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CFD Simulation of Immiscible Liquid Dispersions Srinath Madhavan Department of Chemical Engineering Outline Introduction to liquid-liquid dispersions, Motivation driving the current study, Objectives of the present investigation, Research methodology, Simulation results and discussion, Conclusions and recommendations. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 2 Liquid-Liquid Dispersions Immiscible liquid dispersions are commonly encountered in CPI, For instance in liquid-liquid extraction, emulsification and homogenization, direct contact heat transfer, polymerization etc. Enhanced heat/mass transfer rates are desirable in most processes, These depend on the heat/mass transfer coefficient, the driving force and the interfacial area of contact, It is relatively easier to manipulate the contact area when compared to the driving force or the heat/mass transfer coefficient. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 3 Interfacial Area of Contact For a unit volume of the Liquid-Liquid dispersion, dispersed phase holdup Interfacia l area dispersed phase diameter A combination of smaller drop sizes and larger dispersed phase holdup is usually sought. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 4 Importance of Dispersed Phase Holdup Holdup is a fundamental multiphase characteristic which: Influences the overall performance, Affects the pressure drop, Determines the global residence time, Can significantly modify the flow structure, Is therefore an important design parameter. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 5 Holdup Distribution For improved design and efficient contacting, correlations that relate the system performance to the local flow characteristics need to be developed, While the average dispersed phase holdup can reasonably predict certain parameters such as the pressure drop, it cannot accurately predict local heat/mass transfer rates, It therefore becomes important to carry out experiments to determine the local holdup distribution in the system. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 6 The need for CFD studies Although extensive experiments can provide enough information to develop empirical correlations, there are certain inherent limitations such as: The limited range of application, Simplifying assumptions used in their development, Scale-up issues, Use of intrusive measurement techniques, Inability to develop expressions suited for complex geometries, Time consuming and often expensive, Safety concerns etc. Hence there is a growing need for alternatives to experimental analysis. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 7 Computational Fluid Dynamics (CFD) Accurate simulation of fluid flows by solving the basic conservation equations (mass, momentum and energy) is the primary objective of CFD, Although CFD cannot entirely replace experiments, it features several lucrative advantages when compared to conventional experimental analysis: Low cost, Prompt analysis devoid of any scale-up issues, Simulation of certain situations which cannot be handled experimentally, Advanced visualization of technical results that helps to better understand flow features etc. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 8 CFD and Dispersed Multi- fluid Systems There are quite a few approaches to dispersed Multi- fluid modeling using CFD: Discrete phase (Eulerian-Lagrangian), Two-fluid (Eulerian-Eulerian), Interface tracking (Volume of Fluid), Mixture (Algebraic Slip Mixture Model). Among these, the two-fluid approach is widely used owing to the adequate flow detail it provides (even at high dispersed phase volume fractions) in exchange for a reasonable amount of computation power. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 9 Two-fluid Approach to Multi-fluid CFD Modeling Dispersed phase 1 (e.g. air bubbles) Dispersed phase 2 (e.g. oil drops) Continuous phase (e.g. water) Reality Two-fluid model Realized by averaging the local instantaneous equations (mass, momentum and energy), which reduces computational power requirements, Concept of interpenetrating continua and phasic velocities and volume fractions. