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LAPPEENRANTA UNIVERSITY OF TECHNOLOGYFaculty of TechnologyDepartment of Mathematics and Physics
CFD Simulation of Two-phase andThree-phase Flows in Internal-loop Airlift
Reactors
The topic of this Master’s thesis was approved by the departmental council of the De-partment of Mathematics and Physics on 27th May, 2010.
Supervisors: Professor Heikki HaarioD.Sc. Arto Laari
Examiners: Professor Heikki HaarioD.Sc. Arto Laari
In Lappeenranta August 24, 2010.
Giteshkumar N PatelTeknologiapuistonkatu 2 B 2853850 LappeenrantaPhone: +358466477168Email: [email protected] & [email protected]
Abstract
Lappeenranta University of TechnologyDepartment of Mathematics and Physics
Giteshkumar N Patel
CFD Simulation of Two-phase and Three-phase Flows in Internal-loop AirliftReactorsMaster’s thesis
2010
73 pages, 48 figures, 7 tables
Key words: Airlift reactors, Gas holdup, Scale effect, CFD, Two-phase flow, Three-phaseflow, Solids distribution
Airlift reactors are pneumatically agitated reactors that have been widely used in chemi-cal, petrochemical, and bioprocess industries, such as fermentation and wastewater treat-ment. Computational Fluid Dynamics (CFD) has become more popular approach fordesign, scale-up and performance evaluation of such reactors. In the present work nu-merical simulations for internal-loop airlift reactors were performed using the transientEulerian model with CFD package, ANSYS Fluent 12.1. The turbulence in the liquidphase is described using the κ − ε model. Global hydrodynamic parameters like gasholdup, gas velocity and liquid velocity have been investigated for a range of superficialgas velocities, both with 2D and 3D simulations. Moreover, the study of geometry andscale influence on the reactor have been considered. The results suggest that both, ge-ometry and scale have significant effects on the hydrodynamic parameters, which mayhave substantial effects on the reactor performance. Grid refinement and time-step sizeeffect have been discussed.
Numerical calculations with gas-liquid-solid three-phase flow system have been carriedout to investigate the effect of solid loading, solid particle size and solid density onthe hydrodynamic characteristics of internal loop airlift reactor with different superficialgas velocities. It was observed that averaged gas holdup is significantly decreased withincreasing slurry concentration. Simulations show that the riser gas holdup decreaseswith increase in solid particle diameter. In addition, it was found that the averaged solidholdup increases in the riser section with the increase of solid density. These producedresults reveal that CFD have excellent potential to simulate two-phase and three-phaseflow system.
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AcknowledgementsFirst of all, I wish to convey my gratitude to the Department of Mathematics for thescholarship during my studies at Lappeenranta University of Technology.
I would like to express my sincerest gratitude to Professor Heikki Haario, for his super-vision, advice, and guidance from the very early stage of this thesis as well as giving meall facilities through out the work.
I gratefully acknowledge to D.Sc. Arto laari for his valuable advice and comments,supervision and crucial contribution, which made him a backbone of this thesis. It hasbeen a great pleasure working with and learning from him.
I am extending my heartfelt and very special gratitude to Yogi for her contributions,patience, understanding and love. She provided constant inspiration, constructive ideasand strength to hurdle all the obstacles in the completion this work. Thanks for pushingme to be the best, and giving me a shoulder when times were hard. May GOD blessher in all her endeavors because without her extraordinary support, completion of thisstudy would not have been possible.
I am also grateful to all my friends and colleagues at the Department of Mathematicsfor my memorial stay in Lappeenranta.
Last but not the least, my sincere acknowledgments are addressed to my family and theone above all of us, the omnipresent God, for answering my prayers and for giving methe strength, thank you so much Dear Lord. I am grateful to my dad, Narayanbhai,for giving me the life I ever dreamed and his constant inspiration and guidance kept mefocused and motivated. I can’t express my gratitude for my mom, Sitaben, in words,whose unconditional love has been my greatest strength. The constant love and supportof my elder brother, Rakeshbhai, his wife, Jayshree, and their child Krrish are sincerelyacknowledged. I convey special acknowledgement to my elder sister, Surekhaben, andher family. I would like to dedicate this thesis to my family and Yogi, the most importantperson in my life.
Lappeenranta, August 24, 2010
Giteshkumar N Patel
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Contents
1 Introduction 1
1.1 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Topologies of multiphase flow 4
2.1 Multiphase flow regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Gas-Liquid two phase flow regimes . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Flow patterns in vertical tubes . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Flow patterns in horizontal tubes . . . . . . . . . . . . . . . . . . . 8
2.3 Examples of flow regime maps . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Basic concepts of multiphase flow . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Approaches for numerical calculations of multiphase flow 19
3.1 Eulerian - Lagragian approach . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Eulerian - Eulerian approach . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Volume of fluid method . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Mixture model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.3 Eulerian model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Eulerian model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Volume fraction equation . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Conservation equations . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Interphase momentum exchange 32
4.1 The drag, lift and virtual mass forces . . . . . . . . . . . . . . . . . . . . . 33
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4.2 Interphase exchange coefficients . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.1 Fluid-Fluid exchange coefficient . . . . . . . . . . . . . . . . . . . . 35
4.2.2 Fluid-Solid exchange coefficient . . . . . . . . . . . . . . . . . . . . 36
4.2.3 Solid-Solid exchange coefficient . . . . . . . . . . . . . . . . . . . . 36
5 Two-phase flow simulation in internal-loop airlift reactor 37
5.1 Airlift reactor morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3.1 Comparison between 2D and 3D results. . . . . . . . . . . . . . . . 44
5.3.2 Scale influence on the hydrodynamics of internal-loop airlift reactors. 49
5.3.3 Grid sensitivity study . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.4 Time-step sensitivity study . . . . . . . . . . . . . . . . . . . . . . 56
6 Three-phase flow simulation in internal-loop airlift reactor 59
6.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.1 Effect of solid loading . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.2 Effect of solid particle size . . . . . . . . . . . . . . . . . . . . . . . 65
6.2.3 Effect of solid density . . . . . . . . . . . . . . . . . . . . . . . . . 67
7 Conclusions 70
References 72
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List of Tables
1 Details of computational grids for the Configurations I, II and III . . . . . 40
2 Two-phase simulation settings . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Phase properties used in two-phase simulation . . . . . . . . . . . . . . . . 43
4 Details of grids used in mesh-independence studies . . . . . . . . . . . . . 53
5 Computational time used by each grid for 2D and 3D domains. . . . . . . 56
6 Simulation setup for three phase flow calculation. . . . . . . . . . . . . . 61
7 Phases properties used for three phase flow simulation. . . . . . . . . . . . 61
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List of Figures
1 Multiphase flow regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Flow patterns of multiphase flow . . . . . . . . . . . . . . . . . . . . . . . 6
3 Flow patterns of gas-liquid flow in vertical tubes . . . . . . . . . . . . . . 7
4 Flow patterns in horizontal tubes . . . . . . . . . . . . . . . . . . . . . . . 9
5 Flow regime map for vertical gas-liquid flow . . . . . . . . . . . . . . . . . 10
6 Flow regime map for the horizontal flow of an air/water mixture in a5.1cm diameter pipe (Weisman (1983)) . . . . . . . . . . . . . . . . . . . . 11
7 Overview of CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8 Discretisation of flow in CFD . . . . . . . . . . . . . . . . . . . . . . . . . 16
9 Interface calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10 Different types of ALRs [10]. . . . . . . . . . . . . . . . . . . . . . . . . . 37
11 Schematic overview of computational geometry for Configurations I, IIand III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
12 (a) Boundary conditions applied on computational domain, (b) computa-tional grid, and (c) closer view of the bottom region of reactor. . . . . . . 41
13 Transient approach to steady-state of the gas and liquid velocities at thecentre of the reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
14 Contours of (a) gas velocity, and (b) liquid velocity of Configuration-Iwhen Ug = 0.02 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
15 Gas fraction contours from 2D and 3D cases when Ug = 0.02 m/s. . . . . 45
16 (a) Liquid velocity vector of 2D case and (b) liquid iso-surfaces from 3Dsimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
17 Gas velocity distribution in the riser tube at selected axial location. . . . . 46
18 Liquid velocity distribution at selected axial location. . . . . . . . . . . . . 46
19 Gas holdup distribution in the riser tube at selected axial location. . . . . 47
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20 Radial distribution of gas velocity from 2D and 3D simulations at selectedaxial location in the riser tube. . . . . . . . . . . . . . . . . . . . . . . . . 48
21 Radial distribution of liquid velocity from 2D and 3D simulations at se-lected axial location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
22 Radial distribution of gas holdup from 2D and 3D simulations at selectedaxial location in the riser tube. . . . . . . . . . . . . . . . . . . . . . . . . 48
23 Contours of (a) gas velocity, (b) liquid velocity, and (c) gas holdup forConfiguration-II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
24 Contours of (a) gas velocity, (b) liquid velocity, and (c) gas holdup forConfiguration-III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
25 (a) Gas velocity and (b) liquid velocity distributions of Configuration-I. . 50
26 (a) Gas velocity and (b) liquid velocity distributions of Configuration-II. . 51
27 (a) Gas velocity and (b) liquid velocity distributions of Configuration-III. 51
28 Gas holdup distributions in (a) Configuration-I and (b) Configuration-II. . 52
29 Comparison between average gas holdup in the riser of Configuration-Iand Configuration-II with range of Ug. . . . . . . . . . . . . . . . . . . . . 52
30 Instantaneous snapshots of gas holdup contours with (a) Grid A, (b) GridB, (c) Grid C, and (d) Grid D when Ug = 0.02 m/s. . . . . . . . . . . . . 54
31 Gas velocity distribution with 2D and 3D simulations for different grids. . 55
32 Gas holdup distribution with 2D and 3D simulations for different grids. . 55
33 Liquid velocity distribution with 2D and 3D simulations for different grids. 55
34 Profiles of gas hold produced from each selected grid and time-step size. . 57
35 Profiles of liquid velocity produced from each selected grid and time-stepsize. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
36 Contours of velocity magnitude for (a) liquid, (b) gas, and (c) solid ofthree-phase flow system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
37 Instantaneous snap shot of solid concentration in the reactor when reactoris initially loaded with 5% of solids. . . . . . . . . . . . . . . . . . . . . . . 63
38 Average gas holdup in the riser with different superficial gas velocity. . . . 64
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39 Average solids holdup in the riser with different superficial gas velocity. . . 64
40 Radial distribution of solids volume fraction at the center of the reactorwith different solids loading conditions. . . . . . . . . . . . . . . . . . . . . 65
41 Averaged gas holdup in the riser with different superficial gas velocity. . . 65
42 Average solids holdup in the riser with different superficial gas velocity. . . 66
43 Radial distribution of solids volume fraction at the center of the reactorwith different solid particle size. . . . . . . . . . . . . . . . . . . . . . . . . 67
44 Instantaneous snap shot of solids concentration in the reactor with ρs =2000 kg/m3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
45 Instantaneous snap shot of solids concentration in the reactor with ρs =3000 kg/m3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
46 Average solids holdup in the riser with different superficial gas velocity. . . 69
47 Average gas holdup in the riser section with different superficial gas velocity. 69
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Nomenclature
A cross sectional area [m2]
bd bubble diameter [m]
CD drag coefficient dimensionless
Cl lift coefficient dimensionless
e coefficient of restitution dimensionless
f force [N ]
g gravitational acceleration [m/s2]
h enthalpy J/kg
K interphase exchange momentum [N/m3]
m mass [kg]
N number of nodes
p pressure [n/m2]
Q volumetric flow rate of the phase [m3/s]
Re Reynolds number dimensionless
t time [s]
u velocity [m/s]
Ug superficial gas velocity [m/s]
V volume [m3]
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Greek Letters
α phase volume fraction dimensionless
ε dissipation rate of turbulent kinetic energy [m2/s3]
ε holdup dimensionless
κ turbulent kinetic energy [m2/s2]
µ viscosity [kg/ms]
µt turbulent viscosity [kg/ms]
ρ density [kg/m3]
σ surface tension [N/m]
τk viscous stress tensor [kg/ms2]
∇ gradient operator
Abbreviations
ALR Airlift Reactos
CFD Computational Fluid Dynamics
FDM Finite Difference Method
FEM Finite Element Method
FVM Finite Volume Method
PDE Partial Differential Equation
VOF Volume of Fluid Method
x
Subscript or Superscript
c referring continuous phase
d referring dispersed phase
D drag
G referring to gas
m referring mixture
MT mass transfer
L referring to liquid
LF referring to lift force
LUB referring to lubrication force
S referring to solid
l, s referring to fluid phase and solid phase respectively
p, q referring to phase p and q respectively
TD turbulent dispersion
VM virtual mass
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1 Introduction
The topic of the thesis is CFD simulation of two-phase and three-phase flows in internal-loop airlift reactors. The term airlift reactor (ALR) covers a wide range of gas-liquid orgas-liquid-solid pneumatic contacting devices that are characterized by fluid circulationin a defined cyclic pattern through channels built specifically for this purpose [10]. Airliftreactors are characterized by three distinct regions namely riser, gas-liquid-separator anddowncomer [11]. There are two types of ALR, ‘internal’ and ‘external’ loop ones. Internalloop reactors consist of concentric tubes or split vessels, in which a part of the gas isentrained into the downcomer, whereas external loop reactors are two conduits connectedat the top and the bottom, in which little or no gas recirculates into the downcomer. Thepart in which the sparger is located is called the riser, and the other one, the downcomer.Liquid circulation is induced by injecting gas at the bottom of the riser, thus creatinga density differences between riser and downcomer [12]. In such reactors, the requiredinteraction is provided by the density differences between the gas and the liquid. Thisfact enables extremely interaction and complicated gas-liquid or gas-liquid-solid reactionsto take place in such reactors. The airlift is popular as a gas-liquid contactor because itcan handle large quantities of liquid and gas on a continuous basis. In addition, it hasno moving parts, requires limited amount of energy for its operation and exhibits goodmass and heat transfer characteristics [13]. Due to this, ALR’s are finding increasingapplications in chemical industry, biochemical fermentation and biological wastewatertreatment processes [16][17].
