed flow patterns of R-113 in a main pipe separating into three vertically downward branch tubes and studied the phase separation characteristics. They also measured the pressure drop between the inlet and the outlet of the branch tubes and tried to theoretically explain the mechanism of the pressure drop. The main pipe had a diameter of 13.6 mm and the branch tubes were 11 mm in diameter. The observed flow patterns were slug and froth flow in the main pipe. The flow patterns were not the same in each of the three branches. The liquid slugs were longer in the upstream branch than in the downstream branches. The flow rate of both phases separating into the branch tube is controlled more by the flow rate of liquid in the main pipe than that of the vapor according to their study. The flow rates of both phases and the pressure drop were estimated more accurately by the slip flow model than the homogeneous model. Watanabe et al. [122, 123] conducted an experimental study on R-11 two-phase flow distribution in two different types of multi-pass tubes: one with four vertical upward passes

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attached to a horizontal main pipe and the other with five horizontal passes placed to a vertically oriented main pipe. The passes had an inside diameter of 6 mm and the main pipes had an inside diameter of 20 mm for the vertical type and 6 mm for the horizontal type. Vapor and liquid flow rates at each pass were measured under various flow rates and qualities at the main inlet. The measured values of flow distribution ratio presented a marked distinction between the vertical and horizontal types. The orientation of the main pipe had a great influence on flow distribution. In addition, phase separation and pressure drops at each junction were compared with the experimental data obtained from single T-junction research. It was concluded that the single T-junction data were useful to predict the flow distribution and pressure drops in multi-pass tubes. Prediction of flow distribution based on a simple flow model in the header was attempted and acceptable agreement with the measured data was obtained. Fei et al. [124] studied R-134a two-phase distribution results for the inlet headers of plate evaporators. The test section consisted of a main inlet pipe and 5 branches. The diameter of the main inlet pipe was 10 mm. The flow rates ranged from 20 to 60 g/s for inlet vapor qualities at the inlet header of up to 30%. They measured the distribution of the two-phase refrigerant after the inlet headers and meanwhile visualized the flow patterns. Several conclusions were obtained: (1) The occurrence of dispersed droplets improved the distribution significantly; (2) A homogeneous flow with tiny droplets resulted in a very good distribution because the droplets mostly followed the movement of the vapor even though the inter-phase drag and gravity still played a role. When the droplet was small enough, it followed the vapor flow exactly and filled the header uniformly; (3) For two-phase developing flow, the baffles blocked the expanding flow in the peripheral region and only let the flow in the center to go through. Obstructing the recirculation flow improved the distribution; (4) For the range of 30 to 60 g/s and 10 to 20% quality, the expansion angle of the developing jet did not change much, so the distribution shapes were similar. (5) For most conditions, a large percentage of the flow went into the last branch, because the last baffle destroyed the recirculation of flow. Hence the liquid entering that zone could only flow down into the last branch. Tae and Cho [125] studied the two-phase split of refrigerants at a T-junction. They used the following working fluids: R-22, R-134a and R-410A. They studied the effect of geometric parameters, the direction of the inlet or branch tube and the tube diameter ratio of branch to inlet tube on the two-phase distribution under a wide range of conditions of mass velocity and inlet quality. For the case where the inlet diameter was the same as the branch tube diameter, the tube inner diameters tested were 4.95, 8.12 and 11.3 mm. In further tests, the tube diameter ratio of the branch to the main pipe was varied to 0.72 and 0.44 based on the inlet tube diameter of 11.3 mm. Among the geometric parameters, the branch tube orientation showed the largest sensitivity to the mass flow rate ratio for the gas phase, while the inlet quality showed the largest sensitivity to the mass flow rate ratio among the inlet flow parameters. Furthermore, they proposed a modified model for application to the reduced Tjunction and vertical orientation of tubes. Vist and Pettersen [126] measured R-134a two-phase flow distribution in realistic manifold geometries and under relevant operating conditions. The effect of vapor fraction at the manifold inlet, heating load on the heat exchanger tubes, diameter of the manifold and manifold inlet tube length on flow distribution was studied. From a horizontal manifold, the two-phase refrigerant flow was distributed in 10 parallel heat exchanger tubes. The

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orientation of the test rig could be varied to measure both upward and downward flow out of the manifold. Two-manifolds were used in their study. The test manifold consisted of an inlet main tube with an inside diameter of 16 mm or 8 mm and 10 parallel evaporator tubes with an inside diameter of 4 mm. as show in Fig. 55. Some of their results are presented in here to give examples of the two-phase flow distribution in the manifolds. Figure 56 shows the distribution of the vapor and liquid phases as a function of the manifold inlet mass flow. The inlet vapor fraction was 0.27. The flow ratio in a channel is defined as the liquid (or vapor) flow measured in the channel over the mean liquid (or vapor) flow rate assuming that the liquid (or vapor) distribution is uniformly distributed in all channels, i.e. the actual flow rate with respect to an ideal uniform distribution. Both the vapor and the liquid phase are unequally distributed to the different heat exchanger tubes. The vapor phase is distributed into the tubes adjacent to the inlet and the liquid phase is distributed preferentially to the last tubes encountered Nos. 6 to 10. There is little difference in the two-phase distribution for the three mass flow rates. Figure 57 shows the vapor and liquid phase distribution for the two manifolds at different evaporation temperatures. It can be seen that inlet temperature of water in the water loop has little effect on the flow distribution. Figure 58 shows the vapor and liquid distribution as a function of the inlet vapor fraction in the manifolds. At low vapor quality (x = 0.11), most of the vapor was taken out in the first tube, for both manifolds. With increasing vapor fraction, the vapor was distributed unequally in the first five tubes, while almost only liquid was distributed in tubes No. 6 to 10. Figure 59 shows the two-phase flow distribution for downward flow in the heat exchanger tubes for various inlet vapor fractions at the manifold inlets. Unlike for the upward flow case, the vapor flowed to the end of the manifolds and was distributed in the tubes No. 4 to 10 in the manifold of main tube with a inside diameter of 16 mm, and in tubes No. 3 to 10 in manifold of main tube with an inside diameter of 8 mm. The liquid phase was better distributed than the vapor phase.

Figure 55. Schematic of the two inlet manifolds [126].

Vist and Pettersen [127] also presented similar results for CO2. According to their study, the big difference in density ratios between the two-fluids (CO2 has a density ratio of 4.2 and R-134a has a density ratio of 35.2) has little effect on the two-phase distribution in their study. Vist [128] published the same results for R-134a and CO2 as well. He also compared his data to the existing models for prediction of phase distribution in T-junctions with a horizontal main tube and vertical branch tube and good agreement for the CO2 and R-134a for the manifold of a main tube with an inside diameter of 16 mm was noted while larger

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deviations were obtained for the main tube with an inside diameter of 8 mm. This is the only study on CO2 two-phase distribution. However, since the diameter of the branch tube is still in the macro-channel range, it is not sure if their results can be applied to micro-channels with an inside diameter down to 0.6 mm.

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Figure 56. Influence of mass flow rate on liquid and vapor distribution in the manifold of main tube with an inside diameter of 16 mm and branch tubes of 4 mm [126].

Figure 57. Influence of evaporator water temperature on liquid and vapor distribution during upward flow in the heat exchanger tubes [126].

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Figure 58. Influence of inlet vapor fraction on liquid and vapor distribution during upward flow in the heat exchanger tubes [126].

Cho and Cho [129] studied two-phase flow distribution and pressure drop in microchannel tubes under non-heating and heating conditions using R-22 as the working fluid. This is the only study on a two-phase distributor with micro-channels in the literature so far. The test section consists of inlet and outlet headers with inner diameters of 19.4 mm and 15 or 41 parallel micro-channel tubes. Each micro-channel tube has 8 rectangular ports with a hydraulic diameter of 1.3 mm. For non-heating conditions, the key experimental parameters were orientation of the header, flow direction of refrigerant into the header and inlet quality. For heating conditions, the key experimental parameters were inclination angle, mass flow rate and inlet quality. It was found that the orientation of the header had a great effect on the two-phase flow distribution and pressure drop under the non-heating condition. Two-phase pressure drops through the micro-channel tubes with horizontal header were higher than those of the micro-channel tubes with the vertical header due to the gravitational effect. Under the

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heating condition, the cooling capacities of three prototype evaporators were changed as the mass flow rate and inlet quality were increased due to flow mal-distribution. According to this analysis, it can be concluded that most of the available studies on twophase distributions are so far related to air-water. For the interest of air-conditioning, heat pump and refrigeration systems, only the studies of refrigerant two-phase distributions are presented. Even so, only one study relates to two-phase distribution in an evaporator with micro-channels. Because of the wide diversity in the reporting of the results and observations, no general conclusions and models for two-phase flow distribution for both macro- and micro-channels can be realistically made. Therefore, more systematic knowledge is needed to provide a basis for the design requirements for two-phase distributors. In addition, there is no study on supercritical gas or supercritical gas-oil two-phase flow distributors. Future research should be conducted in this aspect. Specifically, studies should be focused on CO2 and other new refrigerant two-phase distributors with micro-channels to meet the design requirements for automobile air-conditioning, heat pump and refrigeration systems.

