CFD Simulation of Multicomponent Gas Flow Through Porous Media

CFD Simulation of

Multicomponent gas flow through

porous media

Master Thesis

byChethan Mohan Kumar

atLehrstuhl fur Stromungsmechanik

Bergische Universitat Wuppertal

Advisors:

Prof. Dr.-Ing. habil. Uwe Janoske

Dipl.-Math. Markus Burger

11.03.2013

Declaration

Hereby I declare that I have worked out the theme CFD simulation of Multi-

component gas flow through porous media entirely on my own without usingforeign help, that I have only used the indicated sources and utilities and that allcitations, having been taken literally or by content, are designated.

(Date) (Signature)

Abstract

The objective of this thesis is to develop a generic CFD solver to simulate multi-component gas transport involving multiscales. A comparative study between themixture and Eulerian approach for multicomponent flows is done. Eulerian approachwhere every component has its own characteristic velocity is used in this work withreasons stated. Inter-component momentum exchange term has been modelled usingMaxwell-Stefan relations. A semi-heuristic drag term for modelling porous drag isused. Temperature transport for the mixture is developed by considering an ensem-ble averaging method of all components for the mixture. Volume averaged form forall the equations is applied by considering fluid in presence of porous media as apseudo-homogeneous medium. All the equations and terms mentioned are imple-mented in the solver. An open source CFD software OpenFOAM has been used todevelop the solver. Capability of the solver to simulate diffusion dominated masstransfer has been established by using a Loschmidt tube which involves diffusion ofa ternary mixture. Accuracy of 0.5% is observed for the case in comparison withanalytical solution for the problem in one dimension. Validation of porous drag andenergy transport has been done by using a fully developed laminar flow over twoparallel plates and comparing the Nusselt numbers. The Nusselt numbers of 8.85and 9.13 for porous and non-porous zones were observed compared to the literaturevalues of 7.54 and 9.8. The reason for the deviations are stated.

Keywords: multicomponent, multiphase, porous media, Maxwell-Stefan, volumeaveraging, multiphaseEulerFoam, Eulerian, MULES, mixture, Nusselt, ternary, dif-fusion, mass transfer, multicomponentPorousFoam, Darcy, Knudsen, incompressible,Loschmidt tube

TO MY PARENTS

Acknowledgements

I would like to express my deepest gratitude to Professor Uwe Janoske for his con-stant encouragement and guidance throughout this work. His concern and enthusi-asm towards the progress of this thesis has been of great help.

I take this opportunity to thank Markus Burger for his suggestions and guidancewithout which this work would not have been the same. I must mention that hispatience over my supposedly two minute discussions which often extended to pro-longed conversations has been remarkable. His knowledge and approach to problemsolving has influenced me to a great extent.

I extend my gratitude to Shilpa for her constructive criticism over the semantics andwriting style of this thesis and my family and friends for their encouragement andsupport.

Finally, I wish to thank my parents for their unprecedented support throughout mystudy.

CONTENTS

Contents

1 Motivation 1

2 Multi-component gas flow 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Approaches to Multicomponent modelling . . . . . . . . . . . . . . . 5

2.2.1 Volume Of Fluid approach . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Mixture approach . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.3 Eulerian approach . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 Model comparisons . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Mathematical Modelling 14

3.1 Drag modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Effects of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.2 Maxwell-Stefan relations for modelling drag . . . . . . . . . . 16

3.2 Porosity Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Energy transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Numerical Simulation 28

4.1 Open Source CFD: OpenFOAM . . . . . . . . . . . . . . . . . . . . . 29

4.2 Solver: multiphaseEulerFoam . . . . . . . . . . . . . . . . . . . . 29

4.2.1 Solution Method: Phase fraction . . . . . . . . . . . . . . . . 31

4.2.2 Solution Method: Pressure-Velocity coupling . . . . . . . . . . 36

4.2.3 Semi-implicit treatment of Drag term . . . . . . . . . . . . . . 37

4.3 Modified solver: multicomponentPorousFoam . . . . . . . . . . . 38

i

CONTENTS

4.3.1 Source term: Maxwell-Stefan drag . . . . . . . . . . . . . . . . 39

4.3.2 Source term: Porous drag . . . . . . . . . . . . . . . . . . . . 40

4.3.3 Energy transport . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.4 Solution algorithm . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Validation 45

5.1 Case1: Loschmidt tube . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.1 Case Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Case2: Flow through a channel . . . . . . . . . . . . . . . . . . . . . 49

5.2.1 Case Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.2 Validation of temperature transport . . . . . . . . . . . . . . . 51

5.2.3 Validation of porosity effects . . . . . . . . . . . . . . . . . . . 51

5.3 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Conclusions 55

A Loschmidt tube 57

A.1 Scilab Program to calculate 1-D analytical solution . . . . . . . . . . 57

A.2 Animation of initial diffusion of gases . . . . . . . . . . . . . . . . . . 60

B Scilab program to calculate Nusselt number 63

Bibliography 64

ii

LIST OF FIGURES

List of Figures

2.1 Examples of multiphase flows . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Multicomponent flow in porous media . . . . . . . . . . . . . . . . . . 4

2.3 Volume of Fluid approach . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Solution procedure of mixture modelling approach . . . . . . . . . . . 10

3.1 Fundamental types of pore diffusion . . . . . . . . . . . . . . . . . . . 15

3.2 Momentum interaction in multicomponent systems . . . . . . . . . . 16

3.3 Collision Scenario of two molecules . . . . . . . . . . . . . . . . . . . 17

3.4 Interaction of molecules in a control volume . . . . . . . . . . . . . . 18

3.5 Movement of molecules of a single phase in porous region . . . . . . 21

4.1 Flow chart of multiphaseEulerFoam solver . . . . . . . . . . . . . 30

4.2 Schematic to demonstrate MULES convective-only transport solution 33

4.3 Procedure to solve phase volume fraction . . . . . . . . . . . . . . . 34

4.4 Solution procedure of phase continuity equation . . . . . . . . . . . . 35

5.1 Experimetal set up of loschmidt tube . . . . . . . . . . . . . . . . . . 45

5.2 Simulation model of loschmidt tube . . . . . . . . . . . . . . . . . . 46

5.3 Phase fraction ofArgon in left tube: Analytical vs. Calculated . . . . 47

5.4 Phase fraction of CH4 in left tube : Analytical vs. Calculated . . . . 48

5.5 Phase fraction of H2 in left tube: Analytical vs. Calculated . . . . . 48

5.6 Calculation of Nusselt number . . . . . . . . . . . . . . . . . . . . . . 49

5.7 Case set up for a channel flow . . . . . . . . . . . . . . . . . . . . . . 50

5.8 Velocity profile of a non porous zone 2 . . . . . . . . . . . . . . . . . 51

5.9 Temperature profile of a non porous zone 2 . . . . . . . . . . . . . . . 52

5.10 Temperature and velocity variation (non porous zone 2) at x= 0.1m 52

5.11 Velocity profile for a non porous zone 2 . . . . . . . . . . . . . . . . . 53

iii

LIST OF FIGURES

5.12 Temperature profile for a porous zone 2 . . . . . . . . . . . . . . . . . 53

5.13 Temperature and velocity variation (porous zone 2) at x= 0.1m . . . 53

A.1 Initial diffusion of Argon . . . . . . . . . . . . . . . . . . . . . . . . . 60

A.2 Initial diffusion of Methane . . . . . . . . . . . . . . . . . . . . . . . . 61

A.3 Initial diffusion of Hydrogen . . . . . . . . . . . . . . . . . . . . . . . 62

iv

LIST OF TABLES

List of Tables

2.1 Comparison of mixture model and Eulerian model . . . . . . . . . . . 12

4.1 Phase properties and functions to access them . . . . . . . . . . . . . 31

4.2 Member function in multiphaseSystem . . . . . . . . . . . . . . . . . 31

4.3 Computing inter-component drag force by maxwellStefan() function . 39

4.4 Computing porous drag multiplier by porousSource.H . . . . . . . . . 41

4.5 Variables and functions required for energy transport . . . . . . . . . 42

4.6 Solution procedure for temperature of the mixture . . . . . . . . . . . 42

4.7 multicomponentPorousFoam algorithm . . . . . . . . . . . . . . . . . 43

5.1 Initial composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

v

Nomenclature

Nomenclature

Roman Symbols

m Mass transfer kg

~υqx,por Velocity of phase q in porous region ms−1

~υMq Drift velocity of a phase from reference mass of mixture ms−1

~qm Effective conduction heat flux Jm−2s−1

cp Specific isobaric heat capacity(fluid) JKg−1K−1

cs Specific heat capacity(solid) JKg−1K−1

dp Pore diameter m

Fdrag Drag Force N

Fq,por Porous Drag force of a phase q N

Flift,q Lift force N

Fvm,q Virtual mass force N

T Viscous stress tensor Nm−2

A Surface area m2

c Mass fraction 1

D Diffusion coefficient m2s−1

d Particle diameter m

E Energy J

H Height m

h Specific enthalpy JKg−1

K Permeability m−2

k Thermal conductivity Wm−1K−1

kn Knudsen number 1

vi

Nomenclature

L Length m

M Molar mass Kgmol−1

m Mass Kg

n Amount of substance of the gas mol

Nu Nusselt number 1

p Pressure Pa

R Ideal gas constant Jmol−1K−1

T Temperature K

V Volume m3

Greek Symbols

α Volume fraction 1

µ Dynamic viscosity Pa s

ǫ Porosity 1

ρ Density kg m−3

σpq Molecular shock diameter m

τ Tortuosity 1

υ Velocity ms−1

Subscripts

p, q Phases

1,2 Molecules

f Fluid

m Mixture

por Porous region

Superscripts

dar Darcy

i initial

k Knudsen

vii

1. Motivation

1. Motivation

Impact of soot is quite adverse on the environment. There is a need for the reductionof emission which contributes to soot . A recent study by the American GeophysicalUnion in their article “Bounding the role of black Carbon in the climate system:A scientific assessment” [4] gives an objective point of view and sheds light on theimportance of soot reduction. It indicates that the effect of soot on the changein climate is twice as much as the previous estimates. It further states that fullymitigating soot can save 1 − 2 million lives by avoiding diseases caused from sootpollution. Complete statistics and details about the hazardous effects of soot can beviewed in the cited article.

The use of Diesel Particulate Filters (DPF) to reduce emission is widespread in au-tomotive and related industries. A first step towards soot reduction is to thorouglyunderstand various phenomena occuring in a DPF. Diesel filter involves a multicom-ponent flow consisting of Particulate matter, Carbon-Di-Oxide, Oxygen, Nitrogenetc. The mixture encounters the filter which is a porous region. The presence ofporous region further influences the motion of the components in the filter. Henceit is important to model the presence of the porous media with the mixture andcapture the interaction effects among individual components and the componentswith the porous matrix.

Approaches used to calculate gas transport is generally based upon parameterizedand simplified models like the dusty gas model, mean transport model or binaryfriction model which predominantly depend on emperical parameters to model theeffects of porosity. Another approach is to resolve the porous media and to considerthe porosity effects. Though this is quite accurate in predicting actual flow phenom-ena, it is not generic in nature and is a trade off between having a generic modelto represent porous media and computation time. Hence a need for an independantmodel is imminent which does not base itself upon emperical parameters to a largeextent and is generic without compromising the accuracy of the solution.

Simulating the behaviour of a multicomponent gas mixture in porous domains whichis an abstract form of flow phenomena occuring in DPF, using Computational FluidDynamics (CFD) is undertaken in this thesis. The momentum exchange betweenindividual components is modelled. Momentum source due to the presence of porous

1

1. Motivation

structure in the domain is also accounted for. A semi-heuristic term for porousdrag force which considers both Darcy effect and Knudsen diffusion in micro porousdomains is used.

Volume-averaging method of Slattery is used to model porous media. The article byPiesche and Goll [27] gives a direction to the modelling of the required terms. In thiswork, the development of the terms are explained and modelled. The modelled termsare then implemented into the open source field manipulation softwareOpenFOAM

and a solver multicomponentPorousFoam is developed.

Finally, the implementation of modelled terms and capability of the solver to simu-late the flow scenario is substantiated.

2

2. Multi-component gas flow


2.1 Introduction

Transport of more than one fluid is imperative in fluid mechanics given the burgeon-ing applications involving many fluids interacting in a domain. Multiphase flows donot restrict the presence of different phases only in terms of solids, liquids and gasesbut is percieved in a broader sense. Eventually all the different phases are expressedin numbers in terms of density, viscosity, molecular weight, molecular diameter andother properties of individual phases. Hence, it is quite obvious that a group ofparticles having the same properties is represented as one phase rather than abid-ing by the conventional definition. This means that this group has similar dynamicresponses. Some examples of multiphase flows are smoke from a chimney dispersinginto the atmosphere, bubble flow due to aeration and blood flow. Further examplesare shown in the schematic 2.1.

Figure 2.1 gives an inital perspective of multiphase flows. In bubble flow, the pres-ence of bubbles induce drag into the flow domain. In a slurry, the behavior of thedispersed phase is influenced by the continuous medium. Further, in this kind offlow, if the dispersed particles can be divided into two groups which largely differin their respective diameters, then the number of phases must be treated as threeinstead of two since the change in diameter means a change in dynamic response bya group of identical particles. The spray dryer involves a fluid stream comprisingof a solute and a solvent which is sent into a chamber containing hot vapour. Thisresults in the vaporization of the fluid resulting in the separation of solute particleswhich is collected at the bottom of the chamber. These examples give an insightand emphasizes the importance of understanding multiphase flow phenomenon.