Several publications have established the potential of computational fluid dynamics fordescribing the hydrodynamics of bubble columns [14][15]. The design, scale-up, andperformance evaluation of such reactors all require extensive and accurate informationabout the gas-liquid flow dynamics, particularly as CFD has become more popular ap-proach [29] . An important advantage of the CFD approach is that column geometry andscale effects are automatically accounted for. CFD can be used to simulate and optimizemixing, gas hold-up and mass transfer coefficients and distribution of the phases [18].In addition, scale and operating conditions of reactors have significant effects on globalhydrodynamic parameters such as: gas holdup, liquid circulation velocity, overall masstransfer rate in gas-liquid flow. In gas-liquid-solid flow, solid loading, solid particle sizeand solid density significantly affect the hydrodynamics characteristics of internal loopair lift reactor (Snape et al 1995, Vial et al 2000).
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1.1 Objectives of the thesis
The objectives of the thesis have been set as follows:
• To simulate the gas-liquid two-phase flow with internal-loop airlift reactor appli-cation by using Eulerian model in CFD. To investigate the 2D and 3D simulationswith various operating conditions (e.g., superficial gas velocity) and to check theeffect on global hydrodynamic parameters like gas holdup, gas velocity and liquidvelocity. To study the influence of geometry and scale on the reactor hydrodynam-ics. Furthermore, grid and time-step sensitivity are performed to investigate theinfluence of grid refinement and time-step size effect respectively on the results.
• By performing CFD simulation, to simulate gas-liquid-solid three-phase flow ininternal-loop airlift reactor system and to investigate the effects of superficial gasvelocity and particle loading on the gas holdup in the riser section. To study thedistribution of solid phase in the reactor with different operating conditions. Todetermine the effect of solid particle diameter and solid density on averaged gasholdup with the range of superficial gas velocity.
1.2 Thesis structure
This chapter consists of the objectives and the methodology of the thesis work. Thereader is then introduced to the main content of the following chapters.
Chapter 2 gives the brief details of multiphase flow. It describes the division of multi-phase flow regimes. The details information of two-phase flow are provided in it. Somefundamental terms are described there. At the end, CFD is introduced concisely withdiscretisation in solving fluid flow.
Chapter 3 demonstrates the numerical approaches for multiphase flow in CFD. Thedifferent models for solving multiphase flow are explained. The last part contains thedetails of turbulence models for multiphase flow.
Chapter 4 displays the details about the interphase momentum transform. Some infor-mation for interphase exchange coefficients are derived in it.
Chapter 5 presents work done on the two-phase flow with internal-loop airlift reactorsby using Eulerian model with different operating conditions. It shows the comparisonbetween 2D vs. 3D results of global hydrodynamics parameters. Scale effect are explainedin it. Grid and time-step size effect cases are discussed at the end.
Chapter 6 contains the three-phase flow work. In this chapter, the effects of the superficial
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gas velocity and the particle loading on the gas holdup in the riser are discussed. Solidsdistribution in the reactor are studied there. Some results about the effect of solidparticle diameter on gas holdup with the range of superficial gas velocity are explained.
Chapter 7 describes conclusions of the whole thesis. This focuses on the objectives ofthe work and how they are achieved throughout the thesis.
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2 Topologies of multiphase flow
A phase is simply one of the states of matter and it can be either a gas, a liquid, ora solid [19]. Multiphase flow is the simultaneous flow of several phases. Two-phaseflow is the simplest case of multiphase flow [20]. From a practical engineering pointof view one of the major design difficulties in dealing with multiphase flow is that themass, momentum, and energy transfer rates and processes can be quite sensitive to thegeometric distribution or topology of the components within the flow [8]. An appropriatestarting point is a phenomenological description of the geometric distributions or flowpatterns that are observed in common multiphase flows. First, we will discuss about theflow regimes of multiphase flow.
2.1 Multiphase flow regimes
The definition of the flow regime is a description of the morphological arrangement ofthe components, or flow pattern [19]. It is important to appreciate that different flowregimes occur at different fluid flow rates and differences also occur for different materials.Multiphase flow regimes can be grouped into four categories: gas-liquid or liquid-liquidflows; gas-solid flows; liquid-solid flows; and three-phase flows [21]. The following Figure1 represent a schematic diagram of the multiphase flow regimes.
Figure 1: Multiphase flow regimes
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Gas-liquid or liquid-liquid flows
The combined flow of gas and liquid is called a gas-liquid flow. Bubbly flow, slug slow,droplet flow and stratified/free-surface flow, etc. are the flow regimes types of gas-liquidflow. Later on we will get brief details about it.
Gas-solid flows
The simultaneous flow of gas and solid is a gas-solid flow. The following regimes can beseparated as:
• Fluidized bed: Fluidized bed is a vertical cylinder mechanism. It contains thesolid particles (homogeneous material or heterogeneous material). In fluidized beda gas is introduced through a distributor. The gas rising through the bed suspendsthe particles. Depending on the gas flow rate, bubbles appear and rise throughthe bed. It can be seen at CFB (Circulating Fluidized Bed) or BFB (BubblingFluidized Bed) boiler combustion technology.
• Particle-laden flow: This type of flow regime contains discrete particles in a contin-uous gas. Particle-laden flow examples include cyclone separators, air classifiers,dust collectors, and dust-laden environmental flows [21].
• Pneumatic transport: This is a flow regime that depends on factors such as solidloading, Reynolds numbers, and particle properties. Typical patterns are duneflow, slug flow, and homogeneous flow. Pneumatic transport examples includetransport of cement, grains, and metal powders [21].
Liquid-solid flows
The mixture of solid particles in liquid form a liquid-solid flow. Examples of it are slurryflow or sedimentation.
• Slurry flow: This flow is the transport of particles in liquids. The fundamentalbehavior of liquid-solid flows varies with the properties of the solid particles relativeto those of the liquid. Slurry flow examples include slurry transport and mineralprocessing.
• Hydrotransport: This describes densely-distributed solid particles in a continuousliquid. Hydrotransport examples include mineral processing and biomedical andphysio-chemical fluid systems.
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Figure 2: Flow patterns of multiphase flow
• Sedimentation: This describes a tall column initially containing a uniform dispersedmixture of particles. At the top of the column, a clear interface will be appeared.At the bottom, the particles will slow down and form a sludge layer and in themiddle a constant settling zone will exist. Sedimentation examples include mineralprocessing [21].
Figure 2 displays examples of multiphase flow with various flow patterns.
Three-phase flows
Three-phase flows are combinations of the other flow regimes. It means a combinationof gas-liquid-solid or two solid phases and one gas phase, etc. This types of flow canbe seen at petroleum refinery, in chemical separation technology or in combustion. Inthe present work, we will concentrate on both two-phase flow and three-phase flow. Thefollowing section describes the fundamental things about gas-liquid flow.
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2.2 Gas-Liquid two phase flow regimes
The flow of gas-liquid mixtures in pipes and other items of process equipment is commonand extremely important. It is important to appreciate that different flow regimes occursat different gas and liquid flow rates and differences also occur for different materials [22].In this section we will discuss about flow patterns at different position of tubes.
2.2.1 Flow patterns in vertical tubes
When the pipe is oriented vertically, the regimes of gas/liquid flow are illustrated inFigure 3 (see, for example, Hewitt and Hall Taylor 1970, Butterworth and Hewitt 1977,Hewitt 1982, Whalley 1987). The flow patterns are transformed from bubbly flow todisperse flow gradually where ’G’ refers for gas phase and ’L’ for liquid. The sequenceshown is that which would normally be seen as the ratio of gas to liquid flow rates isincreased.
In the bubbly regime, there is a distribution of dispersed bubbles throughout the contin-uous liquid phase. The bubbles may vary widely in size and shape but they are typicallynearly spherical and are much smaller than the diameter of the tube itself. As the gasflow rate increases, the average bubble size increases [22].
Figure 3: Flow patterns of gas-liquid flow in vertical tubes
The next regime occurs when the gas flow rate is increased to the point when manybubbles collide and coalesce to produce slugs of gas or to form larger bubbles. Generallyit is known as slug flow. The gas slugs have spherical noses and occupy almost the entire
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cross section of the tube, being separated from the wall by a thin liquid film. They arecommonly referred to as Taylor bubbles [19]. Between slugs of gas there are slugs ofliquid in which there may be small bubbles entrained in the wakes of the gas slugs.
Increasing the velocity of the flow, the structure of the flow becomes unstable with fluidtraveling up and down in an oscillatory fashion but with a net upward flow. At higherflow rates, a slug flow pattern is destroyed and a chaotic type of flow, generally knownas churn flow. Over most of the cross section there is a churning motion of irregularlyshaped portions of gas and liquid.
Further increase in the gas flow rate causes a degree of separation of the phases, the liquidflowing mainly on the wall of the tube and the gas in the core area. This flow regimesis known as annular flow [23]. The increment in the liquid flow rate, the concentrationof liquid is increased in the gas core. It is known as wispy-annular flow. The maindifferences between the wispy-annular and the annular flow regimes are that in the formerthe entrained liquid is present as relatively large drops and the liquid film contains gasbubbles, while in the annular flow regime the entrained droplets do not coalesce to formlarger drops.
2.2.2 Flow patterns in horizontal tubes
Consider a cocurrent gas-liquid flow in horizontal pipes, which displays similar flowpatterns to those for vertical flow. However, asymmetry is caused by the effect of gravity,which is most significant at low flow rates. The sequence of flow regimes in horizontaltubes as identified by Alves (1954) is shown in Figure 4. In the figure, the ’G’ is gasphase and ’L’ for liquid phase.
In the bubbly regime the bubbles are confined to a region near the top of the pipe. Onincreasing the gas flow rate, the bubbles become larger and coalesce to form long bubbleswhich is known as the plug flow regime. If the gas flow rate as increased then the gasplugs join and form a continuous gas layer in the upper part of the pipe. This type offlow, in which the interface between the gas and the liquid is smooth, is known as thestratified flow regime. Where the gas has lower viscosity and lower density, it will flowfaster than the liquid.
As the gas flow rate is increased further, the interfacial shear stress becomes sufficient togenerate waves on the surface of the liquid producing the wavy flow regime. If the gasflow rate continues to rise, the waves which travel in the direction of flow, grow until theircrests approach the top of the pipe and, as the gas breaks through, liquid is distributedover the wall of the pipe. This is known as the slug flow regime.
At higher gas flow rates an annular flow regime is found as in the vertical flow. At very
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Figure 4: Flow patterns in horizontal tubes
high flow rates the liquid film may be very thin, the majority of the liquid being dispersedas droplets in the gas core. This type of flow may be called as the spray or mist flowregime. It may be noted that similar flow regimes can be seen with immiscible liquidsystems. If the densities of the two liquids are close the flow regimes for the horizontalflow will more nearly resemble those of the vertical flow.
2.3 Examples of flow regime maps
The prediction of the flow regime in gas-liquid two-phase flow is rather uncertain partlybecause the transitions between the flow regimes are gradual and the classification ofa particular flow is subjective [22]. In addition, there are many industrial processes inwhich the mass quality is a key flow parameter and therefore mass flux maps are oftenpreferred.
There are various flow regime maps in the literature which gives the idea of the flowpatterns and behavior, two of which are given in the following Figures 5 and 6. Forvertical flow of low pressure air-water and high pressure steam-water mixtures, Hewittand Roberts (1969) have determined a flow regime map shown in Figure 5. Here, jG andjL denote the volumetric fluxes of the gas and liquid.
9
Figure 5: Flow regime map for vertical gas-liquid flow
For the gas phase the volumetric flux is given by,
jG =QGA
(1)
and for the liquid phase is given by,
jL =QLA
(2)
where QG and QL are the volumetric flow rates of the gas and the liquid, and A is thecross-sectional area. Here in Figure 5 the axes represent the superficial momentum fluxesof the gas and liquid. In addition to allowing the flow regime for a specified combinationof gas and liquid flow rates to be determined, the diagram shows how changes of operatingconditions change the flow regime.
