Addis Ababa University

School of Graduate Studies

Faculty of Technology

Department of Chemical Engineering

Modeling and Simulation of Interface Mass

Transfer with Chemical Reaction

By

Nigus Gabbiye

Advisors

Dr.-Ing Nurelegne Tefera

Dr. D Venkatanarasaiah

A thesis submitted to the Graduate Studies of Addis Ababa University in partial

fulfillment of the Degree of Masters of Science in Chemical Engineering

(Process Engineering)

Addis Ababa

September, 2004

v

Acknowledgement

First of all I would like to thank my God for giving me patience to accomplish my thesis

work successfully. I am most greatful to my advisor Dr.Ing. Nurelegne Tefera for advising

and directing me through out my thesis work. I would like to thank my second advisor Dr.

D .Venkatanarasaiah for his advising and giving comments up on my thesis work.

My thanks are due to my parents and many colleagues for giving me a strong moral and

advice.

Acknowledgment would also be due to the Department of Chemical Engineering and my

colleagues, because after all these years it is from there and them that I got all the

knowledge crucial for this thesis work.

iv

Abstract

Most analytical solutions of interfacial mass transfer is restricted to one

dimensional diffusion equations, while most numerical models make use of the

physically less realistic stagnant film model. The model here has developed

that simulates interface mass transfer with chemical reaction of two

immiscible liquids. It was developed by simultaneously solving the Higbie

Penetration model for the phenomenon of mass transfer accompanying by

chemical reaction. By comparing the model result with other literature results

it was concluded that the model developed for two dimensional diffusion

convection equations simulates the concentration and temperature

distribution satisfactorily.

It has been shown that under some circumstance substantial difference exists

between the literature result and numerical approximation.

vi

Table of contents Pages

Abstract……………………………………………………………………………………..iv

Acknowledgement…………………………………………………………………………..v

Table of contents……………………………………………………………………………vi

List of figures……………………………………………………………………………..viii

List of tables………………………………………………………………………………..x

1. Introduction………………………………………………………………………………1

2. Literature Review………………………………………………………………………. 4

2.1 Mass transfer concept................................................................................................... 4

2.2 Models for mass transfer at Fluid – Fluid interface ..................................................... 5

2.2.1 The Two film theory ............................................................................................. 6

2.2.2 Penetration theory ................................................................................................. 7

2.2.3 Mass transfer Coefficient .................................................................................... 10

2.2.4 Mass transfer with chemical reaction.................................................................. 12

2.3 Numerical Solution Methods ..................................................................................... 14

3. The transport equations in Capillary space……………………………………………...17

3.1 The transport equation in the bulk of the fluid phase................................................. 18

3.1.1 Buoyancy and the Boussinesq Approximation ................................................... 18

3.1.2 Description of the flow in the capillary space..................................................... 21

3.1.3 Vortices transport and poison equation............................................................... 24

3.2 The transport equation at the interface....................................................................... 26

3.2.1 Reyhlogical Properties at the interface ............................................................... 26

3.2.2. The tension equilibrium at the phase interface .................................................. 28

3.2.3 The Vorticity Transport equation at the interface ............................................... 30

3.3 Mass and heat transport equation............................................................................... 31

3.3.1 Mass and heat transport equations in capillary space ......................................... 32

3.3.2 Mass and Heat Transport equations at the phase interface. ................................ 35

4. Numerical Approach of the transport equation………………………………………….39

4.1 The nature of numerical Methods and Discretisation Concept.................................. 39

4.1.1 The nature of Numerical Methods ...................................................................... 39

vii

4.1.2 The discretisation Concept.................................................................................. 39

4.2 Discretisation of Transport equations ........................................................................ 41

4.2.1 Discretisation of temporal changes ..................................................................... 41

4.2.2 The vorticity transport equation in the bulk phase.............................................. 42

4.2.3 The poison equation ............................................................................................ 47

4.2.4 The velocity field ................................................................................................ 47

4.2.5 Mass and heat transport equation in the bulk phases .......................................... 48

4.2.6 Mass transfer equation at the interface................................................................ 58

4.2.7 Heat transfer equation at the interface ................................................................ 63

4.2.8 The vorticity equation at the interface ................................................................ 65

4.3 Discretisation of boundary and initial conditions ...................................................... 70

4.3.1 Horizontal boundary............................................................................................ 70

4.3.2 Interface Boundaries ........................................................................................... 71

4.3.3 Vertical boundaries ............................................................................................. 71

4.3.4 Initial conditions ................................................................................................. 73

4.3.5 Algorithm ............................................................................................................ 78

4.3.6 Solution of the discretised equation .................................................................... 78

5. Results and discussion…………………………………………………………………..85

5.1 Simulation of Concentration Profile .......................................................................... 87

5.1.1 Mass transfer with out chemical reaction............................................................ 87

5.1.2 Mass transfer with chemical reaction.................................................................. 93

5.2 Simulation of temperature Profile.............................................................................. 96

5.3 Validation of the simulation result…………………………………………………100

6. Conclusion and Recommendation……………………………………………………..103

6.1 Conclusion ............................................................................................................... 103

6.2 Recommendation.................................................................................................... 104

Symbols…………………………………………………………………………………..106

Reference…………………………………………………………………………………110

APPENDICES……………………………………………………………………………112

Appendix A: Prediction of physical Properties of substances ....................................... 112

Appendix B: Mathematical manipulation of the Discretisation Equations................... 115

viii

List of figures

Figure 2. 1 Concentration profiles between two phases. ....................................................... 5

Figure 2. 2 Interfacial renewal according to Higbie [5]........................................................ 8

Figure 2. 3 Concentration profile in stagnant fluid element as a function of time ................ 9

Figure 3. 1 Two dimensional representation of the capillary space……………………….17

Figure 3. 2 Velocity profile of laminar flow in capillary tube……………………………..22

Figure 3. 3 the force balance in the x - direction at the interface………………………….28

Figure 4. 1Representation of non uniform mesh.................................................................. 41

Figure 4. 2 Space discretization of vorticity transport equation ......................................... 43

Figure 4. 3Control Volumes for calculation of temperature and concentration distributions

...................................................................................................................................... 48

Figure 4. 4 Control volume for mass and heat transport equations at the interface............. 59

Figure 4. 5 w−ψ grid points at the interface........................................................................ 66

Figure 4. 6 Algorithms for the simulation of mass and heat transport discretisation

equations using implicit scheme. ................................................................................ 79

Figure 5. 1 One dimensional Concentration profiles during mass transfer at the interface as

function of distance from the interface, in the water phase negative, in the

Cyclohexane phase positive, Analytical Solution........................................................ 88

Figure 5. 2 One dimensional Concentration profiles during mass transfer at the interface as

a function of distance from the interface, in the water phase negative, in the

Cyclohexane phase positive, Numerical solutions....................................................... 89

Figure 5. 3 Two dimensional Concentration profile of the diffusing component................ 91

ix

Figure 5. 4 Concentration profiles during mass transfer at the interface as a function of

distance from the interface, in the water phase negative, in the Cyclohexane phase

positive, average value along the horizontal direction for two dimension................... 92

Figure 5. 5 Concentration profiles during mass transfer at the interface as a function of

distance from the interface, in the water phase negative, in the Cyclohexane phase

positive, average value along the horizontal direction for two dimension. ................. 94