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 10 Two-fluid Model: Governing Equations Conservation of mass: n d q q q qvq m pq q q q t p 1 dt For steady-state incompressible flow in the absence of mass transfer this simplifies to: q v q 0 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 11 Two-fluid Model: Governing Equations (2) Conservation of Momentum: q q vq q q vq vq qp q q q g t Fq Flift ,q Fvm, q K pq v p vq m pq v pq n p 1 Again, for steady-state incompressible flow in the absence of mass transfer, external body forces (Fq), and virtual/added mass effects (Fvm), the momentum conservation equation simplifies to: q q v q v q q p q Change of momentum per unit volume Pressure force per unit volume Viscous force per unit volume F K v v q n q q g lift , q p 1 pq p Gravitational force per unit volume Lift force per unit volume Turbulent dispersion & drag force per unit volume 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 12 The closure problem q q v q v q q p q Change of momentum per unit volume Pr essure force per unit volume Viscous force per unit volume K v v q n q q g F lift , q p 1 pq p Gravitational force per unit volume Lift force per unit volume Turbulent dispersion and drag force per unit volume Turbulent stresses (viscous force per unit volume) and interphase forces (drag, lift and turbulent dispersion forces per unit volume) are unknown. In order to obtain a closed set of equations, these terms need to be supplied. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 13 Turbulence Closure Terms Viscous stresses in turbulent flows can be supplied through the specification of a turbulent viscosity calculated using an appropriate turbulence model, In the context of multi-fluid turbulence models, the standard k- turbulence model is most extensively studied. With specific reference to liquid-liquid dispersions, it has been found to be numerically robust and gives reasonable predictions for an affordable computational cost, Turbulence quantities for the dispersed phase can be modeled using Tchen’s theory of dispersion of discrete particles by homogeneous turbulence (TChen, 1947), Effect of dispersed phase on the flow structure of the continuous phase can be accounted for using turbulence modulation. This aspect is nonetheless, still under active research, It is however, a generally accepted fact that more research is required to accurately predict turbulence in multi-fluid systems (Ranade, 2002). 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 14 Interphase Closure Terms Although several interphase forces are encountered in liquid- liquid dispersions, experimental observations indicate that turbulent dispersion, drag and lift forces are the most significant (Farrar and Bruun, 1996; Domgin et al., 1997; Soleimani et al., 1999), With reference to immiscible liquid dispersions, a large number of investigations pertaining to interphase forces (particularly the drag force) are available in the open literature, Nevertheless, there has been no attempt to analyze and evaluate the different expressions for the interphase forces, Non-drag forces such as turbulent dispersion and lift forces dictate the lateral movement of the dispersed phase and thus influence the dispersed phase distribution. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 15 Research Objective The objective of the present study is to identify and quantify the various significant interphase forces encountered in turbulent bubbly flows of immiscible liquid dispersions. The knowledge so gained can be beneficially employed to develop generally applicable CFD guidelines for interphase closure in dispersed liquid-liquid systems. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 16 Overall Approach Selection of a liquid-liquid contactor that can be used to achieve the current research objectives, Review of previous work related to interphase forces in liquid- liquid systems, Selection of data sets for CFD validation, Preliminary simulations of liquid-liquid turbulent bubbly flows to compare and evaluate various formulations for drag and lift forces and turbulent dispersion, Identifying drag, lift and turbulent dispersion coefficient expressions and/or values which yield a good agreement with experimental data, To propose guidelines for inter-phase closure on the basis of the above simulation results. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 17 Choice of L-L Contactor – Vertical pipe Why pipes? Simple hydrodynamics when compared to other contacting units such as stirred tanks or mechanically agitated columns, Turbulence characteristics of the continuous phase are very well investigated, Can be expected to yield accurate predictions of the fundamental two- phase flow characteristics (e.g. local dispersed phase holdup, relative velocity between the phases etc.) without recourse to a large degree of empiricism and know-how. As pipes are ubiquitous in chemical, process and petroleum industries, an extensive database of detailed experimental results is also available. This is particularly true for the case of dispersed liquid-liquid pipeline flow (Foussat and Hulin, 1984; Farrar, 1988; Farrar and Bruun, 1988; Vigneaux et al., 1988; Simonian, 1993; Farrar and Bruun, 1996; Lang and Auracher, 1996; Al-Deen and Bruun, 1997; Ali et al., 1999; Lang, 1999; Soleimani et al., 1999; Fordham et al., 1999; Hamad et al., 2000). 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 18 Review of the Interphase Drag Force In dispersed multiphase systems, the force that opposes the relative velocity between the phases is called the drag Drag force (FD) force, C 3 C D d cVS2 FD Drag force on drops is different from 4 de the drag force on rigid spheres. This is Dispersed attributed to two factors: entity Internal circulation, CA Shape deformation. Drag force on a drop is affected in the A A C presence of adjacent drops. Again, there are two factors responsible for Fluid velocity vectors Direction of this behavior: relative Reduced buoyancy force on the drop, velocity Apparent increase in medium viscosity. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 19 Expressions for the Drag Coefficient of a Single Drop For single rigid spheres, the 0.35 Klee and Treybal expression proposed by Schiller Hu and Kintner Rigid sphere and Naumann (1935) is widely 0.30 Grace et al. Ishii and Zuber used, Schiller and Naumann 0.25 Relative velocity (m/s) For single drops, several expressions for the drag 0.20 coefficient have been proposed: Hu and Kintner (1955) 0.15 Klee and Treybal (1956) 0.10 Grace et al. (1976) Ishii and Zuber (1979) 0.05 Single drops It can be seen that significant differences between the two are 0.00 observed at higher equivalent 0 2 4 6 8 10 12 14 16 drop diameters (i.e. greater Equivalent diameter (mm) than 3 mm). 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 20 Expressions for the Drag Coefficient of a Drop in the Presence of Adjacent Drops 0.16 medium = μc medium > μc 0.14 Relative velocity (m/s) ρmedium = ρc ρmedium < ρc 0.12 us 0.1 0.08 0.06 0.04 um 0.02 de = 5 mm us 0 um 0 0.1 0.2 0.3 0.4 0.5 0.6 Dispersed phase holdup (-) drop = μd drop = μd Ishii and Zuber (Dense fluid particles) ρdrop = ρd ρdrop = ρd Ishii and Zuber (corrected w ith Rusche and Issa (2000) Kumar and Hartland Ishii and Zuber (Single drop) If (drop > μc) and (ρdrop < ρc) => um < us 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 21 Review of the Interphase Lift Force – Inviscid Lift Axis When a dispersed phase entity moves through a non-uniform flow field, it will Fluid velocity vectors experience a lift force due to the vorticity or shear in the continuous phase field, A C The lift force acts on the dispersed entities B Dispersed entity in a direction perpendicular to the relative motion between the two phases. Wall Flift C L q p v q v p v q Axis Low velocity High Pressure A B CA C C Inviscid Inviscid Lift force CB Lift force CA CB Wall High velocity Low pressure Calculation of Relative Velocity 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 22 Review of the Interphase Lift Force – Vortex-shedding Lift Recent studies indicate that the inviscid lift force may not be the only Axis lift force experienced by a dispersed entity in shear flow (Taeibi-Rahni and Loth, 1996; Loth et al., 1997; Moraga et al., 1999), Wake-induced Lift force Larger dispersed entities moving Wake much faster than the fluid shed Inviscid Lift force vortices as they move, An asymmetric wake behind the Wall dispersed entity can give rise to significant lateral forces that oppose the inviscid lift force. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 23 Expressions for the Lift Coefficient (CL) Constant lift coefficient, 0.7 The expression for lift 0.5 coefficient proposed by Moraga et al. (1999), 0.3 An approach similar to 0.1 that of Moraga et al. (1999) in which validity CL (-) -0.1 limits for the lift Moraga et al. (1999) coefficient expression CL = 0.0 -0.3 have been modified in accordance with the Troshko et al. (2001) recommendations made ReReÑ = 183897 -0.5 by Troshko et al. (2001). Moraga et al. (1999) applied to drops and bubbles -0.7 1 100 10000 1000000 100000000 Re Re 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 24 Review of Turbulent Dispersion A pseudo-force