Figure 59. Influence of evaporator vapor fraction water temperature on liquid and vapor distribution during downward flow in the heat exchanger tubes [126].

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7. Conclusions

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Various issues related to sub- and supercritical CO2 two-phase flow and heat transfer in macro- and micro-channels are presented and discussed in this chapter. The available experimental studies on the relevant topics are analyzed and summarized. Especially, our newly developed two-phase flow pattern map for CO2 evaporation, flow pattern based flow boiling heat transfer and pressure drop models are presented. Furthermore, some simulation results for electronic chips cooling using CO2 evaporation are presented and discussed. The research needs in the future are identified. The main conclusions are as follows: (1) Different flow boiling heat transfer and two-phase flow behaviors of CO2 have been shown for high and low reduced pressures in the available studies. The boiling heat transfer and two-phase flow of CO2 at saturation temperatures ranging from 0 to 25°C show different characteristics from those of conventional refrigerants due to the significant differences in physical properties. Generally, CO2 has much higher flow boiling heat transfer and much lower pressure drops than other low-pressure refrigerants. One feature is the dominance of the nucleate boiling at low/moderate vapor qualities prior to dryout. Another feature is that the dryout in CO2 flow boiling occurs much earlier (at relatively lower vapor qualities) than conventional refrigerants. Furthermore, the effect of the saturation temperature on heat transfer coefficient is more noticeable. At high saturation temperatures, nucleate boiling is more pronounced and plays an important role at low vapor quality. (2) The experimental data from the different independent studies show somewhat different heat transfer trends at similar test conditions. It is difficult to understand the differences at similar conditions or the parameter effects on heat transfer behavior. Therefore, more accurate experimental data are needed in both macro- and micro-channels through careful experiments. (3) The available studies have shown different heat transfer behaviors at lower saturation temperatures from those at higher saturation temperatures. It is difficult to explain the experimental results in some studies at low temperatures. So far, there are several studies of CO2 at low temperatures in the literature but still very limited information available. Considering the big differences among the available data, it is recommended that more and accurate experimental data at low temperatures are needed by careful and well designed experiments. (4) Overall, the updated general flow pattern based flow boiling heat transfer model predicts the overall database quite well. However, for the dryout and mist flow regimes with partially or all dry perimeters, the heat transfer model is only partially satisfactory. Therefore, more careful experiments are needed in these two regimes to provide more accurate heat transfer data, with attention to also determine the transitions xdi and xde, because they are typical working conditions in the microscale channels of extruded multi-port aluminum tubes used for automobile airconditioners that operate over a wide range of mass velocities up to as high as 1500 kg/m2s. (5) The CO2 two-phase flow pressure drop model based on flow patterns predicts the CO2 pressure drop database better than the existing methods. Due to the very few

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(7)

(8)

(9)

609

and less accurate experimental data in micro-scale channels currently available, the new CO2 pressure drop model does not predict these data satisfactorily. It is suggested that additional, more accurate experimental CO2 pressure drop data be obtained to further test or improve the model in the future. The simulations of electronic chip cooling using CO2 have shown that CO2 has much better thermal performance and much lower pumping power requirements than R236fa. It should however be pointed out that very high heat flux cooling is beyond the applicable conditions of the CO2 heat transfer and pressure drop models. Therefore, further experiments are needed for CO2 flow boiling in microchannels to extend the heat transfer model to extremely high heat fluxes and to understand the physical mechanisms which might be quite different from those in macro-scale channels. Furthermore, it must be realized that the operational pressure of CO2 is much higher than R236fa. Therefore, CO2 also presents a great technological challenge just like any other new cooling technology. Oil has significant effects on CO2 flow boiling heat transfer and two-phase pressure drops. In general, very small amounts of the lubricating oil (< 0.5% wt) seem to have little effect while larger concentrations (> 1% wt) tend to dramatically reduce flow boiling heat transfer coefficients. For two-phase pressure drops, the effect of lubricating oil only occurs at vapor qualities larger than around 0.6. So far, the research on these aspects is very limited. Therefore, further experimental studies on flow boiling and two-phase flow of CO2-immiscible oil mixtures in both macroscale and micro-scale channels be conducted over a wide range of test conditions to provide more experimental data on these aspects. In order to understand the heat transfer mechanisms of flow boiling with CO2 and immiscible oil, flow visualization and local measurement of oil concentration are also suggested to be done. Due to the immiscibility of lubricant oil and CO2, whether the available methods for miscible oil can be applied to CO2-lubricant oil or not still need to be further verified. Therefore, extensive experimental data are needed to verify the available methods or to develop new methods for CO2. So far, there are no correlations for the oil effect on heat transfer and pressure drop for cooling of supercritical CO2. In fact, only a few studies have been performed. More experimental studies are necessary for studying the influence of oil before a general correlation can be developed. The physical mechanisms of the effects of oil on heat transfer and pressure drop should also be understood through flow visualization. Most of the available studies on two-phase distributions are so far related to airwater. For the interest of air-conditioning, heat pump and refrigeration systems, only the studies of refrigerant two-phase distributions are presented. Even so, only one study relates to two-phase distribution in an evaporator with micro-channels. Because of the wide diversity in the reporting of the results and observations, no general conclusions and models for two-phase flow distribution for both macro- and micro-channels can be realistically made. Therefore, more systematic knowledge is needed to provide a basis for the design requirements for two-phase distributors. In addition, there is no study on supercritical distributors. Future research should be conducted in this aspect. Specifically, studies should be focused on CO2 and other

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Lixin Cheng and John R. Thome new refrigerant two-phase distributors with micro-channels to meet the design requirements for automobile air-conditioning, heat pump and refrigeration systems. (10) Overall, a lot of research work is needed to provide complete knowledge in twophase flow and heat transfer of CO2. Further study should be done to explore the application of CO2 evaporation in electric chip cooling.

References [1] [2] [3] [4] [5] [6] [7]

[8]

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[9] [10]

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Nomenclature A AL ALD AV AVD CO D e FrL FrV,Mori, f G g

cross-sectional area of flow channel, m2; cross-sectional area occupied by liquid-phase, m2 dimensionless cross-sectional area occupied by liquid-phase cross-sectional area occupied by vapor phase, m2 dimensionless cross-sectional area occupied by vapor phase Confinement number internal tube diameter, m thickness of channel base, m liquid Froude number [G2/(ρL2gDeq)] vapor Froude number [G2/(ρv(ρL -ρv)gDeq)] defined by Mori et al. [42] friction factor total vapor and liquid two-phase mass flux, kg/m2s gravitational acceleration, 9.81 m/s2

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H h hL hLD hLV k L M m N Pi PiD PL PLD Pr PV PVD p pr q Re ReLO ReM ReV Reδ S T t u W WeL WeV x Y

height of channel, m heat transfer coefficient, W/m2K vertical height of liquid, m dimensionless vertical height of liquid latent heat of vaporization, J/kg thermal conductivity, W/mK channel length, m molecular weight, kg/kmol parameter defined by Eq. (79) channel number perimeter of interface, m dimensionless perimeter of interface perimeter of tube wetted by liquid, m dimensionless perimeter of tube wetted by liquid Prandtl number [cpμ/k] perimeter of tube in contact with vapor, m dimensionless perimeter of tube in contact with vapor pressure, bar reduced pressure [p/pcrit] heat flux, W/m2 homogeneous Reynolds number {(GDeq/μV )[x+ρV/ρL(1-x)]} Reynolds number considering the total vapor-liquid flow as liquid flow [GDeq/(μL)] Reynolds number [GDeq/(μH)] defined in mist flow vapor phase Reynolds number [GxDeq/(μVε)] liquid film Reynolds number [4G(1-x)δ/(μL (1-ε))] nucleate boiling suppression factor temperature, oC thickness of fin, m mean velocity, m/s width of channel, m liquid Weber number [G2Deq/(ρLσ)] defined by Eq. (17); [ρL uL2Deq/σ] defined by Eq. (51) vapor Weber number [G2Deq/(ρvσ)] defined by Eq. (22) vapor quality correction factor

Greek Symbols

Δp δ ε εIA η μ

pressure drop, Pa liquid film thickness, m cross-sectional vapor void fraction vapor void fraction at x = xIA fin efficiency dynamic viscosity, Ns/m2

Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Two Phase Flow and Heat Transfer of Sub- and Super-critical CO2…