Figure 5.6 Concentration profiles during mass transfer at the interface as a function of

distance from the interface, in the water phase negative, in the Cyclohexane phase

positive, average value along the horizontal direction for two dimension................... 95

Figure 5. 7 One dimensional temperatures profile at the interface as function of distance

from the interface, in the water phase negative, in the Cyclohexane phase positive ,for

Uo = 5 w m-1 k-1 ............................................................................................................ 97

Figure 5. 8 Two dimensional temperature distribution of the system, for Uo = 5 w m-1 k-1 98

Figure 5. 9 Temperatures profile at the interface as function of distance from the interface,

in the water phase negative, in the Cyclohexane phase positive for kmwU o /5= ,

Average value of temperature along the horizontal direction for two dimensions. ..... 99

Figure 5. 10 Concentration profile as a function of distance from the interface, Comparison

of numerical solution with analytical solution ........................................................... 101

Figure 5. 11 Temperature distributions as function of distance from the interface,

Comparison of numerical solution with other literature [13]..................................... 102

x

List of tables Table 3. 1Summary of the transport equations in the capillary flow. .................................. 37

Table 4.1The function ( )PeA for different schemes …..…... …………………………… 55

Table 5. 1 Parameter used for the simulation....................................................................... 86

1

1. Introduction

Reaction that occurs at the interface between two immiscible liquids is of intrinsic

fundamental significance in heterogeneous catalysis and separation processes. E.g. in

solvent extraction process, reactive distillation and phase transfer catalysis .This interfacial

Phenomena’s are associated with momentum, energy and mass transfer at the phase

interface and they are attributed to local gradients of interfacial tension and called

marangoni effects. Gradients in interfacial tension are caused because of gradients in

reactant and product or surfactant concentration and temperature changes with in the

interface.

A phenomenon of mass transfer with chemical reaction takes place whenever two phases

which are not at chemical equilibrium with one another are brought in to contact [1]. Such

phenomena are made up of a number of elementary steps, which may be summarized as

follows.

i. Diffusion of reactants from the bulk of phases one to interface between the

two phases. Physical equilibrium may be assumed prevail at the interface;

whenever the concentration of the reactant at the interface is finite in one

phase, it is also finite in the other.

ii. Diffusion of reactants from the interface to wards the bulk of phase two.

iii. Chemical reaction with in phase two.

iv. Diffusion of reactants initially present with in phase two, and/or of reaction

products, with in phase two itself, due to concentration gradients which are

set up by the chemical reaction.

2

The two phases involved in chemical reaction may be either both or one fluid or the other

solid. If phase two is fluid the problem to be considered is far more complicated, because

the fluid mechanics of that phase need to be considered. In view of this consideration,

attention will be focused in this work on the case where the fluids are liquids.

Liquid-Liquid systems are hydro-dynamically unstable leading to formation of wave’s

slugs and pulses, and possibly atomization. In addition the interfacial instability generated

near the gas-liquid or liquid-liquid interface results in the appearance of convective flows

which increase significantly the transfer coefficients

Knowledge about the parameter affecting the hydro-dynamic instability in the fluid-fluid

systems helps us in better design of reactors and separation columns. The parameters

affects the hydro-dynamic instability are density difference and surface tensional changes

due to concentration gradients. Concentration gradient deepens with simultaneous mass

transfer with chemical reaction.

Concentration gradients in the liquid-liquid system can be estimated using the following

experimental techniques namely, potentiometer or amperometric, optical measurements

(Laser induced fluorescence (LIF) etc. These experimental methods are economically

expensive and time consuming to get the concentration gradients at the liquid-liquid

interface. In addition these measurements can be made only in the bulk of the system with a

great care and it is very difficult to measure the interfacial composition. Therefore with the

help of high-speed computers we can adopt numerical simulation method to simulate the

concentration changes at the liquid-liquid interface.

Majority of the recent publications on numerical simulation of multi-phase flow

phenomena are used finite volume method with Euler-Euler or Euler- LaGrange

3

approaches. Numerical difference method gives us pure mathematical solution where as

finite volume method involves the physical significance of the problem.

In this study, both one and two-dimensional computer code for simulation of convection-

diffusion model are described. The code is implemented in mat lab. The partial differential

equation is converted to a system of linear equations with the finite volume method. The

system of equations is solved with an iterative line by line method, Tri-Diagonal matrix

Algorithm.

4

2. Literature Review

2.1 Mass transfer concept Mass transfer is the net movement of a component in a mixture from one location to

another location where the component exists at different concentration. Often the transfer

component takes place between phases across an interface [2].

Mass transfer occurs by two basic mechanisms:

1. Molecular diffusion: by random and spontaneous microscopic movement individual

molecules in a gas, liquid or solid is transferred as a result of thermal motion.

2. Eddy (turbulent) diffusion by random microscopic fluid motion.

Molecular and / or eddy diffusion frequently involves the movement of different species in

opposite direction. When a net flow occurs in one of these directions, the total rate of mass

transfer of individual species is increased or decreased by this bulk flow or convection

effects, which is a third mechanism of mass transfer. Molecular diffusion is extremely slow,

whereas eddy diffusion, when it occurs is order of magnitude more rapid.

Fig ure3.1 shows typical concentration gradient in the bulk of the two phases.

5

Figure 2. 1 Concentration profiles between two phases.

2.2 Models for mass transfer at Fluid – Fluid interface

Of greater interest in separation processes is mass transfer across an interface between a gas

and a liquid or between two liquid phases. Such interface exists in absorption, extraction,

and stripping. Because no materials accumulate at the interface, the rate of transfer on each

sides of the interface must be the same, and therefore the concentration gradient

automatically adjust themselves so that they are proportional to the resistance to transfer in

the particular phase. In addition, if there is no resistance to transfer at the interface, the

concentration on each side will be related to each other by the phase equilibrium

relationship. The following theoretical models have been developed to describe mass

transfer from a fluid to such an interface [2].

6

2.2.1 The Two film theory

Many different theories have been developed to calculate mass transfer coefficient .The

difficulty of the particular model lies in the consideration of the properties of the fluid

phase at the interface which highly influences the mass transfer rate. Simple theoretical

model for turbulent mass transfer to or from a fluid phase boundary was suggested in 1904

by Nernst [2] who postulated that the entire resistance to mass transfer in a given turbulent

phase is in thin, stagnant region of that phase at the interface, called the film. The two film

theory states laminar sub layer at both side of the interface so that mass transfer process can

be described by diffusion. The main transport resistance lies between the two turbulent

mixing bulk phases. I.e. there is no resistance to solute transfer across the interface

separating the phases, Whitman [3]. The equation of the two film theory is given by:-

( ) ( )int22int11 .. CnDCnD ∇=∇ (2.1)

This describes the continuity of the diffusive mass transfer of both phases in the normal

direction (n).

By the help of the diffusivity and film thickness over the changes of the concentration

between the bulk phases and the interface one gets the mass transfer coefficients.