Multiphase flows and multicomponent flows are two terminologies that varies ina trifle manner but demands to be distinguished. Multiphase flows involve fluidsmixed above molecular level whereas multicomponent flows comprises fluids mixed atthe molecular level. Multiphase flows require interface capturing. Multicomponentflows do not require interface capturing since various components are mixed at themolecular level. Exhaust gas treatment which involves the presence of diesel filterdefined as a porous matrix defines the scope of this work.

3


Air

Hot vapour

Spray dryer

Slurry

Dried powderWater

Bubble column Slurry

Figure 2.1: Examples of multiphase flows

Porous media

Atmosphere

Particulate Matter

+

CO

+

O2

+

N2

Figure 2.2: Multicomponent flow in porous media

The figure 2.2 abstracts the whole process that occur in the treatment of exhaustgases. A multitude of gases and particulate matter mixed at the molecular levelenters a channel. This multicomponent mixture encounters the porous media. Thecharacterisitc property of a porous media is a randomised distribution of pores alongspace and varying pore diameters. At this juncture, it is important to note allthe components flow through as one mixture and there is only a small flux due tomolecular diffusion that drifts from the main mixture. “This is because of Knudsennumbers(kn) being greater than 1× 10−3 because of elevated temperatures, low oper-ating pressures and pore sizes on the microscale”[27]. The methods used to modelmultiphase or multicomponent flow is addressed in the following section.

4


2.2 Approaches to Multicomponent modelling

The two distinct treatment of multiphase flows are Lagrangian and Euler-Euler.The former is a particle tracking technique used when there is a dispersed phaseinteracting with a continuous media. It can largely be remarked that this methodworks well when there is a one way coupling between the interacting phases. Considern particles with negligible mass being carried on by a fluid stream. This results ina one way coupling since the flow of fluid influences the particles but not the viceversa. We can go ahead and state that there is a momentum lost from the fluidthrough drag to the particles but not the other way around. Thus the whole processis controlled by the dominant continuous phase. But in flow domains where no phaseis dominant, a need arises to formulate a method where all the phases involved areequally weighed.

The Euler-Euler method models multiple phases with this approach. As formerlymentioned, the interest of this work deals with multicomponent modelling, we candirectly decide to model it with the Euler-Euler method since there is no phase whichis dominant when the components are mixed at the molecular level. The couplingbetween the phases is achieved by volume fraction. The Volume fraction of phasesvaries across space and over time. Simultaneous occupation of a phase by two fluidsis not possible. Hence closure is obtained by a condition that the sum of volumefractions of all the phases to be unity.

2.2.1 Volume Of Fluid approach

αAir=1

αwater=1

Figure 2.3: Volume of Fluid approach

Volume of Fluid (VOF) model is a surface-tracking technique applied to a fixedEulerian mesh. “It is designed for two or more immiscible fluids where the positionof the interface between the fluids is of interest”[1]. Surface tracking is done byevaluating the values at individual contol volumes. Consider two phases interactingas shown in the figure 2.3. A cell is considered to be completely filled by air whenthe phase fraction is one and completely empty when the phase fraction is zero.

5


When the volume fraction is between unity and zero in a particular cell, then it ismarked with the presence of interface. This demands an equation in order to solvethe phase fraction of individual phases which is given by:

1

ρq

[

∂

∂t(αqρq) +∇ · (αqρq ~υq) = Sαq

+n

∑

p=1

(mpq − mqp)

]

. (2.1)

The above equation 2.1 contains the substantive derivative of mass of individualphases. mpq is the mass transfer from phase p to phase q and mqp is the masstransfer from phase q to phase p. Sαq

is source terms for mass of individual phases.Only volume averaged field variables are solved. Hence the individual phases sharethis common field variables proportionate to their volume fraction in all the cells.The volume averaged momentum equation is:

∂

∂t(ρ~υ) +∇ · (ρ~υ~υ) = −∇p+∇ · [µ(∇~υ +∇~υT )] + ρ~g + ~F . (2.2)

The energy equation shared by the phases is:

∂

∂t(ρE) +∇ · (~υ(ρE + p)) = ∇ · (keff∇T ) + Sh . (2.3)

Energy and temperature are treated as mass averaged variables in the above equation2.3. Sh is the heat source term. Energy shared by individual phases is proportionateto its mass fraction in the whole mixture. Surface tension can also be modelled byadding a source term to the momentum equation. Thus the VOF approach modelsthe required effects in case of immiscible fluids where the need of interface trackingexists.

2.2.2 Mixture approach

The mixture modelling is one of the methods to model multiphase systems. In thisapproach, a single continuity equation and momentum equation is solved for themixture. The mixture is assumed to be the averaged field properties of the fieldvariables. Though the averaging is quite similar to the VOF approach, it has tobe noticed that there was no equation solved for the conservation of mass in VOFapproach since the interest was mainly in the interface.

Contrastingly, in the mixture approach, the continuity equation for the mixture issolved. Hence it can model phases moving with different velocities. It is assumed thatthe individual phases with different velocities diffuse away from the whole mixturewith a drift flux. This drift is represented as the relative velocity between the mixtureand the individual phases. The generic equations for the mixture approach ([1],[18])is explained subsequently .

The continuity and momentum equations for a single phase q can be written as:

6


∂

∂t(αqρq) +∇ · (αqρq ~υq) = Γq , (2.4)

∂

∂t(αqρq ~υq) +∇ · (αqρq ~υq ~υq) = −αq∇pq +∇ · [αq(τq + τTq)] + αqρqg +Mq , (2.5)

where Γq represents the rate of mass generation of phase q at the interface. Mo-mentum exchange between the interface is represented by the source term Mq. τq isthe average viscous stress tensor and τTq is the turbulent stress tensor. To obtainthe equations for the mixture as a whole, all the equations of individual phases areadded. The continuity equation for the mixture is:

∂

∂t

n∑

q=1

(αqρq) +∇ ·n

∑

q=1

(αqρq~υq) =n

∑

q=1

Γq . (2.6)

The mass in the whole system is conserved. Hence, the sum of Γq over individualphases must be zero;

n∑

q=1

Γq = 0 . (2.7)

The continuity equation thus becomes:

∂

∂t(αmρm) +∇ · (αmρm~υm) = 0 , (2.8)

where the mixture density is:

ρm =n

∑

q=1

αqρq , (2.9)

and mixture velocity is:

~υm =1

ρm

n∑

q=1

αqρq~υq . (2.10)

The drift flux needs to have a reference velocity which is represented by the mixturevelocity at the mass centre. The momentum equation for the mixture is obtainedby adding the momentum equation of all phases:

7


∂

∂t

n∑

q=1

(αqρq~υq) +∇ ·n

∑

q=1

(αqρq~υq~υq) =n

∑

q=1

−αq∇pq +∇ ·n

∑

q=1

[αq(τq + τTq)]

+n

∑

q=1

αqρqg +n

∑

q=1

Mq . (2.11)

The second term on the left hand side equation 2.11 can be rewritten in terms ofmixture density and mixture velocity:

∇ ·n

∑

k=1

αqρq~υq~υq = ∇ · (ρm~υm~υm) +∇ ·n

∑

q=1

αqρq~υMq~υMq . (2.12)

Here ~υMqis the drift velocity of a phase from the reference mass of the mixture;

~υMq = ~υq − ~υm . (2.13)

The momentum equation in terms of mixture variables is then:

∂

∂t(ρm~υm) +∇ · (ρm~υm~υm) = −∇pm +∇ · ((τm + τTm)]−

∇ ·n

∑

q=1

αqρq~υMq~υMq + ρmg +Mm , (2.14)

where τm, τTm represent the average viscous and turbulent stress because of the slipof a phase. The pressure of the mixture is:

∇pm =n

∑

q=1

αq∇pq . (2.15)

But more often the partial pressure and the mixture pressure is taken as the samebarring a few cases. The term Mm is the contribution of the sum of surface tensionfrom individual phases.

The formulation for relative(slip) velocity is needed since the continuity and momen-tum equations are expressed in terms of it following which the continuity equationfor phase fraction can be solved. In this approach, an algebraic formulation for slipvelocity is constructed. Density differences of individual phases has a major influ-ence on particles with different physical properties resulting in a varied dynamicresponse. The relative velocity between two phases q and j is further coupled to thedrift velocity by:

8


~υMq = ~υqj −n

∑

i=1

(ci~υji) , (2.16)

where ci is the mass fraction of a phase;

ci =αiρiρm

. (2.17)

The algebraic formulation for the relative velocity(slip) between two phases generallyconsiders that the mass of a particle in the mixture is very small and that the particlereaches a terminal velocity over a short distance. In short, a local equilibriumis assumed. The additional force due to the relative velocity of the particle withrespect to a fluid is represented by the drag force on the particle by the fluid. Thus,the relative velocities of a particular phase is obtained which can be used in thephase continuity and momentum equations subsequently.

The expression of Manninen et al.[18] is generally used:

~υqj =τq

Fdrag

(

ρq − ρmρq

~α

)

, (2.18)

where α is the secondary phase particle acceleration:

~α = ~g − (~υm · ∇)~υm −∂~υm∂t

(2.19)

and τq is the particle relaxation time

τq =ρqdq

2

18µq

, (2.20)

where dq is the particle diameter of phase q.

The continuity equation for a phase which is incompressible and for a process whereno phase changes occur is:

∂

∂t(αq) +∇ · (αq~υm) = −∇ · (αq~υMq) . (2.21)

An overview of the complete solution procedure is shown in figure 2.4.

Thus, the mixture approach sums up the continuity and momentum equations ofindividual phases and circumvents solving for the velocity of phases by algebraicformulation for the relative velocity between phases and coupling drift velocity torelative velocity. It assumes local equilibrium over short spatial scales.

9


timen

find slip velocities from

(2.18)

find the phase fraction

of all phases

Update

mixture density

(2.9)

find the drift velocity from

drift velocity-slip velocity

coupling (2.16)

solve for velocity-pressure

using momentum(2.14) and

continuity(2.8) equations

iteratively

End time

Figure 2.4: Solution procedure of mixture modelling approach

2.2.3 Eulerian approach

Eulerian approach relies on solving the momentum and continuity equations for allthe phases. “In the Euler-Euler approach, the different phases are treated math-ematically as inter-penetrating continua”[1]. The coupling between the phases isestablished by an interaction term in the momentum equation of each phases whichcontains the relative velocity of a phase with respect to all the other phases. Unlikethe former approaches, a number of phases can be modelled which can move witha particular velocity. The concept of volume fraction is considered and a particu-lar space cannot be occupied by two phases. The volume occupied by a phase isproportionate to its phase fraction similar to the mixture theory approach.

Though continuity and momentum equations are solved for individual phases, theeffect of all these phases has to be considered in the multiphase framework. Hence,characteristic properties of viscosity and density of a particular phase have to bemodified to fit the multiphase framework. It is straightforward to predict that theeffect of physical property from a phase in the multiphase domain is proportional toits phase fraction. Hence the effective density used in the equations are:

10


ρq = αqρq (2.22)

and viscosity

µq = αqµq . (2.23)

If all terms related to mass transfer across interfaces are set to zero, the continuityequation for a phase is

∂

∂t(αqρq) +∇ · (αqρq ~υq) = 0 . (2.24)

The momentum balance equation is:

∂

∂t(αqρq~υq) +∇ · (αqρq~υq~υq) = −αq∇p+∇ ·T+ αqρq~g +

n∑

p=1

(~Rpq)

+(~Fq + ~Flift,q + ~Fvm,q) . (2.25)

The interaction term for ~Rpq is modelled as an interphase drag term which is afunction of relative velocity of the phases:

n∑

p=1

~Rpq =n

∑

p=1

Kpq(~υp − ~υq) . (2.26)

Kpq here is the interphase momentum exchange coefficient. Lift and virtual massforces are not in the scope of this work and will not be discussed exhaustively.Various models of drag, heat transfer or molecular diffusion between the phases canbe modelled using the generic interphase momentum transfer term ~Rpq. The phasefraction is solved from the continuity equation and the effective densities of phasesare updated.

Eulerian modelling is used in complicated flow phenomenon where the formerlystated approaches do not work. The complexity involved is the courtesy of allowingthe phases to have its individual velocities and maintaining the boundedness ofphase fractions. Additionally, it is computationally expensive since n continuityand momentum equations are solved in every iteration and demands larger memoryrequirements. The equations in this section are referred from [27] and [1].

11


Feature Mixture

model

Eulerian Model Remark

Velocity mixture velocity individual phasevelocities

Pressure mixture pressure shared by all thephases

mixture pressureequal to partialpressure in mostcases

Coupling

between

phases

relative velocitybetween phases,drift velocity,mixture velocity

interphase momen-tum exchange termwhich accounts forthe slips betweenphases

solving for phasevelocity circum-vented and neversolved in mixtureapproach

Wall slip

modelling

mixture slip atwall

phase slip at wall velocity of phaseknown in Eulerianmodel

Computation

cost

reasonably sameas single phasemodel

increases with thenumber of phases

Eulerian model de-mands large mem-ory requirements

Table 2.1: Comparison of mixture model and Eulerian model

2.2.4 Model comparisons

A suitable multiphase modelling approach has to be selected which suffices andrepresents all the processes that occur in case of flow through the porous media.It can directly be analysed that VOF approach is out of the purview of our worksince interface tracking in not an important requirement. Now we are left with twoapproaches which needs diligent comparison. Table 2.1 shows the capabilities of thetwo modelling approaches.