θdry θ*dry θstrat θ*strat θwet ρ σ

dry angle of tube perimeter, rad dimensionless dry angle [θdry/(2π)] stratified flow angle of tube perimeter, rad dimensionless stratified flow angle [θstrat/(2π)] wet angle of the tube perimeter, rad density, kg/m3 surface tension, N/m

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Subscripts

A B B cb crit de di dry dryout eq f g H h I IA i iD in L LD LO LV M m nb pc out Slug SW sat si static strat strat(x≥xIA)

annular flow bubbly flow base convective boiling critical dryout completion dryout inception dry dryout region equivalent frictional gas homogeneous hydraulic intermittent flow intermittent flow to annular flow transition liquid-vapor interface interface in cross section tube inlet liquid liquid in cross section of the tube considering the total vapor-liquid flow as liquid flow liquid-vapor mist flow momentum nucleate boiling pseudocritical tube outlet slug flow stratified-wavy flow saturation silicon static stratified flow stratified flow at x≥ xIA

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619

620

Lixin Cheng and John R. Thome stratified flow at x< xIA total two-phase flow vapor vapor in cross section of the tube wall wavy flow wet perimeter

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strat(x psat

if

(20)

pH2O < psat

if

where kc and ke are the vapor condensation and evaporation rate constants, xH2O is the mole fraction of water vapor, and psat is the water vapor saturation pressure calculated using:

psat = 10 −2.1794 + 0.02953T −9.1837×10

−5

T 2 +1.4454×10 −7 T 3

(21)

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Assuming that the water exists only in the liquid phase with no net water source in the membrane, then the liquid water transport equation in the membrane can be expressed:

⎛ ⎛ α d M H2O G ⎞ ⎞ ⎛ M H2O ρdry ⎞ im ⎟ λ − ⎜ Dλ ⎟ ∇λ ⎟⎟ = 0 ∇ ⋅ ⎜⎜ ⎜ ⎠ ⎝ Mm ⎠ ⎝⎝ F ⎠

(22)

where αd is a constant equal to 2.5/22, ρdry is the membrane dry density, Mm is the membrane equivalent weight and Dλ is the diffusivity expressed as a function of the membrane water content [31]:

⎛ 1 ⎞⎞ ⎛ 1 − ⎟⎟ Dλ = Dλref exp ⎜ 2416 ⎜ ⎝ 303 T ⎠ ⎠ ⎝ Dλref = 10−10

λ

(1 + 0.0126λ ) a (17.81 − 79.7a + 108a 2 ) Dλ′

⎧λ 4 ⎪ Dλ′ = ⎨0.5 + 3.25 ( λ − 2 ) 4 ⎪ ⎩3.75 + 4 ( λ − 26 ) 15

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λ≤2 λ≤6 λ >6

(23)

(24)

(25)

632

Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

where a is the water activity. At the interface between the membrane and the CL, the water content is related to the water activity as [31]:

0.043 + 17.18a − 39.85a 2 + 36.0a 3

0 < a ≤1

λ = 14 + 1.4 ( a − 1)

03

16.8 The water concentration in the membrane is defined as:

CH 2 O =

ρdry λ

(27)

Mm

The overpotentials are defined as:

η = Φs − Φ m

on the anode side

(28)

η = Φ s − Φ m − Voc

on the cathode side

(29)

where Voc is the open-circuit potential defined as

Voc = 1.23 − 0.9 × 10−3 (T − 298 ) + 2.3

RT log pH2 2 pO2 4F

(

)

(30)

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The fuel cell voltage is given by:

Vcell = Voc − ηa − ηc − η m

(31)

where ηm is the Ohmic overpotential in the membrane. The boundary conditions at the anode and cathode flow channels are that the inlet flow rates are constant, the inlet gas compositions are constant, and the flows are fully developed at the anode and cathode flow channel outlets. The solid walls are no slip with zero flux boundary conditions. At the interfaces between the gas channels, the GDLs, the catalyst layers, and the PEM, the velocities, mass fractions, momentum fluxes, and mass fluxes are all assumed to be equal. All of the fixed parameters used in the model are listed in Table 1. The generalized convection-diffusion equation can be expressed in conservative form as:

∂ ( ρφ ) + ∇ ⋅ ( ρ uφ − Ξφ ∇φ ) = Sφ ∂t

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(32)

Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design… where t is time and

633

φ is a general dependent variable, such as the velocity components and

concentrations, Ξφ is the exchange coefficient such as the viscosity or diffusion coefficient,

G Sφ is the source term such as the pressure gradient, u is the velocity vector and ρ is the

density. Table 1. Key model parameters Quantity εchannel

1

εGDL

0.4 or 0.5

εCL

0.4

εMEM

0.28

kp,channel

∞

kp,GDL

1.76×10-10 m2 or 1.76×10-11 m2

kp,CL

1.76×10-11 m2

kp,MEM

1.8×10-18 m2

τchannel

1

τGDL

1.5

τCL

1.5

τMEM

Dagan model

ref Aj0,a Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Value

(C ) ref H2

0.5

ref Aj0,c COref2

9.227×108 (A m-3)/(m3 kg-1 mol H2)0.5 1.05×106 (A m-3)/(m3 kg-1 mol O2)

αa/αc on the anode side

0.5/0.5

αa/αc on the cathode side

1.5/1.5

kc

100 s-1

ke

100 atm-1 s-1

σ

0.0625 N m-1

σs

5000 S m-1

σm,CL

4.2 S m-1

ρl

1000 kg m-3

ρdry

1980 kg m-3

μl

3.65×10-4 Pa s

Mm

1.1 kg mol-1

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634

Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

When Eq. (32) is integrated over a control volume, the finite-volume equation can be written as

aPφP = aEφE + aWφW + aNφN + aSφS + aHφH + aLφL + b where

(33)

φP is the value of φ at node P in a control volume, φE "φL are the values of φ at

nodes in neighboring control volumes, aP " aL are coefficients in the discretized equations and b is the source term in the discretized equation for

φ.

The SIMPLE（ Semi Implicit Method for Pressure-Linked Equation） algorithm, developed by Patankar [57], was employed to solve the governing equations. A combination of the continuity and momentum equations gives the pressure correction equation as:

aP P 'P = aE P ' E + aW P ' W + aN P ' N + aS P 'S + aH P ' H + aL P 'L + b

(34)

where P P is the value of P at node P in a control volume, PE ⋅⋅⋅ PL are the values of P at '

'

'

'

'

nodes in neighboring control volumes, aP ⋅⋅⋅ aL are coefficients of P in the pressure '

correction equation and b is the source term. The pressure correction, PP′ , is used to correct the velocities and pressures. The coupled set of equations was solved iteratively, with the solution considered to be converged when the relative error in each field between two consecutive iterations was less than 10-6. G rid (I) G rid (II) G rid (III)

4800 4700 2

Iy ( A/m )

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

4900

4600 4500 4400 4300 0.000

0.005

0.010

0.015

0.020

x (m)

Figure 2. Influence of number of elements on the local current densities.

The model used non-uniformly distributed elements in the x, y and z directions. The grid independence was examined in preliminary test runs for all the cells discussed in sections 3-4. For example, for a parallel flow field PEM fuel cell with dimensions of 23 mm×23 mm×2.845 mm as shown in Figure 4(a), three non-uniformly distributed grid configurations were evaluated with (I) 70×70×25, (II) 93×93×33, and (III) 116×116×41 elements in the x, y and z directions. The influence of the number of elements on the local current densities in the

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Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design…

635

middle cross-section in the PEM is shown in Figure 2. Grid (II) was chosen for the simulations as a tradeoff between accuracy and execution time.

experim antal results predicted results

1.0

Vcell (V)

0.8

0.6

0.4

0.2

0

1000

2000

3000

4000

5000

6000

7000

2

I ( A/m )

Figure 3. Comparison of experimental and predicted polarization curves.

15 01 0. = y

m y

15 .01 =0

m

z y Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

x

(a)

(b)

0 0. y=

11

5m

(c) Figure 4. Continued on next page.

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636

Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

5 11 .0 0 y=

m y

(d)

25 .01 =0

m

(e)

Figure 4. Schematics of the PEM fuel cells with various cathode flow field designs. (a) parallel flow field, (b) Z-type flow field, (c) serpentine flow field, (d) interdigitated flow field and (e) Z-type flow field with baffles.

The numerical model was validated by comparing the present predictions with previous experimental results [58]. Figure 3 shows that there is only a small difference between the calculated polarization curve and experimental data for a fuel cell with the parallel flow field design and an active area of 141×141 mm2. Hence the model is adequate for analyzing the influence of the different flow field designs on the PEM fuel cell performance.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

3. Optimal Flow Field Designs Water flooding normally occurs at the cathode electrode of the PEM fuel cell because the electrochemical reaction on the cathode side produces water vapor. If the partial pressure of the water vapor is higher than the saturation pressure, water vapor condenses to form liquid water. When a large amount of liquid water accumulates in the porous layer pores, the oxygen transport resistance increases and the oxygen mass flow rate decreases. Therefore, the cathode flow field design is a key factor for enhancing reactant and product transport and for removing the liquid water. Typical flow field designs for PEM fuel cells include the parallel flow field, interdigitated flow field, and serpentine flow field. This section focuses on the optimization of these three flow fields on the cathode side.