( )

( )bDh

aDh

D

D

2.2

2.2

2

22

1

11

δ

δ

=

=

7

The two film theory, which is easy to understand and apply, is often criticized because it

appears to predict that the rate of mass transfer is directly proportional to the molecular

diffusivity. This dependency is at odd with the experimental data, which indicates

a dependency of nD , where n ranges from about 0.5 to 0.75. Especially a strong increase

of mass concentration at the interface can not be explained correctly .However if δ

ABD is

replaced with Dh , which is then estimated from the Chilton - Colburn analogy [2] one can

obtain Dh proportional to ABD 3/2 , which is in better agreement with experimental data.

Regardless of whether the criticism of the film theory is valid, the theory has seen and

continuous to be widely used in the design of mass transfer equipment.

Next to film theory the other models considers convective transport of near the interface

with the bulk phase as discussed below.

2.2.2 Penetration theory

The first penetration model originates from Higbie [4] where the mass transfer process is

described as follows: the interface is assumed to be covered with small stagnant fluid

elements and mass transfer occurs in to the element. After the contact time θ with the

interface, the element return to the well mixed bulk of the fluid from which they came,

while new element originating from the bulk take their place at the interface. Coming from

the bulk, all stagnant elements have a uniform initial concentration KC . At the interface

mass transfer occurs by molecular diffusion. This diffusion process is essentially transient.

Figure 2:3 is a qualitative sketch of the concentration profile versus the distance from the

interface for several times. The depth of penetration of k in to the element is assumed to be

small with respect to the thickness of the elementδ .

8

Non-stationary diffusion takes place at the phase interface during the residence

time 01 tt −=θ [5]. As only the direction perpendicular to the interface is significant, Fick’s

second law for non-stationary one-dimensional diffusion applies:

Figure 2. 2 Interfacial renewal according to Higbie [5]

2

2

yCD

tC K

KK

∂∂=

∂∂

(2.3)

Subject to the conditions:

Initial conditions: KK CCandt =>= ,00 (2.4a)

Boundary conditions

( )( )cCCyandt

bCCyandt

kK

KiK

4.20

4.2,00

=∞=>

==>

The last boundary condition can be used because the penetration depth is much smaller than

the thickness of the element. So a semi - infinite medium can be assumed.

9

The solution found for the above equation becomes

=

−−

tDyerf

CCCC

kkai

kki

2 (2.5)

Differentiating the above equation gives the molar flux of component k at the interface:

( )kkik

y

kkk CC

tD

yC

DJ −=

∂∂

−== π0

(2.6)

t

Dh k

D π= Mass transfer coefficient.

This shows that the mass transfer rate decreases with increasing time, which is caused by a

decrease of y

Ck

∂∂

at the interface. The rate of mass transfer can be lumped in the

parameterθ , which is called the contact time or the penetration time.

t

c

tt

t > t > t

y

1

2

3

3 2 1

Figure 2. 3 Concentration profile in stagnant fluid element as a function of time (θ = maximum time: contact time in the Higbie penetration model)

10

The modification of the Higbie penetration model has been developed by Danckwerts [4].

Instead a constant contact timeθ , he assumed that the chance of a surface element being

replaced with in a given time is independent of its age. Thus, each element has an equal

chance of being replaced during the next time unit. The fraction of elements that is replaced

per unit time is assumed to be S ( ).1−s Thus, the following can be derived.

( )kkikk CCSDJ −= (2.7)

The penetration model of Danckwerts is known as the surface renewal model. There are

also theories that describes mass transfer models; Film-surface renewal theory Dobbins,

and Surface-stretch theory Lightfoot and his Coworkers [3]. In this study the penetration

model is used. This is because it considers both molecular diffusion and bulk diffusions at

the interface.

The Penetration theory is most useful when mass transfer involves bubbles or droplets, or

flow over random packing [2].

2.2.3 Mass transfer Coefficient

On the bases of the theories discussed above, the rate of mass transfer in the bulk flow is

directly proportional to the deriving force expressed as a molar concentration difference,

and therefore

( )KoKiDK CChN −= (2.8)

Dh is the mass transfer coefficient. In the two - film theory Dh is directly proportional to

the diffusivity and inversely proportional to the square root of the film thickness. According

to the penetration theory, it is proportional to the square root of the diffusivity and, when all

surface elements are exposed for equal time, it is inversely proportional to the square root

11

of the time of exposure; when random surface renewal is assumed, it is proportional to the

square root of the rate of renewal.

In the process where mass transfer takes place across a phase boundary, the same

theoretical approach can be applied to each of the phases, though does not follow that the

theory is best applied to both phases. For example, the film model might be to one phase

and the penetration model to the other.

When the film theory is applicable to each phase (the two - film theory), the process is

steady state through out and the interface composition does not then vary with time. For

this case the two film coefficient can be readily be combined. Because material does not

accumulate at the interface, the mass transfer on each side of the phase boundary will be the

same and for two phases it follows that

( ) ( )222111 KoKiDKiKoDA CChCChN −=−= (2.9)

If there is no resistance to transfer at the interface 1KiC and 2KiC will be corresponding

values in the phase equilibrium relationship. Usually the value of the concentration at the

interface is not known and the mass transfer coefficient is considered for the over all

process. Over all transfer coefficients are then defined by:-

( ) ( )222111 KoKeDKeKoDK CChCChN −=−= (2.10)

where 1KeC is the concentration in phase one in equilibrium with 2KoC in phase two, and

2KeC is the concentration in phase two in equilibrium with 1KoC in phase one. If the

equilibrium relation is linear

2

1

2

1

2

1

Ko

Ke

Ke

Ko

Ki

Ki

CC

CC

CC

N === (2.11)

Where N is proportional constant (distribution coefficient).

12

The relationship between the various transfers coefficients are obtained as follows.

211

11

DD hN

hK+= (2.12)

Similarly 212

111

DD hNhK+= (2.13)

And hence 21

1KN

K= (2.14)

These relationships between the various coefficients are valid provided that the transfer rate

is linearly related to the deriving force (concentration gradient) and that the equilibrium

relationship is a straight line [6]. They are therefore applicable for the two - film theory,

and for any instant of time for the penetration theory. In general, application to time -

average coefficients obtained from the penetration theory is not permissible because the

condition at the interface will be time - dependent unless all of the resistance lies in one of

the phases.

2.2.4 Mass transfer with chemical reaction

The theoretical under standing of mass transfer at the fluid interface with chemical reaction

is described by Hatta [4]. Hatta theory is based on the film theory model. He assumed that a

fast irreversible chemical reaction at the interface.

CBA CBA υυυ →+ (2.15)

The transfer component A reacts with B in the absorbing phase to produce C at the

interface. The molar concentration of A and B decreases from its original values in the bulk

phase towards the interface. This simple model leads

+= ∞,int, B

B

A

A

BAADA C

DDChn

ννν (2.16)

13

Equation (2.16) equals the general equation developed for mass transfer with out reaction

equation (2.1). One gets therefore after dividing equation (2.16) by interface

concentration int,kC , the mass transfer coefficient becomes as follows:-

+=

∞

∞

,

,1A

B

AA

BBDDr C

CDDhh

υυ (2.17a)

+=

∞

∞

,

,1A

B

AA

BB

D

Dr

CC

DD

hh

υυ (2.17b)

This ratio is called the reaction factor. It describes the increase of mass transfer coefficient

if the mass transport is associated with chemical reaction. The reaction factor is dependent

on concentration and has the value of one if there is no chemical reaction. In the absorbing

phase calculation based on the surface renewal theory leads to a similar expression for the

reaction factor.