Though the mixture approach seems an attractive option for our flow scenario, itassumes local equilibrium between the phases and the relaxation time of a particlemust be lesser than the timescale of the problem. This means that a strong couplingbetween the phases must be present. A sudden acceleration of a few phases mayoccur as the flow encounters porous media which might weaken the coupling.

Another drawback of using mixture approach for our flow scenario is the require-ment of the slip boundary condition for velocity at the wall. “The usual boundarycondition, that the velocity of a gas bounded by a wall should be equal to that of thewall at every point of its surface, is not exactly fulfilled when there exists a velocitygradient perpendicular to the wall, a temperature gradient parallel to the wall or-incase of a mixture- a concentration gradient parallel to the wall”[17].

Both temperature and concentration gradients exist parallel to the wall as a resultof porous media in the flow domain. The porous media has its characteristic ther-mal conductivity which is different than the multiphase mixture. Hence there isa temperature gradient over the porous domain parallel to the wall. Existence ofconcentration gradient parallel to the wall can be attributed to the permeability ofthe porous media. Because of the permeability and a knudsen number greater than

12


0.01 in the porous domain amounts to the presence of varied mass across its sides.There is relatively more mass towards the inlet than the outlet.

The aforementioned reasons establishes the fact that the mixture theory though withits lower computation time fails to fulfill the present requirements. Though with itsconsiderably higher computation time, we proceed with the Eulerian multiphasemodel as it is capable of modelling the desired effects.

2.3 Closure

The three main approaches of multiphase modelling; VOF, mixture theory and Eule-rian were introduced in this chapter. A broader perspective of considering multiphasemedia and the slight distinction between multiphase and multicomponent flows werediscussed. Though this work concerns multicomponent flows, the term multiphasehas been used to explain the approaches in order to retain the semantics of prevalentliteratures on the same. Further, the characteristics of all the approaches were eval-uated and compared. The Eulerian multiphase modelling was selected for modellingour flow problem with reasons stated.

13

3. Mathematical Modelling


Mathematical modelling is imperative in order to express a physical system in amaterialized form. A mathematical representation of systems or processes generatesa possibility to study and analyse the influence of different components in the system.It starts with transforming the various processes and interactions into governingequations. The governing equations have to be modelled in such a way that, it mustposses the capability to express all the important phenomena that may occur. Theequations must also convene the application of boundary conditions to realisticallysimulate the physics of the system.

In fluid systems, the continuity or the mass conservation, momentum conservationand the energy conservation are the three fundamental forms of equations that re-quire mathematical modelling. Differential equations are used quite often in orderto express such processes where there is a continuous variation of a quantity overspace and time. A solution of the differential equation in terms of time and do-main thus results in a possibility of monitoring the behaviour of a quantity overthe whole domain and its variation with time. Most of the mathematical systemsresults in non-linear equations. The non-linearities pose some difficulties in findingan analytical solution to the modelling systems.

The Navier-Stokes equation models the mass and momentum conservation of a fluidsystem and will be used in this work. The momentum interaction and porous dragterms are developed and are considered as source terms in the momentum equation.Energy equation for the pseudo homogeneous medium is modelled where the mixtureand the porous matrix is assumed to have the same temperature and a local thermo-dynamic equilibrium exists between the individual phases. The subsequent sectionsdeals with the development of mathematical models to express the flow phenomenaof our system.

3.1 Drag modelling

3.1.1 Effects of Diffusion

Diffusion is a prime mechanism responsible for a variety of processes involving masstransfer. This broad spectrum involves bulk gas or liquid diffusion, knudsen diffusion

14


Micropore diffusion

Knudsen diffusion

Bulk diffusion

Figure 3.1: Fundamental types of pore diffusion

in pores and molecular diffusion inside micro pores. The driving force for diffusion isascribed to the concentration gradients that exist in space. The tendency is alwaysto flow from a region of higher concentration to a region of lower concentration. Thesubject of interest here is porous diffusion. To distinguish between different kindof porous diffusion mechanisms, we need to first distinguish between the differentkinds of pores which are macro pores(size > 50nm), micro pores(size < 2nm) andmeso pores(size between 2nm-50nm). The three types of diffusion mechanisms areshown in the figure 3.1 referred from [24].

In high pressure systems the number of molecules are so high that collision occursmore frequently amongst the molecules than with the wall. Large pore sizes alsoaccommodate for such free molecular collisions. This kind of diffusion is termed asBulk diffusion. Knudsen diffusion arises when the mean free path of a molecule islesser than the charateristic length of the domain. Hence, the molecule travels lesserthan the diameter in case of a circular tube before collision which results in a highprobability of molecule-wall collisions. Surface diffusion occurs primarily in micropores when the molecular species are adsorbed on the surface of porous media. “Bulkand Knudsen diffusion occur together and it is prudent to take both mechanisms intoaccount rather than assume that one or other mechanism is ‘controlling’ ”[24].

The physical meaning of interaction between molecules of different components in amulticomponent system can be related to drag effects. The interpretation is that,molecules travelling with different velocities in a system exchange momentum amongthem due to relative velocity. It is visualised in the figure 3.2. The inter component

15


friction 1-2

friction 1-

friction 2-

U2

U1

1

2

Figure 3.2: Momentum interaction in multicomponent systems

momentum exchange is directly proportional to the relative velocity. A suitablemodel which considers the stated effects of diffusion and replicates the appropri-ate interaction effects between components must be selected and integrated to themulticomponent framework. Diffusion of ideal gas mixtures is considered in thiswork.

3.1.2 Maxwell-Stefan relations for modelling drag

Consider a collision scenario of two molecules with mass m1 and m2 travelling withvelocities u1 and u2 respectively. The assumption of collision is completely elastic.There is no loss of energy during the process. The other important assumption isthat the molecules are spherical in shape. An example of such a process is collisionbetween two billiard balls.

The two molecules carry a momentum of m1υ1 and m2υ2. If υ1′ and υ2

′ are velocitiesafter the collision and the path traced by the particles before and after collision isassumed to be the same, then according to the law of conservation of momentum:

m1~υ1 +m2~υ2 = m1~υ′

1 +m2~υ2′ , (3.1)

m1(~υ1 − ~υ′

1) = −m2(~υ2 − ~υ′

2) . (3.2)

16


Plane of collision

11 2 2

V1'V1

Path of collision

Before collision Before collisionAfter collision After collision

V2'

V2

Figure 3.3: Collision Scenario of two molecules

m1(~υ1−~υ′

1) is the momentum transferred from molecule 1 to 2. The average velocityafter collision υ′

1 in such conditions is the velocity of the center of mass of the pairof molecules[25].

uc =m1~υ1 +m2~υ2m1 +m2

= ~υ′

1 . (3.3)

The momentum transferred from molecule 1 to molecule 2 is:

m1(~υ1 − ~υ′

1) = m1~υ1 −

(

m1~υ1 −m2~υ2m1 +m2

)

=m1m2(~υ1 − ~υ2)

m1 +m2

. (3.4)

The inference from equation 3.4 is that momentum is transferred only betweenmolecules of different components. The amount of momentum transfer is propor-tional to the relative velocity between components. In short no intra-componentbut only inter-component momentum transfer takes place. With the mechanics ofcollision understood, we move on to the Maxwell-Stefan relations.

In the control volume shown in the schematic 3.4, movement of molecules of type 1is obstructed by molecules of type 2 which can be interpreted as the drag effect. Mo-mentum transfer between the phases take place in such interactions. The applicationof Newton’s second law implies that

Sum of forces ∝ Rate of change of momentum in the system

The momentum exchange between molecules depends on the frequency of their col-lisions. It is obvious that the probability of molecules of two components increases

17


dz

Figure 3.4: Interaction of molecules in a control volume

with the number of molecules present. For example, if 10 moles of Oxygen, 20 molesof Nitrogen and 4 moles of Argon is present in a contol volume, then the frequency ofcollision between Oxygen and Nitrogen is more extensive than other pairs of collision.The collision frequency thus depends on the phase fraction of the two components.

Collision frequency between 1− 2 ∝ α1α2

Assuming that the control volume moves with the flow at the same velocity, the netflux becomes zero. Then in a system of unit volume in unit time, for molecules 1and 2 the rate of change of momentum of molecule 1 is proportional to the productof average momentum exchanged in one collision and the collision frequency.

Applying force balance on molecules of type 1 in the control volume and neglectinggravity forces and forces due to the gradient of velocity and considering only forcedue to pressure on the system:

−∇p1 ∝ α1α2(~υ1 − ~υ2) , (3.5)

where p is the absolute pressure. The proportionality constant which is the dragcoefficient f12 is introduced to the equations:

−∇p1 = f12α1α2(~υ1 − ~υ2) . (3.6)

f12 is defined as an inverse drag coefficient and D12 = p/f12

18


−∇p1 = pα1α2(~υ1 − ~υ2)

D12

= −pα1α2(~υ2 − ~υ1)

D12

. (3.7)

System pressure is often taken as constant. Hence,

−∇p1 = −p∇α1 (3.8)

since

p1 = α1p (3.9)

The final relation becomes:

− p∇α1 = −pα1α2(~υ2 − ~υ1)

D12

. (3.10)

The force balance for component 2 is,

− p∇α2 = −pα2α1(~υ1 − ~υ2)

D21

. (3.11)

Extending the relation to a multi component system,

− p∇αq = −pn

∑

p=1

αqαp(~υp − ~υq)

Dqp

. (3.12)

p∇αq = pn

∑

p=1

αqαp(~υp − ~υq)

Dqp

. (3.13)

Thus to implement Maxwell-Stefan drag into the multicomponent model, equation3.13 has to be added to the momentum equation. The pressure terms cancel witheach other but it is convenient to have it in this form to synchronize with dimensionalunits in the momentum equation. The procedure of drag modelling in the section isreferred from [25] and further details can be found in the same.

19


3.2 Porosity Modelling

The presence of porous media in the domain offers resistance to the flow because ofdrag effects. Porous structures are characterized by voids or pores through whichthe fluid flows, labyrinth or tortousity factor which defines interconnectivity of theporous structure and porosity which is the ratio of the amount of fluid present inthe porous structure to the total volume including fluid and solid volumes. It canbe directly deduced that the drag force increases with increase in the surface area ofthe porous matrix. The surface area of the matrix can further be divided into twoparts. Area normal to the fluid flow and internal surface area of the matrix structurewhich induces viscous drag on the fluid flowing through it.

In the experiment conducted by Darcy, it was realized that the bulk shear stresseswere dominant as the length of the porous media and hence the internal surface usedwas many times greater than its normal surface area. Darcy then found the velocityof the fluid in the porous matrix ~υdar

qx,por by measuring flux of an incompressible

fluid flowing through a cylindrical tube at the outlet ρA~υdarqx,por and dividing it by

ρA. Pressure difference was the driving force which was directly proportional tothe flux and hence the velocity in the porous region ~υdar

qx,por . He realized that theproportionality constant could be expressed in terms of permeabilityK of the porousmedia and viscosity µ of the fluid. The darcy relation is [20]:

∂p

∂x= −

µ

K~υdarqx,por , (3.14)

which is rewritten as:

~υdarqx,por = −

K

µ

∂p

∂x. (3.15)

“Permeability accounts for the interstitial surface area, the fluid particle path asit flows through the matrix and other related hydrodynamic characteristics of thematrix”[20]. In micro porous matrices, the flux at the outlet is generally more thanthose measured by Darcy’s experiment since an additional Knudsen diffusion fluxappears because of slip flow of the molecules with the porous structure walls. Hencethe additional knudsen flux has to be modelled to appropriately represent the flowcondition.

Figure 3.5 shows different forms of transport of molecules in a porous matrix. Asseen, momentum is lost by the molecules due to their interaction with the surfaceof the porous matrix. This is termed as the Darcy drag. A few molecules whichmove in micro pores experience knudsen slip as mentioned earlier contributing to theadditional Knudsen flux. The remaining molecules retrace their path because of theinter connected pores in the porous matrix. This is accounted for by the tortuosityterm.

Consider a fluid q flowing through a porous matrix. From the kinetic theory of gases,the mean molecular velocity for the component can be written as [20]:

20


Px Px+dx

Molecules that undergo Darcy drag from

internal surface area of porous matrix

Molecules that are transported due to Knudsen slip

Molecules that are retraced due to

labyrinth/tortuosity factor of the porous matrix

Figure 3.5: Movement of molecules of a single phase in porous region

~υqm,por=

√

8RT

πMq

. (3.16)

R is the universal gas constant, T is the temperature in Kelvin and Mq is themolecular weight in Kg mol−1. Considering a flow through a tube which is theidealised form of a pore with diameter dp and applying momentum conservation,Shear stress at the radius of the pore is[20]:

τ(dp/2) =3

4nqkBT

~υkqx,por

~υqm,por

= −dp4

dp

dx. (3.17)

From the above equation, an expression for ~υkqx,por is obtained,

~υkqx,por = −

dp4

dp

dx

4

3nqkBT~υqm,por

. (3.18)

Ideal gas behavior is assumed:

pq = nqkBT . (3.19)

Substituting ~υqm from equation 3.16:

~υkqx,por

= −dp3pq

√

8RT

πMq

dp

dx, (3.20)

21


which can be rewritten as,

~υkqx,por = −

Dqk

pq

dp

dx, (3.21)

where

Dqk =

dp3

√

8RT

πMq

(3.22)

is the Knudsen diffusion coefficient of a gas component.