In particular it can be seen that the sequence of flow regimes described above is producedby increasing the gas momentum flux and/or reducing the liquid momentum flux. If themomentum flux of gas and liquid are relatively low then the flow regime can be churnand for higher rate of momentum flux, it will be wispy annular.
Figure 6 displays the flow regime map for horizontal gas-liquid flow patterns. It wasgiven by Weisman (1983). It shows the occurrence of different flow regimes for the flowof an air/water mixture in a horizontal, 5.1cm diameter pipe. Here GL and GG, denote
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Figure 6: Flow regime map for the horizontal flow of an air/water mixture in a 5.1cmdiameter pipe (Weisman (1983))
the superficial mass fluxes of the liquid and the gas. For the liquid phase,
GL =ML
A= jLρL (3)
and for gas phase it is given by,
GG =MG
A= jGρG (4)
Where jL and jG denote the volumetric fluxes of the liquid and gas respectively. Densityof liquid phase is ρL and for gas is ρG.
2.4 Basic concepts of multiphase flow
This section introduces a few definitions that are fundamental to multiphase flows. Forconvenience, the term discrete or dispersed phase will be used for the particles, droplets,or bubbles, while carrier or continuous phase will be used for the carrier fluid. Thedispersed phase is not materially connected.
11
Dispersed Phase and Separated Flows
Dispersed phase flows are flows in which one phase consists of discrete elements, such asdroplets in a gas or bubbles in a liquid. The discrete elements are not connected. Wherein a separated flow, the two phases are separated by a line of contact. An annular flowis a separated flow in which there is a liquid layer on the pipe wall and a gaseous core.In other words, in a separated flow one can pass from one point to another in the samephase while remaining in the same medium.
Volume fraction and Densities
The volume fraction of the dispersed phase is defined as
αd = limδV→V 0
δVdδV
(5)
where δVd is the volume of the dispersed phase in volume δV . The volume δV 0 isthe limiting volume that ensures a stationary average. Unlike a continuum, the volumefraction cannot be defined at a point. Equivalently, the volume fraction of the continuousphase is
αc = limδV→V 0
δVcδV
(6)
where δVc is the volume of the continuous phase in volume. This volume fraction issometimes referred to as the void fraction and in the chemical engineering literature, thevolume fraction of the dispersed phase is often referred to as holdup [24]. By definition,the sum of the volume fractions must be unity, i.e.,
αd + αc = 1 (7)
The bulk density (or apparent density) of the dispersed phase is the mass of the dispersedphase per unit volume of mixture or, in terms of a limit, is defined as
ρd = limδV→V 0
δMd
δV(8)
where δMd is the mass of the dispersed phase. The bulk density is related to the materialdensity ρd by
12
ρd = αdρd (9)
The sum of the bulk densities of the dispersed and continuous phases is the mixturedensity
ρd + ρc = ρm (10)
Superficial and Phase Velocities
For multiphase flow, the superficial velocity of each phase is the mass flow rate Mof thatphase divided by the cross sectional area A and material density or in other words, thesuperficial velocity is nothing but the velocity of a fluid in a pipe, conduit, column etc inthe absence of packing or obstruction. Like in packed columns the actual velocity of thefluid through it is actually the volumetric flow rate divided by the cross sectional area.so the velocity achieved by the same fluid in the same column in absence of the packingis called superficial velocity. The superficial velocity for the dispersed phase is
Ud =Md
ρdA(11)
The phase velocity u is the actual velocity of the phase. The superficial velocity and thephase velocity are related by the volume fraction
Ud = αdud (12)
In addition, the same relations hold for the carrier phase.
Quality, Concentration, and Loading
Another parameter important to the definition of dispersed-phase flows is the dispersed-phase mass concentration, which is the ratio of the mass of the dispersed phase to thatof the continuous phase in a mixture as,
C =ρdρc
(13)
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This parameter will sometimes be referred to as the particle or droplet mass ratio. Some-times dispersed-phase volume fraction is designated as concentration. The quality of aliquid–vapor mixture where the liquid in the dispersed phase is
x =ρdρm
(14)
Another term in common use in multiphase flows is loading, which is the ratio of massflux of the dispersed phase to that of the continuous phase:
z =md
mc(15)
However, loading has also been used to denote concentration.
2.5 Computational Fluid Dynamics
Fluid dynamics is the science of fluid motion. Fluid flow is commonly studied in one ofthree ways:
• Experimental fluid dynamics
• Theoretical fluid dynamics
• Numerically: computational fluid dynamics (CFD)
CFD is one of the branches of fluid mechanics [25]. It is the science of predicting fluidflow, heat transfer, mass transfer, chemical reactions, and related phenomena by solvingthe mathematical equations which govern these processes using numerical methods andalgorithms. In order to provide easy access to their solving power all commercial CFDpackages include sophisticated user interfaces to input problem parameters and to exam-ine the results. Hence all CFD codes contains three main elements: (i) a pre-processor,(ii) a solver and (iii) a post-processor [26].
• Pre-processor: pre-processing consists of the input of a flow problem to a CFDprogram by means of an operator-friendly interface and the subsequent transfor-mation of this input into a form suitable for use by the solver [26]. The regionof fluid to be analysed is called the computational domain and it is made up of anumber of discrete elements called the mesh (or grid). The users need to definethe properties of fluid acting on the domain before the analysis is begun; theseinclude external constraints or boundary conditions, like pressure and velocity toimplement realistic situations.
14
• Solver: a program that calculates the solution of the CFD problem. Here thegoverning equations are solved. This is usually done iteratively to compute theflow parameters of the fluid as time elapses. Convergence is important to producean accurate solution of the partial differential equations.
• Post-processor: used to visualize and quantitatively process the results from thesolver. In a contemporary CFD package, the analysed flow phenomena can bepresented in vector plots or contour plots to display the trends of velocity, pressure,kinetic energy and other properties of the flow.
Figure 7: Overview of CFD
When solving fluid flow problems numerically, the surfaces, boundaries and spaces aroundand between the boundaries of the computational domain have to be represented in aform usable by computer. This can be achieved by some arrangement of regularly andirregularly spaced nodes around the computational domain known as the mesh. Basically,the mesh breaks up the computational domain spatially; so that calculations can becarried out at regular intervals to simulate the passage of time, as numerical solutionscan give answers only at discrete points in the domain at a specified time. The process
15
of transforming the continuous fluid flow problem into discrete numerical data which arethen solved by the computer is known as discretisation. Generally, there are three majorparts of discretisation in solving fluid flow
• Equation discretisation
• Spatial discretisation
• Temporal discretisation
The following figure shows the procedures of performing discretisation on a typical fluidflow problem.
Figure 8: Discretisation of flow in CFD
Equation discretisation
As mentioned above, the governing equations consist of partial differential equations.Equation discretisation is the translation of the governing equations into a numericalanalogue that can be solved by computer. In CFD, equation discretisation is usuallyperformed by using the finite difference method (FDM), the finite element method (FEM)or the finite volume method (FVM) [26].
The FDM employs the concept of Taylor expansion to solve the second order partialdifferential equations (PDE) in the governing equations of fluid flow. This method isstraightforward, in which the derivatives of the PDE are written in discrete quantitiesof variables resulting in simultaneous algebraic equations with unknowns defined at thenodes of the mesh. FDM is famous for its simplicity and ease in obtaining higher orderaccuracy discretisation. However, FDM only applies to simple geometries because it
16
employs a structured mesh.
Unlike FDM, unstructured mesh is usually used in FEM. The computational domain issubdivided into a finite number of elements. Within each element, a certain number ofnodes are defined where numerical values of the unknowns are determined. In FEM,the discretisation is based on an integral formulation obtained using the method ofweighted residuals, which approximates the solutions to a set of partial differential equa-tions using interpolation functions. FEM is famous for its application around complexgeometries because of the application of unstructured grid. But numerically, it requireshigher computer power compared to FDM. So the finite volume method (FVM), whichis mathematically similar to FEM in certain applications, but requires less computerpower, is the next consideration in CFD applications.
In FVM, the computational domain is separated into a finite number of elements knownas control volumes. The governing equations of fluid flow are integrated and solvediteratively based on the conservation laws on each control volume. The discretisationprocess results in a set of algebraic equations that resolve the variables at a specified finitenumber of points within the control volumes using an integration method. Through theintegration on the control volumes, the flow around the domain can be fully modelled.FVM can be used both for the structured and unstructured meshes. Since this methodinvolves direct integration, it is more efficient and easier to program in terms of CFDcode development. Hence, FVM is more common in recent CFD applications comparedto FEM and FDM.
Spatial discretisation
Spatial discretisation divides the computational domain into small sub-domains makingup the mesh. The fluid flow is described mathematically by specifying its velocity at allpoints in space and time. All meshes in CFD comprise nodes at which flow parametersare resolved. Three main types of meshes commonly used in computational modellingare structured, unstructured and multi-block structured mesh.
A structured mesh is built on a coordinate system, which is common in bodies with asimple geometry such as square or rectangular sections. However, a structured meshperforms badly when the geometry is complex, which is quite common in industrialapplications. In the view of this, unstructured meshes were introduced.
In an unstructured mesh, the nodes can be placed accordingly within the computationaldomain depending on the shape of the body, such that different kinds of complex compu-
17
tational boundaries and geometries can be simulated. An unstructured mesh works wellaround complex geometries but this requires more elements for refinement compared toa structured mesh on the same geometry, leading to higher computing cost. To com-pensate between computing cost and flexibility, we turn our attention to the multi-blockstructured mesh.
In a multi-block structured mesh, the computational domain is subdivided into differentblocks, which consists of a structured mesh. A multi-block structured mesh is morecomplicated to generate compared to a structured and an unstructured mesh but itcombines the advantages of both. It is more computer efficient than an unstructuredmesh and yet provides ease of control in specifying refinement needed along certainsurfaces or walls, especially for meshing around complex geometries.
Temporal discretisation
The third category of discretisation is the temporal or time discretisation. Generally,temporal discretisation splits the time in the continuous flow into discrete time steps.In transient or time-dependent formulations, we have an additional time variable t inthe governing equations compared to the steady state analysis. This leads to a systemof partial differential equations in time, which comprise unknowns at a given time as afunction of the variables of the previous time step. Thus, unsteady simulation normallyrequires longer computational time compared to a steady case due to the additional stepbetween the equation and spatial discretisation.
Either explicit or implicit method can be used for unsteady time-dependent calculation.In an explicit calculation, a forward difference in time is taken when calculating thetime tn+1 by using the previous time step value (n denotes state at time t and n + 1at time t + ∆t ). An explicit method is straight forward, but each time step has to bekept to a minimum to maintain computation stability and convergence. On the otherhand, implicit method computes values of time step tn+1 at the same time level in asimulation at different nodes based on a backward difference method. This results ina larger system of linear equations where unknown values at time step tn+1 have to besolved simultaneously. The principal advantage of implicit schemes compared to explicitones is that significantly larger time steps can be used, whilst maintaining the stabilityof the time integration process . A smaller time step ∆t in an explicit method implieslonger computational running time but it is relatively more accurate [9].
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3 Approaches for numerical calculations of multiphase flow
A multiphase flow system consists of a number of single phase regions which are boundedby moving interfaces. In principle, a multiphase flow model could be formulated in termsof the local instant variables relating to each phase and matching boundary conditionsat all phase interfaces. It is very complicate to obtain solution of multiphase systemdirectly or in other words it is almost impossible to solve directly. As a starting point forderivation of macroscopic equations which replace the local instant description of eachphase by a collective description of the phases.
For the formulation of the multiphase flow, averaging procedures can be classified intothree main groups (Ishii 1975), the Boltzmann averaging, the Lagrangian averaging andthe Eulerian averaging. These groups can be further divided into sub-groups based onthe variable with which a mathematical operator or averaging is defined. Here we willdiscuss about two numerical approaches for solving multiphase flows in CFD.
3.1 Eulerian - Lagragian approach
This approach is applicable to continuous-dispersed systems and is often referred to as adiscrete particle model or particle transport model. The primary phase is continuous andis composed of a gas or a liquid. The secondary phase is discrete and can be composedof particles, drops or bubbles.
In the Eulerian–Lagrangian (E–L) approach, the continuous phase is treated in an Eu-lerian framework (using averaged equations) [27]. Its continuous-phase flow field is com-puted by solving the Navier-Stokes equations. The dispersed phase is represented bytracking a small number of representative particle streams. For each particle stream,ordinary differential equations representing mass, momentum and energy transfer aresolved to compute its state and location. The two phase are coupled by inclusion ofappropriate interaction terms in the continuous-phase equations.
In this approach the volume displaced by the dispersed phase is not taken into account.So, this approach is applicable for low-volume fractions of the dispersed phase. Thisapproach is applicable for situations in which the discrete phase is injected as a continuousstream into the continuous phase. A force balance equation based on Newton’s secondlaw of motion is solved to compute the trajectory of the discrete phase.