By performing a reaction through a liquid interface a concentration difference of the

product forms near the interface. It has been shown that in many cases, those gradients lead

to hydrodynamic instabilities, which then breaks non - linearly in to a pattern on sets slow

convection [7].

Although interfacial effects accompanying solute transfer have been subject of several

analyses [8], no generalized model has yet evolved which would enable the effect of

interfacial activities up on transfer rates to be predicted quantitatively. Such analysis are

complicated by superimposed effects such as convective flow induced by density gradients.

Berg and Moring (1969) [8] demonstrated that such gradients might retard or enhance the

convection generated by the marangoni effect. A further complication may rise from the

14

heat of solution accompanying the transfer of solute across an interface, since this can

influence both interfacial tension and any density gradients.

2.3 Numerical Solution Methods

A lot of works have been done for solving convective-diffusion problems for determined

velocity field. However, except in some very special circumstances, it is not possible to

specify the flow field; rather, one must calculate the local velocity component and the

density fields from the appropriate governed equation [9]. The velocity components are

governed by the momentum equations.

There are three distinct streams of numerical solutions techniques: finite difference, finite

element and spectral methods [10].The main difference between the three separate streams

are associated with the way in which the transport variables are approximated and with the

discretisation processes.

Finite difference methods: - Finite difference methods describe the unknowns transport

equation by means of point samples at the node points of a grid of coordinate lines.

Finite element methods: - Finite element methods use simple piecewise functions (e.g.

linear or quadratic) valid on elements to describe the local variations of unknowns transport

variableφ .

Spectral methods: - Spectral methods approximate the unknowns by means of truncated

Fourier series or series of Chebyshev polynomials. Unlike the finite difference or finite

element approach the approximations are not local but valid throughout the entire

computational domain

Finite volume methods: - The finite volume method was originally developed as a special

finite difference formulation. It is the most well - established and thoroughly validated

15

general purpose CFD techniques. It is central to four of the five main commercial available

CFD codes: PHOENICS, FLUENT, FLOW3D AND STAR.CD. The numerical algorithm

consists of the following steps.

• Formal integration of the governing equations of fluid flow over all the control

volumes of the solution domain.

• Discretisation involves the substitution of a variety of finite - difference - type

approximations for the term in the integrated equation representing flow processes

such as convection, diffusion and sources. This converts the integral equation in to a

system of algebraic equations.

• Solution of the algebraic equations by an iterative method.

The first step, the control volume integration, distinguishes the finite volume method from

all other CFD techniques. The resulting statements express the (exact) conservation of

relevant transport properties for each finite size cells. This clear relationship between the

numerical algorithm and the underlying physical conservation principles forms one of the

main attractions of the finite volume method and makes’ its concepts much more simple to

understand by engineers than finite or spectral methods.

The conservation of a general transport variable φ with in a finite control volume can be

expressed as a balance between the various processes tending to increase or decrease it.

+

+

=

volumecontrol

theinsideofcreationofrateNet

volumecontrolthe

toindiffusiontodue

offluxNet

volumecontrolthe

toinconvectiontodue

offluxNet

timetorespectwithvolumehecontrlofchangeofRate

φ

φφ

φint (2.18)

16

The above sets of equations are complex so an iterative solution approach is required. The

most popular solution procedures are TDMA line - line solver of the algebraic equations

and the SIMPLE algorithm to ensure correct linkage between pressure and velocity [10].

It is aimed in the present study to develop a mathematical model for interface mass transfer

with chemical reaction in liquid-liquid systems, incorporating the changes due to density

and surface tension on the transport coefficients It is also aimed to go for numerical

simulation using finite volume method (FVM) and to develop a general computer code.

One dimensional as well as two-dimensional computer code for solution of convection -

diffusion is described. The code is implemented in mat lab. The partial differential equation

is converted to a system of linear equations with the finite volume method. The system of

equations is solved with an iterative method.

17

3. The transport equations in Capillary space

In developing the mathematical model of the transport equations in the capillary space it

has been considered that the flow is caused by natural convection. Natural convection is

generated mainly due to density variation combined with gravity. Density variation and

buoyancy driven flows can result from solute concentration or temperature changes. The

convective terms of mass and heat transport equations are dependent on the velocity

generated by buoyancy driven flow.

( ) 0, =∂

∂y

tHC

( ) 0,0 =∂

∂y

tC

Figure 3. 4 Two dimensional representation of the capillary space

18

As shown in figure 3.1 two layers of liquids are superimposed one another. The upper layer

is cyclohexane and the bottom layer is water solution. Initially the bottom layer of depth h

contains only water solution containing one molar solution of sodium hydroxide no acetic

acid ( )0=oC , and the upper layer of depth hH − , contains acetic acid ( )oCC = . Mass

transfer is therefore from the upper phase to the lower phase. Thus the transfer mechanism

has been modeled mathematically using micro balance model.

3.1 The transport equation in the bulk of the fluid phase

As it has been discussed above there are two phases for this particular study. Initially the

two phases are separated by hypothetical stretch (layer). Hence it is possible to treat the two

bulk phases independently by the usual differential equations of mass, heat and momentum

balances. But as time goes the interface of the two phases behaves differently. Hence the

interface of the phases needs a special treatment. All about these are discussed in the

subsequent sections.

3.1.1 Buoyancy and the Boussinesq Approximation

The variation in the fluid density which underlie free convention preclude the use of

Nervier stokes equation [11]. We begin instead with the Cauchy forms of conservation of

momentum, written as

τρρ ⋅∇++∇−= gDtDV P (3.1)

With forced convection, it is often desirable to work with dynamic pressure p instead of

the actual pressureP .

gp oρ−∇=∇ P (3.2)

19

oρ is the constant density in some arbitrarily chosen reference state.

The pressure and gravitational terms in the momentum equation can be written as

gpg oo )( ρρρ −+∇=−∇ P (3.3)

Using equation (3.3) in equation (3.1) the momentum equation becomes

τρρρ ⋅∇+−−∇−= gpDtDV

o )( (3.4)

The term ( )oρρ − may view as a force per unit volume due to buoyancy. In essence, it is

the force responsible for free convection. If oρρ = every where, the pressure and

gravitational terms reduce P∇− , as forced convection. This suggests that we associate the

gradient in dynamic pressure with forced convection.

In free convection problems involving a pure, non isothermal fluid, the unknowns are the

velocity, pressure, temperature and density [11]. The basic equations are continuity,

conservation of momentum, and conservation of energy (all with variable ρ ), together with

an equation of state for the fluid. The equation of state, which is of the form ( )Tp,ρρ = ,

makes the fluid dynamic problem dependant on the heat transfer problem, even if all other

fluid properties are constant. For an isothermal mixture in which the density depends on the

local composition, the energy equation is replaced by one or more species composition.