3.1. Performance Comparison of Various Flow Field Designs Cells with different flow field designs will have different performance characteristics because the various flow field designs provide different reactant transport rates to the porous electrodes and different liquid water removal rate from the porous electrodes and flow channels. Therefore, before the optimization of the geometric parameters for the various flow field designs, we first analyze the effects of the flow field type on the cell performance. Five flow field designs, parallel, parallel with baffles, serpentine, Z-type, and Z-type with baffles as shown in Figure 4, were compared in this section.

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Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design…

637

For a fair comparison, all cells have the same dimensions of 23 mm×23 mm×2.845 mm, the inlet and outlet cross-sectional areas of each flow channel are 1×1 mm2, and the rib width and heights are all 1 mm. All cells have the same 0.4 mm thick GDL, 0.005 mm thick CL, and 0.035 mm thick proton exchange membrane. The anode flow channels in all the cells are assumed to be in parallel with the channel and rib. All cells operate at the same conditions. The fuel cell temperatures was are all assumed to be 323 K, the reactants on the anode side are assumed to be hydrogen and water vapor with a relative humidity of 100%, and the reactants on the cathode side are assumed to be oxygen, nitrogen, and water vapor with a relative humidity of 100%. The inlet flow rate on the anode side is 260 cm3/min and the inlet flow rate on the cathode side is 700 cm3/min. The five flow field designs can be divided into conventional and interdigitated types. The conventional flow fields include the parallel, serpentine, and Z-type designs which and the interdigitated flow fields include the parallel with baffles and Z-type with baffles designs. The parallel flow field has 12 inlet flow channels with each channel 23 mm in length. The Z-type flow field has 6 inlet flow channels with a total flow channel length approximately one and a half times that of the parallel flow field. In addition, a bisection design is added to the flow channel to increase the diffusion area for the reactants to enter the fuel cell to fully utilize the supplied reactants. The serpentine flow field has 4 inlet flow channels with a total flow channel length approximately three times that of the parallel flow field, with twice as many turns as the Z-type flow field. The fuel consumption and the polarization loss inside the fuel cells differ due to the different numbers of channels, flow channel lengths and numbers of turns in the three flow field designs. Figure 5 shows the current density distributions on the middle cross-section in the PEM at the operating voltage of 0.3 V for the conventional flow field designs. In these three flow fields, the maximum current densities all appear at the flow channel inlets and decrease along the channels to the outlets as the oxygen is gradually consumed and liquid water accumulates due to the electrochemical reaction along the flow channels. In the parallel flow field, the parallel configuration of the flow channels leads to similar current density distributions along each flow channel. However, the reactant flow rate in a single channel is lowest in this configuration because it has the largest number of inlet flow channels. Hence, it provides the worst cell performance among the three flow field designs. The Z-type flow field has half as many inlet flow channels as the parallel flow field, so the fuel flow rate in each channel is twice as much and, consequently, more oxygen enters the cathode GDL. Moreover, at the bends in the flow channels, the shear stresses generated by the recirculating flow cause more oxygen to reach the reactive surfaces which significantly enhances the local current density is enhanced significantly. In generally, the Z-type flow field provides better performance than the parallel flow field. The serpentine flow field has the smallest number of inlet flow channels among the three designs and the highest fuel flow rate. The fuel experiences several turns along the way to the outlet and the flow channel length is several times longer than in the parallel flow field, which ensures that the oxygen diffuses to the reactive surfaces more completely. Moreover, the serpentine flow field has twice as many bends as the Z-type flow field so it benefits more from the shear stress effect which increases the liquid water removal. Thus, the serpentine flow field design provides the best cell performance if the pressure losses in the flow channel are not considered.

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638

Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

(a)

(b)

(c)

Figure 5. Current density distributions (A/m2) on the middle cross-section in the PEM at the operating voltage of 0.3 V. (a) parallel flow field, (b) Z-type flow field and (c) serpentine flow field.

(a)

(b)

(c)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 6. Liquid water distributions at the cathode GDL-CL interface at the operating voltage of 0.3 V. (a) parallel flow field, (b) Z-type flow field and (c) serpentine flow field.

The effects of the conventional flow field designs on the liquid water distributions at the cathode GDL-CL interface at the operating voltage of 0.3 V are shown in Figure 6. The maximum concentrations of liquid water for all three flow fields occurs under the ribs at the flow channel outlets where the local convection effect is not significant and the liquid water can not be efficiently removed, which results in a large amount of liquid water accumulation. Figure 6(a) indicates that in the parallel flow field the liquid water removal due to the convection effect is the least efficient because of the low fuel flow rate in each flow channel. In addition, without the corner effect, the porous material is almost completely blocked by the liquid water. In the Z-type flow field, the liquid water removal due to the convection effect is much more efficient, which leads to a highest liquid water concentration decrease between the inlet and the outlet. The serpentine flow field has the least amount of liquid water with only a small amount of liquid water accumulating at the flow channel outlets because the serpentine flow field has the highest flow rate per each channel, stronger convection and more corners in the flow field, which helps remove the liquid water. The interdigitated flow field designs discussed in this paper include the parallel flow field with baffles and the Z-type flow field with baffles. In the parallel flow field with baffles design, the baffles are added to the conventional parallel flow field to create inlet flow channels and outlet flow channels separated as shown in Figure 4(d). The purpose of the baffles is to improve

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Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design…

639

the low liquid water remove efficiency and the water concentration losses in the conventional parallel flow field at high current densities. In the Z-type flow field with baffles design, the baffles are included in the Z-type flow field in the split flow channels as shown in Figure 4(e) which increases the amount of oxygen reaching the reactive surfaces by forced convection in addition to diffusion and removes the liquid water generated by the electrochemical reaction.

(a)

(b)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 7. Current density distributions (A/m2) on the middle cross-section in the PEM at the operating voltage of 0.3 V. (a) parallel flow field with baffles and (b) Z-type flow field with baffles.

Figure 7 shows the local current density distributions on the middle cross-section in the PEM at the operating voltage of 0.3 V for the interdigitated flow fields. In the parallel flow field with baffles, the baffle effect maximizes the current density underneath the inlet flow channels, with lower current densities along the ribs on both the lef and right sides, and the minimum current densities underneath the outlet flow channels. In the Z-type flow channel with baffles where the baffles are located in the middles of the flow channels, the cell performance enhancement by the forced convection is less than that in the parallel flow field with baffles. The current densities are highest underneath the split flow channels. Generally speaking, the parallel flow field with baffles has stronger convection than the Z-type flow field with baffles so the parallel provides better cell performance.

(a)

(b)

Figure 8. Liquid water distribution at the cathode GDL-CL interface at the operating voltage of 0.3 V. (a) parallel flow field with baffles and (b) Z-type flow field with baffles.

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Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

Figure 8 shows the liquid water distributions at the cathode GDL-CL interface for the interdigitated flow fields at the operating voltage of 0.3 V. There is only a small amount of liquid water in the parallel flow field with baffles with the maximum liquid water concentration at the outlet flow channels. In the Z-type flow field with baffles, the forced convection is stronger in the middle of the flow channels with very good liquid water removal there. In the upstream and downstream flow channels in the Z-type flow field with baffles, the liquid water generated by the electrochemical reaction is removed mainly by diffusion which results in liquid water accumulation under the ribs. In the conventional flow fields, the fuel transport from the flow channels to the GDL and the CL is primarily driven by diffusion and recirculation at the corners. At low operating voltages, the strong electrochemical reaction causes excessive water accumulation at the cathode which may block the porous layers and reduce the reactant transport to the cathode GDL and CL. The interdigitated flow fields insert baffles into the conventional flow fields, with the reactants entering the cathode GDL mainly by forced convection induced by the baffles. In the upstream flow channels, the reactants are transported to the GDL by diffusion. As the reactants move toward the baffles, they are obstructed by the baffles and forced to flow forwards the reactive surfaces. Furthermore, the shear stresses generated by the forced convection helps remove the excessive water in the GDL which reduces the water flooding at the cathode and mass transport losses in the fuel cell. Figure 9 shows the I-Vcell polarization curves for fuel cells with the conventional and the interdigitated flow field designs. For operating voltages greater than 0.7 V, the flow field design has little effect on the cell performance because the electrochemical reaction is very mild and only a limited amount of oxygen is consumed. However, the situation is quite different for operating voltages lower than 0.7 V. The cell performance increases in the following order: the parallel flow field, the Z-type flow field, the serpentine flow field, the Z-type flow field with baffles and the parallel flow field with baffles. The cell performance with the parallel flow field with baffles is 94% better than with the basic parallel flow field, while the Z-type flow field with baffles provides 46% higher cell performance than the basic Z-type flow field. Therefore the results show that adding baffles to the conventional flow fields effectively reduces the mass transport losses, increases the limiting current density and enhances the cell performance.