The Eulerian-Lagrangian approach is suitable to unit operations in which the volumefraction of the dispersed phase is small, such in spray dryers, coal and liquid fuel combus-tion, and some particle-laden flow [21]. This approach provides complete information onthe behavior and residence time of individual particles. Interaction of individual particle
19
streams with turbulent eddied and solid surfaces such as walls can be modeled.
3.2 Eulerian - Eulerian approach
Eulerian-Eulerian approach is the most general approach for solving multiphase flows. Itis based on the principle of interpenetrating continua , where each phase is governed bythe Navier-Stokes equations [21]. The phases share the same volume and penetrate eachother in space and exchange mass, momentum and energy. Each phase is described by itsphysical properties and its own velocity, pressure, concentration and temperature field.The interphase transfer between phases is computed using empirical closure relations.The Eulerian-Eulerian approach is applicable for continuous-dispersed and continuous-continuous systems.
For continuous-dispersed systems, the velocity of each phase is computed using theNavier-Stokes equations. The dispersed phase can be in the form of particles, dropsor bubbles. The forces acting on the dispersed phase are modeled using empirical corre-lations and are included as part of the interphase transfer terms. In addition, drag, lift,gravity, buoyancy and virtual-mass effects are some of the forces that might be actingon the dispersed phase. These forces are computed for an individual particle and thenscaled by the local volume fraction to account for multiple particles.
There are three different Euler-Euler multiphase models available:
• Volume of fluid method (VOF)
• Mixture model
• Eulerian model
3.2.1 Volume of fluid method
In computational fluid dynamics, the Volume of fluid method is one of the most wellknown methods for volume tracking and locating the free surface. The motion of allphases is modelled by solving a single set of transport equations with appropriate jumpboundary conditions at the interface [1]. It can model two or more immiscible fluids bysolving a single set of momentum equations and tracking the volume fraction of eachof the fluids throughout the domain. Typical applications include the motion of largebubbles in a liquid, the motion of liquid after a dam break, the prediction of jet breakup,and the steady or transient tracking of any liquid-gas interface.
In general, the steady or transient VOF formulation relies on the fact that two or more
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fluids (or phases) are not interpenetrating. During the numerical calculation in eachcontrol volume, the sum of the volume fractions of all phases remains to unity. In ad-dition, the fields for all properties and variables are shared by the phases and representvolume-averaged values, as long as the volume fraction of each of the phases is known ateach location. Thus in any given cell, the properties and variables are either purely rep-resentative of one of the phases, or representative of a mixture of the phases, dependingupon the volume fraction values. Also in other words, if pth fluid’s volume fraction inthe cell is denoted as αp , then the following three different conditions are possible:
• αp = 0: The computational cell is empty of the pth fluid.
• αp = 1: The computational cell is full of the pth fluid.
• 0 < α < 1: The computational cell contains interface between pth fluid and one ormore other fluids available.
Volume fraction equation
In VOF method, the interface(s) tracking between the phases is established by gettingthe solution of a continuity equation for the volume fraction of one (or more) of thephases [1] . For the pth phase, this equation has the following form:
1ρp
∂∂t
(αpρp) +∇ · αpρp−→υ p = S +n∑q=1
(mqp − mpq)
(16)
Where ρp is the density of the pth fluid. Also mpq is the mass transfer from phase p tophase q and mqp is the mass transfer from phase q to phase p. This volume fractionequation will be solved for the secondary phase. It will not be solved for the primaryphase. The primary-phase volume fraction will be calculated based on the followingconstraint:
n∑p=1
αp = 1 (17)
The volume fraction equation may be solved either through implicit or explicit timediscretization scheme.
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Momentum equation
In the VOF method, a single set of momentum equation is solved throughout the wholecomputational domain. The resulting velocity field is shared among the phases. Wherethe momentum equation is dependent on the volume fractions of all the phases throughthe density ρ and viscosity µ. The momentum equation is as follow:
∂
∂t(ρ−→υ ) +∇ · (ρ−→υ −→υ ) = −∇p+∇ ·
[µ(∇−→υ +∇−→υ T )
]+ ρ−→g +
−→F (18)
Surface tension
The VOF method can also include the effects of surface tension along the interfacebetween each pair of phases. The model can be augmented by the additional specificationof the contact angles between the phases and the walls. The continuum surface tensionforce (CSF) of Brackbill et al. [2] have been widely used to model surface tension inmultiphase flow in volume of fluid (VOF), level-set (LS) and front tracking (FT) methods[3]. The solver will include the additional tangential stress terms that aries due to theavariation in surface tension coefficient. The effect of variable surface tension are usuallyimportant only in zero/near-zero gravity conditions.The surface tension is a force, acting only at the surface, that is required to maintainequilibrium in such instances. Surface tension aries as a result of attractive forces betweenmolecules in a fluid. For example, consider an air bubble in water. Within the bubble,the net force on a molecule due to its neighbors is zero. At the surface, the net forceis radially inward, and the combined effect of the radial components of force across theentire spherical surface is to make the surface contract, thereby increasing the pressureon the concave side of the surface. It acts to balance the radially inward intermolecularattractive force with the radially outward pressure gradient force across the surface. Inregions where two fluids are separated, but one of them is not in the form of sphericalbubbles, the surface tension acts to minimize free energy by decreasing the area of theinterface [4].
Reconstruction based schemes
For the special interpolation treatment to the computational cells that lie near the inter-face between two phases, there are two reconstruction based schemes as Geo-Reconstructand Donor-Acceptor.
Figure 9 shows an actual interface shape along with the interfaces assumed during com-putation by these two methods.
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Figure 9: Interface calculation
• The Geometric Reconstruction Scheme
Whenever a cell is completely filled with one phase or another, the geometric recon-struction approach, the standard interpolation schemes that are used to obtain the facefluxes. When the cell is near the interface between two phases, the geometric recon-struction scheme is used. The geometric reconstruction scheme represents the interfacebetween fluids using a piecewise-linear approach. In ANSYS FLUENT this scheme isthe most accurate and is applicable for general unstructured meshes. It assumes thatthe interface between two fluids has a linear slope within each cell, and uses this linearshape for calculation of the advection of fluid through the cell faces. We can see in theabove figure. The procedure of the geometric reconstruction approach is as follow,
• To calculate the position of the linear interface relative to the center of eachpartially-filled cell, based on information about the volume fraction and its deriva-tives in the cell
• To calculate the advecting amount of fluid through each face using the computedlinear interface representation and information about the normal and tangentialvelocity distribution on the face
• To calculate the volume fraction in each cell using the balance of fluxes calculatedduring the previous step
• The Donor-Acceptor Scheme
When the cell is near the interface between two phases, a “the Donor-Acceptor” schemeis used to determine the amount of fluid advected through the face [4]. This scheme
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identifies one cell as a donor of an amount of fluid from one phase and another (neighbor)cell as the acceptor of that same amount of fluid, and is used to prevent numericaldiffusion at the interface. The amount of fluid from one phase that can be convectedacross a cell boundary is limited by the minimum of two values: the filled volume in thedonor cell or the free volume in the acceptor cell. The orientation of the interface is alsoused in determining the face fluxes. It is either horizontal or vertical, depending on thedirection of the volume fraction gradient of the pth phase within the cell, and that ofthe neighbor cell that shares the face in question. The flux values are obtained by pureupwinding, pure downwinding, or some has a combination of the both. In addition itdepends on the interface’s orientation as well as its motion.
3.2.2 Mixture model
The mixture model is a simplified multiphase model that can be used in different ways.The mixture model can apply to model multiphase flows where the different phases moveat different velocities and also it is applicable to model homogeneous multiphase flowand to calculate non-Newtonian viscosity.The mixture model can model n phases (fluid or particulate) by solving both the con-tinuity equation and the momentum equation for the mixture, where mixture can bea combination of continuous phase and the dispersed phase. In addition, the mixturemodel solves the energy equation for the mixture, and the volume fraction equation forthe secondary phases, as well as algebraic expressions for the relative velocities (if thephases are moving at different velocities). Also it allows us to select the granular phasesand we can calculate the different properties for granular phases. It is applicable in theparticle-laden flows with low loading, and bubbly flows where the gas volume fractionremains low, cyclone separators, sedimentation and in liquid-solid flows [21]
Continuity equation for the mixture
First of all, we can write the continuity equation as follow from Ishii (1975),
∂
∂t(αkρk) +∇ · (αkρk−→υ k) = Γk (19)
where αk is the volume fraction of the kth phase and the term Γk represents the rateof mass generation of phase k at interface. From the above equation(19), we obtain bysumming over all phases,
24
∂
∂t
n∑k=1
(αkρk) +∇ ·n∑k=1
(αkρk−→υ k) =n∑k=1
Γk (20)
Because of the mass conservation the right hand side of the equation (20) must be vanish,so
n∑k=1
Γk = 0 (21)
From it, we obtain a continuity equation of the mixture as
∂ρm∂t
+∇ · (ρm−→υ m) = 0 (22)
Here the mixture density and the mixture velocity are defined as,
ρm =n∑k=1
αkρk (23)
−→υ m =1ρm
n∑k=1
(αkρk−→υ k) =n∑k=1
ck−→υ k (24)
The mixture velocity um represents the velocity of the mass centre. Also ρm variesalthough the component densities are constants. The mass fraction of phase k is definesas
ck =αkρkρm
(25)
equation(22) has the same form as the continuity equation for the single phase flow.
Momentum equation for the mixture
The momentum equation for kth is written as,
∂
∂t(αkρk−→υ k) +∇ · (αkρk−→υ k−→υ k) = −αk∇pk +∇ · [αk(τk + τTk)] + αkρkg + Mk (26)
where Mk is the average interfacial momentum source for the phase k. τk is the averageviscous stress tensor and τTk is the turbulent stress tensor.
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Now the momentum equation for the mixture follows (26) by summing over the phases
∂
∂t
n∑k=1
αkρk−→υ k +∇ ·
n∑k=1
αkρk−→υ k−→υ k
= −n∑k=1
αk∇pk +∇ ·n∑k=1
αk (τk + τTk) +n∑k=1
αkρkg +n∑k=1
Mk (27)
By using the definitions (23) and (24) of the mixture density ρm and the mixture velocity−→υ m, the second term of equation(27) can be written as
∇ ·n∑k=1
αkρk−→υ k−→υ k = ∇ · (ρm−→υ m−→υ m) +∇ ·
n∑k=1
αkρk−→υ Mk
−→υ Mk (28)
where −→υ Mk is the diffusion velocity, i.e.,the velocity of the kth phase relative to thecentre of the mixture mass
−→υ Mk = −→υ k −−→υ m (29)
The momentum equation takes the form in terms of the mixture variables as,
∂
∂tρm−→υ m +∇ · (ρm−→υ m−→υ m) = −∇pm +∇ · (τm + τTm) +∇ · τDm + ρmg + Mm (30)
Where τm, τTm, and τDm are the stress tensors.
Volume fraction equation for the secondary phases
We can obtain the volume fraction equation for secondary phase p from the continuityequation of the secondary phase p as follow:
∂
∂t(αpρp) +∇ · αpρp−→υ m = −∇ · αpρp−→υ dr,p +
n∑q=1
(mqp − mpq) (31)
In the mixture model some terms as relative (slip) velocity and drift velocity, interfacialarea concentration, solid pressure and granular properties like collisional viscosity, kineticviscosity, granular temperature are also important.
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3.2.3 Eulerian model
Eulerian model is the most general model for solving multiphase flows. In the presentwork, we are using Eulerian model to simulate two-phase and three-phase flow. TheEulerian model is the most complex of the multiphase models. It solves a set of nmomentum and continuity equations for each phase. In the next section, there are moredetails about Eulerian model.
3.3 Eulerian model theory
As we discussed above, in the Eulerian approach all the phases are treated as continuum.Eulerian model solves continuity, momentum and energy equations for each phase.
3.3.1 Volume fraction equation
The description of multiphase flow as interpenetrating continua incorporates the conceptof the phasic volume fractions which is denoted as αq. The volume fractions representthe space occupied by each phase and the laws of conservation of mass and the con-servation of momentum are satisfied by each phase individually. The derivation of theconservation equations can be done by averaging the local instantaneous balance for eachof the phases or by using the mixture theory approach.
The volume of the phase p is denoted by Vp and is defined by,
Vp =∫VαqdV (32)
Where the total volume fraction in the computational cell remains one as,
n∑p=1
αp = 1 (33)
The effective density of the phase p is defined by,
ρp = αpρp (34)
where ρp is the physical density of the phase p.The volume fraction equation may be solved by either implicit time discretization or
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explicit time discretization.
3.3.2 Conservation equations
The volume-averaged mass and momentum conservation equations in the Eulerian frame-work are given by following way.
1. Conservation of mass
The continuity equation for the phase q is written as follow,
∂
∂t(αqρq) +∇ · (αqρq−→υ q) =
n∑p=1
(mpq − mqp) + Sq (35)
In the above equation, the velocity of phase q is given by −→υ q and mpq is charac-terized by the mass transfer from the pth to qth phase and mqp is the mass transferfrom phase q to phase p. The last term of the right hand side in Equation (35), Sq,is the source term. In addition, it can be specified as a constant or by user-definefunction for each phase.