Clearly, the analysis of buoyancy - driven flow may be extremely complicated. The

solution to the general set of equations is so difficult that almost all published works has

employed what is called the Boussinesq Approximation [11]

Generally there are two parts for the Boussinesq Approximation

i. The first is the assumption that ρ varies linearly with temperature and/or

concentration

20

( ) ( )oiCioTo

oCioTo

o CCC

TTT

−

∂∂+−

∂∂+=

,,

ρρρρ (3.5)

( ) )(1 oiioo

CCTT −+−+= ββρρ (3.6)

CioToo

iCioToo CT ,,

1,1

∂∂−=

∂∂−= ρ

ρβρ

ρβ

iand ββ are the thermal and the solutal expansion coefficient respectively.

ii. The second part of the Boussinesq approximation is the assumption that the variable

density ρ can be replaced every where by the constant value oρ , except in the

term ( )goρρ − . This is crucial, because it leads to the continuity equation for constant

density fluid and allows the viscous stress to be evaluated as in the Navier stokes equation.

For Liquids the necessary condition is .10

<<∆ρρ Hence it is possible to use equation (3.1)

provided that the density is constant except in gravitational terms. In this paper the second

approximation of the Boussinesq Approximation is applied due to its simplicity.

Using the Boussinesq Approximation the continuity and momentum equations for pure

Newtonian fluids with constant µ becomes

0=⋅∇ V (3.7)

( ) gpVVVtV

oo ρρ

ρν +∇−∇=∇⋅+

∂∂ 12 (3.8)

Actually this approach yields what is still a difficult set of coupled differential equations,

one which is much more amendable to analysis. Especially the second approximation

simplifies the dependency of the momentum equation on temperature.

21

3.1.2 Description of the flow in the capillary space

The flow in the capillary is considered to be two dimensional for this particular model.

Hence the z component of the velocity w is zero due to the geometrical configuration of

the system. It is assumed that the flow is laminar. To proof the above assumption, the

Reynolds number at thermal convection in an infinite length of the capillary tube

( ∞→yx, ) and mmZ 1= is considered. The fluid particle shall have temperature

difference of kT 10=∆ compare to the surrounding fluid. The volume expansion

coefficient 1310 −−≈ kTα (Benzene) the gravitational force 210 −= msg , neglect the effect of

the fluid viscosity over the distance of ,1.0 my =∆ the velocity of the fluid particle will be

11 14.010001.0101.02

2−− ≈××××=

∆∆=

msms

Tgy TαV (3.9)

The characteristic length can be determined by the hydraulic diameter hd

x

ZZ

xZZx

UAdh

+=

+==

12

2244 (3.10)

Where U = perimeter, A Area, x and Z are side of the capillary tube. If ∞→x , the

hydraulic diameter will be

∞→

=×==x

mmZd h 002.0001.022lim (3.11)

The capillary is hydraulic equivalent with a pipe of Zdh 2= . The kinetic viscosity of a

fluid is given by ( 127108.6 −−×= smv ). This will give a rough estimate of the Reynolds

number Re

412108.6

002.014.0Re 7 ≈××== −ν

hdV (3.12)

22

Since the Reynolds number is less than 2300, the flow considered above in the capillary

space is laminar.

Figure 3. 5 Velocity profile of laminar flow in capillary tube

In the absence of an end effect, the equation of motion for the liquid film in fully developed

laminar flow in the x - direction is given by

pgzu ∆==

∂∂ ρµ 2

2

(3.13)

Usually in fully developed flow, u is independent of the distance x.

Boundary condition

ZandZzatu21

21,0 −== (No slip condition at solid surface)

0,0 ==∂∂ zat

zu

Integrating equation (3.13) twice with the given boundary conditions

−= 2

241ˆ)(Zzuzu (3.14)

23

u is the maximum velocity at 0=z . The average velocity is obtained by integrating

equation (3.14) in the interval of ZandZ21

21−

udzzuZ

u

Z

Z

ˆ32)(1 2

2

∫−

== (3.15)

The two velocity component therefore

−= 2

2412

3)(Zzuzu (3.16a)

and with the same analogy the y - component becomes

−= 2

2412

3)(ZzVzV (3.16b)

From equation (3.8) a two dimensional viscous flow problems at constant fluid properties

are generally governed in the Cartesian system by the following three equations.

0=∂∂+

∂∂

yV

xu (3.17a)

xy

uxu

yuV

xuu

tu

o ∂∂−

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ ρ

ρν 1

2

2

2

2

(3.17b)

yoo

gxy

VxV

yVV

xVu

tV

ρρρ

ρν +

∂∂−

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ 1

2

2

2

2

(3.17c)

The flow is only the function of x, y and t. Substituting equation (3.16) in equation (3.17)

and integrating each term in terms of z in the interval of

−2

,2

ZZ , the following space

average equations can be obtained.

24

x

uZy

uxu

yuV

xuu

tu

o ∂∂−−

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ ρ

ρνν 112

56

22

2

2

2

(3.18a)

and

yoo

gx

VZy

VxV

yVV

xVu

tV

ρρρ

ρνν +

∂∂−−

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ 112

22

2

2

2

(3.18b)

The three unknowns’ u, v and p often termed as “primitive variables.” The above system

equations are still very complicated, and an important simplification is obtained by

employing the stream functionψ , defined in the next sub section. In the following section

the average velocities are used and no “bar” symbol is used “u ”.

3.1.3 Vortices transport and poison equation

The stream function is a mathematical constructs which is extremely useful for solving

incompressible flow problems where only two non vanishing components and two special

components are involved [11]).

For two dimensional flow described by rectangular coordinates, the stream function

( )yx,ψ is defined by

y

u∂∂= ψ ,

xV

∂∂−= ψ (3.19)

And the vorticity ω defined by

xV

yu

∂∂−

∂∂=ω (3.20)

Where ω is the vorticity, being only the z component of zω for the above two dimensional

flow.

The main advantage of these choices is seen by substituting equation (3.19) in a

corresponding continuity equation for incompressible fluid

25

=∂∂+

∂∂

yV

xu 0=

∂∂−

∂∂+

∂∂

∂∂

xyyxψψ (3.21)

Thus the stream function is defined in such away that ( )yx,ψ can be determined for the

given flow, the continuity equation will be satisfied automatically.

Having defined the stream and vorticity functions, it is now possible to solve the transport

equation (3.18). But the pressure term should be eliminated. The pressure term is

eliminated by differentiating each term in equation (3.18a) in terms of y and each term in

equation (3.18b) in terms of x. Then subtracting one from the other, and using the vorticity

(ω ) from equation (3.20) the final equation becomes

xg

ZyxyVw

yV

xuw

xu

t o

y

∂∂−−

∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

+∂∂ ρ

ρωνωωνωωω

22

2

2

2 1256

(3.22)

The terms )()(yVand

xu

∂∂

∂∂ ωω can be eliminated using the continuity equation (3.7), and

Equation (3.22) becomes

xg

ZyxyV

xu

t o

y

∂∂−−

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ ρ

ρωνωωνωωω

22

2

2

2 1256

(3.23)

The convective term y

Vandx

u∂∂

∂∂ ωω in equation (3.23) can be calculated using stream

function (3.19).

2

2

2

2

yx ∂∂+

∂∂= ψψω (3.24)

26

3.2 The transport equation at the interface

3.2.1 Reyhlogical Properties at the interface

Components of any solution behave differently at an interface than they do with in the bulk

solution. One example is that their concentration at the interface is different from with in

the bulk phase. This different, called adsorption at the liquid interface manifests itself by a

change in the capillary properties of the interface [12].