Thus for flows in 1-dimension and from 3.21 , 3.15 we have:

~υqx,por = ~υdarqx,por + ~υk

qx,por , (3.23)

~υqx,por = −Dq

k

pq

dp

dx−

K

µ

∂p

∂x, (3.24)

and extending it to 3-dimensions,

~υq,por = −[Dq

k]

pq∇p−

[K]

µ∇p . (3.25)

In this work, porosity is modelled for non deformable and isotropic media.

The permeability expression for permeabilityK is modelled according to the velocityprofile of a Hagen-Poiseuille flow[27]. It is dependant on the diameter of the poreand is:

K =dp

2

8. (3.26)

The dimensionless tortuosity or the labyrinth factor τ is introduced into the equa-tions to compensate the extended path a fluid travels when it encounters closedpaths due to interconnectivity of the porous media.

Thus the expression for pressure gradient which is the driving force for the flows inporous media is:

−∇p = τ 2[

Dqk

pq+

K

µ

]−1

~υq,por . (3.27)

22


In order to model this momentum source due to porous structure in the fluid equa-tions, local volume averaging is used. It is a technique introduced by Whitaker[28]and Slattery[13]. The approach is to define a quantity φ which is in a fluid volumeVf by averaging over a volume V. The condition for averaging is that the volumeover which φ is averaged must neither be large compared to the flow domain nor verysmall to represent the geometry of the porous structure. “The local representativeelementary volume is chosen such that the smallest differential volume that resultsin statistically meaningful local average properties (such as local porosity) and whenthis volume is appropriately chosen, adding extra pores(extra volume) around it willnot result in changes to the values of these local properties”[20].

A penalty approach is used in order to define the void distribution fuction or thesolid distribution function. Let us name this distribution function as gp defined as,

gp(x) =

1 if x is in void region,Vf

0 if x is in solid region,Vs(3.28)

Local porosity is defined as:

ǫ(x) =1

V

∫

V

gp(x)dV =Vf

V(3.29)

where V = Vf + Vs.

Volume average of a quantity φ is,

< φ >=1

V

∫

Vf

φdV = ǫφ . (3.30)

A comprehensive explanation of volume averaging including the general form ofReynolds transport theorem can be found in [20] and will not be explicitly mentionedhere.

Thus by applying the stated averaging technique to the modelled source porosityterms in equation 3.30 the final term that is added to the fluid momentum equationis:

Fq,por = −ǫ∇p− gpτ2

[

Dqk

pq+

K

µ

]−1

~υq,por . (3.31)

The first term on the right hand side can replace −∇p in the Eulerian momentumequation since ǫ is 1 in the absence of porous media and accounts for both porousregions and non porous regions.

23


3.3 Energy transport

Energy transport is modelled by the mixture approach. In case of non-isothermalflow conditions, the change of temperature over space and time has to be considered.Local thermodynamic equilibrium is assumed so that a single energy equation issufficient for the gas mixture as a whole[22].

Consider the general Reynold’s transport equation for a variable φ for a specificmass:

∂(ρφ)

∂t+∇ · (ρ~υφ) = ∇ · (γ∇φ) + Sφ . (3.32)

It comprises the unsteady term, convective term, diffusive and source terms. Thevariable in case of energy transport is the enthalpy. Since we are solving the energyequation for the mixture, the equation becomes[27]:

∂(ρmhm)

∂t+∇ · (ρmhm~υm) = −∇ · (~qm) +

∂p

∂t. (3.33)

Heat conduction is written normally in terms of mass averaged velocity of themixture[29]:

~υm =1

ρm

n∑

q=1

ρq~υq . (3.34)

The effective heat conduction of the mixture is:

~qm = −km∇T +n

∑

q=1

ρqhq(~υq − ~υm) . (3.35)

km is the molecular thermal conductivity of the gas mixture[22] and is obtained by:

km =n

∑

q=1

αqkq . (3.36)

αq and kq are the phase fraction and thermal conductivity of a component respec-tively. Now substituting 3.35 in 3.33:

∂(ρmhm)

∂t+∇ · (ρmhm~υm) = ∇ · (km∇T )−∇ ·

n∑

q=1

(ρqhq~υq) (3.37)

+∇ · (ρmhm~υm) +∂p

∂t

24


n∑

q=1

(ρqhq) =n

∑

q=1

(αqρqi)(αqhq

i) = ρmhm (3.38)

The above equation then reduces to the following form:

∂(ρmhm)

∂t+∇ ·

n∑

q=1

(ρqhq~υq) = ∇ · (km∇T ) +∂p

∂t. (3.39)

Further the enthalpy can be related to temperature by:

dhq = Cp,qdT , (3.40)

where Cp,q is the specific heat of a component. Then equation 3.39 transforms intoa scalar transport equation by using the relation from equation 3.40:

∂(ρmCp,mT )

∂t+∇ ·

n∑

q=1

(ρqCp,q~υqT ) = ∇ · (km∇T ) +∂p

∂t. (3.41)

“When porous media is considered, in the absence of local heat sources or high fre-quency temperature changes, local thermodynamic equilibrium between gas and solidcan be assumed” [20]. This in the concept of volume averaging implies that in theaveraged volume, fluid and solid both have the same temperature.

To describe the transport of energy in such cases, a single equation is setup whichconsiders the pseudo homogeneous medium. The temporal changes of local enthalpyand heat conduction have to be expressed in terms of pseudo-homogeneous gas/solidmedium [27]:

< ρCpT >=1

Vf

∫

Vf

(ρmCp,mT )dV +1

Vs

∫

Vs

(ρsCsT )dV

= [ǫ(ρmCp,m) + (1− ǫ)(ρsCs)]T . (3.42)

By the application of volume averaging of Slattery’s theorem[20] of volume averaging:

< ∇ · (k∇T ) >= ∇· < k∇T > +km − ks

V

∫

Af,s

~nT ′dA . (3.43)

Af,s is the fluid-solid interface area. The second part on the right hand side of theabove equation demands information on geometry of porous structure and is henceneglected following [15, 19] and [2]. The first term of equation 3.43 is then similarlytransformed by following equation 3.42:

25


< k∇T >=< k > ∇T = [ǫkm + (1− ǫ)ks]∇T (3.44)

The final equation for the pseudo-homogeneous medium is:

∂

∂t[ǫ(ρmCp,m) + (1− ǫ)(ρsCs)]T +∇ ·

[

n∑

q=1

(ρqCp,qT~υq)

]

= ∇ · [ǫkm + (1− ǫ)ks]∇T + ǫ∂p

∂t. (3.45)

Equation 3.45 is used to model energy transport in the present work. Following thesolution of the equation, temperature dependant properties of viscosity µ and Diffu-sion coefficient Dpqin the system are updated using the Chapman-Enskog relation.

µq =5

16

1

σ2qΩV

√

kBMqT

πNA

. (3.46)

νq is the viscosity of a component, σq is the molecular shock diameter, NA is theAvagadro number and kB is the Boltzmann constant. ΩV is the collision integralfor viscosity based on Lennard-Jones-Potenzial[3]. Dpq is the diffusion coefficientbetween two components:

Dpq =RT

32

p

3

8σ2pqΩD

√

kBπMpqNA

. (3.47)

In equation 3.47, ΩD is the collision integral for diffusion. Mpq and σpq are theeffective molar mass and molecular shock diameter of two components respectively.They are given by:

Mpq = 2(Mp−1 +Mq

−1)−1

, (3.48)

σpq =σp + σq

2. (3.49)

The density of the component can also be updated by using the equation of statefor ideal gases:

ρq =pMq

RT. (3.50)

Thus the energy transport model takes into account all the properties that changewith temperature. The implementation of the equation into the CFD solver will bediscussed in the following chapter. It can be summarized that though the momen-tum and continuity equations are solved using the Euler-Euler approach for reasonsmentioned earlier, energy equation is solved by the mixture approach in this work.

26


3.4 Closure

In this chapter, the effects of diffusion were stated and the drag term was modelledusing Maxwell-Stefan relations. A flow scenario of molecules in a control volumewas used to model the additional driving force due to concentration gradients of aphase and the drag term. Both terms were modelled and the final term Fdrag fromequation 3.13 has to be added to the momentum equation of the Eulerian model.

The effect of porous media in the flow domain was treated by considering both theviscous drag or Darcy drag effects and the addition Knudsen flux due to slip of themolecules with walls of micro porous domains. The modelled porosity term canrepresent macro properties of micro porous domain. Further, volume averaging wasused in order to represent averaged properties of the pseudo-homogeneous mediaof solid and gas. Permeability and Knudsen diffusion coefficient are considered forisotropic porous media though the derived momentum source terms due to porouseffects Fq,por equation 3.30 is inversed and can be easily extended to other types ofporous structures. The Knudsen diffusivity Dq

k depends on the ideal pore diameterdp (3.24). Permeability K which also depends on the pore diameter and tortuosityis modelled semi-heuristically.

Finally, the energy transport for the whole domain is modelled by volume-averaging(3.45). The mixture approach is used for solving the equation by assuming localthermodynamic equilibrium between the phases. The solid and liquid phase presentin the averaged volume that represent porous media share the same temperature.Temperature dependance of viscosity (3.48), diffusion coefficient (3.49) and density(3.50) are also considered in the modelling framework.

27

4. Numerical Simulation


Computational methods for Fluid Dynamics (CFD) offers many possibilities andflexibility to envision the influence of various parameters on the flow. Experimentalmethods though sturdy, do not offer this and takes a longer time and higher costs.“While the first computers were built in 1950s performed only a few operations persecond, machines are being designed to produce 1012 floating point operations persecond”[14]. With computing power increasing, it is not a surprise that CFD hasfound its popularity for flow simulations.

The previous chapter on mathematical modelling contained many differential equa-tions which are mainly used to describe transport phenomena. Analytical solutionto these differential equations is restricted to simple flows. Hence numerical methodsmust be used to discritize and solve complicated differential equations.

The three main approaches for discritization are Finite Element Method(FEM), Fi-nite Difference Method(FDM) and Finite Volume Method(FVM). All three methodsfollow the saying “How do you eat an elephant - one bite at a time”, by dividing thewhole domain into smaller parts and solving the discritized form of the differentialequation for every part and in turn finding the solution for the whole domain.

FVM is the most commonly used discritization technique in CFD. The small “bites”are defined by a control volume(C.V) and two C.Vs share one common face amongthem. The calculation domain is divided into a number of non overlapping controlvolumes so that there is one control volume surrounding each grid point[21]. Thisworks well for flow simulations as volume based properties like pressure, velocity canbe distinguished with face centered properties such as flux. The discritized equationconstructed in each C.V is of the form:

A~x = ~b (4.1)

where A is a coefficient matrix, ~x is the unknown vector and ~b is the source vector.The value of unknown variable is known at all locations of the domain by interpo-lation. The interpolation schemes can be linear or higher order schemes based onrequirements.

28


4.1 Open Source CFD: OpenFOAM

OpenFOAM is an open source software used for Field Operation And Manipula-tion. It has a good potential to solve CFD problems as it houses many algorithmsand interpolation schemes to solve discritized equations and interpolation schemes.Standard Dirichlet and Neumann along with more commonly used boundary condi-tions are built in the software and can be directly used. Coupling betwen pressureand velocity is handled by algorithms such as SIMPLE, PIMPLE and PISO whichuse the iterative approach for solution.

Object Oriented Programming is the language used for programming and hence givesus enormous possibilities to extend and customize or build even a solver to suit ourrequirements. “Majority of fluid dynamics can be described using the tensor calculusupto rank 2, i.e. scalars, vectors and second-rank tensors” [9]. Tensors are storedas lists such as Field<scalar>, Field<vector>, Field<Tensor> which are termed asscalarField, vectorField and tensorField for easy identification. Volume centred andsurface centred properties can be further distinguished by adding a prefix ‘vol’ or‘surface’ such as volScalarField in case of pressure and surfaceScalarField incase of flux. Storing a variable as a list offers an easy way to operate or manipulatevariables with each other.

Focusing on developing a solver in OpenFOAM for our flow scenario, the solvermultiphaseEulerFoam in OpenFOAM 2.1.1 which solves for systems with manyincompressible phases is identified as a good starting point.

4.2 Solver: multiphaseEulerFoam

multiphaseEulerFoam in OpenFOAM 2.1.1 is based on the Eulerian method forsolving multiphase flows. It treats individual phases as incompressible. The defaultmechanism used for the transport of individual phases is the single phase transportmodel based on the viscosity model. Unsteady terms are included in the equa-tions and hence transient flow cases can be simulated. The presence of individualphases and the averaged mixture in the system demands a separate considerationduring programming to deal with the properties of phases and mixture. multi-

phaseEulerFoam handles this requirement by two distinct libraries, phaseModeland multiphaseSystem. Many models for drag, heat transfer, kinetic theory are al-ready present and can be used by invoking them from the case directory accordingto the requirements.

Additionally, it handles every phase/component in an idealized way by defining itas spherical particles and the diameter model is used for this purpose. Turbulencefor the mixture can be solved by using LES method. MULES algortihm solvesfor phase fraction. Pressure-velocity coupling is dealt by PIMPLE algortihm usinga small variation to projection methods. “Methods which first construct velocityfield that does not satisfy the continuity equation and then correct it by subtractingsomething (usually a pressure gradient) are known as projection methods”[14]. Thevariation and the method will be discussed in detail subsequently. Flow chart in 4.1shows the main parts and working of multiphaseEulerFoam solver.