2. Conservation of momentum
The momentum balance for the phase q is given by the following,
∂
∂t(αqρq−→υ q) +∇ · (αqρq−→υ q−→υ q) = −αq∇p+∇ · τ q + αqρq
−→g +
n∑p=1
(−→R pq + mpq
−→υ pq − mqp−→υ qp) + (
−→F q +
−→F lift,q +
−→F vm,q) (36)
Where−→F q is an external body force,
−→F lift,q is a lift force and
−→F vm,q is a virtual
mass force.−→R pq is an interaction force between phases and the pressure, which is
shared by all phases, is given by p.−→υ pq is the interphase velocity which can be defined as follows.
• If mpq > 0(i.e., phase p mass is being transferred to phase q), −→υ pq = −→υ p• If mpq < 0(i.e., phase q mass is being transferred to phase p), −→υ qp = −→υ q• If mqp > 0, −→υ qp = −→υ q• If mqp < 0, −→υ qp = −→υ p
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In addition, the interaction force between phases−→R pq depends on the friction,
pressure, cohesion, and other effects. In literature, the simple form of interactionterm is given by following way,
n∑p=1
−→R qp =
n∑p=1
Kpq(−→υ p −−→υ q) (37)
Where Kpq(= Kqp) is the interphase momentum exchange coefficient.In Equation (36), τ q is the stress-strain tensor of the qth phase and it can be writtenas follow,
τ q = αqµq(∇−→υ q +∇−→υ Tq ) + αq(λq −23µq)∇ · −→υ qI (38)
In the above equation, µq and λq express the shear and bulk viscosity of the phaseq.
3. Conservation of energy
The conservation of energy in Eulerian multiphase modeling approach is describedas a separated enthalpy equation for each phase as follow,
∂
∂t(αqρqhq)+∇·(αqρq−→υ qhq) = αq
∂pq∂t
+τ q : ∇−→υ q−∇·−→q q+Sq+n∑p=1
(Qpq+mpqhpq−mqphqp)
(39)Where hq is the specific enthalpy of the qth phase, −→q q is the heat flux, Sq is a sourceterm, which is a source of enthalpy due to chemical reaction or radiation. The termQpq displays the intensity of heat exchange between the pth and qth phases, andhpq is the interphase enthalpy.
3.4 Turbulence models
In the study of the multiphase flow, it is important to include turbulence in it. To solveand to describe the effects of turbulent fluctuations of velocities and scalar quantitiesof flow, there are various closure models of turbulence are available. In comparisonto single-phase flows, the number of terms to be modeled in the momentum equationsin multiphase flows is large, and this makes the modeling of turbulence in multiphasesimulations extremely complex([4]) .
In the present work, we have used ANSYS FLUENT 12.1. It provides three methodsfor modeling turbulence in multiphase flows within the context of the κ − ε models. Inaddition, there are two turbulence options within the context of the Reynolds stressmodels (RSM). The κ− ε turbulence model has the following options,
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• mixture turbulence model
• dispersed turbulence model
• turbulence model for each phase
And the RSM turbulence model contains the following options,
• mixture turbulence model
• dispersed turbulence model
The choice of model depends on the importance of the secondary phase turbulence inyour application.
κ− ε turbulence models
There are three turbulence model options in the context of the κ−εmodels as the mixtureturbulence model, the dispersed turbulence model, or a each phase turbulence model. Inthe present work we have used κ− ε Mixture Turbulence Model in both applications.
κ− ε mixture turbulence model
The model equations of κ and ε are as follows:
∂
∂t(ρmκ) +∇ · (ρm−→v mκ) = ∇ ·
(µt,mσκ∇κ)
+Gκ,m − ρmε (40)
∂
∂t(ρmε) +∇ · (ρm−→v mε) = ∇ ·
(µt,mσε∇ε)
+ε
κ(C1εGκ,m − C2ερmε) (41)
where the mixture density and velocity, ρm and −→υm are calculated from
ρm =N∑i=1
αiρi (42)
and−→υ m =
∑Ni=1 αiρi
−→υ i∑Ni=1 αiρi
(43)
The turbulent viscosity, µt,m and production of turbulence kinetic energy are computedfrom the following equations w.r.to ,
µt,m = ρmCµκ2
ε(44)
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Gκ,m = µt,m(∇−→υ m + (∇−→υ m)T ) : ∇−→υ m (45)
The model constants are as follow,
Cµ = 0.09; σκ = 1.00; σε = 1.30; C1ε = 1.44 and C2ε = 1.92 (46)
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4 Interphase momentum exchange
The conditional averaging process used to derived the two-phase or three-phase flowequations necessarily results in a loss of detailed information about the local flow condi-tions and the introduction of additional unknown correlation terms. These extra termsrepresent the effect of the local flow details on the averaged properties of the flow. Thelost information must be restored by the appropriate modeling of these terms so that theycan be calculate from known or previously calculated quantities. In the phase momen-tum equations, two extra terms are introduced which represent the effect of inter-phasemomentum transfer and the effects of turbulence.
In the present work we are specifically interested in dispersed two-phase and three-phaseflows where phase or phases exist in discrete regions of space, each completely surroundedby the continuous phase. The additional terms resulting from the conditional averag-ing process comprise the interfacial momentum transfer terms and the phase Reynoldsstresses. The phase Reynolds stress terms relate to the effects of velocity fluctuations onthe mean transport of the phases.
On the other hand, the interfacial terms relate to the transfer of momentum betweenone phase and the other due to to their relative motion. These terms are not directlyrelated to turbulence, although they may be affected by it.
Interphase momentum transfer, Mαβ , occurs from the interfacial forces acting on eachphase α, due to interaction with another phase β. The total force on phase α due tointeraction with other phases is denoted Mα, and is given by:
Mα =∑β 6=α
Mαβ (47)
The total interfacial force acting between two phases may arise from several independentphysical effects [20]:
Mαβ = MDαβ +MLF
αβ +MLUBαβ +Mp
αβ +MVMαβ +MTD
αβ +MMTαβ (48)
The forces indicated above respectively represent the interphase drag force, lift force, walllubrication force, interfacial pressure force, virtual mass force, turbulence dispersion forceand force due to mass transfer between the phases.
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4.1 The drag, lift and virtual mass forces
The relative motion between a submerged body and the surrounding fluid gives rise tothe so-called drag force. Generally, a drag is a resistance force which is a force thatslows the motion of a body moving through a fluid. The drag force has two components;skin friction and form drag. The former is related to the shear stress at the surfaceof the submerged body while the latter aries due to the non-uniform surface pressuredistribution caused by the relative motion.
It is usual to consider the total drag force (i.e., the skin friction and the form drag)together rather than the individual components since it is the total drag which is usuallyof interest. The drag force on a single droplet is expressed in terms of its projected area,the relative velocity and a dimensionless drag coefficient Cd as,
−→F d =
12ρcACd|−→υ r|−→υ r (49)
where A is the projected area of the droplet and ur is the relative velocity between it andsurrounding fluid. The total averaged drag force per unit volume of the dispersed phasefollows from consideration of the average number of droplets per unit volume, given interms of the phase fraction, and the corresponding averaged area and relative velocity.
While drag forces act in the direction of the freestream fluid flow, another force, knowas lift, is generated perpendicular to the fluid flow. These lift forces act on a particlemainly due to velocity gradients in the continuous phase flow field. The lift force actingon a dispersed phase d in a continuous phase c is computed from
−→F lift = −0.5ρcαd(−→υ d −−→υ c)× (∇×−→υ c) (50)
Where lift force−→F lift will be added to the right-hand side of the momentum equation
for the both phases.
The virtual mass or added mass is the inertia added to a system because an acceleratingor decelerating body must move some volume of surrounding fluid as it moves throughit, since the object and fluid cannot occupy the same physical space simultaneously. Forsimplicity this can be modeled as some volume of fluid moving with the object.
For example drag force takes into account the interaction between liquid and bubblesin a uniform flow field under nonaccelerating conditions. If, however, the bubbles areaccelerated relative to the liquid, part of the surrounding liquid has to be accelerated aswell. This additional force contribution is called the added mass force or virtual massforce [7].
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−→F vm =
αdρc2
(dc−→υ cdt− dd
−→υ ddt
)(51)
Where ρc is the mass density of phase c, αd is the volume fraction of the phase d and dcdt
is the material time derivative form.The virtual mass effect is significant when the discrete phase density is much smallerthan the continuous phase density (e.g., for a transient bubble column).
4.2 Interphase exchange coefficients
In multiphase flow, another way of presenting the momentum exchange between thephases is based on the use of the fluid-fluid exchange coefficient Kpq and, for granularflows, the fluid-solid and solid-solid exchange coefficients Kls. Before we discuss aboutinterphase exchange coefficients, we take a short look on momentum equations of fluid-fluid system and fluid-solid system.
Fluid-Fluid momentum equations
The conservation of momentum for a fluid phase q is∂
∂t(αqρq−→υ q) +∇ · (αqρq−→υ q−→υ q) = −αq∇p+∇ · τ q + αqρq
−→g +
n∑p=1
(Kpq(−→υp −−→υq) + mpq−→υ pq − mqp
−→υ qp) + (−→F q +
−→F lift,q +
−→F vm,q) (52)
Where −→g is the acceleration due to gravity and other symbols are discussed in equation36.
Fluid-Solid momentum equations
In the Fluid-Solid momentum equations, the conservation of momentum for a fluid phaseis same as equation 52. The momentum equation for sth, solid phase is solved by thefollowing equation,∂
∂t(αsρs−→υ s) +∇ · (αsρs−→υ s−→υ s) = −αs∇p−∇ps +∇ · τ s + αsρs
−→g +
N∑l=1
(Kls(−→υl −−→υs) + mls−→υ ls − msl
−→υ sl) + (−→F s +
−→F lift,s +
−→F vm,s) (53)
Where ps is the solid pressure of sth, Kls = Ksl is the momentum exchange coefficientbetween fluid or solid phase l and solid phase s, N is the total number of phases.
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4.2.1 Fluid-Fluid exchange coefficient
In general, for fluid-fluid system each secondary phase is assumed to form bubbles ordroplets. The exchange coefficient for these types of bubbly, liquid-liquid or gas-liquidmixtures can be written in the following general form as,
Kpq =αqαpρpf
τp(54)
In the above equation, f indicates the drag function and τp is the particulate relaxationtime, is defined as
τp =ρpd
2p
18µq(55)
where dp is the diameter of the bubbles or droplets of phase p.
Drag function f depends on a drag coefficient, CD that is based on the relative Reynoldsnumber, Re. It is this drag function that differs among the exchange-coefficient models.For all these situations, Kpq should tend to zero whenever the primary phase is notpresent within the domain. To enforce this, the drag function f is always multiplied bythe volume fraction of the primary phase q, from Equation 54.
There are some models for the drag correlation in ANSYS FLUENT 12.1. In the presentwork, we have used the Schiller and Naumann drag correlation.
f =CDRe
24(56)
where CD is calculated with following conditions,
CD =
{24(1 + 0.15Re0.678)/Re Re ≤ 10000.44 Re > 1000
(57)
In the above equation, Re is the relative Reynolds number. The relative Reynoldsnumber for the primary phase q and secondary phase p is obtained from,
Re =ρq|−→υ p −−→υ q|dp
µq(58)
The relative Reynolds number for secondary phase p and r is obtained from,
35
Re =ρrp|−→υ r −−→υ p|drp
µrp(59)
where µrp = αpµp + αrµr is the mixture viscosity of the phases p and r.
There are many correlations for drag force available for fluid-fluid and fluid-solid systems.Some of them are the Morsi and Alexander model, the symmetric model, etc.[21]. Allmodels are applicable with certain conditions and for certain application.
4.2.2 Fluid-Solid exchange coefficient
The general form of the fluid-solid exchange coefficientKsl can be written in the followingform,
Ksl =αsρsf
τs(60)
where f is the drag function and, τs, is the particulate relaxation time defined as
τs =ρsd
2s
18µl(61)
Here ds indicates the diameter of particles of phase s. In Equation 60, f include a dragfunction which is based on the relative Reynolds number (Res). There are some wellknown correlations for drag function such as the Syamlal-O’Brien model, the Wen andYu model, the Gidaspow model, etc.[21].
4.2.3 Solid-Solid exchange coefficient
The solid-solid exchange coefficient Kls has very complicated form which is given asfollow,
Kls =3(1 + els)
(π2 + Cfr,ls
π2
8
)αsρsαlρl(dl + ds)2g0,ls
2π(ρld3l + ρsd3
s)|−→υl −−→υs| (62)
where,els = coefficient of restitutionCfr,ls = coefficient of friction between the lth and sth solid-phase particlesdl = diameter of the particles of solid lg0,ls = radial distribution coefficient
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5 Two-phase flow simulation in internal-loop airlift reactor
Before two-phase flow simulation for internal-loop airlift reactor is discussed in detail,take a short look for some basic facts of airlift reactor system.