The liquid droplet has a tendency to contract as if it is surrounded by a hypothetical

stretched “skin”. If the drop of liquid is uninfluenced by external forces, such as gravity, it

will adopt a truly spherical shape.

Young first explained surface tension as resulting from the attraction between the

neighboring molecules (Young, 1805) [12]. These forces are isotropic in the interior of the

bulk phase but molecules at the interface are attracted less completely than in the bulk.

Because the free energy of a system tends to a minimum the surface of a pure phase or the

interface between two immiscible liquids will always tend to contract spontaneously.

The interfacial tension is always positive (Landau and Lifshitz, 1958) [12]. If the interfacial

tension, 0<σ the interface between two immiscible liquids will tend to increase without

limit, and the phase would mix and cease to exist separately. If 0>σ , then the interface

become as small as possible for a given volume of the two phases.

The interfacial tension can be altered by adding a surface active solute. The molecule of a

surfactant tends to concentrate at the interface rather than in the bulk phase and typically

form monomolecular layer.

Gibbs published his classic treaties about adsorption and capillarity in 1875 [12] but his

pioneering was only appreciated year later, and experimental application of the Gibbs

27

equation of adsorption becomes important only in the beginning of this century (Gibbs,

1928). The Gibbs equation is expressed by

dCd

RTc σ

−=Γ (3.25)

Where Γ represents the excess number of adsorption particles in a portion of solution

containing a unit area in a comparison with some homogenous solution having exactly the

same volume.

The tension at the interface of between two phases is dynamic in nature. The value of the

tension is dependent on the velocity of deformation of the interface.

The equation of the dynamic surface tension at the interface wants to know the viscosity of

the fluid ( )εκ and .These interfacial shear viscosity ( )ε and interfacial dilatation viscosity

( )κ describe the viscous condition of the interface. These viscosities determine the

movement of the molecules between the two phases.

The dynamic surface tension dynσ depends also on the coordinate system.

For Newtonian fluid εκ and are constant. Therefore for flat interface parallel to the x

and z direction the equation of the dynamic interfacial tension can be formulated as follows

( )

∂∂−

∂∂+

∂∂+

∂∂++=

zw

xu

zw

xu

equdyn εεκσσ (3.26a)

and in the z direction

( )

∂∂−

∂∂+

∂∂+

∂∂++=

zu

xw

zw

xu

equdyn εεκσσ (3.26b)

where, u and w are velocity components in x and z direction respectively

28

3.2.2. The tension equilibrium at the phase interface

Along the phase interface if there is concentration and temperature gradient, due to this

there will be also interfacial tension gradient. There exist expansion and contraction of the

interface due to the variation of the interfacial tension at the interface. Normal to the

interface there is a gradient parallel to the velocity component which means at the interface

it generate an impulse. The impulse is transferred to the neighboring liquid phase by the

viscous forceτ .

zxin δδτ *

zσδ zxx

δδσσ

∂∂+

e

xδ

Figure 3. 6 The force balance in the x - direction at the interface.

The mathematical description of the interface is discontinuous. At the interface the

density ( )ρ , the viscosity ( )µ and the viscous stress ( )τ changes suddenly. Therefore the

balance of the force at the interface can be expressed by

0* =∂∂+− zx

xzxzx inin δδσδδτδδτ (3.27)

Multiplying the above equation by the reciprocal of the zxδδ

0* =∂∂+−

xininσττ (3.28)

zxin δδτ

29

Forx∂

∂σ , the x - differentiation of dynamic interfacial tension given in equation (3.26) is

used. If fast diffusing component with low adsorption is assumed, then the interface

viscosities εκ and can be neglected. The interfacial tension is dependent on the local

temperature and concentration. The total differential of the interfacial tension is therefore

xT

TxC

Cxequequ

∂∂

∂∂

+∂∂

∂∂

=∂∂ σσσ

(3.29)

The terms T

andC

euqequ

∂∂

∂∂ σσ

are measurable at equilibrium in two phase systems.

The viscous stress τ and *τ can be expressed in Newton’s law of viscosity.

∂∂+

∂∂=

xV

yuητ (3.30a)

*

**

∂∂+

∂∂=

xV

yuητ (3.30b)

Along a flat locally fixed interface the normal velocity component equals zero ( 0=V ).

Hence equation (3.30) can be simplified to

intint

∂∂=

yuητ (3.31a)

*intint **

∂∂=

yuητ (3.31b)

The viscous stress can be also expressed in terms of the Vorticity equation )20.3( and

stream function equation ( )19.3 as usual. When 0int =V the Vorticity equation becomes

intint yu

∂∂=ω (3.32a)

30

*intint

*

yu

∂∂=ω (3.32b)

Substituting equation (3.32) in equation (3.31)

intint ωητ = (3.33a)

*int

*int ωητ = (3.33b)

The force balance becomes

0** intint =∂∂+−

xσηωωη (3.34)

In terms of the stream function

0* 2

2

int2

2

=∂∂+

∂∂−

∂∂

xyyσψηψη (3.35)

At the interface there are two unknown parameters *ωω and . To determine these

parameters, the equilibrium tension balance should be satisfied and the Vorticity transport

equation is needed.

3.2.3 The Vorticity Transport equation at the interface

The derivation of the viscous stress at the interface shows that the Vorticity force along the

interface is not generally constant. Therefore the Vorticity equation requires the value of

the two terms ( )*ωωand and it is also important to consider the sudden change of the

physical properties of the fluid ( ).,σρ

To formulate those equations at the interface the average of the contribution of both phases

to the impulse balance is required. The momentum equation derived for the bulk phase

using Boussinesq Approximation is not applicable for the interface, because the

31

formulation of the average of the two terms, which is only purely kinematics, violates

conservation equation.

So to get the correct equation we start with the Vorticity equation

xg

ZyyxyV

xu

t yo ∂∂−−

∂∂

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ ρωηωηωηωωωρ 22

2 1256 (3.36)

The density and viscosity variation in y - direction is considered normal to the interface.

When the velocity component 0=V , then 0=∂∂

yV ω , therefore equation (3.36) will be

xg

Zyyxxu

t yoo ∂∂−−

∂∂

∂∂+

∂∂=

∂∂+

∂∂ ρηωωηωηωρωρ 22

2 125

6 (3.37)

∂∂+

∂∂=

∂∂

ttt ooo**

21 ωρωρωρ (3.38)

∂∂+

∂∂=

∂∂

xxx ooo**

21 ωρωρωρ (3.39)

∂∂+

∂∂=

∂∂

2

2

2

2

2

2 **21

xxx oooωρωρωρ (3.40)

( )**21 ωηηωη +=w (3.41)

∂∂+

∂∂=

∂∂

xxx*

21 ρρρ (3.42)

Together with the tension equilibrium equation (3.34), we have two equations with two

unknown variables, therefore one can solve simultaneously.

3.3 Mass and heat transport equation

The mass and heat transport equation contains both conductive and convective terms like

momentum transport. The single conductive transport term in multi-component system is

32

independent to each other. Thermal diffusion (mass transport due to temperature gradient),

the heat flow due to concentration gradient, cross diffusion (mass transfer due to

concentration of other components), are neglected in a capillary flow.