which induces a diffusive flux that accounts for dispersion (or spread) of dispersed phase entities due to the random influence of the turbulent eddies present in the continuous phase. Model proposed by Simonin and Viollet In the In the (1990) absence of presence of Dp Dq Turbulent Turbulent v dr p q Dispersion Dispersion pq q pq p Very high DPN Very low DPN Dispersion Prandtl Number (DPN) K pq v p v q K pq U p U q K pq v dr Drag force Turbulent dispersion force per unit volume per unit volume 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 25 Data Sets used for CFD Validation 1 mm de 5 mm Continuous phase superficial Dispersed phase superficial Average dispersed phase Data Set Data Point velocity (m/s) velocity (m/s) holdup (-) F20 0.4935 0.1363 0.1912 78 mm ID Farrar and Bruun (1996) F25 0.4634 0.1637 0.2275 F30 0.4263 0.1972 0.2783 H10 0.5855 0.0651 0.0873 Hamad et al. (2000) H20 0.5855 0.1464 0.1764 A5 0.5441 0.0286 0.0493 Al-Deen and Bruun A10 0.5441 0.0605 0.0917 (1997) A20 0.5441 0.1360 0.1872 A30 0.5441 0.2332 0.2992 L20A 0.4000 0.1000 0.1851 16 mm ID L20B 1.2000 0.3000 0.1809 Lang (1999) L40 0.3000 0.2000 0.3692 L60 0.2000 0.3000 0.5474 200 mm ID V5 0.2268 0.0119 0.0323 V10 0.2149 0.0239 0.0661 Vigneaux et al. (1988) V30 0.1671 0.0716 0.2350 V50 0.1194 0.1194 0.4308 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 26 Comparative Evaluation of Drag Coefficient Expressions for Single Entities (using CFD Simulations) Experimental conditions of Al- Deen and Bruun (1997) were Expression for CD0 de = 2 mm de = 5 mm de = 8 mm used as an example – phase ratio of the dispersed phase = Schiller and Naumann (1935) 0.04429 0.03894 0.03651 5 %, – Rigid sphere All expressions for single entities predict similar holdups Ishii and Zuber (1979) 0.04405 0.04095 0.04094 at low equivalent diameters (de 2 mm), As the equivalent diameter Grace et al. (1976) 0.04405 0.04023 0.04055 increases, the single drop holdup predictions start to Hu and Kintner (1955) 0.04432 0.04014 0.04032 deviate from the rigid sphere predictions, The drag model proposed by Klee and Treybal (1956) 0.04428 0.04208 0.04204 Ishii and Zuber (1979) was chosen as a representative for single drops. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 27 Comparative Evaluation of Drag Coefficient Expressions that account for the presence of other Drops (using CFD Simulations) Experimental conditions of Al-Deen and Bruun (1997) were used as an Expression for CDM de = 2 mm de = 5 mm example – phase ratio of the dispersed phase = 30 %, Ishii and Zuber - Dense fluid All expressions predict similar particles (1979) 0.2855 0.2751 holdups at de = 2 mm and at de = 5 mm, Kumar and Hartland (1985) 0.2890 0.2797 When compared to the average holdup as reported in the experiment ( 29 %), it is seen that Ishii and Zuber (1979) drag accounting for the presence of expression for single drops, adjacent entities results in a slightly modified to account for the presence 0.2902 0.2812 better prediction, of adjacent drops using the correction factor proposed by The expression proposed by Kumar Rusche and Issa (2000) and Hartland (1985) suitably accounts for the presence of Ishii and Zuber (1979) drag adjacent drops as its holdup expression for single drops 0.2824 0.2707 predictions lie between the other two approaches. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 28 Comparative Evaluation of Lift Coefficient Expressions/Values (using CFD Simulations) Experimental conditions of 0.5 Farrar and Bruun (1996) are Phase ratio of the chosen as an example, 0.45 de = 5 mm dispersed phase = 30 %, Dispersed phase holdup (-) The expression for lift 0.4 no turbulent dispersion coefficient proposed by 0.35 Moraga et al. (1999) was found to give numerical 0.3 instabilities and/or unphysical 0.25 predictions, Troshko et al. (2001) CL = + 0.01 Positive constants for CL 0.2 predict wall peaks whereas 0.15 CL = + 0.005 CL = + 0.001 negative constants predict ‘coring’ and/or ‘near-wall 0.1 CL = 0.0 CL = - 0.001 peaking’ trends, 0.05 CL = - 0.005 CL = - 0.01 All constant lift coefficients and the expression proposed 0 by Troshko et al. (2001) 0 0.2 0.4 0.6 0.8 1 predict non-zero volume fractions at the wall. Normalized radius (-) Drag coefficient expression used: Kumar and Hartland (1985) 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 29 Comparative Evaluation of Turbulent Dispersion Coefficient Values (using CFD Simulations) Turbulent dispersion effects 0.4 were simulated using the