5.1 Airlift reactor morphology
First of all, the term airlift reactor (ALR) covers a wide range of gas-liquid or gas-liquid-solid pneumatic contacting devices that are characterized by fluid circulation in a definedcyclic pattern through channels built specifically for this purpose. In ALRs, the contentis pneumatically agitated by a stream of air or sometimes by other gases. In those cases,we can say the name as gas lift reactors. In addition to agitation, the gas stream has theimportant function of facilitating exchange of material between the gas phase and themedium; oxygen is usually transferred to the liquid, and in some cases reaction productsare removed through exchange with the gas phase. The main difference between ALRsand bubble columns lies in the type of fluid flow, which depends on the geometry of thesystem.The bubble column is a simple vessel into which gas is injected, usually at thebottom, and random mixing is produced by the ascending bubbles.
Figure 10: Different types of ALRs [10].
In the ALR, the major patterns of fluid circulation are determined by the design of thereactor, which has a channel for gas-liquid upflow-the riser-and a separate channel forthe downflow. Figure 10 shows the various types of ARL’s
37
The two channels are linked at the bottom and at the top to form a closed loop. Thegas is usually injected near or from the bottom of the riser. The extent to which thegas looses at the top, in the section termed as the gas separator, is determined by thedesign of this section and the operating conditions. The fraction of the gas that doesnot disengage, but is entrapped by the descending liquid and taken into the downcomer,has a significant influence on the fluid dynamics in the reactor and hence on the overallreactor performance.
Generally, airlift reactors can be divided into two main types of reactors on the basis oftheir structure as [10],
• external loop vessels: In which, circulation takes place through separate anddistinct ducts.
• baffled (or internal-loop) vessels: In which baffles placed strategically in asingle vessel create the channels required for the circulation.
All ALRs, regardless of the basic configuration (external loop or baffled vessel), comprisefour distinct sections with different flow characteristics:
• Riser: Riser is the middle part of the reactor. The gas is injected at the bottomof this section, and the flow of gas and liquid is predominantly upward.
• Downcomer: Downcomer is connected to the riser at the bottom and at the topregion and is parallel to the riser. The flow of gas and liquid is predominantlydownward.
• Base: Usually believed that the base does not significantly affect the overall be-havior of the reactor, but the design of this section can influence gas holdup, liquidvelocity, and solid phase flow [10].
• Gas separator: It is a connector of the riser and the downcomer.
5.2 Computational details
In the present work, Eulerian model have been chosen to simulate two-phase flow andthree-phase flow in internal-loop airlift reactor application. In addition, we have studiedthe grid and time-step sensitivity on the results. We have used an unsteady approachfor all simulations. The brief details of simulations are as follow:
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Computational domain
In the two-phase flow case, three different configurations of an internal-loop airlift reactorare investigated. Following Figure 11 shows a schematic overview of Configuration I, IIand III. Initially there is a liquid in the whole reactor and gas is injected from the bottomregion. In the figure, section C shows the bottom region. The bottom region has 3 inletholes of 0.02 m length. The riser tube is fixed at 0.11 m from the bottom of the reactorin all configurations. The grid details are specified in Table 1. It shows that the totalheight of all configurations are fixed. Configuration I, consists of a riser tube with aninner diameter of 0.10 m. The riser wall has 0.005 m thickness and the total downcomerarea is 0.09 m.
Figure 11: Schematic overview of computational geometry for Configurations I, II andIII.
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Table 1: Details of computational grids for the Configurations I, II and III
Type A B C D E F G
Confi.I 0.2 m 0.75 m 0.09 m 0.4 m 0.045 m 0.1 m 0.045 m(40 cells) (152 cells) (40 cells) (80 cells) (9 cells) (20 cells) (9 cells)
Confi.II 0.24 m 0.75 m 0.09 m 0.4 m 0.065 m 0.1 m 0.065 m(48 cells) (152 cells) (48 cells) (80 cells) (13 cells) (20 cells) (13 cells)
Confi.III 0.2 m 0.75 m 0.09 m 0.4 m 0.025 m 0.14 m 0.025 m(40 cells) (152 cells) (40 cells) (80 cells) (5 cells) (28 cells) (5 cells)
In Configuration II, a riser diameter of 0.10 m (inner diameter) and height of 0.40 mare assumed. The total reactor diameter is 0.24 m. So, it has a larger downcomer areathan Configuration I and III. The other dimensions and gas injection region are identicalto the Configuration I and III. Configuration III has a riser tube with 0.14 m diameterinside a reactor of 0.20 m. Riser diameter is bigger than in Configuration I and II. So, itis clear that Configuration III has smaller downcomer area than Configuration I and II.Bottom region has 3 inlet holes with 0.02 m length. In addition, for the grid sensitivitystudy and time-step study, Configuration I has been used in both 2D and 3D cases.
Boundary conditions
In all the cases of two-phase flow simulation, the reactor is initially full with liquid aswe discussed before. Gas is feeded from the bottom region of the reactor. The bottomregion contains 3 gas inlet holes with velocity inlet boundary condition. Following Figure12 shows the details about the boundary conditions. A pressure boundary condition isapplied to the top of the reactor with an average static reference pressure of 0 Pa. Therest of the boundaries for reactors walls ( i.e., front, back and others) and riser wallsare standard no-slip boundary condition where velocity increases from zero at the wallsurface to the free stream velocity away from the surface for both phases.
40
Figure 12: (a) Boundary conditions applied on computational domain, (b) computationalgrid, and (c) closer view of the bottom region of reactor.
Meshing
For the grid generation purpose, we have used ANSYS GAMBIT 2.4. package. Astructured quadrilateral mesh is employed for 2D domains and structured hexahedralmesh for 3D. In the grid and time-step sensitivity study we have used four different gridsin both 2D and 3D cases. The grid is uniform in the upper region of the reactor except atthe lower region. Figure 12b displays the mesh generation in the computational domain.Figure 12c shows the closer view of the bottom region with gas inlet holes. It shows thatthere are three inlet holes of 2cm× 1cm for gas phase.
Simulation setup
A commercial CFD package ANSYS Fluent 12.1. is used for the numerical solution ofthe flow. This CFD code is based on a finite volume approach. In this section the variousCFD code settings and options are summarized. For all cases, unsteady and pressurebased solver are used. A least square cell based method is used to calculate gradients.
41
In this work, the Euler-Euler method based on a two-fluid system is employed with aκ − ε model for the turbulence. The pressure-velocity coupling is obtained using thePhase Coupled SIMPLE algorithm. In addition, Boundary conditions and different dis-cretization schemes are used depending on the turbulence model. They are summarizedin Table 2. We have considered convergence criteria 1.0× 10−5 for these simulations.
Table 2: Two-phase simulation settings
Settings Choice
Simulation type 2D and 3D, UnsteadySolver Double precision, Pres-
sure based and implicitTemporal discretization 1st orderMultiphase model Eulerian, 2-phase, im-
plicitTurbulence model κ− ε, mixture modelPressure-velocity coupling Phase Coupled SIMPLEMomentum 1st order upwindVolume fraction 1st order upwindTurbulent kinetic energy 1st order upwindTurbulent dissipation rate 1st order upwindConvergence criteria 1.0× 10−5
Boundary conditions:
Inlet Velocity inletOutlet Pressure outletRiser walls No-slip wallReactor walls No-slip wall
Liquid is the primary phase for all simulations. We have assumed water as a liquid phase.Air is considered as a gas phase which is the secondary phase. The properties of bothphases are given in Table 3. Surface tension is chosen with 0.0728 N/m value. Moreover,interactions between phases are taken into account by means of a momentum exchange,or drag, coefficient based on the Schiller-Naumann drag correlation. The contributionsof the added mass and lift forces are both ignored in the present work. For all the casesbubble size, db, is fixed to 1mm. The range of superficial gas velocities have been usedfrom 0.01 m/s to 0.05 m/s.
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Table 3: Phase properties used in two-phase simulation
Phase propertiesLiquid Gas(Water) (Air)
ρ = 998.2 [kg/m3] ρ = 1.225 [kg/m3]
µ = 0.001003 [Pa · s] µ = 1.7894e− 05 [Pa · s]
5.3 Results and Discussions
As mentioned before, Eulerian model has been used to simulate two-phase flow and tostudy the hydrodynamics parameters in ARL’s. A typical transient approach to steady-state of the gas and liquid velocities, at the centre of the column, are shown in Figure 13for Configuration I operating at a superficial gas velocity Ug = 0.02 m/s. Here, Grid Cfor both 2D and 3D simulation are used for this purpose where the detail about grids aregiven in Table 4. Figure 13 shows that the simulation reaches steady state after 18-20 s.By considering this fact, for all the simulations we have taken the time averaged valuesfor 20 to 30 s time interval for all parameters. In addition, for better convergence, wehave fixed 20 iterations per time step. To prevent start-up problems, the whole systemwas initialized by setting the gas holdup to 0.5 %.
Figure 13: Transient approach to steady-state of the gas and liquid velocities at thecentre of the reactor.
43
5.3.1 Comparison between 2D and 3D results.
Two- and three-dimensional simulations of internal-loop reactor at different operatingconditions have been performed for studying the effect on hydrodynamic parameters. Thefollowing Figure 14 shows the gas velocity and water velocity profiles of 2D simulation atUg = 0.02 m/s for Configuration-I. In Figure 14a and b, all the walls of the reactor andriser have zero velocity because of no-slip boundary condition. In addition, the liquidvelocity magnitude in the downcomer region is almost same as in riser. The driving force,based on the static pressure difference, or the mixture density difference, between the riserand the downcomer generates the loop liquid circulation. Figure 14b, clearly shows thehigh velocity at the bottom part of riser tube. It means high turbulence kinetic energyin this region. In addition, high turbulent kinetic energy results in significant energydissipation, which in turn affects the hydraulic resistance and the liquid circulation flux[29].
Figure 14: Contours of (a) gas velocity, and (b) liquid velocity of Configuration-I whenUg = 0.02 m/s.
The results from 2D and 3D simulations have been compared for Configuration-I withthe range of Ug. Figure 15 represents gas fraction contours of 2D and 3D operating atUg = 0.02 m/s. Gas enters the reactor from the bottom part. As the time passes thevolume fraction gas grows up, and it becomes stable after a while. Gas bubbles spiralupwards due to the density difference between gas and liquid phases. Figure 15 shows
44
good agreement between 2D and 3D results. Figure 16 displays the velocity vectors andiso-surfaces of liquid phase.
Figure 15: Gas fraction contours from 2D and 3D cases when Ug = 0.02 m/s.
Figure 16: (a) Liquid velocity vector of 2D case and (b) liquid iso-surfaces from 3Dsimulation.
45
The radial velocity profiles, from 2D simulations, of gas phase are shown in Figure 17 forConfiguration-I at selected location in riser tube. Figure 17a and b demonstrate that forlow superficial gas velocities,Ug, gas phase can be considered to be virtually in plug flow.With increasing superficial gas velocities, gas phase loses their plug flow character andthe velocity profile assume a parabolic shape. The parabolic velocity profile becomesmore prominent with increasing superficial gas velocities.
Figure 17: Gas velocity distribution in the riser tube at selected axial location.
Figure 18: Liquid velocity distribution at selected axial location.
Same physics can be applied to the liquid phase. Figure 18 shows the radial liquidvelocity profiles at selected location in the whole reactor. In figure, dark black linesindicate the riser wall. Liquid velocity increases from zero at the wall surface to thefree stream velocity away from the surface due to no-slip BC. Figure displays that withincreasing superfacial gas velocities, the liquid velocity will increase in downcomer wherethe negative value indicates the downward direction of the flow. Moreover, radial velocityprofiles for both phases are almost axi-symmetric.
46
Figure 19: Gas holdup distribution in the riser tube at selected axial location.
The radial distribution of gas holdup, εg, in riser tube, obtained from 2D simulationsare shown in Figure 19 for Configuration-I. It exhibits that the gas holdup shows apronounced increase in the gas holdup at the centre of the riser tube with increasing Ugvalues, in this case the liquid recirculation tend to move gas towards the centre of theriser tube. Also, there is an increment in gas holdup near the riser wall. The reason forthis is that the larger sized liquid circulation at the riser bottom tends to move the gastowards the wall too.
In order to check the effect of 2D and 3D simulations, we have used Grid C of Configuration-I. The time-averaged values for gas velocity, liquid velocity and gas holdup have beencalculated at selected position. Figure 20 a and b show the gas velocity distribution inthe riser tube for 2D and 3D simulations respectively. In both cases, it can be seen thatthe parabolic profile of gas velocity becomes more prominent with increasing superficialgas velocities and 3D cases predict higher value than 2D. In addition, it is clear that gasvelocity profiles from 3D simulations are more parabolic compared to 2D.
Figure 21 displays the radial distribution of liquid velocity in whole reactor from 2D and3D cases. Figure indicates that 3D estimates higher value for liquid velocity to the 2Din the riser tube. 3D cases lead more parabolic profile in the riser tube. It is noted thatin the downcomer area of 2D simulations gives higher value than 3D.