The required attributes and methods are intialized with phaseModel, multiphas-eSystem and createFields. Following initialization, the pressure-velocity coupling is

29


multiphaseEulerFoam

Interfacial Models:

Library for drag and

heat transfer models

phaseModel:

Library to define phase

properties and functions

to operate on

individual phases

multiphaseSystem:

Library with functions to

operate on fluid

properties or calculations

involving more than one phase

Pimple loop for Pressure Velocity correction:

1. Update mixture turbulence

2. Solve phase fraction using phase continuity

equation:fluid.solve()

3. Update mixture density

4. Calculate phase volumes of zones when cell

zones are present in the domain

5. Set up the velocity equation for individual phases

6. Solve for pressure using pressure

corrector pimple loop

7. Reconstruct phase velocities

Update substantive derivative

for individual phases

End

Start

KineticTheoryModels:

Library for kinetic theory models

createFields:

Define mixture properties

Figure 4.1: Flow chart of multiphaseEulerFoam solver

handled by a PIMPLE loop. Turbulence for the mixture is first corrected and up-dated. Then, the phase fraction for individual phases is solved and the phase fractionfor the current time step is updated. The phase continuity equation is solved usingthe old values of velocity. Density of the mixture which depends on phase fraction isthen updated. Next step in the loop involves calculating phase volumes of a cell zonewhen required. The velocity for individual phases is now set up which do not satisfythe continuity equation. Then pressure is solved by a PIMPLE loop and velocityis reconstructed. The reconstructed velocities now satisfy the continuity equation.After executing the PIMPLE loop, the substantive derivative of individual phases isupdated.

Table 4.1 and 4.2 lists the attributes and methods available in the solver to operateon component and mixture based parameters. The list defines the operating spaceand potential of the solver and gives a direction to extend the variables and functions

30


required for our flow case. The solver is already validated for a few cases which canbe found in the tutorials of OpenFOAM 2.1.1 and will not be discussed here. Adetailed veiw on the solution method of the solver will be discussed next.

Variable

Name

Description Access

function

Field Type Units

nu Kinematic Viscos-ity

nu() dimensionedScalar m2s−1

kappa Thermal Conduc-tivity

kappa() dimensionedScalar kgms−3K−1

Cp Heat capacity Cp() dimensionedScalar m2s−2K−1

rho Density rho() dimensionedScalar kgm−3

U Velocity U() volVectorField ms−−1phiAlpha Flux phiAlpha() surfaceScalarField m3s−1

alpha Volume fraction alpha() volScalarField 1

Substantive Derivative: DDt

= ∂∂t

+ ~υ · ∇All variables in the table are phase specific and always appears with the phase subscript. forexample, nuq,ρq.Viscosity of a phase q can be accessed by q.nu()

Table 4.1: Phase properties and functions to access them

Name Return type Function Actual term

phases() Pointer Dictio-nary

All the phases canbe iterated

rho() volScalarField Returns the mix-ture density

ρm =∑n

q=1 αqρq

nu() volScalarField Returns the mix-ture laminar viscos-ity

νm =∑n

q=1 αqnuq

Cvm(phase) volScalarField Returns virtualmass coefficient fora given phase

Cvmq =∑n

i=1 Cvm(q, i) ∗αi

Svm(phase) volVectorField Returns the virtualmass- source for thegiven phase

Svmq =∑n

i=1 Cvm(q, i)αiαq

Cvm(q,i) is user defined value obtained from the case directory.To access the properties, an instance of the class multiphaseSystem is created (mulit-phaseSystem fluid) and then accessed (fluid.rho()).Surface tension and drag model properties are not mentioned as it is out of the frameof this study.

Table 4.2: Member function in multiphaseSystem

4.2.1 Solution Method: Phase fraction

Phase fractions are solved from the phase continuity equation.The solution procedurebases itself on the two fluid model developed by Weller [7]. Assuming there is nomass transfer across phases, the equation is of the form[27]:

31


∂

∂t(αqρq) +∇ · αqρq~υq = 0 . (4.2)

Since each phase is incompressible, the operating equation can be rewritten as:

∂αq

∂t+∇ · αq~υq = 0 . (4.3)

Continuity is enforced by the phase fraction of each components and demands astrong boundedness property for phase fractions. In his work, Weller[7] introducedthe conservative form of the equation for a system of two fluids a and b:

∂αa

∂t+∇ · αa~υ +∇ · (~υrαa(1− αa)) = 0 , (4.4)

where ~υ = αa~υa +αb~υb and ~υr = ~υa − ~υb. This approach also couples the two phasesmore implicitly through the presence of relative velocity ~υr in the third term[26].The discritised form of the equation is:

[

∂[αa]

∂t

]

+[

∇ ·[

αaf(φ,S)

]

φ]

+[

∇ ·[

αaf(φra ,S)

]

φra

]

= 0 , (4.5)

where φra = αbf(−φr,S)φr and φr = φa − φb.

The general approach to solve the phase fractions of a multiphase system in mul-

tiphaseEulerFoam is similar but there are some differences in handling the equa-tions, or it can be said that it is “evolving”. MULES algorithm handles the bounded-ness property by first limiting the flux transport and then solves for phase fraction.The limiting of flux transport is important since large transport of fluxes from a cellmay drive the volume fraction in a particular cell below zero during one time step.Hence the flux is at first limited and then is passed on to the MULES solver to solvefor phase fraction. The constructor of MULES function is given below:

MULES::explicitSolve(const RhoType& rho,volScalarField& psi,

const surfaceScalarField& phiPsi,const SpType& Sp,const SuType& Su)

It is seen that density has to be passed as an argument to the constructor. In caseof incompressible phases, it is passed as geometricOneField() which is a unit valuefield. The second argument psi is the variable to be solved which is phase fractionin our case. The limited normal convective flux is the next argument which is earlier

32


solved explicitly by MULES:limiter. The next two terms are the explicit and implicitsource terms in the continuity equation which arise when mass transfer across thephases or reaction source terms exist. In our case we pass both the arguments aszeroFields() since there is no mass transfer. MULES algorithm solves for phasefraction with explicit consideration of convective flux of phase fraction. It must alsobe mentioned that, the transport mechanism considered is convective-only.

Consider equation 4.3. The flux of phase fraction is written as:

φq = αq~υq . (4.6)

Equation 4.3 in numerical form is written as:

αqnew − αq

old

δt+∇ · φq = 0 , (4.7)

αqnew = αq

old − (∇ · φq)δt . (4.8)

But in equation 4.8, the right hand side has two different types of fields and thesurfaceScalarField of flux has to be converted to an equivalent volScalarField

in order to perform algebraic operations.

(αqUq)in (αqUq)out(αq)

old

E

Figure 4.2: Schematic to demonstrate MULES convective-only transport solution

Figure 4.2 shows a cell volume in 2 dimensions. Let us consider the flow dominantonly in one direction. At the new time step, we need to know the effective amountof αq, phase fraction added or removed from the cell. If the mechanism of transportover the cell faces is only convective, then the quantity of αq at each of the cell facesgoing out or coming in from the neighbouring cell is αq~υq. The normal convectivefluxes are already known from the MULES:limiter function. Hence the divergenceof flux is known, which means that the effective amount of αq added or removedis known. Applying the Gauss divergence theorem and from [11], the divergenceoperator can be transformed into a sum of all faces by:

33


∫

Vp

∇ · αqdV =∑

f

(∫

f

dS · αq

)

, (4.9)

(∇ · αq)Vp =∑

f

S · αqf . (4.10)

The theorem can be used to convert a surfaceScalarField back to a volScalarField.This means that by knowing the convective fluxes, precisely limited convective fluxes,we can find αq added for the new time. Thus phase fraction is updated for the currenttime using equation 4.8

solveAlphas()

Calculate the flux of the

mixture : Φc

q=0

for all pha

Initialize

phiAlpaq=0

Initialize phiAlphaCorr phiAlpaq)

for all pha

Calculate relative flux be

phir=phiq-phip

Update phiAlphaCorr

phiAlphaCorr= phirapaq

Limit the flux tran e

bounde MULES:limit()

Initialize =0

for all pha

Initialize limited bounded flux

phiAlpha=phiAlphaCorr

Interpola

Solve for

MULES:explicitSolve()

phiAlphaq += phiAlpha

+= q

End

Figure 4.3: Procedure to solve phase volume fraction

The solution procedure is shown in figure 4.3 In the first step, relative fluxes of aphase with respect to all other phases are found out. Then fluxes for individual

34


phases are limited to ensure boundedness. Next step involves passing the argumentsof limited fluxes(MULES:limit) formerly found out and phase to solve the phasefraction for the current time step using MULES:explicitSolve(). Th interpolation offluxes from centre to face is done by Upwind Differencing(UD). Using UD might bequite diffusive, however using a higher order scheme instead reduces the numericaldiffusion , but might compromise the boundedness of the solution[26]. Finally, allthe phase fractions are solved for the current time step and sum of all the phases iscalculated.

alpha correction

for all pha

Set old alpha value αqold

Initialize q=0

while na

Solve αq and q

for all pha

= q

end

nalpha

)

Figure 4.4: Solution procedure of phase continuity equation

Solution for phase fraction is invoked by fluidSolve() function. The schematic of thefunction operation is shown in figure 4.4. Correctors for phase fraction is read fromthe pimple directory in the case file. If the number of correctors are greater than one,then phase fraction at old time is stored and fluxes for all phases are set to zero.Non-orthogonality for phase fractions is solved by the function solveAlphas()4.3.Then the total contribution of fluxes from individual phases is calculated. If the

35


number of non orthogonal correctors is zero, then solve for phase fraction and totalflux by calling solveAlphas().

4.2.2 Solution Method: Pressure-Velocity coupling

Projection methods are used to handle the pressure velocity coupling. The methodinvolves setting up an equation for velocity by projecting out pressure related terms.At this stage, the velocities do not satisfy the continuity equation. Equation forpressure is then set up using the continuity equation and then solved. Since pressureis now known, the contribution of pressure term is removed from velocity to find thecorrect velocities which satisfy the continuity equation.

This method works for an arrangement where pressure is stored at cell centres andvelocity at cell faces; in case of staggered grids. But since OpenFOAM uses collo-cated grid arrangement where both pressure and velocities are stored at cell cen-tres, appropriate changes have to be made for the procedure to work. In his work,Weller[8] reformulated the solution procedure for collocated grid arrangement wherevelocities are reconstructed from flux fields. The approach of momentum predictionand pressure correction transforms to flux prediction and pressure correction. It isbecause flux, which is stored at the cell face now becomes the primary variable inthe solution procedure which represents velocity. “The cell centred velocity is merelyregarded as a secondary variable which is used in the construction of momentumequation” [26]. The solution procedure of Weller[8] is as follows:

Consider the semi-discritized momentum equation:

AUU = b − ∇p , (4.11)

where b is a vector of all source terms except pressure gradient. AU is the coefficientmatrix of veclocity vector. If it is written as sum of diagonal and off diagonalelements as,

AU = AD + AN . (4.12)

Considering the off diagonal elements explicitly, the discritized equation is of theform,

ADU = AH −∇p , (4.13)

where AH = b − ANU . The velocity equation is:

U =AH

AD

−∇p

AD

. (4.14)

The velocity at the face is then:

36


Uf =

(

AH

AD

)

f

−

(

∇p

AD

)

f

. (4.15)

Flux is obtained by multiplying velocity with the area of cell face.

Hence velocity at the face is obtained by:

φ = |S|Uf . (4.16)

Equation for flux is then set up in agreement with equation 4.15 as:

φf = φf∗ −

(

∇p

AD

)

f

. (4.17)

Applying the continuity equation on equation 4.17:

∇ ·

[

(

AH

AD

)

f

−

(

∇p

AD

)

f

]

= 0 . (4.18)

The final poisson equation for pressure is:

(

∇2p

AD

)

f

= ∇ ·

(

AH

AD

)

f

. (4.19)

Since the pressure is solved at this stage, flux is corrected using equation 4.17 andvelocities are reconstructed using equations 4.16 and 4.15

4.2.3 Semi-implicit treatment of Drag term

Drag force by and large is a function of relative velocity. If ~υp and ~υq are velocitiesof phases p and q, drag force is written as:

FDrag = C(~υp − ~υq) . (4.20)

The above expression can be treated in three ways; explicit, semi-implicit and fullyimplicit. In multiphaseEulerFoam ,it is treated as semi- implicit source term. Ifn phases are interacting, the drag term for phase q is:

FDrag,q =n

∑

p=1

Cpq(~υp − ~υq) . (4.21)

37


Cpq is the known coefficient of the drag term between two interacting phases. Theabove equation is rewritten as:

FDrag,q =n

∑

p=1

Cpq(~υp)−n

∑

p=1

Cpq(~υq) , (4.22)

FDrag,q = (FDrag,q)explicit + (FDrag,q)implicit . (4.23)

Hence while setting up the velocity equation 4.15, the terms are split into two asshown above into fully explicit and fully implicit. The semi-implicit treatment of thewhole drag term, does not affect the diagonal dominance of the coefficient matrixbut in fact enhances it.

4.3 Modified solver: multicomponentPorousFoam

Source terms required to model multicomponent flows through porous media werediscussed and derived in chapter 3. Maxwell-Stefan relation is used for modellingmomentum transfer between different components. An additional source term tomodel porous drag has been added. Volume-averaged energy equation needs to beimplemented in the solution procedure. Modifications and extensions to the existingsolver needs to be done.

The following are the equations of continuity and momentum in the existing mul-

tiphaseEulerFoam :

∂αq

∂t+∇ · (αq~υq) = 0 , (4.24)

∂αq~υq∂t

+∇ · (αq~υq~υq) = −αq∇p+∇ ·T+ αqρq~g .