Vcell (V)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

640

1.0

P arallel Flow Field Z-type Flow Field Serpentine Flow Field

0.8

Parallel Flow Field w ith B affle Z-type Flow Field w ith B affle

0.6 0.4

B ase C ase A node Fuel=H 2 C athode Fuel=Air A node Flow R ate=260cc/m in C athode Flow R ate=700cc/m in

0.2 0.0

0

5000

10000

15000

20000

25000

30000

2

I ( A/m )

Figure 9. Polarization curves of PEM fuel cells with the conventional and interdigitated flow field designs.

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Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design…

641

The flow field design influences not only the cell performance but also the pressure drop in the fuel cell. Large pressure drops in the fuel cell mean that more power is needed to pump the reactant gases. Thus, the pressure drop is a significant issue to be considered in choosing the flow field designs in addition to the polarization curve. The cathode pressure drop is related to the power densities:

Wp =

ΔpAchannelV Atotal

(35)

where Wp represents the cathode pressure drops, Δp is the total cathode pressure drop across the fuel cell, Achannel is the cathode cross-sectional inlet flow area, V is the reactant velocity at the cathode inlet, and Atotal is the reaction area. The calculated cathode pressure drops and output power for the conventional and interdigitated designs are listed in Table 2. Although the pressure drops for the interdigitated flow field designs are larger than for the conventional flow field designs, the pressure drops are far less than the cell output power for these miniature fuel cells, so the pressure drops are not significant. For this reason, the PEM fuel cell using the parallel flow field with baffles is the best choice of five flow field designs. Table 2. Estimated pressure drops at an operating voltage of 0.3 V for the five flow field designs Δp (Pa)

Wcell (W/m2)

Wp (W/m2)

Wnet (W/m2)

Parallel

107.92

13876

0.198341

13875.80

Serpentine

406.97

17900

2.243848

17897.76

Z-type

155.78

15255

0.5726

15254.43

Parallel with baffles

210.10

26890

0.772264

26889.23

Z-type with baffles

453.10

22257

1.665459

22255.33

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Flow field type

3.2. Effect of Width Ratio of Flow Channel to Rib This section discusses how the ratio of the channel to rib cross-sectional areas affects the PEM fuel cell performance and the reactant and liquid water distributions for various flow field designs. The parallel, interdigitated and serpentine cathode plates flow fields (shown in Figure 10 are analyzed. The cell performance is simulated for a small x×y×z= 23×23×2.845 mm3 cell. The anode flow field for all the cells has 12 parallel 1 mm wide channels and 11 1 mmwide ribs. The diffusion layer is 0.4 mm thick, the CL is 0.005 mm thick, and the PEM is 0.035 mm thick. For the parallel flow field, the cathode flow field has 12 channels and 11 ribs, for the interdigitated flow field, the cathode flow field has 6 inlet channels, 6 outlet channels and 11 ribs, and for the serpentine flow field design (here a triple serpentine flow field), the cathode flow field has three serpentine loops. All channels and ribs are 1 mm wide.

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Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

(a)

(b)

(c)

Figure 10. Schematics of the PEM fuel cell cathode flow fields, (a) parallel flow field; (b) interdigitated flow field; (c) triple serpentine flow field.

The flow channel and rib widths are defined as Wc and Wr. A non-dimensional parameter, the flow channel area ratio, Ar, defined as the ratio of the cathode flow channel area to the total reaction area in a cell, is used to describe the effect of the channel to rib cross-sectional areas area ratio. Ar is taken as 0.3, 0.4, 0.522, 0.6 and 0.7 for the parallel and interdigitated flow fields and as 0.4, 0.522 and 0.6 for the serpentine flow field, where Ar=0.522 denotes that the widths of the flow channels and ribs are equal (1 mm). The widths of the flow channels and ribs for the various flow channel area ratios are listed in Table 3. The operating conditions for all the fuel cells were the same as in section 3.1.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Table 3. The real widths of flow channel and rib for the various flow channel area ratios Ar 0.3 0.4 0.522 0.6 0.7

Lc (mm) 0.575 0.767 1.000 1.150 1.342

Lr (mm) 1.464 1.255 1.000 0.836 0.627

The effects of Ar on the I-Vcell polarization curves for parallel, interdigitated and serpentine flow fields are presented in Figs. 11 (a-c). At operating voltages greater than 0.7 V, the flow field type and flow channel area ratio have little effect on the cell performance because the electrochemical reaction rate is slow and only a limited amount of oxygen is consumed with only a small amount of liquid water produced. Therefore, the oxygen transport rates for these three flow field types and all the flow channel area ratios are sufficient to maintain the reaction rates. However, for low operating voltages, the flow field design significantly affects the cell performance. For the parallel flow field design, since the reactant transport into the porous layers is mainly by diffusion, the large flow channel area ratio, which represents a large contact area between the reactant and the GDL, allows more reactant to directly diffuse into the CL and participate in the electrochemical reaction, resulting in improved cell performance. However, for the interdigitated flow field design,

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Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design… (a)

643

A r=0.3

1.0

A r=0.4 A r=0.522 A r=0.6

0.8 Vcell (V)

A r=0.7

0.6

0.4

0.2

0

3000

6000

9000

12000

15000

2

I ( A/m ) (b)

A r=0.3

1.0

A r=0.4 A r=0.522 A r=0.6

Vcell (V)

0.8

A r=0.7

0.6

0.4

0.2

0

4000

8000

12000 16000 20000 24000 28000 2

I ( A/m ) (c)

Α r=0.4

1.0

Α r=0.6

0.8 Vcell (V)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Α r=0.522

0.6

0.4

0.2

0

5000

10000

15000

20000

2

I ( A/m )

Figure 11. Effects of the flow channel area ratio on the performance of PEM fuel cell with various flow field designs. (a) parallel flow field; (b) interdigitated flow field; (c) serpentine flow field.

because the forced convection produced by the baffles efficiently increases the fuel transport into the porous layers, the flow channel area ratio is not important and has relatively little effect on cell performance. Figure 11(b) shows that the effect of the flow channel area ratio gradually increases for the interdigitated flow field design as the operating voltage is reduced. For example, at an operating voltage of 0.4 V, the current densities for the five area ratios

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Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan

differ by 560 A/m2, while at 0.3 V, the difference is 1100 A/m2. For the interdigitated flow field design, the cell with Ar=0.7 has the worst performance while the cell with Ar=0.4 has the best performance. Figure 11(c) shows that for the triple serpentine flow field, at the lower operating voltages, the cell performance improves slightly as the flow channel area ratio increases, indicating that the cell performance is also not strongly related to the flow channel area ratio. For the serpentine flow field, as for the interdigitated flow field, the reactants enter the GDL by forced convection at the flow channel bends due to the corners. In addition, there is also under-rib convection between adjacent flow channels (see section 3.3). The forced convection and the under-rib convection significantly enhance the local oxygen mass flow rates in the porous layers, so the flow channel width ratio has less effect on the serpentine flow field design. Since the reactant transport and liquid water removal rates for the interdigitated and serpentine flow field designs are similar, we will only focus on comparing the parallel and interdigitated flow fields to understand the effect of the flow channel area ratio on the cell performance. 28000 24000

(a) □

0.3 V A r=0.3,

○

A r=0.4,

0.7 V △

A r=0.522, ▽

A r=0.6,

◇

A r=0.7

2

Ix ( A/m )

20000 16000 12000 8000 4000

0.005

0.010

0.015

0.020

x (m)

(b) 40000

□

0.3 V A r=0.3,

○

A r=0.4,

0.7 V △

A r=0.522, ▽

A r=0.6,

◇

A r=0.7

30000 2

Ix ( A/m )

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

0 0.000

20000

10000

0 0.000

0.005

0.010

0.015

0.020

x (m)

Figure 12. Effects of the flow channel area ratio on the local current density distribution at y=11.5 mm on the middle cross-section in the PEM. (a) parallel flow field design; (b) interdigitated flow field design