In z - direction both temperature and concentration gradient in the capillary are also

neglected

=∂∂=

∂∂ 0

zC

zT . The density change in z - direction, which is dependent on

temperature and concentration are also neglected

=∂∂ 0

zρ . In the convective term, the

velocity is taken to be the average value of the capillary space.

3.3.1 Mass and heat transport equations in capillary space

Problem involving chemical reaction are handled most easily with molar units, so that kC

is generally preferred. The molar flux of component k relative to fixed coordinates is kN

[11]. Thus, the basic form of the conservation equation for component k is

vkkk rNt

C +∇−=∂∂

(3.43)

The source term vkr represents net rate of formation or consumption of species k by

chemical reaction per unit volume. Reactions which appear in this manner (i.e., ones which

occur through out the volume of the gas or liquid solution) are called homogeneous. When

there is fluid flow the preferred way to write the total flux is

kkk JCN += V (3.44)

Where, Jk is the molar flux of k relative to the mass average velocity.

33

Several forms of Fick’s law for binary mixtures were presented in Transport analysis by

William M.Deen [11]. For a binary mixture of component k and B at constant density, the

molar flux of k relative to the mass - average velocity is given by

kkBk CDJ ∇−= (3.45)

This is an excellent approximation for most liquid solutions, less accurate for gas mixtures

[11]. Although strictly valid only for a binary system, equation (3.45) can be applied also

to certain multi - component mixtures. In multi - component liquid solution or gas mixtures

there are diffusion interactions between each pair of components. However, if all

component but one are present in small amounts, interaction among the minor component

tends to be negligible, and only the binary diffusivity of each minor component k with the

abundant component is important. For this pseudobinary situation at constant density, the

flux equation for each minor component is give by

kkk CDJ ∇−= (3.46)

Equation (3.46) is the most frequently used flux equation for liquid solutions, even when

there are more than two components [11].

The differential equation of the mass transport with chemical reaction in a control volume

for component k is described in a conservation form for constant kDandρ as follows.

( ) ( ) Vkkkkkk rMCDVC

tC

ν+∇⋅∇=⋅∇+∂∂

(3.47)

For two dimensional problems and constant diffusion coefficient kD equation (3.47) can be

written as follows

Vkkkk

kkkk rM

yC

xC

Dy

CV

xC

ut

C ν+

∂∂

+∂∂

=∂∂

+∂∂

+∂∂

2

2

2

2

(3.48)

34

vr is the volume base reaction velocity (production rate per unit volume per unit time).

Through multiplication with the Stoichiometry coefficient kν and the molar mass kM one

can get the source term for mass concentration of component k. kν is negative for reactants

and positive for products.

The heat transport equation is also described in conservation form as shown below.

( ) ( ) ( ) vRPP HrTVTc

tTc

∆−Φ+∇⋅∇=⋅∇+∂

∂ λρρ (3.49)

HR∆ is the molar reaction enthalpy and Φdissipation function.

The reaction rate velocity vr is dependent on the temperature and concentration of the

system. Both equations (3.48) and (3.49) are interrelating to each other by the reaction rate.

The dissipation function Φ considered is the conversion of kinetic energy in to heat energy

through the friction of the fluid. The estimation of this function shows that the viscous

dissipation due to the given velocity is neglected. The shear force at the surface

±= Zz21

had attained its maximum value. Taking the flow velocity of 11.0 −= msu in the middle of

the capillary, the viscosity of the fluid mPas1=µ , the width of the capillary mmZ 1= ,

the maximum shear force at the interface will be

PaPauZZzz

u 4.01.0001.0

001.044

2

max =

××==±=∂

∂= ηητ (3.50)

Assume the temperature difference KT 10=∆ in the capillary for length cmL 1=∆ , and the

thermal conductivity 115.0 −−= Kwmλ . The ratio of dissipation energy to heat conduction

becomes

35

6max 106.8

01.0105.0

1.04.0 −×=×

×=

∆∆

LTu

λ

τ (3.51)

The viscous dissipation energy is 610− smaller than the heat flow that is expected; hence it

is possible to neglect it.

In two dimensional Cartesian coordinate with constant λ and neglected viscous dissipation,

the heat transport equation becomes

( ) ( ) ( )vR

ppP HryT

xT

yVTc

xuTc

tTc ∆−

∂∂+

∂∂=

∂∂

+∂

∂+

∂∂

2

2

2

2

λρρρ

(3.52)

Through both walls of the capillary, there is a heat lost to the surrounding. This can be

described based on the reaction volume of the capillary and width Z by

( )TTZUq s

o −−=2 (3.53)

Where sT is surrounding temperature, oU is overall heat transfer coefficient given by

iowo hs

U11 +=

λ (3.54)

For constant specific heat capacity pc and density oρ and using Boussinesq approximation

and continuity equation, the heat transport equation becomes

( )

−−∆−

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ TT

ZU

Hrcy

TxTa

yTV

xTu

tT

so

rRPo

212

2

2

2

ρ (3.55)

The temperature coefficient (thermal diffusivity) is given by ( )poca ρλ=

3.3.2 Mass and Heat Transport equations at the phase interface.

In two film theory (2.2), we only considered the conductive transport of heat and mass. But

now we are considering both types of transport (conductive and convective) of mass and

36

heat. These requires convection - diffusion equation similar to equation (3.48) and (3.55) in

the vicinity of the interface.

The concentration gradient of the transport component at the interface is in general

unsteady. The different in concentration gradient at the interface is the result of different

solubility’s of the transport components between the two liquid phases. The interface

concentrations intint* CandC are only in equilibrium if the interface itself does not resist

the mass transport. Based on tow - film theory, Whteman, [3], the relation between the

concentrations can be described by the following equation

int

*int

CC

N = (3.56)

The distribution coefficient N is a function of temperature and concentration.

The transport resistance grows for a component that tends to precipitate at the interface.

Therefore the interface becomes broader. At the interface the adsorbed component hinders

the diffusion of a certain component. In this study it is assumed that the two phases are in

equilibrium and the transport resistance at the interface is neglected.

The second equation that is necessary to calculate the unknown interface concentration is

the mass transport equation. Due to the sharp variation of the concentration, the variation of

the diffusion coefficient is taken to be normal to the interface therefore the equation

becomes the arithmetic mean of the two phases.

kkkk

kk

kkkk rM

yC

Dyx

CD

yC

Vx

Cu

tC

ν+

∂∂

∂∂+

∂∂

=∂∂

+∂∂

+∂∂

2

2

(3.57)

∂∂+

∂∂

=∂∂

xC

xC

tC kkk

*

21 (3.58)

37

x

Cux

Cu

xC

u kkk

∂∂+

∂∂

=∂∂ *

21 (3.59)

∂∂+

∂∂

=∂∂

2

*2*

2

2

2

2

21

xCD

xC

DxC

D kk

kk

kk (3.60)

( )vvv rrr *

21 += (3.61)

The temperature at the interface is steady, i.e. int*

int TT = . With in the fluid phase, there

shall be a constant value of *** ,,,, λρλρ opop candc . Under this condition we built the

differential equation for heat transport equation at the interface as follows

( )TTZkrH

yT

yxT

yTVc

xTu

tTc uvRpopo −+∆−

∂∂

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ 2

2

2

λλρρ (3.62)

Where ( )popopo ccc *

21 ρρρ += (3.63)

and

( )*

21 λλλ += (3.64)

Table 3. 1 Summary of the transport equations in the capillary flow. Momentum Mass Heat

Bulk Phase

Interface

(3.19), (3.23), (3,24)

(3,19), (3,34), (3,35),(3.37)

(3.48)

(3.56), (5.57)

(3.55)

(3.62)

Generally the following assumption are used in deriving the model equations

• Flat interface

• Boussinesq Approximation is applied

38

• Two dimensional transports with parabolic velocity profile.