The energy transport equation was developed in section 3.3 and is rewritten here:

∂

∂t[ǫ(ρmCp,m) + (1− ǫ)(ρsCs)]T +∇ ·

[

n∑

q=1

(ρqCp,qT~υq)

]

= ∇ · [ǫkm + (1− ǫ)ks]∇T + ǫ∂p

∂t. (4.25)

Modifications and extension to the momentum equation along with the implemen-tation of the complete energy equation is done in the new solver multicomponent-

PorousFoam .

38


4.3.1 Source term: Maxwell-Stefan drag

The terms added to the momentum equation for modelling drag is derived in section3.1.2 and is rewritten here:

Fdrag = −p∇αq + pn

∑

p=1

αqαp(υp − υq)

Dqp

= Sp1 + Sp2 . (4.26)

Both the source terms Sp1 and Sp2 are not present in the existing equations 4.25of the solver. Hence they are added as source terms to the momentum equations.Equation 4.26 is added to the momentum equation as:

− p ∗ fvc :: grad(αq) + fluid.maxwellStefan(phase,p,Tfluid) . (4.27)

Both terms are considered explicitly in this work though the second term can beeasily extended to a semi-implicit treatment as shown in section 4.2.3. As seen in4.27, the calculation of drag force is done by the function maxwellStefan.() which isa method of the multiphaseSystem and can be accessed by instances of the multi-phaseSystem class (fluid.multiphaseSystem).

The constructor takes three arguments which is the phase name, pressure and tem-perature. The execution of the function is enumerated below for phase q.

1. Read phase name, pressure and temperature.

2. Set FDrag,q = 0

3. Loop over all other phases p = 1 : n excluding q

• Calculate temperature and pressure dependant diffusion coefficientfor two interacting phases Dpq from equation 3.47.

• Calculate Relative velocity between p and q

• Calculate drag for two interacting phases F(Drag,pq)

• Calculate total drag for phase q , F(Drag,q) = F(Drag,q) + F(Drag,pq),Sp2 in equation 4.26

4. Return total drag F(Drag,q)

Table 4.3: Computing inter-component drag force by maxwellStefan() function

The source terms added creates an explicit dependance of velocity on total pressureof the system. It is not a cause for concern since the drag term in our case islinearly related to relative velocity unlike the usual non-linear dependance. Thesolution divergence is controlled using lesser value for time steps to avoid instabilities.

39


Lessening the time steps may act as an under-relaxation factor for the iterativemethod. The presence of phase fraction ∇αq poses another restriction. Phase-intensive formulation of momentum equations suggested by Weller[7] is not used andhence cases involving sudden reductions of phase fractions to zero must be avoidedwhich may drive the momentum equation to singularity or cause the presence ofexceptionally large numbers due to high gradients of phase fractions resulting infloating point exceptions.

4.3.2 Source term: Porous drag

Momentum source term due to porous media derived in section 3.2 is rewritten here,

Fq,por = −ǫ∇p− gpτ2

[

Dqk

pq+

K

µ

]−1

~υq,por = Sp3 + Sp4 . (4.28)

Before proceeding further, three fields are introduced in multicomponentPorous-

Foam ; porosity(ǫ), tortuosity(τ) and porousPlug(gp). The three fields facilitateswitching on the required terms when porous media is used in the solver and turningoff porous drag effects when an open solid is used. All three fields are dimensionlessand must be present in the starting conditions folder in the operating case file. Thefields mentioned are set by using setFields application in the flow domain similar tothe setting up of phase fractions and must adhere to the following conditions,

porosity =

Vf

(Vf+Vs)in porous region,

1 in non-porous region,(4.29)

tortuosity =

tortuosity in porous region,1 in non-porous region,

(4.30)

porousPlug =

1 in porous region,0 in non-porous region,

(4.31)

In order to introduce porosity effects in the equations, the momentum equation 4.25is multiplied by porosity (ǫ) along with source terms Sp1 and Sp2 added due to drag.To accommodate path elongations in porous media, the source term Sp2 is multipliedby tortuosity(τ). The porous drag term Sp4 is added to the momentum equation.The generic momentum equation that accomodates both porous and non-porouseffects is then:

ǫ(∂αqρq~υq

∂t+∇ · (αqρq~υq~υq) = −αq∇p+∇ · T + αqρq~g + (4.32)

Sp1 + τSp2 + gpSp4

ǫ)

40


All terms are implemented into equation 4.33 and can represent the desired effects.It can be verified that, in case of a non-porous region, according to 4.29 , 4.31 and4.31, the equation represents a flow in non-porous region and the multiplicity of ǫ isalso marginalized as it becomes 1.

Sp4 is modelled implicitly in the solver as:

−(gpǫ

)

∗ fvm :: Sp(pow(tortuosity, 2) ∗ phase.porousDragMultiplier(), Uq) . (4.33)

porousDragMultiplier() is a function that returns the coefficient of porous drag forceof the phase. Coefficient or multiplier is the drag force term without the velocity ofphase since the porous drag force term is considered implicitly, velocity is not knownand has to be calculated for the current time. It is programmed as a phase propertysince it does not depend on many fluid properties which was in case of maxwell-Stefan() function. It is invoked as shown in 4.33 when setting up the momentumequation for all the phases q. The porous drag contribution of all the phases are cal-culated after solving the energy equation since the knudsen diffusion coefficient (Dq

k)is temperature dependant. The calculations are done in the file porousSource.H andstored for all the phases at current time. The computing procedure is shown in table4.4.

1. Begin porousSource.H

2. Loop over all phases q.

• Calculate knudsen diffusion coefficient Dqk from 3.24

3. Calculate permeability K from 3.26

4. Loop over all phases q

• Calculate the porous drag force multiplier Fmq,porfrom 3.31 without

considering ~υq

• Store Fmq,poras q.porousDragMultiplier()

5. Exit porousSource.H

Table 4.4: Computing porous drag multiplier by porousSource.H

Thus the terms required for a generic momemtum equation which can switch rolesto represent a porous or a non-porous zone is implemented into multicomponent-

PorousFoam .

4.3.3 Energy transport

The implementation of equation 3.45 is done in Tfluid.H file and is as follows:

41


( fvm::ddt(Coefficient1,Tfluid)+fvm::div(Coefficient2,Tfluid) (4.34)

== fvm::laplacian(Coefficient3,Tfluid)+porosity*fvc::ddt(p));

The variables and functions needed by the above equation 4.35 is listed in table 4.5.

Variable Field<type> Calculation form Invoking func-

tion

Cp volScalarField∑n

q=1 αqCq fluid.Cp()

kappa volScalarField∑n

k=1 αqkq fluid.kappa()Coefficient1 volScalarField ρmCp+(1−ǫ)(ρpor−Cpor) —Coefficient2 volScalarField ǫk + (1− ǫ)kpor fluid.Coefficient2()Coefficient3 surfaceScalarField

∑nq=1 ρqφqCq —

k : kappa, Thermal conductivity ρm: Density of the mixturesubscript por: porous material

Table 4.5: Variables and functions required for energy transport

The algorithm to solve for temperature of the mixture is shown in table 4.6 and isas follows,

1. Calculate thermal conductivity kappa for the mixture

2. Calculate specific heat capacity of the mixture

3. Calculate Coefficient1

4. Calculate Coefficient2 by invoking the function Coefficient2()

5. Calculate Coefficient3

6. Set up the temperature transport equation for the mixture 4.35

7. Relax the temperature equation

8. Solve the temperature equation

9. Update viscosity of individual components 3.46

Table 4.6: Solution procedure for temperature of the mixture

Only transport of temperature is solved in multicomponentPorousFoam . Thedensities of individual phases can be updated after solving the temperature equationbut is out of the scope of this work.

4.3.4 Solution algorithm

The solution process of multicomponentPorousFoam is listed in table 4.7. Solv-ing the temperature of the mixture results in updating temperature dependant prop-erties in the flow. A condition at the start of setting up a case is that the porous

42


related fields have to be initialized in order to ditinguish between porous and non-porous zones which initiates appropriate consideration of source terms in the mo-mentum equation as discussed earlier. The remaining steps involved can be seen inthe solution algorithm.

a Set fields for phase fractions, porosity, porousPlug and tortuosity

b runtime++

1 Start Pimple loop

2 Update mixtue turbulence

3 Solve for phase fractions using fluid.solve() 4.4

4 Update mixture density

5 Solve for mixture temperature 4.25

– Update viscosity of individual components 3.46

– Update Knudsen diffusion coefficient of all components 3.24

– Update diffusion coefficients of two interacting components 3.47

6 Calculate porous drag multipliers 4.4

7 Set momentum equations for all components 4.33

8 Pressure corrector loop

– Solve for pressure

– Correct flux

– Reconstruct velocities from corrected flux

9 Update Substantive derivative for all components

10 End Pimple loop

c End time loop

Table 4.7: multicomponentPorousFoam algorithm

4.4 Closure

A brief introduction and the essence of numerical simulation to solve fluid flowproblems were discussed. Reasons for working with OpenFOAM were then stated.

Characteristics of the existing solver multiphaseEulerFoam which solves multi-phase flows by the Eulerian approach were then analyzed. The solution method usedby the solver to solve phase fractions were then discussed and explained with appro-priate schematics and equations. The MULES algorithm used to limit and solve aconvective only transport system of a phase was also explained. Further, handling ofpressure-velocity coupling by flux prediction and pressure correction was discussedand demonstrated with the help of discritized equations. The semi-implicit treat-ment of drag force by the solver was explained.

43


Development of multicomponentPorousFoam, an extended and modified solverfor flow in porous media was then implemented. The modifications to the momen-tum equation and development of a generic equation for the same which representsboth porous and non-porous zones with the help of certain fields; porosity, porous-Plug and tortuosity is briefly explained and implemented. Volume-averaged energyequation to solve for temperature transport of the mixture is developed which isalso generic and accommodates porous and non-porous zones. Solution Algorithmof multicomponentPorousFoam was then enumerated 4.7 .

44

5. Validation

5. Validation

5.1 Case1: Loschmidt tube

Validation of our solver for diffusion dominated flows involving mass transfer isundertaken in this section. Such flow scenarios are quite often validated by usingdiffusion tubes [27, 25, 23]. We use the Loschmidt tube for validation.

Duncan and Toor conducted a few experiments to examine diffusion in an idealternary gas mixture containing Hydrogen, Nitrogen and Carbon-di-oxide [24]. Itbasically consists of two bulbs connected by a capillary as shown in figure 5.1. Amongthe three gases, one gas is made stationary and two gases are set in motion. Thisis achieved by adding almost equal quantities of one gas in both the bulbs (q2)resulting in a very small or in fact a negligible gradient of the gas rendering italmost stationary. The remaining two gases are added one in each bulb resulting inlarge gradients thus driving the two gases in motion when the stop cock is removedin order to achieve homogeniety.In our simulation, Loschmidt tube with Methane(CH4), Argon (Ar) and Hydrogen (H2) is simulated.

q1

q2

q3+

q1+

q2

q3-

Bulb 1 Bulb 2

Figure 5.1: Experimetal set up of loschmidt tube

In such a scenario, the gas with a small gradient (q2) behaves in a certain way. Toorpredicted the behaviour and named them; Osmotic diffusion, Reverse diffusion andDiffusion barrier [10]. It is important to elaborate these terms which occur in aternary mixture.

45

5. Validation

1. Osmotic diffusion: At time t = a , though the gas has negligible driving force,it moves.

2. Reverse Diffusion: During a time interval a < t < b, the gas moves in adirection opposite to its existing trifle gradient thus defying Fick’s law whichstates that, gases diffuse in a direction normal to their concentration gradients.

3. Diffusion barrier: Between b < t < c, the gas does not diffuse even though alarge gradient exists. The diffusion flux is zero despite a large driving force[24]

The experimental results have been compared with the Maxwell-Stefan equations byDuncan and Toor [12] and is in good agreement. Hence the analytical solution ofthe Maxwell-Stefan equation is obtained by assuming a 1−D flow and verified withthe results generated by the solver.

5.1.1 Case Setup

In order to simulate the Loschmidt tube, the whole process has to be abstracted toa simple model. The first abstraction done is the consideration of only the capillaryand eliminating the two bulbs. Since the whole system is closed and entites ofinterest are the phase fractions, it is justified. The geometry used for simulation isshown in figure 5.2

Wall

Left tube

- -

Figure 5.2: Simulation model of loschmidt tube

The length of the capillary is reduced to 100µm. High gradients prevail initially atthe division of left tube and right tube. In order to resolve them, edge lengths of10−6 are required [27]. To save computation time, the length of the whole capillaryis reduced. The experiment is conducted at atmospheric pressure of 101.3kPa andunder isothermal conditions. Composition of the three gases in both left and righttube is given in table 5.1

It is observed that Ar behaves as gas q2 explained earlier and must undergo the statedosmotic diffusion, reverse diffusion and diffusion barrier effects. All the boundarypatches are considered as walls ensuring a closed surface. The internal field is setto 101.3kPa. The pressure at the walls are set to zero gradient condition. Zerogradient means that there is no change of a quantity in normal direction. Velocity

46

5. Validation

Component Left tube Right tube

CH4 0.295 0.405H2 0.4 0.3Ar 0.305 0.295

Table 5.1: Initial composition

at the wall is set to slip condition for all components. The slip condition ensuresthat the tangential component of velocity is zero gradient as explained earlier andthe normal velocity component near the wall is zero [5].