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Figure 12 shows the local current density distribution at y=11.5 mm on the middle crosssection in the PEM at operating voltages of 0.3 V and 0.7 V for the parallel and interdigitated flow field designs. At the higher operating voltage of 0.7 V, the flow field type and the flow channel area ratio have almost no impact on the local current densities. The local current densities are all 5000 A/m2 under the flow channels and under the ribs for both the parallel and the interdigitated flow field designs, far less than at the lower operating voltage. Hence, the cell performance is not dependent on the flow field design at higher operating voltages. At lower operating voltages, the flow field design significantly affects the cell performance. At 0.3 V, the local current densities for the two types of flow field designs are quite different, while they are quite similar for the various flow channel area ratios for the same flow channel type. The local current densities for the interdigitated flow field design are much higher than those for the parallel flow field design, indicating that the interdigitated flow field design significantly improves the cell performance. For the parallel flow field design, the current densities under the flow channels are larger than those under the ribs since the diffusion path for oxygen arriving in the CL under the ribs is longer than that for under the flow channels and the stronger shear stress under the flow channels enhances the liquid water removal. As the flow channel area ratio increasing, the local current densities under both the flow channels and the ribs increase for the parallel flow field design. Therefore, the overall cell performance improves as the flow channel area ratio increases. The baffles at the ends of the flow channels divide the flow channels in the interdigitated flow field design into inlet flow channels and outlet flow channels, with their local current density distributions differing from those in the parallel flow field design. The local current density distribution along the x direction shows that: the maximum occurs along the inlet flow channel (peaks in Figure 12(b)), with decreases under the rib on the right side of the inlet flow channel and further decreases to a minimum under the outlet flow channel (valleys in Figure 12(b)). The current densities then increase again under the rib on the right side of the outlet flow channel to the maximum under the next inlet flow channel. Although the flow channel area ratio has relatively little effect on the cell performance for the interdigitated flow field design, Figure 12(b) shows that the local current densities under the outlet flow channels for the cell with Ar=0.4 are higher than those for the cell with Ar=0.3 and are almost equal to the other three flow channel area ratios, and that the local current densities under the inlet flow channels are very similar by highest for Ar=0.3, so Ar=0.4 gives the best performance. The local oxygen concentrations and liquid water distributions inside the fuel cell directly affect the local current density distributions and the cell performance. Figures 13 and 14 show the effects of the flow channel area ratio on the local oxygen mass flow rates and liquid water distributions at y=11.5 mm on the cathode GDL-CL interface at operating voltages of 0.3 V and 0.7 V for the parallel and interdigitated flow field designs. At the higher operating voltage of 0.7 V, for both the parallel and interdigitated flow field designs, the local oxygen mass flow rates and liquid water concentrations for the various flow channel area ratios are all less than those at the lower operating voltage of 0.3 V, indicating that the oxygen consumption and the liquid water production rates are both less due to the slower chemical reaction rate at the higher operating voltage. At the lower operating voltage of 0.3 V, both the flow channel type and the flow channel area ratio significantly affect the local oxygen mass flow rates and liquid water distributions along the cathode GDL-CL interface. The oxygen mass flow rates are lower and the liquid water concentrations are higher for the parallel flow field design than for the interdigitated flow field design, so the parallel channel cell

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performance is worse. For the parallel flow field design, the oxygen mass flow rates are larger under the flow channels but rapidly decrease under the ribs, while the liquid water concentration displays the opposite trend. As the flow channel area ratio increases, the oxygen mass flow rates increase under both the flow channels and the ribs, while the liquid water concentrations decrease. For the interdigitated flow field design, the oxygen mass flow rates and liquid water distributions are quite different from the parallel flow field design, while they are quite similar for the various flow channel area ratios. The oxygen mass flow rate reaches a maximum under the inlet flow channel, gradually decreases under the rib on the right side of the inlet flow channel, and then further decreases to a minimum under the outlet flow channel. The oxygen flow rate then gradually increases again under the next rib to a maximum under the next inlet flow channel. The liquid water distributions are lowest under the inlet flow channel, increase under the rib on the right side of the inlet flow channel, and continue to increase under the neighboring outlet flow channel to a maximum. The liquid water distributions then decrease under the neighboring rib to a minimum under the next inlet flow channel. 0.0030

(a) □

○

A r=0.4,

0.7 V △

A r=0.522, ▽

A r=0.6,

A r=0.7

◇

mO2( kg/m s)

0.0025

0.3 V A r=0.3,

2

0.0020 0.0015

.

0.0010 0.0005 0.005

0.010

0.015

0.020

x (m)

(b) 0.004

□

0.3 V A r=0.3,

○

A r=0.4,

0.7 V △

A r=0.522, ▽

A r=0.6,

◇

A r=0.7

0.003

2

mO2( kg/m s)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

0.000

.

0.002

0.001

0.000

0.005

0.010

0.015

0.020

x (m)

Figure 13. Effects of the flow channel area ratio on the local oxygen mass flow rates at y=11.5 mm on the cathode GDL-CL interface. (a) parallel flow field design; (b) interdigitated flow field design.

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Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design…

0.25

(a) □

0.3 V A r=0.3,

○

A r=0.4,

647

0.7 V △

A r=0.522, ▽

A r=0.6,

◇

A r=0.7

0.20

s

0.15

0.10

0.05

0.00 0.000

0.005

0.010

0.015

0.020

x (m)

0.12 (b) □

0.3 V A r=0.3,

○

A r=0.4,

0.7 V △

A r=0.522, ▽

A r=0.6,

◇

A r=0.7

s

0.09

0.06

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0.03

0.00 0.000

0.005

0.010

0.015

0.020

x (m)

Figure 14. Effects of the flow channel area ratio on the local liquid water distributions at y=11.5 mm on the cathode GDL-CL interface. (a) parallel flow field design; (b) interdigitated flow field design

Comparison of Figs. 12-14 indicates that the local current density distributions are dependent on the local oxygen mass flow rates and liquid water distributions. The high oxygen mass flow rates along the cathode GDL-CL interface mean that more oxygen diffuses into the CL to participate in the electrochemical reaction and more liquid water is generated, resulting in a higher local current density. For the parallel flow field design, since the fuel enters the porous layers mainly by diffusion, as the flow channel area ratio increases, the contact area between the oxygen and the diffusion layer increases so more oxygen directly diffuses into the porous layers and participates in the electrochemical reaction and the cell performance is improved. For the interdigitated flow field design, the oxygen is forced to enter the diffusion layer at the end of the inlet flow channel due to the baffles, thus the oxygen utilization efficiency is increased. Meanwhile, the higher stresses produced by the forced convention help remove the liquid water. Therefore, the cell performance of the interdigitated flow field design is higher than that of the parallel flow field design. Although

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more liquid water is produced by the electrochemical reaction in the GDL-CL interface(?) under the flow channels, high shear forces carry the liquid water to under the adjacent ribs and the outlet flow channels, thus providing higher oxygen mass flow rates in the diffusion layer and CLs under the flow channels. Under the ribs and the outlet flow channels, the oxygen mass flow rates are lower and the electrochemical reaction is weaker with less liquid water production. However, the liquid water from under the inlet flow channels is carried there, so a large amount of liquid water accumulates under the outlet flow channels. Excessive liquid water in the porous layers blocks the porous media pores and reduces the oxygen transport and the electrochemical reaction rates. Therefore, at lower operating voltages, the liquid water distribution strongly affects the oxygen transport rates and the local current density distributions, which ultimately affect cell performance. 0.0048

Α r=0.4

Α r=0.522

Α r=0.6

V cell=0.7 V

Α r=0.4

Α r=0.522

Α r=0.6

0.0032

2

mO2 ( kg/m s)

0.0040

V cell=0.3 V

.

0.0024 0.0016 0.0008 0.0000 0.000

0.005

0.010

0.015

0.020

Figure 15. Oxygen mass flow rates at y=11.5 mm on the cathode GDL-CL interface at operating voltages of 0.3 V and 0.7 V for the triple serpentine flow fields with various flow channel area ratios. 0.16 0.14

V cell=0.3 V

Α r=0.4

Α r=0.522

Α r=0.6

V cell=0.7 V

Α r=0.4

Α r=0.522

Α r=0.6

0.12 0.10 s

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

x (m)

0.08 0.06 0.04 0.02 0.00 0.000

0.005

0.010

0.015

0.020

x (m)

Figure 16. Liquid water concentrations at y=11.5 mm on the cathode GDL-CL interface at operating voltages of 0.3 V and 0.7 V for the triple serpentine flow fields with various flow channel area ratios.

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Similarly, Figs. 15 and 16 show the oxygen mass flow rates and liquid water distributions at y=11.5 mm on the cathode GDL-CL interface at the operating voltages of 0.3 V and 0.7 V for the triple serpentine flow fields with the various flow channel area ratios. The liquid water distribution is opposite to the oxygen mass flow rate distribution. At the higher operating voltage of 0.7 V, the oxygen mass flow rates and liquid water concentrations are lower than those at the lower operating voltage of 0.3 V, with their values almost equal to each other for the three flow channel width ratios, so the cell performance is not dependent on the flow channel width ratio. At the lower operating voltage of 0.3 V, as the flow channel width ratio increases, the oxygen mass flow rates increase slightly, while the liquid water concentrations decreases slightly, so the cell performance improves a little. 1100 0.3 0.7 0.3 0.7 0.3 0.7

1000 900 800

Δp (Pa)

700

V, V, V, V, V, V,

interdigitated interdigitated parallel parallel serpentine serpentine

600 500 400 300 200 100 0

0.3

0.4

0.5

0.6

0.7

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Ar

Figure 17. Effects of the cathode flow channel area ratio on the pressure losses in PEM fuel cells with various flow field designs.