The gas holdup distribution in the riser tube for 2D and 3D simulations are shown inFigure 22 a and b. The predicted values of gas hold up from both 2D and 3D cases areclose to one another. Near the riser walls, 2D simulations give little bit higher value than3D.
47
Figure 20: Radial distribution of gas velocity from 2D and 3D simulations at selectedaxial location in the riser tube.
Figure 21: Radial distribution of liquid velocity from 2D and 3D simulations at selectedaxial location.
Figure 22: Radial distribution of gas holdup from 2D and 3D simulations at selectedaxial location in the riser tube.
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5.3.2 Scale influence on the hydrodynamics of internal-loop airlift reactors.
In view of the encouraging results obtained above with CFD simulations, we have beenattempted to investigate the influence of scale on the hydrodynamics of internal-loopairlift reactors. In fact, the geometry of the reactor has a strong influence on the hy-drodynamics parameters [28]. For this purpose, we have been chosen three differentconfigurations. The brief details about all configurations are given in Table 1. It isclear that Configuration-II has a larger downcomer area than Configuration-I and III.Configuration-III has a largest riser diameter than Configuration-I and II.
Figure 23 and Figure 24 show the contours gas velocity, liquid velocity and gas holdupfor Configuration-II and Configuration-III respectively. In order to make a comparisonbetween Configuration-I and Configuration-II, where both have same riser diameter butConfiguration-II has a larger downcomer area. Larger downcomer means smaller fric-tion losses and therefore a higher volume of liquid is recirculated. The higher liquidrecirculations will affect on the gas holdup.
In case of Configuration-I and Configuration-III, both have different riser diameter.Configuration-III have a smaller downcomer area than Configuration-I. So, Configuration-III has more friction losses which influence on the liquid recirculation.
Figure 23: Contours of (a) gas velocity, (b) liquid velocity, and (c) gas holdup forConfiguration-II.
49
Figure 24: Contours of (a) gas velocity, (b) liquid velocity, and (c) gas holdup forConfiguration-III.
To determine this scale effect, The phase velocities and gas holdup have been calculatedat the middle of all configurations. The Figure 25, 26 and 27 represent the radial profilesof phase velocities for Configuration-I, II and III respectively.
Figure 25: (a) Gas velocity and (b) liquid velocity distributions of Configuration-I.
50
Figure 26: (a) Gas velocity and (b) liquid velocity distributions of Configuration-II.
Figure 27: (a) Gas velocity and (b) liquid velocity distributions of Configuration-III.
The figures show that the gas and liquid velocities are higher for Configuration-II compareto Configuration-I and III. It pronounces that Configuration-II has smaller friction lossesand the amount of rotational liquid is more where the draft tube is farther from the wall.Gas velocity profiles of Configuration-II is flatter than Configuration-I and III. Moreover,gas velocities of Configuration-III have more parabolic shape compared to Configuration-I. From Figures 25 and 26, it can be seen that Configuration-II has lower phases velocitiesthan Configuration-I. The smaller the distance between the riser tube and the reactorwall, the smaller the liquid recursive movement space would be. All these things showthat CFD simulations are able to pick up these geometry effects very well.
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Figure 28: Gas holdup distributions in (a) Configuration-I and (b) Configuration-II.
Figure 29: Comparison between average gas holdup in the riser of Configuration-I andConfiguration-II with range of Ug.
Figure 28a and b display the radial distribution of gas holdup for Configuration-I and IIat the middle in the riser tube. It is clear that gas holdup is increased with incrementin superficial gas velocity for both configurations. Both Configuration-I and II have thesame riser diameter. Configuration-II has a larger downcomer are; the friction losses aresmaller and therefore a higher volume of liquid is recirculated. The higher liquid recircu-lations cause a smaller slip velocity between the gas and liquid phases, and consequentlya smaller gas holdup. In order to establish this effect, the average gas holdup have beencompared in the riser tube for both configurations. Figure 29 displays the comparisonbetween average gas holdup of Configuration-I and II. It shows that as downcomer areaincreases, the average gas holdup decreases or as the downcomer area decreases, theaverage gas holdup increases in the riser tube.
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5.3.3 Grid sensitivity study
Grid independence tests have been performed to investigate the influence of grid re-finement on the solution and representative results are presented here for four of thecomputational grids used with 2D and 3D cases. Table 4 provides some details of thegrids, including the total number of nodes and the numbers of nodes in x−, y− and z−directions. In table Nx, Ny and Nz represent the number of nodes in x−, y− and z−directions respectively. Total cell size of the computational domain is given by Ntotal.
Table 4: Details of grids used in mesh-independence studies
2D 3DGrid Nx Ny Ntotal Nx Ny Nz Ntotal
A 10 39 390 10 39 2 780
B 20 76 1520 20 76 2 3040
C 40 152 6080 40 152 4 12160
D 80 304 24320 80 304 8 194560
By using these grids, transient simulations have been performed to check the sensitivityof grid. Following Figure 30 shows the gas holdup contours with four different gridswhen Ug =0.02m/s. From figure, it can be observed that there is a variation in the gasholdup profiles when we refine the grid from Grid A to Grid D. During simulations withGrid A and Grid B both have reached to steady state quickly than Grid C and GridD for both 2D and 3D cases. Gas is injected from the bottom part of reactor and itsprofile grow up as the time passes. The driving force, which is based on density differenceor static pressure difference, between the riser and the downcomer generates the liquidrecirculation. During loop circulation of liquid, it carries some amount of gas in theriser rube through downcomer. Figure shows that Grid A and B predict gas phase inthe whole system while Grid C and Grid D give different profiles. Moreover, Figure 30demonstrates that Grid C and Grid D predict gas phase around the riser walls. As thegrid size is increased, the small details have lost and uniform distribution have seen.
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Figure 30: Instantaneous snapshots of gas holdup contours with (a) Grid A, (b) Grid B,(c) Grid C, and (d) Grid D when Ug = 0.02 m/s.
In order to check the grid influence on the local hydrodynamic parameters, the time-averaged value for gas holdup and velocities of both phases have been calculated at themiddle of the reactor. The following Figures 31, 32 and 33 display the gas velocity, gasholdup and liquid velocity distribution with different grids when Ug = 0.02 m/s. Figure31 describes that grids have strong influence in both 2D and 3D cases. Grid A and Bare not able to capture small details of the flow while Grid C and D give almost sameprofiles. Moreover 3D simulations are much sensitive compare to 2D with respect toGrid A to D. Grid A predicts almost 50% less value than Grid D at the centre of theriser tube in 3D case.
Figure 32 presents the radial distribution of gas holdup with selected grids. Gas holdupprofiles have been affected with grids. It can be observed that gas holdup profiles becomemore parabolic when go toward Grid A to Grid D for both 2D and 3D results. Radialdistribution of the liquid velocity can be seen in Figure 33. It shows that the griddependency in the 3D cases seems to be more significant than 2D. It is fact that variationbetween predicted value from Grid D in 50% more compare to Grid A and B in 3D cases.In downcomer area, grids play a vital role to simulate the liquid velocity. From the abovestudy, it is clear that geometry grid size has a strong influence on the hydrodynamics ofinternal-loop airlift reactor.
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Figure 31: Gas velocity distribution with 2D and 3D simulations for different grids.
Figure 32: Gas holdup distribution with 2D and 3D simulations for different grids.
Figure 33: Liquid velocity distribution with 2D and 3D simulations for different grids.
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Computational time
CFD simulation depends very much on the availability of computer power. Moreovergrid size of the computational domain is the important parameter which effect on thetotal CPU time. All the simulations of the present work were carried out at CSC —IT Center for Science, Espoo with eight quad-core AMD Opteron 8360 SE (Barcelona)2.5GHz processors.
Table 5 presents the computational time required to simulate 30 second case with singleCPU for each of the grids. For each simulation, we have fixed ∆t = 0.001. From table itis clear that as total number of cell increased, it required higher CPU time. Moreover,3D simulations have been used almost three time more CPU time compared to 2D.
Table 5: Computational time used by each grid for 2D and 3D domains.
Grid Total CPUTime(hrs)
A 10.82D B 22.8case C 42.7
D 78.0A 30.0
3D B 65.4case C 118.64
D 1368.2
5.3.4 Time-step sensitivity study
In order to check the effect of time-step, ∆t on the results we have chosen three differenttime-step as 0.1s, 0.01s and 0.001s. In this section, all the simulations have been per-formed by adopting Configuration-I with Grid A to D in 2D. The radial distribution ofgas holdup and liquid phase velocity have been calculated at the middle of the reactor.The following Figure 34 describes the local gas holdup in the riser section with Grid Ato D using three different ∆t. From figure it is fact that there are very small variationbetween the results produced from each time-step. However, the effect of time step sizeon gas holdup profile is not as significant as grid size in this case.
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Figure 34: Profiles of gas hold produced from each selected grid and time-step size.
Liquid velocity profiles with different grids and different time-steps can be seen in Figure35. Figure shows the sensitivity of ∆t on the selected grids. There are not much differencebetween simulated values in the both section, a riser and a downcomer.
From the above study, it is pronounced that time-step sizes have been produced smalleffect on liquid velocity distribution. While the prediction of gas holdup is independentto ∆t.
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Figure 35: Profiles of liquid velocity produced from each selected grid and time-step size.
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6 Three-phase flow simulation in internal-loop airlift reac-tor
In the previous chapter, CFD simulation of gas-liquid two-phase flow has been discussedand investigated in internal-loop airlift reactors. In the present chapter, we will discussabout gas-liquid-solid flow. Airlift reactors are pneumatically agitated devices with nu-merous applications as gas-liquid-solid three phase reactors. For example, in the chemicalindustry, gas-liquid reactions requiring heterogeneous catalysis, such as catalytic hydro-genations [30]. In the three phase system the solid phase consists of fine catalyst particleswhich are used to improve the process. They are suspended in the liquid phase due to thegaseous reactants introduced into the reactor from the bottom often through a sparger.It is a very interesting and complex system. The generated recirculation determines, e.g.the residence time of the gas phase. Furthermore, it controls the spatial distributionof the catalyst particles that is of importance for the performance of the reactor (Lain,Broder and Sommerfeld, 1999 and Cockx, Roustan, Do-Quang and Lazarova, 1997).There are various approaches to the mathematical and physical modelling of multiphaseflows. The most widely used methods for CFD are the Eulerian-Eulerian (Sokolichin andEigenberger, 1999) and the Eulerian-Lagrangian approaches (e.g. Lain et al., 1999) bothwith their advantages and disadvantages and their range of applicability. In the Eule-rian method, both the continuous and dispersed phase(s) are mathematically modelledas interpenetrating fluids, represented by sets of mass, momentum and energy balances.In the Lagrangian approach, a large number of particles is tracked individually, whilethe liquid phase is treated as a continuum. The interaction between the particles andthe liquid show up as a source term in the momentum equations. The advantage of theEulerian method over the Lagrangian is seen when the void fraction of the dispersedphase is high.In gas-liquid-solid flow, solid loading, solid particle size and solid density have signifi-cant effect on the hydrodynamic characteristics of internal loop airlift reactor (Snape etal 1995, Vial et al 2000). In the present work, with the help of CFD, the three-phasesystem in internal-loop airlift reactor has been simulated by using the Eulerian-Eulerianapproach.
6.1 Computational details
For the three-phase flow study, Eulerian model have been chosen to simulate the flow.The system contains liquid-solid mixture and gas is feeded from the bottom region ofthe reactor. There are three main purposes of this study: (a) to check the effect of solidloading on the gas holdup, (b) to study the solid particle size influence on the gas holdup,and (c) to clarify the solid density effect. In addition, it has been studied how the solids
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will distribute in the system in different conditions. An unsteady approach has been forall simulations. The details of the simulations are as follow:
Computational domain and boundary conditions
For the three phase flow simulation, Configuration-I was chosen for all cases. The briefdetails of the Configuration-I are given in Table 1. It has three gas-inlet holes at thebottom part of the reactor. Initially the reactor is filled with liquid phase and solid phaseis distributed in the whole system with different loading from 5% to 20%. In addition,the solid particle diameter have been considered from 0.1 mm to 1 mm with the samedensity. All the details about the computational domain and boundary conditions arementioned in Figure 12. A pressure boundary condition is applied to the top part of thereactor with an average static reference pressure of 0 Pa. The rest of the boundaries forreactors walls and riser walls are standard no-slip boundary condition for all phases.
Simulation setup
As a CFD code, ANSYS Fluent 12.1. package has been used for the numerical solutionof the system. For all the cases, unsteady and pressure based solver are used. A leastsquare cell based method is fixed to calculate gradients. In this work, the Euler-Eulermethod based on a two-fluid system is employed with a κ− ε model for the turbulence.The standard single-phase parameters Cµ = 0.09, σκ = 1.00, σε = 1.30, C1ε = 1.44,and C2ε = 1.92 are used for κ − ε model. Here, the momentum equation is solvedwith 2nd order upwind scheme. The pressure-velocity coupling is obtained using thePhase Coupled SIMPLE algorithm. A fully implicit upwind differencing scheme hasbeen used for the time integration. The brief details of simulation setup are summarizedin Table 6. We have fixed 1.0× 10−4 convergence criteria for all simulations. Moreover,interactions between phases are taken into account by means of a momentum exchange,or drag, coefficient based on the Schiller-Naumann drag correlation. This correlation isapplicable for catalyst solid spheres [31]. The effects of the added mass and lift forces areignored in the present work. Moreover, Coalescence Kernal and Breakage Kernel havebeen specified by hibiki-ishii model.