approach proposed by Simonin Phase ratio of the and Viollet (1990), which 0.35 dispersed phase = 30 % accounts for the response of Dispersed phase holdup (-) 0.3 drops to turbulent eddies in the continuous phase, 0.25 The experimental conditions of Farrar and Bruun (1996) are Drag coefficient expression used: 0.2 used as an example. A ‘data Kumar and Hartland (1985) point’ featuring a ‘near-wall’ 0.15 peak was chosen to demonstrate the effect of turbulent 0.1 DPN 0.075 DPN 0.75 dispersion, The expression for lift coefficient 0.05 DPN 7.5 DPN 0.0075 as proposed by Troshko et al. (2001) was used, 0 High DPN values (e.g. 7.5) 0 0.2 0.4 0.6 0.8 1 decrease the degree of turbulent Norm alized radius (-) dispersion and vice-versa. de = 5 mm 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 30 Summary of Simulation Details CFD package: Pre-processor: Gambit 2.1.2, Solver and Post-processor: Fluent 6.1.22, Hardware: GNU/Linux workstation (Pentium IV 2.53 GHz CPU, 1 GB DDR SDRAM, 1 GB swap space) running Red Hat Linux 9, Simulation time (20 minutes to 5 hours), Computation grid: Axisymmetric structured grid with 1:1 cell aspect ratios (6,000 to 80,000 cells), Near-wall treatment: Y+ 30 for Standard wall functions and Y+ 5 for Enhanced wall treatment, Solver configuration: Eulerian multiphase model, k-ε turbulence model for the continuous phase, TChen (1947) theory of dispersion by homogeneous turbulence for the dispersed phase, Mono-dispersed drop sizes in the range (1 to 5 mm), Drag coefficient expression proposed by Kumar and Hartland (1985), Lift coefficient expression proposed by Troshko et al. (2001) and constant (negative) lift coefficients, Turbulent dispersion using the Simonin and Viollet (1990) approach, Steady state solution approach. RSS (Residual sum of sqaures) was used to compare simulations with experimental data wherever possible. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 31 Data set of Farrar and Bruun (1996) 0.4 Experimental conditions: Pipe ID = 78 mm, 0.35 Length = 1.5 m ( 20 pipe diameters), 0.3 Dispersed phase holdup (-) QT = 0.00308 m3/s, 0.25 phase ratios of the dispersed phase (20, 25 0.2 & 30%), 0.15 Simulation conditions: F20 - SIM F20 - EXP de = 5 mm, 0.1 F25 - SIM F25 - EXP Lift coefficient proposed by Troshko et al. (2001), 0.05 F30 - SIM F30 - EXP DPN = 7.5. 0 0 0.01 0.02 0.03 0.04 Radial position (m) 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 32 Data set of Hamad et al. (2000) 0.4 Experimental conditions: Pipe ID = 78 mm, 0.35 H10 - SIM H10 - EXP Length = 4.2 m ( 53 pipe diameters), Dispersed phase holdup (-) 0.3 QT = 0.00310 (H10) and H20 - SIM H20 - EXP 0.00348 (H20) m3/s, 0.25 phase ratios of the dispersed phase (10, 0.2 20%), 0.15 Simulation conditions: Lift coefficient proposed 0.1 by Troshko et al. (2001), H10: de = 3.25 mm & 0.05 DPN = 0.01, H20: de = 3.50 mm & 0 DPN = 0.075. 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Radial position (m) 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 33 Data set of Al-Deen and Bruun (1997) Experimental conditions: 0.6 Pipe ID = 78 mm, A5 - SIM A5 - EXP Length = 1.5 m ( 20 A10 - SIM A10 - EXP pipe diameters), 0.5 QT = 0.00272 – 0.00369 A20 - SIM A20 - EXP Dispersed phase holdup (-) m3/s, phase ratios of the A30 - SIM A30 - EXP dispersed phase (5, 10, 0.4 20 & 30%), 0.3 Simulation conditions: Lift coefficient proposed by Troshko et al. (2001), 0.2 A5: de = 3.0 mm & DPN = 0.024, 0.1 A10: de = 3.5 mm & DPN = 0.075, A20: de = 4.0 mm & 0 DPN = 0.412, 0 0.01 0.02 0.03 0.04 A30: de = 4.0 mm & Radial position (m) DPN = 7.5. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 34 Data set of Lang (1999), Pipe ID: 16 mm, Length 65 pipe diameters 0.3 0.9 0.8 L20A - SIM L20A - EXP 0.25 Dispersed phase holdup (-) 0.7 Dispersed phase holdup (-) L40 - SIM L40 - EXP 0.2 0.6 0.5 0.15 L20B - SIM 0.4 0.1 0.3 0.2 0.05 L20B - EXP 0.1 QT = 0.00030 m3/s QT = 0.00010 m3/s 0 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0 0.002 0.004 0.006 0.008 Radial position (m) Radial position (m) L20A (DPN = 0.75; CL = -0.075; de = 1 mm) (DPN = 0.06; CL = -0.050; de = 2 mm) L40 (DPN = 3.00; CL = -0.050; de = 1 mm) 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 35 Data set of Vigneaux et al. (1988) Experimental conditions: 0.8 Pipe ID = 200 mm, Length V5 - SIM V5 - EXP = 14 m ( 70 pipe 0.7 diameters), V10 - SIM V10 - EXP QT = 0.0075 m3/s, phase V30 - SIM V30 - EXP Dispersed phase holdup (-) 0.6 ratios of the dispersed V50 - SIM V50 - EXP phase (5, 