Figure 17 shows the effect of the flow channel area ratio on the cathode side pressure drop in the cells with the three flow field designs. Figure 17 indicates that the cell pressure drops are almost independent of the operating voltages, but depend strongly on the flow channel area ratio. As the flow channel area ratio increases, the cell pressure drops decrease. Since the cathode oxygen flow rates are the same in all the simulations, the larger flow channel area ratios result in lower fuel inlet velocities and reactant flow resistances. The pressure drops can be calculated from Eq. 35. Again, the pressure drops can be ignored because the pressure drops are far less than the cell output power for these miniature fuel cells. Therefore, the pressure drops for these miniature PEM fuel cells will not be discussed further except for some special cases.

3.3. Effect of Flow Channel Aspect Ratio This section discusses the effect of the flow channel aspect ratio on the PEM fuel cell performance for the parallel, interdigitated and serpentine flow field designs. Miniature fuel cells with the same dimensions of the anode flow field, two GDLs, two CLs, and membrane

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in section 3.1 were analyzed. Figure 18 shows side views of the cells with various cathode aspect ratios. All the cells included 12 flow channels with 11 ribs on the cathode side all 1 mm wide and with various cathode flow channel heights of 0.5, 0.75 1.00, 1.25, 1.50 and 2.00 mm. The operating conditions for all cells were the same as in section 3.1.

(a) Λ=0.500

(b) Λ=0.75

(c) Λ=1.00

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

(d) Λ=1.25

(e) Λ=1.50 z

x

(f) Λ=2.00

Figure 18. Side views of PEM fuel cells with various cathode flow channel aspect ratios.

Section 3.2 illustrated the importance of the under-rib convection with the serpentine flow field design. This section analyzes why the under-rib convection occurs in the serpentine flow field design. When the reactants are pumped through the flow channel from the inlet to the outlet, the local pressure in the channel can be approximated as

p ( L ) = pin −

128μ Q L 4 πd eff

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(36)

Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design…

651

where pin is the inlet pressure of the reactants, p(L) is the local pressure of the reactants, μ is the reactant viscosity, Q is the mass flow rate of the reactants, deff is the effective hydraulic diameter of the flow channel, and L is distance along the channel measured from the inlet. Equation (36) indicates that the pressure decreases downstream due to viscous losses. For the single and triple serpentine flow field designs shown in Figure 19, each serpentine loop proceeds through several 180° turns to form 12 channels and 11 ribs. For each rib, the two channels on the left and right sides may belong to the same loop, or may belong to two different loops; therefore, the pressure differences at each location between the two channels on the left and right sides differ for the various cases. For example, for the triple serpentine flow field design shown in Figure 19(b), the two channels on the left and right sides of rib 1 belong to loops AA' and BB'. The inlet pressures of the reactants in these two channels are almost equal; thus, the pressure difference between points a and b is nearly zero. However, the two channels on the left and right sides of rib 2 belong to the same loop, CC'; therefore, the pressure at point c is higher than that at point d. The pressure difference across the rib, i.e., between points c and d, is expressed as:

Δp =

128μ Q ( Ld − Lc ) 4 πDeff

(37)

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Therefore, the pressure difference between points c and d is proportional to the distance between the points, Ld-Lc. When the two points are near the channel inlet, the pressure difference is maximized, but is minimized near the channel bend. When the pressure gradient across the rib is much larger than the pressure gradient along the channel direction, considerable cross flow of reactants from point c to point d occurs between adjacent channels (referred to as under-rib convection). According to Darcy’s law, the average superficial velocity in the porous layer across the rib can be expressed as:

vaver =

128kpQ 4 πDeff Wr

( Ld − Lc )

(38)

where kp is the permeability of the reactants through the GDL and Wr is the rib width. Therefore, when the inlet flow rate of the reactants, the channel and rib dimensions, and permeability of the GDL remain constant, vaver is proportional to the distance between points c and d, Ld-Lc. Therefore, the under-rib convection is stronger near the inlet of loops CC' and weaker at the channel bend. The under-rib convection induces strong convection in the electrode, bringing the reactants to the CL and removing the liquid water from the reaction sites and porous electrodes; therefore, this under-rib convection significantly influences the PEM fuel cell performance with the serpentine flow field design. For the single serpentine flow field, the under-rib convection occurs between each two adjacent flow channels (under all the ribs), while for the multi-serpentine flow field, it occurs only under certain ribs, for example, for the triple serpentine flow field design, under-rib convection also occurs under ribs 2 and 3 as well as under rib 4. Due to the different under-rib convection flow rates, the flow channel aspect ratio is expected to have different effects on the cell performance for the

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single and triple serpentine flow fields. Therefore, this section considers the single and triple serpentine flow fields.

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Figure 19. Schematics of the single and triple serpentine flow fields on the cathode side of the PEM fuel cells. (a) single serpentine; (b) triple serpentine.

First, consider the effect of the flow channel aspect ratio on the PEM fuel cell performance with the parallel and interdigitated flow fields. Figure 20 presents the I-Vcell polarization curves for the parallel and interdigitated designs with various flow channel aspect ratios, Λ. For the parallel design with operating voltages greater than 0.7 V, the polarization curves almost coincide with each other for the various aspect ratios as shown in Figure 20(a), indicating that the cell performance is independent of the aspect ratio. However, at operating voltages less than 0.7 V, the flow channel aspect ratio strongly affects the cell performance, with the effect increasing with decreasing operating voltage. Furthermore, the cell performance is improved as the flow channel aspect ratio decreases. In this design, the reactant transport to the GDL and the CL is mainly by diffusion with smaller aspect ratios giving larger inlet reactant velocities at the same inlet flow rate, which enhances the liquid water removal. A lower water content in the porous layers then reduces the oxygen transport resistance, which improves the cell performance. For the interdigitated design, baffles are inserted at the end of the specified channels so that the reactant transport to the porous layers is by forced convection instead of diffusion. This significantly increases the reactant transport to the porous layers and also produces larger shear forces which enhance liquid water removal. Therefore, the aspect ratio has less effect on the reactant transport, liquid water removal, and cell performance. Thus, for the interdigitated design, even at an operating voltage of 0.5 V, the cell performances for the various aspect ratios are still almost the same. At operating voltages less than 0.5 V, the aspect ratio has a relatively smaller effect on the cell performance compared with the parallel design. Table 4 lists the mean current densities at

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an operating voltage of 0.3 V, which shows that the aspect ratio is not linearly related to the cell performance. As the aspect ratio increases from 0.50 to 1.00, the mean current densities increase, while as the aspect ratio increases from 1.00 to 2.00 the mean current densities decrease. Thus, the optimal cell performance occurs at an aspect ratio of 1.00. (a)

Λ =0.50 Λ =1.25

1.0

Λ =0.75 Λ =1.50

Λ =1.00 Λ =2.00

Vcell (V)

0.8

0.6

0.4

0.2

0

3000

6000

9000

12000

15000

2

I ( A/m ) Λ =0.50 Λ =1.25

(b)

1.0

Λ =0.75 Λ =1.50

Λ =1.00 Λ =2.00

Vcell (V)

0.8

0.6

0.4

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

0.2

0

5000

10000

15000

20000

25000

2

I ( A/m )

Figure 20. Polarization curves of PEM fuel cells with parallel and interdigitated flow fields for various cathode flow channel aspect ratios. (a) parallel flow field; (b) interdigitated flow field.