An analytical solution to the problem assuming a 1−D transient problem is devel-oped in [25] (pages 110 − 115). A sci-lab program written based on the solutionmethod which solves the phase fraction of a three component mixture from its ini-tial composition for any time t in the left tube can be found in appendix A.1. Agraph is generated for each component using the program to plot the variation of itsphase fraction with time.

5.1.2 Results

0.3

0.301

0.302

0.303

0.304

0.305

0.306

0.307

0.308

0.309

0.31

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014

calculated alphaAranalytical alphaAr

b c

a

Figure 5.3: Phase fraction ofArgon in left tube: Analytical vs. Calculateda− b : Reverse diffusion. b− c :Diffusion barrier. > c :Normal diffusion

Figure 5.3 shows the behaviour of Argon. From time a until time b, reverse diffusiontakes place. Though the left tube has a higher concentration, there is an incomingflux to the left tube(0.305) from right tube(0.295) and hence an increase and decreaseof phase fraction on left and right respectively. Between time b and c, a diffusionbarrier exists resulting in stagnation of flux even though a considerable gradientexists. After c, Argon diffuses normally. This assures the physical reality of solution.The other two component gases begin to start homogenizing following the Fick’s lawof diffusion and diffuse normally, down their concentration gradients. The animationof the intial stage of the other all the gases can be seen in appendix A.2

47

5. Validation

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014

calculated alphaCH4analytical alphaCH4

Figure 5.4: Phase fraction of CH4 in left tube : Analytical vs. Calculated

Methane behaves normally and obeys Fick’s law of diffusion. The initial compositionof Methane in the left tube was comparatively lower. Hence phase fraction graduallyincreases until homogenization as shown in figure 5.4.

0.35

0.355

0.36

0.365

0.37

0.375

0.38

0.385

0.39

0.395

0.4

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014

calculated alphaH2analytical alphaH2

Figure 5.5: Phase fraction of H2 in left tube: Analytical vs. Calculated

There is not much of a surprise in case of Hydrogen too. It diffuses normally downits gradient. The initial composition of Hydrogen being high in left tube in compar-ison to right tube reduces in order to effect the homogenization process as seen infigure 5.5. The end time shown in the graph is not the time required to completelyhomogenize. There is no further change in the behaviour till the homogenization ofboth tubes.

48

5. Validation

From figures 5.3, 5.4 and 5.5, it is observed that analytical results and the result ofphase fraction of all three components calculated from the solver is in good agree-ment. Thus the solver is validated for diffusion dominated flows involving masstransfer.

5.2 Case2: Flow through a channel

Validation of volume averaged energy transport and porous drag source terms isbased on calculation of heat transfer coefficients in thermally fully developed laminarflows. The dimensionless Nusselt number is finally calculated from the velocitytemperature profile when fully developed. There is no influence of mass flow rate,gas properties and absolute temperature on Nusselt number when the temperatureprofile is fully developed. It then depends exclusively on the flow profile [31]. Henceit serves as a good criteria for validation since, in case of the energy equation andporous media. Nusselt number for a flow in a rectangular duct with infinite lengthin z-direction and with a height H is given by,

H

∂(Tfluid)

∂n

Tfluid v

Twx

y

Figure 5.6: Calculation of Nusselt number

Nu =2H

Tw − T

(

∂T

∂n

)

w

(5.1)

T is the mean temperature over the cross section of the duct. Based on the velocityprofile, it can be determined by [6],

T =

∫ H

0(υT )dy

∫ H

0υdy

(5.2)

Figure 5.6 is a schematic depicting the determination of the required values of tem-perature and velocity of the fluid at a fixed location on the x-axis. The variation ofthe values along the y-direction and the normal gradient of temperature at the wallmust be considered as shown in the figure. Thus the Nusselt number is calculatedusing equation 5.2 and equation 5.1.

49

5. Validation

5.2.1 Case Setup

A duct which is partially cooled by walls as shown in figure 5.7 is used for validation.Two zones 1 and 2 exist in the geometry. Zone 1 is surrounded by adiabatic walls andzone 2 by cooling walls. A homogeneous mixture of Oxygen and Nitrogen is then fedinto the channel. The component fractions of Oxygen and Nitrogen are maintainedat 0.21 and 0.79 respectively. The mixture enters through the inlet at 373.15K. Thewalls in zone 2 are set to 323.15K. The starting field of the two components are alsoset homogeneous.

0.04m

0.12m

Adiabatic Walls Tw=323.15

Tfluid=373.15K

1 2

Figure 5.7: Case set up for a channel flow

The mass flow specified at inlet is 4× 10−3kgs−1 to generate a laminar flow profile.It results in a parabolic flow profile at the outlet. The velocities of individual com-ponents are then calculated proportional to their phase fraction and set in the casedirectory.

Two variations for zone 2; non-porous and porous zones is set. For the non-porouscase, the porosity is 1. In the second case, zone 2 is filled by a porous region with aporosity of 0.9. Further, the density of porous media is taken as 5900kgm−3, specificheat capacity of 500 Jkg−1K. Tortuosity factor of the porous media is 1.1. Diameterof pores is assumed to be 100µm.

A symmetry plane is used for the case specified making the top plane as the planeof symmetry. When a steady state solution is sought, a solution symmetric to theplane of symmetry exists [14]. Since the fully developed temperature profile is thesolution that we seek, the use of symmetry is justified.

Dirichlet boundary condition [5] is used for pressure at the outlet and is 0. Velocityof both components are given at the inlet from the known flux. A no-slip conditionfor velocity is used at the walls.

For temperature of the mixture (Tfluid ), Neumann boundary condition [5] withgradient as zero is set at outlet. At inlet, a temperature of 373.13K is fixed andfor the walls in zone 2, 323.15K is prescribed. For adiabatic walls in zone 1, a zerogradient boundary condition is used.

In order to find the Nusselt number from equation 5.1, the velocity and tempertureprofile in y-direction is probed at a location of x= 0.1m. A program in Sci-lab isused (appendix B) to calculate the Nusselt number using the equations 5.2 and 5.1.A comparison of the Nusselt number generated by the program and the literaturevalues of Nusselt number for the two kinds of profiles generated gives the proximityof calculations by the solver to accurate values.

50

5. Validation

5.2.2 Validation of temperature transport

Velocity profile developed when zone 2 is non porous is shown in figure 5.8. The flowis fully developed and is as expected. The temperature profile since it is transportedby flux develops a similar profile of velocity. There is no loss of heat in the adiabaticzone 1 and a boundary layer is developed in zone 2 where a temperature gradientexists.

Figure 5.8: Velocity profile of a non porous zone 2

Variation of temperature when zone 2 is non porous is shown in figure 5.9.

Variation of velocity and temperature along y-axis at x= 0.1m is shown in figure5.10. The average velocity developed is 0.16714 ms−1. In order to calculate thegradient normal to the wall, a point on the y−axis has to be selected. Since thegradient of temperature normal to wall is required, the value of temperature samplednext to the wall at a y coordinate 4.20168 × 10−5m is chosen. The temperature atthis point is 323.661. The value of temperature gradient normal to the wall is thus12161.802Km−1. The mean temperature is 350.633 K. Thus the calculated Nusseltnumber is 8.85. The literature value of Nusselt number for a parabolic flow profileis 7.54 [30]. A relative error is observed between the values and the reasons arediscussed subsequently.

5.2.3 Validation of porosity effects

The velocity profile for a porous zone is shown in figure 5.11. There is a marginalincrease in maximum velocity.

Temperature contour is shown in figure 5.12. It is observed that there is a changein the contour developed due to the presence of porous media. As the temperature

51

5. Validation

Figure 5.9: Temperature profile of a non porous zone 2

320

325

330

335

340

345

350

355

360

0 0.001 0.002 0.003 0.004 0.005

Tfluid

0

0.05

0.1

0.15

0.2

0.25

0 0.001 0.002 0.003 0.004 0.005

U

Figure 5.10: Temperature and velocity variation (non porous zone 2) at x= 0.1m

gradient reduces with positive x-direction, the lesser it diffuses thus generating thecontour shown.

Variation of velocity and temperature along y-axis at x= 0.1m is shown in 5.10. Thereduced values of temperature can be observed. The temperature profile follows thevelocity profile. The average velocity developed in case of a porous zone 2 is 0.16718ms−1. Temperature is sampled at the same point considered for the previous case.The value of temperature here is 323.223K. This is because of the block profile caseof a porous zone with a porosity value of 0.9. The value of temperature gradientnormal to the wall is 1737.4003Km−1. The mean temperature is 326.95619 K. Nusseltnumber for this case is 9.13. The literature value of Nusselt number for a block flowprofile is 9.80 [31].

52

5. Validation

Figure 5.11: Velocity profile for a non porous zone 2

Figure 5.12: Temperature profile for a porous zone 2

323

323.5

324

324.5

325

325.5

326

326.5

327

327.5

328

0 0.001 0.002 0.003 0.004 0.005

Tfluid

0

0.05

0.1

0.15

0.2

0.25

0 0.001 0.002 0.003 0.004 0.005

U

Figure 5.13: Temperature and velocity variation (porous zone 2) at x= 0.1m

5.3 Closure

The Loschmidt tube considered as the validation case for Maxwell-Stefan sourceterms was set up and explained. Abstraction of experimental set up for CFD simu-

53

5. Validation

lations were also shown. The behaviour patterns of a ternary mixture was explained.Since it is a closed geometry, the pressure value used here was 101.325KPa. Thelarge value of pressure significantly reduced the time step to obtain a stable simula-tion. The time step observed for the simulation was 10−10 seconds. The analyticalsolution developed as a Scilab program for a transient 1 dimensional case was usedto verify the results. The phenomenon observed for comparison was the variationof phase fraction of all three gases in the left tube of the capillary. By observingthe variation of phase fraction, it is indeed equivalent to the observation of variationof flux of each phase. The simulation results are in very good agreement with theanalytical results over the complete observation period. Additionally, the variouskinds of diffusion behavior were also captured in the simulation.

The second case of fully developed flow over parallel plates was used for the validationof temperature transport and porous drag effects. The case setup involves a suddencooling of a mixture entering into a region cooled by the walls. Nusselt number wasused as the validation criteria and compared with values available in the literature.In order to validate both the stated effects, the cooling zone was considered non-porous for the first case and as porous zone with a porosity of 0.9 for the second.

For non-porous region, the Nusselt number calculated was 8.85 compared to theliterature value of 7.54. For the porous case, the Nusselt number calculated was 9.13compared to the literature value of 9.8. Since the solver deals with incompressiblephases, instabilities were observed when total pressure was used due to the presenceof large numbers which may increase cumulative errors. Hence only the relativepressure or pressure fluctuation values were used for this case. The terms related byideal gas law were kept intact by adding a value 105 Pa to the pressure terms. Thismay be the cause for the observed deviation. The temperature profile developed forthe two cases are realistic and rational.

54

6. Conclusions

6. Conclusions

Scope of this work and its implications have been extensively discussed in the pre-vious chapters. A few important conclusion and further potential improvements arestated here:

• A circumspective attitude is generally taken towards the Eulerian approach tosolve for transport of multicomponent gases. The reason being the complexityand computation costs expended for the approach. For flow scenarios whichrequire slip velocities of individual components, the mixture approach whichis the most chosen alternative to Eulerian approach cannot be used since itcircumvents solving individual component velocities. Hence the Eulerian ap-proach is chosen and models the mulitcomponent flow scenario accurately.

• Maxwell stefan relations models diffusion dominated problems involving masstransfer. In multicomponent flows, inter-component and not intra-componentmomentum tranfer takes place which is established in section 3.1.2. The phe-nomena of osmotic diffusion, reverse diffusion and diffusion barrier is accuratelymodelled by Maxwell-Stefan relations which the Fick’s law fails to predict.

• The intercomponent momentum transfer implemented to the modified solvermulticomponentporousFoam simulates ternary diffusion in a Loschmidt tubeaccurately. The results are in perfect agreement with the analytical solutionfor the case solved as transient one dimensional flow. The graph of variationof phase fractions can be seen in figures 5.4 5.5 and 5.3.

• Phase intensive momentum equations are not used in the solver. Hence flowsinvolving sudden changes of phase fraction of a component must be avoided.The presence of the gradient term of phase fraction is responsible for thisshort coming which cause floating point exceptions when it encounters largegradients. The use of upwind interpolation scheme to solve for phase fractionsrestricts the use of large time steps. It is not a major problem for transientflows due to the presence of a small time step. When a steady state solutionis sought, the demand for a small time step may be a drawback.

55

6. Conclusions

• Volume averaged equations are implemented in the solver. The presence ofporososity, porousPlug and tortuosity fileds provides a generic operating formfor the solver. The porous drag terms and the effect of porosity can be switchedon in a particular zone by setting the mentioned fields appropriately. Currently,isotropic and non deformable porous media have been modelled. Extensionsto orthotropic media can be done by adding appropriate tensorial terms in thesolver. The equations are inversed for this purpose. The results generated bythe solver is in close agreement.

• Transport of temperature of the mixture assuming local thermodynamic equi-librium between components accurately predicts the transfer of temperaturein the pseudo homogeneous media. The difference between the contour of tem-perature in a non porous domain and porous domains can be seen in figures5.9 and 5.12.

• The deviation in Nusselt number is attributed to the use of only the relativepressure values for simulation. But it does not largely affect the simulationsince all the terms which are related by ideal gas properties are properly ac-counted by adding a value of 105Pa to the relative pressure solved for incom-pressible flows. The calculation of gradients at the wall in order calculateNusselt number is also speculated to be the cause for the error. The probelocation for calculation of gradient at a wall can vary continuously in the celladjacent to the wall. The temperature also varies accordingly resulting inmany possibilities to calculate temperature gradient at the wall.And even asmall variation in its value affects the final solution quite largely.