10, 30 & 50%), 0.5 Simulation conditions: 0.4 V5: de = 2.00 mm, CL = -0.05 & DPN = 1.593, 0.3 V10: de = 2.75 mm, CL = -0.05 & DPN = 0.328, 0.2 V30: de = 5.00 mm, Troshko et al. (2001) & 0.1 DPN = 4.125, V50: de = 5.00 mm, 0 Troshko et al. (2001) & 0 0.02 0.04 0.06 0.08 0.1 DPN = 4.125. Radial position (m) 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 36 Conclusions from CFD Simulations Following conclusions can be drawn based on the numerical study of liquid-liquid up-flows in vertical pipes spanning a wide range of experimental conditions: Liquid-Liquid bubbly up-flows in vertical pipes typically feature ‘Wall peaking’, ‘Near-wall peaking’ and ‘Coring’ trends for the dispersed phase holdup distribution. In order to successfully predict such inhomogeneous phase distributions, accounting for drag and lift forces and turbulent dispersion is imperative, An analysis of several drag coefficient expressions clearly reveals that the drag on drops differs significantly from the drag on rigid spheres, particularly at larger equivalent drop diameters. Also, at high dispersed phase holdups, accounting for the presence of adjacent drops yields slightly better predictions. However, the drag force alone cannot predict the local holdup accurately. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 37 Conclusions from CFD Simulations (2) Non-drag lateral forces such as the lift force and turbulent dispersion dictate the overall phase distribution: The expression for lift coefficient proposed by Troshko et al. (2001) yields very good predictions when bubble Reynolds numbers greater than 250 are encountered. However, for bubble Reynolds numbers lower than 250, constant (negative) values for the lift coefficients were found to yield the best predictions, Turbulent dispersion is found to be more significant at lower dispersed phase holdups when compared to higher dispersed phase holdups, The equivalent drop diameter (de) has to be increased as the dispersed phase holdup increases. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 38 Interphase Closure Guidelines for Liquid-Liquid Systems The following closure guidelines are recommended for application in dispersed liquid-liquid flows: Drag force: The drag coefficient expression proposed by Kumar and Hartland (1985) should be used to account for the drag force in immiscible liquid dispersions, Lift force: The expression for lift coefficient proposed by Troshko et al. (2001) should be used when bubble Reynolds numbers greater than 250 are encountered. At lower bubble Reynolds numbers, constant (negative) lift coefficients in the range (-0.05 to -0.075) are recommended, Turbulent dispersion: For the model proposed by Simonin and Viollet (1990), Dispersion Prandtl Numbers (DPN) in the range 0.01 DPN 0.075 are recommended for use at low dispersed phase holdups (< 10%) and Dispersion Prandtl Numbers in the range 0.075 DPN 7.5 are recommended for use at high dispersed phase holdups (i.e. up to 50%). 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 39 Recommendations for Future Work For dispersed flows featuring low Reynolds numbers (Re 250), expressions which directly estimate the lift coefficient based on local flow properties should be identified and tested, Various approaches to account for turbulent dispersion should be analyzed. In particular, models such as the one proposed by Lopez de Bertodano (1998) where the dispersion coefficient is expressed as a function of the locally evaluated, turbulent Stokes number should be tested, The effect of turbulence modulation (both enhancement and suppression) should be included in future simulations, The ability of the models to predict turbulence intensities in the continuous phase should then be tested, The effect of accounting for a dynamic size distribution of drops should also be investigated. Various drop breakage and coalescence models should be reviewed, and selected models ought to be suitably coupled to the multi-fluid CFD framework in order to study this effect properly. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 40 Acknowledgements Thanks to: Supervisors Dr. Al Taweel and Dr. Murat Koksal. Guiding committee members: Dr. Gupta, Dr. Dabros and Dr. Chuang. Fellow colleagues at the Mixing and Separation Research Laboratory. Guidance from the 'Academic Support Team' at Fluent is gratefully acknowledged. 6th July 2005 CFD Simulation of Immiscible Liquid Dispersions 41

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