Table 4. Current densities at an operating voltage of 0.3 V for the interdigitated flow field design for various aspect ratios Λ I (A/m2)

0.50 25908

0.75 26428

1.00 26619

1.25 26596

1.50 26562

2.00 26536

To further understand the effect of the aspect ratio on cell performance, the local transport characteristics were analyzed at two representative operating voltages of 0.3 and 0.7 V for aspect ratios of 0.50, 1.00, and 2.00. Figure 21 shows the local current density distributions at y=11.5 mm in the middle cross-section of the PEM for the parallel and interdigitated designs. At the operating voltage of 0.7 V, the local current densities for the various aspect ratios are all about 5000 A/m2, far less than at the operating voltage of 0.3 V. Consequently, the cell

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performance is not dependent on the aspect ratio. For the parallel design, the maximum local current densities occur at the centers of the flow channels, while the minimum local current densities occur at the centers of the ribs because the oxygen diffusion path to the CL under the ribs is longer than under the channels. At the operating voltage of 0.3 V, for the parallel design the peaks and valleys of the local current densities all increase as the aspect ratio decreases and the cell performance increases with decreasing aspect ratio. For the interdigitated design, the channels are divided into inlet and outlet flow channels with the local current density distributions markedly different from those for the parallel design. The maximum local current densities occur at the centers of the inlet flow channels, while the minimum local current densities occur at the centers of the outlet flow channels. At the operating voltage of 0.3 V, the peak local current densities for the various aspect ratios are almost the same, while the minimums for Λ=0.5 are the lowest, then better for Λ=2.0, and highest for Λ=1.0. Therefore, the best cell performance occurs at Λ=1.0, and the worst is at Λ=0.5. (a)

30000

V cell=0.3 V Λ =0.50 Λ =2.00

2

Ix ( A/m )

25000

V cell=0.7 V Λ =1.00

Λ =0.50 Λ =2.00

Λ =1.00

20000 15000 10000 5000 0.000

0.005

0.010

0.015

0.020

x (m) (b)

42000 36000

V cell=0.3 V Λ =0.50 Λ =2.00

V cell=0.7 V Λ =1.00

Λ =0.50 Λ =2.00

Λ =1.00

30000 2

Ix ( A/m )

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

48000

24000 18000 12000 6000 0.000

0.005

0.010

0.015

0.020

x (m)

Figure 21. Local current density distributions at y=11.5 mm on the middle cross-section in the PEM for various flow channel aspect ratios. (a) parallel flow field; (b) interdigitated flow field.

Figure 22 shows the water concentration distributions at y=11.5 mm along the cathode GDL-CL interface for the two designs. The liquid water concentrations at the lower operating voltages are higher than those at the high operating voltages. Therefore, limitation of the mass transport by water is a key factor affecting the cell performance at low operating voltages. For

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the parallel design, the minimum water concentrations occur at the centers of the channels, while the maximum water concentrations occur at the centers of the ribs because the shear forces produced by the reactant flow in the porous layers under the channels are larger than those under the ribs. As the aspect ratio decreases, the reactant inlet velocity increases and the water trapped in the porous layers is efficiently removed. Therefore, the water concentrations decrease with decreasing aspect ratio. For the interdigitated design, the minimum water concentrations occur at the center of the inlet flow channels, while the maximum water concentrations occur at the centers of the outlet flow channels. The baffles force the reactants to enter the porous layers by forced convection, so the liquid water distributions are not sensitive to the aspect ratio. Figure 22(b) shows that at the operating voltage of 0.3 V, the water concentration is the highest at Λ=0.5, better at Λ=2.0, and lowest at Λ=1.0. 0.27

(a)

V cell=0.3 V

0.24

Λ =0.50 Λ =2.00

0.21

V cell=0.7 V Λ =0.50 Λ =2.00

Λ =1.00

Λ =1.00

0.18 s

0.15 0.12 0.09 0.06 0.03 0.000

0.005

0.010

0.015

0.020

x (m)

0.10

(b) V cell=0.3 V Λ =0.50 Λ =2.00

V cell=0.7 V Λ =1.00

Λ =0.50 Λ =2.00

Λ =1.00

0.06 s

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0.08

0.04

0.02

0.00 0.000

0.005

0.010

0.015

0.020

x (m)

Figure 22. Liquid water distributions at y=11.5 mm on the cathode GDL-CL interface for various flow channel aspect ratios. (a) parallel flow field; (b) interdigitated flow field.

Figure 23 shows the oxygen mass flow rates at y=11.5 mm along the cathode GDL-CL interface for the two cases. Higher oxygen mass flow rates along the cathode GDL–CL interface mean that more oxygen enters the CL to participate in the electrochemical reaction per unit time, resulting in higher current densities and better cell performance. Comparison of Figs. 21 and 23 shows that the local current density distributions are similar to the oxygen mass flow rate distributions. Comparison of Figs. 22 and 23 shows that the local oxygen mass

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flow rate distributions are opposite to the water concentration distributions. For the parallel design, as the aspect ratio decreases the liquid water removal capability increases, so the oxygen mass flow rates increase and the cell performance improves. However, for the interdigitated design, since the water is removed mainly by forced convection produced by the baffles, the effect of the aspect ratio on the liquid water removal and the oxygen mass flow rates is less significant. 0.0035 (a)

V cell=0.3 V

0.0030

2

mO2 ( kg/m s)

0.0025

V cell=0.7 V

Λ =0.50 Λ =2.00

Λ =1.00

0.005

0.010

Λ =0.50 Λ =2.00

Λ =1.00

0.0020 0.0015

.

0.0010 0.0005 0.000

0.015

0.020

x (m) 0.0048

(b) V cell=0.3 V Λ =1.00

0.005

0.010

Λ =0.50 Λ =2.00

Λ =1.00

0.0032

2

mO2 ( kg/m s)

0.0040

V cell=0.7 V

Λ =0.50 Λ =2.00

0.0024

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

.

0.0016 0.0008 0.000

0.015

0.020

x (m)

Figure 23. Oxygen mass flow rates at y=11.5 mm on the cathode GDL-CL interface for various flow channel aspect ratios. (a) parallel flow field; (b) interdigitated flow field.

The flow channel aspect ratio also affects the performance of the serpentine flow field PEM fuel cell. Figure 24(a) shows the polarization curves for the single serpentine flow field with various flow channel aspect ratios. At operating voltages higher than 0.7 V, all the cells have the same performance, indicating that the flow channel aspect ratio has almost no effect on the cell performance. As the operating voltage decreases, the electrochemical reaction rates increase with more oxygen consumption and liquid water production. Since the various flow channel aspect ratios have significantly different liquid water removal and oxygen transport rates to the CL, the flow channel aspect ratio will affect the cell performance at lower operating voltages. Figure 24(a) shows that for the single serpentine flow field, as the flow channel aspect ratio decreases the cell performance improves. However, for the flow channel aspect ratios of 0.5 and 0.75, the cell performance is almost the same, indicating that further reduction of the flow channel aspect ratio has less effect on the cell performance.

Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Two-Phase Model of PEM Fuel Cells for Optimal Flow Field Design… (a)

Λ =2.00 Λ =1.50 Λ =1.25 Λ =1.00 Λ =0.75 Λ =0.50

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Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 24. Polarization curves for PEM fuel cells with various cathode flow channel aspect ratios. (a) single serpentine flow field; (b) triple serpentine flow field.

Figure 24(b) shows the polarization curves for the triple serpentine flow field for various flow channel aspect ratios. As with the single serpentine flow field, for operating voltages greater than 0.7 V, the cell performance for the triple serpentine flow field design is independent of the flow channel aspect ratio, while for operating voltages less than 0.7 V, the flow channel aspect ratio affects the cell performance with stronger effects at lower operating voltages. As the flow channel aspect ratio increases, the cell performance for the triple serpentine flow field improves more than with the single serpentine flow field, indicating that the flow channel aspect ratio has a stronger effect on the triple serpentine cell performance. To understand the effect of the flow channel aspect ratio on the cell performance for the single and triple serpentine flow fields, the local transport characteristics were analyzed for cells with flow channel aspect ratios of 0.5, 1.0 and 2.0. Figure 25(a) shows the local current density distributions at y=11.5 mm on the middle cross-section in the PEM for the single serpentine flow field. For the operating voltage of 0.7 V with the lower electrochemical reaction rates, the oxygen transported to the CL for all the various flow channel aspect ratios satisfies the reactions, so the local current density distributions are almost the same. For the operating voltage of 0.3 V, the electrochemical reaction rates are markedly higher, so the various flow channel aspect ratio designs provide different amounts of oxygen to the CL,

Multiphase Flow Research, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

658

Xiao-Dong Wang, Wei-Mon Yan and Yuan-Yuan Duan (a)

35000 30000

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Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 25. Local current density distributions at y=11.5 mm on the middle cross-section in the PEM for various flow channel aspect ratios. (a) single serpentine flow field; (b) triple serpentine flow field.

which leads to very different current density distributions for the various flow channel aspect ratios. As the flow channel aspect ratio decreases, the maximums and minimums of the local current densities all increase, so the cell performance improves. With increasing x, the local current densities decrease because the oxygen is gradually consumed by the electrochemical reactions along the flow direction. Figure 25(b) shows the local current density distributions at y=11.5 mm on the middle cross-section in the PEM for the triple serpentine flow field. As with the single serpentine flow field, for the operating voltage of 0.7 V the local current density distributions are the same for the various flow channel aspect ratios. However, for the operating voltage of 0.3 V, the current density distributions are significantly different from those for the single serpentine flow field. For the single serpentine flow field, there is a relatively little difference in the current densities under the flow channels and ribs because the under-rib convection occurs between each two adjacent flow channels, which provides additional oxygen to the CL under the ribs, so the current densities under the ribs significantly increase. However, for the triple serpentine flow field, the under-rib convection occurs only under rib 1 (5 mm