We have assumed water as a liquid phase and it is the background phase for all simula-tions. Air is considered as a gas phase and with sauter mean diameter is used for bubble.Solids are dispersed in the reactor initially. The properties of all the phases are given inTable 7.
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Table 6: Simulation setup for three phase flow calculation.
Settings Choice
Simulation type 2D, UnsteadySolver Double precision, Pres-
sure based and implicitTemporal discretization 1st orderMultiphase model Eulerian, 3-phase, im-
plicitTurbulence model κ− ε, mixture modelPressure-velocity coupling Phase Coupled SIMPLEMomentum 2nd order upwindVolume fraction 1st order upwindTurbulent kinetic energy 2nd order upwindTurbulent dissipation rate 2nd order upwindConvergence criteria 1.0× 10−4
Boundary conditions:
Inlet Velocity inletOutlet Pressure outletRiser walls No-slip wallReactor walls No-slip wall
Table 7: Phases properties used for three phase flow simulation.
Phase properties
Liquid (Water) Gas (Air) Soild
ρ = 998.2 kg/m3 ρ = 1.225 kg/m3 ρ = 1000 kg/m3
µ = 0.001003 Pa · s µ = 1.7894e− 05 Pa · s µ = 0.001003 Pa · s
— sauter mean diameter ds = 0.1, 0.5 and 1 mm
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6.2 Results and Discussions
First of all, it was tried to verify the effect of solid loading on the gas holdup of the reactor.Moreover, it is interesting to see the solid phase distribution in the whole system withdifferent operating conditions.
6.2.1 Effect of solid loading
Initially, the reactor is full with liquid where solid is dispersed in the whole reactor. Fromthe bottom part, gas is feeded with certain Ug. Catalyst solid particles start to suspendin the liquid phase due to the gas introduced into the reactor. To check the influenceof increased solid loading on the gas fraction in the riser, four initial solid loading levelhave been used. The solids loading ranges from 5% to 20% and the diameter of thesolid particles is considered 100 µm for each simulation. The operating superficial gasvelocity in the range of 0.01 to 0.05 m/s is used. The following Figure 36a, b and cdisplay the contours of velocity magnitude for liquid, gas and solid respectively. Thesevelocity profiles are produced from 5% solid loading initially and gas is introduced withUg = 0.02 m/s.
Figure 36: Contours of velocity magnitude for (a) liquid, (b) gas, and (c) solid of three-phase flow system.
Figure 37 represents instantaneous snap shot of solid concentration in the reactor wheninitial solid loading was 5% and Ug = 0.02 m/s. Profile of gas phase grow up as the
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time passes due to this liquid and solid phase start to circulate between the riser and thedowncomer. It can be observed that solid concentration at 5 sec is dispersed in the wholereactor. The solid concentration will decrease when time goes because of the pressureoutlet boundary condition at the top. After some time, system gets to steady state.Figure shows that there is higher concentration of the solid particles on the riser walls.
Figure 37: Instantaneous snap shot of solid concentration in the reactor when reactor isinitially loaded with 5% of solids.
The influence of the solids loading on the total gas holdup for varying superficial gasvelocities are shown in Figure 38, where solid loading considered with in the range from5% to 20%.
Figure 38 indicates that gas holdup increases with an increment in the superficial gasvelocity. Increment in the average gas holdup is almost linearly with increasing superficialgas velocity in the system for each solid loading case from 5% to 20%.
Moreover, it is clear that adding of more solid particles to the system results in a lower gasholdup. It indicates that presence of the solids particles enhances the bubble coalescenceresulting in higher rise velocities. Increased rise velocity of the small bubbles signifieslarger bubble diameters caused by increased bubble coalescence. therefore, the total gas
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holdup decreases with increasing particle concentration.
Figure 38: Average gas holdup in the riser with different superficial gas velocity.
Figure 39 reveals the effect of superficial gas velocity on the calculated averaged solidholdup, εs in the riser section. Averaged solid concentration is increased with incrementin the initial solid loading. Figure represents that the solid holdup decreases with increasein superficial gas velocity up to certain level. This corresponds to the expected behaviourbecause, when the gas flow rate is low, solid cannot leave the system. Consequently,increasing the gas flow rate, increases the rising force. Due to this, higher amount ofsolid settle down in the downcomer area and some of them leave the system since pressureoutlet boundary condition.
Figure 39: Average solids holdup in the riser with different superficial gas velocity.
The radial distribution of solid phase at the middle of the reactor with the different solidsloading conditions are shown in Figure 40. It is observed that the local solids holdupin the riser section decreased with the increase of the superficial gas velocity. Moreover,figure reveals that the higher concentration of the solid particles are on the riser walls
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due to liquid recirculation. The solid concentration increases at the riser walls with theincrement in solid loading initially. At the center of the reactor, solid concentration islower due to gas phase.
Figure 40: Radial distribution of solids volume fraction at the center of the reactor withdifferent solids loading conditions.
6.2.2 Effect of solid particle size
In order to check the influence of the solid particle size in the three-phase flow systemof airlift reactor, three different particle diameter as 0.1 mm, 0.5 mm and 1 mm havebeen considered. In addition, averaged gas holdup is calculated of the riser section withdifferent operating conditions. In these cases, initially the reactor is loaded with 15%solid particles.
Figure 41: Averaged gas holdup in the riser with different superficial gas velocity.
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Figure 41 describes the profiles of averaged gas holdup of the riser with different super-ficial gas velocity when different sized solid loading.
It is observed from the figure that the riser gas holdup decreases with increase in particlediameter. It indicates that the surface area per unit volume of the solid bed increasesfor smaller particle size and hence, gas bubbles break and the residence time of the gasbubble increases on the riser and hence riser gas holdup increases.
Moreover, when particle size increases, it results in higher solids holdup in the riser tube.This fact is revealed from Figure 42.
Figure 42: Average solids holdup in the riser with different superficial gas velocity.
Figure 42 shows the effect of superficial gas velocity on the calculated averaged solidholdup in the riser section with different particle size.
In order to check the radial distribution of solid phase at the middle of the reactor withthe different particle size, Ug = 0.03 and 0.05 m/s can be used. Initially, system isconsidered with 15% solid loading for all cases. Figure 43 displays that the local solidsholdup in the riser section decreased with the increase of the superficial gas velocity. Itcan be seen that the higher concentration of solid particles are on the both riser walls.The solid concentration increases at the riser walls with the increment in solid size.Particles with 1 mm diameter produce higher concentration in the system compare toothers. At the center of the reactor, solid concentration is lower due to gas phase.
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Figure 43: Radial distribution of solids volume fraction at the center of the reactor withdifferent solid particle size.
6.2.3 Effect of solid density
Solid density is also important parameter which can influence on the hydrodynamic ofairlift reactor. By choosing three different density, ρs as 1000, 2000 and 3000 kg/m3,simulations have been performed with 15% of initial solid loading in the whole reactor.Figure 44 and 45 display the contours of solid distribution when Ug = 0.05 m/s in thesystem with ρs = 2000 kg/m3 and 3000 kg/m3 respectively .
Initially, solid particles are uniformly distributed in the whole reactor. The solids motionis governed by the liquid, gravity (resulting in slip velocity of solid between the liquid)and the turbulent dispersion mechanism. When time goes solid starts to sediment inthe system due to the effect of gravity. The system gets to the stable state after 25 sec.Figure 45 displays that the solids fraction is relatively constant, but it is higher in thelower part of the reactor.
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Figure 44: Instantaneous snap shot of solids concentration in the reactor with ρs =2000 kg/m3.
Figure 45: Instantaneous snap shot of solids concentration in the reactor with ρs =3000 kg/m3.
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To ensure the effect of solids density on overall solids holdup of in the riser section, aver-aged solid holdup have been calculated. Three superficial gas velocities have been usedfor each case. The influence of solids density on solid holdup in the riser is representedin Figure 46 with the range of Ug.
Figure 46 reveals that averaged solids holdup increases with the increase of solids density.Initially, solids are dispersed in the system. Due to higher density, solids start to settledown and it results in high solids holdup. Moreover, the results show that for the highdensity case, superficial gas velocity have no affect in solids holdup.
Figure 46: Average solids holdup in the riser with different superficial gas velocity.
The influence of the solids density on the averaged gas holdup for varying superficialgas velocities is shown in Figure 47. The gas holdup significantly increases with theincreasing solid particle densities and it decreases with lower solid density. For lowerdensity case, bubble rise velocity is higher. Due to this, the small bubbles signifies largerbubble diameters caused by increased bubble coalescence. A change in the density ofthe slurry phase helps the bubbles to rise faster, hence gas holdup decreases.
Figure 47: Average gas holdup in the riser section with different superficial gas velocity.
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7 Conclusions
In the present work, the CFD simulations of gas-liquid two-phase flow in internal-loopairlift reactors have been performed. Transient Eulerian-Eulerian approach was usedto predict phase distribution and phases velocities. Moreover, simulation have beenconducted to investigate the comparison between 2D and 3D results, scale effect on thehydrodynamics and influence of grid and time-step size. The results of the present workof two-phase flow yields the following major conclusions:
• The Eulerian model is able to simulate two-phase flow in internal-loop airlift re-actors. In order to check, with increment in superficial gas velocities, gas phaseloses their plug flow character and the velocity profile assume a parabolic shape.The parabolic velocity profile becomes more prominent with increasing superficialgas velocities in both 2D and 3D cases. The results from 2D and 3D simulationsreveal that the assumption of 3D leads to radial profiles of phases velocities thathave more parabolic character than that for 2D simulations. The predicted localgas holdup is found to be very good agreement with 2D and 3D.
• The important point in designing airlift reactors is to determine the distance be-tween the riser tube and the reactor walls. Three configurations of internal-loopairlift reactors were used to determine the scale influence on the hydrodynamics.The results show that as the downcomer area increases, the mixing of the gas andliquid increases and the gas holdup decreases. While downcomer area decreases,the mixing decrease and the gas holdup increase.
• In the present work, grid sensitivity have been studied with 2D and 3D cases. Therefinement of the grid has a strong influence on the hydrodynamics of internal-loopairlift reactor for both 2D and 3D cases. It can be observed that gas holdup profilesbecome more parabolic when go toward coarse grid to fine grid for both 2D and 3Dresults. The results demonstrated that the grid refinement in the 3D cases seemsto be more significant than 2D.
• Time-step size is an important parameter for transient simulation. To determinethe influence of time-step on the hydrodynamics of internal-loop airlift reactor,three various time-step have been used. Influence of time-step study reveals thatgas holdup has very less sensitivity with changes in the ∆t than liquid velocity.
In the present work, CFD simulations for internal-loop airlift reactor with gas-liquid-solid three-phase flow system have been carried out by using Eulerian model where thecontinuous and dispersed phase(s) are mathematically treated as interpenetrating fluids.Numerical calculations were performed to investigate the effect of solid loading, solid
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particle size and solid density on the hydrodynamic characteristics of internal loop airliftreactor. Moreover, study of the solid distribution in the system with different operatingconditions have been discussed. The major conclusions emerging from three-phase flowsimulation work are listed below:
• For each solid loading cases from 5% to 20%, results shows that the superficialgas velocity looks to be the most effective parameter on the gas holdup, wherethe averaged gas holdup in the riser section increases with the increase in thesuperficial gas velocity. The averaged gas holdup is significantly decreased withincreasing slurry concentration. Overall solid concentration in the riser section isincreased with increment in the initial solid loading. For each solid loading cases,higher concentration of the solid particles is observed on the riser walls.
• By using different solid particle sizes varies from ds = 0.1 mm to 1 mm, simula-tions have been performed. The averaged gas holdup in the riser section increasessignificantly with the increase in the superficial gas velocity for each cases. It wasobserved that the riser gas holdup decreases with increase in solid particle diam-eter. The averaged solid holdup in the riser section is significantly increased withincreasing solid particle size and the averaged solid holdup is reduced with incre-ment in superficial gas velocity. The solid concentration increases at the riser wallswith the increment in solid particle size.
• By choosing three different solid densities, ρs, as 1000, 2000 and 3000 kg/m3
simulations have been carried out to investigate the influence on the hydrodynamicof internal-loop airlift reactor. It had found that the averaged solid holdup increasesin the riser section with the increase of solid density. For lower solid density,the averaged solid holdup decreases with increment in superficial gas velocity butsuperficial gas velocity have not affect to solid holdup when density of solid ishigher. In addition, it have been observed that the averaged gas holdup in the risersignificantly increases with the increasing solid particle densities and it decreaseswith lower solid density.
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