• Temperature dependance of viscosity and diffusion coefficients have been mod-elled in the solver. Density is considered as a scalar in the solver. This restrictsits dependance on temperature. It is also predicted to be a source of error inthe final solution. A large dependance of variables exist on density in thesolver. Hence to convert it into a scalarField may prove detrimental since theaspect of considering individual components as incompressible may be violatedand cause mass conservation problems.

• Source terms which consists of pressure terms are modelled explicitly. Pro-jecting pressure dependant source terms out of the momentum equation andsolving for them in the poisson equation which is set for pressure improvesthe accuracy and may even reduce the small time step requirement. But thepresence of velocity terms simultaneously poses a problem in doing so sincepressure-velocity is solved using a segregated approach.Thus, using a coupledsolution procedure would be suitable for handling the source terms which con-tain pressure and velocity terms. The article [16] which implements a coupledpressure based solution using a block matrix structure may be taken as a cuefor establishing a coupled solution procedure. Further, Maxwell Stefan termsare added explicitly in the solver. A semi-implicit treatment of the terms mayimprove accuracy in flows involving large pressure gradients.

• Reaction terms can be easily added to the phase continuity equation in thesolver. The MULES algorithm which solves the equation contains explicit andimplicit source terms as its arguments. Thus reaction can be modelled bypassing the source term field to the function.

56

A. Loschmidt tube

A. Loschmidt tube

A.1 Scilab Program to calculate 1-D analytical

solution

f unc t i on t e r na r yD i f f u s i on ( )

x1m = input ( ”Enter x1 in l e f t h a l f ”) ; //x1 : Methanex1p = input ( ”Enter x1 in r i g h t h a l f ”) ;x2m = input ( ”Enter x2 in l e f t h a l f ”) ; //x2 : Argonx2p = input ( ”Enter x2 in r i g h t h a l f ”) ;x3m = input ( ”Enter x3 in l e f t h a l f ”) ; //x3 : Hydrogenx3p = input ( ”Enter x3 in r i g h t h a l f ”) ;l = input ( ”Enter the l ength o f the d i f f u s i o n tube ”) ;d12 = input ( ”D i f f u s i o n 1−2 ”) ;d13 = input ( ”D i f f u s i o n 1−3 ”) ;d23 = input ( ”D i f f u s i o n 2−3 ”) ;time = input ( ” I n t e r e s t e d probe time ”) ;deltaT=input ( ” de l t a t ”) ;

x1=(x1m+x1p ) /2 ;x2=(x2m+x2p ) /2 ;x3=(x3m+x3p ) /2 ;

S=x1∗d23+x2∗d13+x3∗d12 ;

D11=d13 ∗ ( ( x1∗d23+(1−x1 )∗d12 ) )/ S ;D12=(x1∗d23 ∗( d13−d12 ) )/ S ;D21=(x2∗d13 ∗( d23−d12 ) )/ S ;D22=(d23 ∗( x2∗d13+(1−x2 )∗d12 ) )/ S ;

D=[D11 D12 ,D21 D22 ] ;data = [ ] ;

57

A. Loschmidt tube

//Eigen Valuesa=1;b=−(D11+D22 ) ;c=D11∗D22−D21∗D12 ;

D1=(−b+sq r t (bˆ2−4∗a∗c ) ) / 2 ;D2=(−b−s q r t (bˆ2−4∗a∗c ) ) / 2 ;

d i sp (D1 ,D2 ) ;

//Modal matrix P

p11=1;p12=1;p21=(D1−D11)/D12 ;p22=(D21 )/ (D2−D22 ) ;

P=[p11 p12 ; p21 p22 ] ;d i sp (P) ;

Pinv=inv (P) ;r =60;t=0p i l 2 =60;f 1 =(1/2)−((4/(% pi ˆ 2 ) ) . .∗exp(−(%pi ˆ2∗D1∗ t )/(4∗ l ˆ2 ) ) ) ;

f 2 =(1/2)−((4/(% pi ˆ 2 ) ) . .∗exp(−(%pi ˆ2∗D2∗ t )/(4∗ l ˆ2 ) ) ) ;

//Endf =[ f 1 0 ; 0 f 2 ] ;

// d i sp ( f ) ;xD i f f =[x1p−x1m ; x2p−x2m ] ;PinvxDi f f=Pinv∗ xDi f f ;fP invxDi f f=f ∗PinvxDi f f ;PfPinvxDi f f=P∗ fP invxDi f f ;

x i n i t i a l =[x1m ;x2m ] ;x= PfPinvxDi f f+x i n i t i a l ;xnormal=x ( 2 ) ;

// Funct ionsf o r t=0: deltaT : time

p i l 2 =60;f1 secondTerm = 0 ;f2 secondTerm = 0 ;f o r k=0:1:2

58

A. Loschmidt tube

m=k+0.5;f1 secondTerm = f1 secondTerm . .

+ (1/(mˆ 2 ) ) . .∗exp(−(mˆ2∗%pi ˆ2∗D1∗ t )/ ( l ˆ2) ) ;

f2 secondTerm = f2 secondTerm . .+ (1/(mˆ 2 ) ) . .∗exp(−(mˆ2∗%pi ˆ2∗D2∗ t )/ ( l ˆ2) ) ;

end

f1=(1/2)−(1/%pi ˆ2)∗ f1 secondTerm ;

f2=(1/2)−(1/%pi ˆ2)∗ f2 secondTerm ;

//End

f =[ f 1 0 ; 0 f 2 ] ;

// d i sp ( f ) ;

xD i f f =[x1p−x1m ; x2p−x2m ] ;PinvxDi f f=Pinv∗ xDi f f ;fP invxDi f f=f ∗PinvxDi f f ;PfPinvxDi f f=P∗ fP invxDi f f ;

x i n i t i a l =[x1m ;x2m ] ;

x= PfPinvxDi f f+x i n i t i a l ;// p l o t ( ( ( t∗d12 )/ ( l ˆ 2 ) ) , ( x (2)/ xnormal ) , ’ ∗ ’ ) ;p l o t ( t , x ( 2 ) , ’ ∗ ’ ) ;// data ( s i z e ( data , ’ r ’)+1 ,1)=( t∗d12 )/ ( l ˆ 2 ) ;

// data ( s i z e ( data , ’ r ’ ) , 2 )=x (2)/ xnormal ;data ( s i z e ( data , ’ r ’)+1 ,1)= t ;data ( s i z e ( data , ’ r ’ ) , 2 )=x ( 1 ) ;

data ( s i z e ( data , ’ r ’ ) , 3 )=x ( 2 ) ;data ( s i z e ( data , ’ r ’) ,4)=1−x(1)−x ( 2 ) ;

end

d i sp ( data ) ;fp r in t fMat ( ’ data . dat ’ , data ) ;

endfunct ion

59

A. Loschmidt tube

A.2 Animation of initial diffusion of gases

t=0.5 x 10-6 s

t=2.5 x 10-6 s

t=4 x 10-6 s

t=6 x 10-6 s

t=9.5 x 10-6 s

Figure A.1: Initial diffusion of Argon

60

A. Loschmidt tube

t=0.5 x 10-6 s

t=2.5 x 10-6 s

t=4 x 10-6 s

t=6 x 10-6 s

t=9.5 x 10-6 s

Figure A.2: Initial diffusion of Methane

61

A. Loschmidt tube

t=0.5 x 10-6 s

t=2.5 x 10-6 s

t=4 x 10-6 s

t=6 x 10-6 s

t=9.5 x 10-6 s

Figure A.3: Initial diffusion of Hydrogen

62

B. Scilab program to calculate Nusselt number

B. Scilab program to calculate

Nusselt number

a1 = fscanfMat ( ” f i l ename . xy ”) ;m=s i z e ( a1 ) ;sum ut=0;sum u=0;f o r i =1:1 :m(1)

sum ut=sum ut+a1 ( i , 2 )∗ a1 ( i , 3 ) ;sum u=sum u+a1 ( i , 3 ) ;

endd i sp ( sum ut ) ;T mean=sum ut/sum u ;d i sp (T mean ) ;gradT=(a1 (2 ,2)−a1 ( 1 , 2 ) ) / ( a1 (2 ,1)−a1 ( 1 , 1 ) ) ;d i sp ( gradT ) ;Nu=(0.02∗gradT )/(T mean−323 .15) ;d i sp (Nu) ;d i sp ( a1 ( 1 , 2 ) , a1 ( 1 , 1 ) ) ;

The above code reads a .xy file where the values of temperature and magnitude ofvelocity is stored as columns. The file can be generated by sampleDict utility inOpenFOAM where sampling for Tfluid and magU can be set for a particular time.Values of magU can be calculated by the utility foamCalc. Sampling must be donefor the time when the temperature profile is fully developed. The code is specificfor the case explained in section 5.2.1. In case of change in geometry or boundaryconditions, height of the parallel plates and temperature at the wall in zone 2 mustbe changed appropriately.

63

BIBLIOGRAPHY

Bibliography

[1] ANSYS FLUENT Theory Guide 12.0.

[2] ANSYS FLUENT User’s Guide 12.0.

[3] R.C.Reid et al. The properties of gases and liquids, 4 ed. McGraw-Hill, NewYork, 1988.

[4] T.C.Bond et.al. Bounding the role of black Carbon in the climate system: Ascientific assessment. American Geophysical Union, doi:10.1002/jgrd.50171,2013.

[5] OpenFOAM Programmer’s guide 2.1.1.

[6] K.Stephan H.D.Baehr. Heat and mass transfer , first ed. Springer, 1998.

[7] H.G.Weller. Derivation, modelling and solution of the conditionally averagedtwo-phase flow equations.

[8] H.G.Weller. Alternative pressure-velocity algorithm. Private Communication,1999.

[9] H.G.Weller and C. Fureby G.Tabor, H.Jasak. A tensorial approach to com-putational continuum mechanics using object-oriented techniques. AmericanInstitute of Physics, 1998.

[10] H.L.Toor. Diffusion in three component gas mixtures. A.I.Ch.E.J. 3, 198-207,1957.

[11] Hrvoje Jasak. Error Analysis and Estimation for the Finite Volume Methodwith Applications to Fluid Flows. PhD thesis, Imperial college, University ofLondon, 1996.

[12] J.B.Duncan and H.L.Toor. An experimental study of three component gas dif-fusion. A.I.Ch.E.J. 3, 198-207, 1957.

[13] J.C.Slattery. Momentum, Energy and Mass Transfer in continua. McGraw-Hill,New York, 1972.

[14] Milovan Peric Joel H.Ferziger. Computational Methods for Fluid Dynamics,Second Edition. Springer, 1999.

[15] C.L.Tien K. Vafai. Boundary and inertia effects on flow and heat transfer inporous media. int. J. Heat Mass Transfer 24, 1981.

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[16] Hrvoje Jasak Steffen Schutz Karsten Urban Manfred Piesche Kathrin Kissling,Julia Springer. A Coupled pressure based solution algorithm based on the volumeof fluid approach for two or more immicible fluids. V European conference onComputational Fluid Dynamics, ECCOMAS CFD, 2010.

[17] H.A Kramers and J.Kistemaker. On the slip of a diffusing gas mixture along awall.

[18] V. Taivassalo M. Manninen and S. Kallio. On the mixture model for multiphaseflow. VTT Publications 288,Technical Research Centre of Finland, 1996.

[19] M.Kaviany. Laminar flow through a porous channel bounded by isothermal par-allel plates. int. J. Heat Mass Transfer 28 (4), 1985.

[20] M.Kaviany. Principles of Heat Transfer in Porous Media. Mechanical Engi-neering Series, Springer, 1991.

[21] Suhas V. Patankar. Numerical Heat Transfer and Fluid Flow. HemispherePublishing Corporation, 1980.

[22] Edwin N. Lightfoot R.Byron Bird, Warren E. Stewart. Transport Phenomena,Second Edition.

[23] R.Krishna. Problems and pitfalls in the use of Fick formulation for intraparticlediffusion. Chem. Eng. Sci. 48(5),(845-861), 1993.

[24] J.A.Wesselingh R.Krishna. The Maxwell-Stefan approach to mass transfer. El-sevier Science Ltd.

[25] R.Krishna Ross Taylor. MULTICOMPONENT MASS TRANSFER. ElsevierScience Ltd, 1996.

[26] Henrik Rusche. Computational fluid dynamics of dispersed two-phase flows athigh phase fractions.

[27] Manfred Piesche Stephan Goll. Multi-component gas transport in micro-porousdomains:Mutltidimensional simulation at the macroscale. International Journalof Heat and Mass Transfer.

[28] S.Whitaker. Advances in theory of fluid motion in porous media. Ind.Eng Chem.61(12), 1969.

[29] S.Whitaker. Role of species the species momentum equation in the analysis ofthe Stefan diffusion tube. Ind. Eng. Chem. Fund.30, 1991.

[30] VDI-GesellschaftVerfahrenstchnik und Chemieingenieurwesen (Ed.). VDI HeatAtlas. second ed., Springer, 2010.

[31] H.C.Perkins W.M.Kays. Forced Convection, internal flows in ducts, inJ.P.Hartnett, W.M.Rohsenow(Eds.), Handbook of Heat transfer. McGraw-Hill,New York, p.124, 1973.

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CFD Simulation of Multicomponent Gas Flow Through Porous Media

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