• Conductive transport are independent to each other

• Equilibrium is assumed at the interface.

• Viscous dissipation is neglected

The model can be further simplified if only a single component is transported to the

interface. This single component under go chemical reaction immediately it passes the

interface. The second reactant in the second phase should be present in large amount, so

that the concentration does not affect the reaction rate. The reaction release heat in vicinity

of the interface. The reaction considered here is very fast; hence the reaction takes place at

the interface. The component that reaches as reactant to the interface will leave as a

product. Having this simplification we do not have to consider the chemical reaction of the

component so that the reaction term in the mass transport equation (3.57) will be zero. In

the heat transport equation at the interface equation (3.62), the reaction term can be

substituted with the source term. The value of the source term is proportional to the local

mass flux at the interface. Away from the interface the heat source in the transport equation

(3.55) is zero. The model equations are solved using numerical discretisation techniques

discussed in the following chapter.

39

4. Numerical Approach of the transport equation

4.1 The nature of numerical Methods and Discretisation Concept

4.1.1 The nature of Numerical Methods

A numerical solution of a differential equation consists of a set of numbers from which the

distribution of the dependent variable φ can be constructed. In this sense, a numerical

method is akin (similar) to a laboratory experiment, in which a set of instrumental reading

enables us to establish the distribution of the measured quantities in the domain under

investigation [9]. Most numerical methods employ methods that give a value of φ at a

finite number of locations called grid points in the calculation domain. The method includes

the tasks of providing a set of algebraic equations for those unknowns and of prescribing an

algorithm for solving the equation.

4.1.2 The discretisation Concept

In focusing attention on the values at the grid points, one can replace the exact solution of

the differential equation with discrete values. One can thus discretise the distribution ofφ ,

and it is appropriate to refer to this class of numerical methods as discretisation methods.

The Algebraic equations involving the unknown values of φ at chosen grid point, which

we shall now name discretisation equation, are derived from the differential equation

governingφ .

For a given differential equation, the required discretisation equation can be derived in

many ways. Some of them are Taylor series formulation, Methods of weighted average,

40

Variation formulation and the control volume formulation (Finite volume method). Here

central - difference control volume (Finite volume) method is employed.

The control-volume finite-difference (CVFD) method is widely used in the numerical

simulation of fluid dynamics, heat transfer and combustion. Several commercial CFD

(Computational fluid dynamics) codes are based on this method. Though there is no one

ultimate numerical approximation scheme, the CVFD method has numerous desirable

features. It naturally maintains conservation of species when applied to conservation laws.

It readily handles material discontinuities and conjugate heat transfer problems. It also has

the pedagogical advantage that only simple calculus is required to derive the CVFD

approximation to common conservation equations.

The form of the calculation area is designed based on the given problem. The investigated

capillary is discretised through orthogonal grids. As the mesh size becomes smaller the

solution becomes similar to the exact solution. To optimize saving space and calculation

time the distance of the gridline should be chosen in a way that only those areas where we

expect a great gradient of the transport equation should be finer. For the given problem

therefore the area of the interface requires a finer grid in the vertical direction. The distance

yδ is calculated recursively and through geometric series.

1+= jyFy δδ (4.1)

F is the factor to make the grid finer and it has the value smaller than one. For the

horizontal boundaries the distance reaches its maximum value maxyδ . So in the x - direction

the mesh size is constant X∆ , fig 4.1

41

X∆

y∆

Interface

y

z x Figure 4. 1 Representation of non uniform mesh

4.2 Discretisation of Transport equations

4.2.1 Discretisation of temporal changes

The transport equation described the temporary change of the transport values in

differential volume element as a balance of those streams that cross the boundary of the

volume element. For the general function φ this change is expressed in transit formt∂

∂φ .

The real temporal development of φ at the grid point is approximated through the

calculation at the grid time point mt . We also have to approximate t∂

∂φ based on the time

42

discrete value. Often t∂

∂φ can be discretised using first order back ward differentiation

scheme.

1

1

−

−

−−≈

∂∂

mm

mP

mPm

p tttφφφ (4.2)

The linear profile does not necessary display the real profile of the temporal change of the

transport value, especially for periodical changes. A better approximation can be achieved

by using a second order differentiation. For simplicity and to save computer storage first

order backward differentiation is used for this study.

4.2.2 The vorticity transport equation in the bulk phase

The vorticity transport equation is discretised through finite central different method. The

differential quotient at the grid point is approximated through central difference.

The vorticity transport equation as derived in chapter three is given by

x

gZyxy

Vx

ut o

y

∂∂−−

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ ρ

ρωνωωνωωω

22

2

2

2 1256 (4.3)

The time function is discretised at the grid point P. It is discretised using a first order

backward differencing scheme.

tt

mP

mPm

P δωωω 1−−

≈∂∂

(4.4)

In equation (4.4) superscript 1−m refers to vorticity at time tt ∆− and m refers to vorticity

at time t .

The diffusion term is discretised using central finite difference in x and y direction.

( ) ( )

−−−+

+−≈

∂∂+

∂∂

yyy

xyxSSPNPNEPW

δδωωδωω

δωωωνωων 22

2

2

2 2 (4.5)

43

Where ( )SN yyy δδδ +=21

For the wall shear term it is discretised at the grid point P.

PP ZZωνων

22

1212 ≈ (4.6)

Figure 4. 2 Space discretisation of vorticity transport equation The notation of the grid points are according to the compass notation. The two arrows at the

grid points are representing the two velocity components ( Vu & ).

The density term is discretised at the grid point P

x

gx

g we

o

yP

o

y

δρρ

ρρ

ρ−

⋅≈∂∂ (4.7)

The density distribution can be calculated by an appropriate mass law out of temperature

and concentration profiles. Density is calculated at the grid point of temperature and

concentration calculation area.

44

To get a stable and realistic solution the chosen discretisation scheme should have to show

some fundamental properties of real streams. The important ones are:

• Conservativeness

• Tran sportiveness

• Bounded ness

To ensure conservation of φ for the whole solution domain the flux of φ leaving a control

volume across a certain face must be equal to the flux of φ entering the adjacent control

volume through the same face [9]. To achieve this flux through a common face must be

represented in a consistent manner - by one and the same expression in adjacent control

volume.

Diagonal dominance is a desirable feature for satisfying the boundedness criterion .This

states that in the absence of source term, the internal nodal values of the property φ should

be bounded by its boundary value. Another essential requirement for boundedness is that all

coefficients of the discretised equation should have the same sign (usually all positive) [10].

The transportiveness property states that the transport value is basically transported to the

stream down ward. That means the value at the grid point is determined by the stream up

wards.

In order to satisfy the above properties it is important to select the best formulation to

discretise the give transport equations. Here the upwind differencing scheme is used in the

discretisation of the convective terms of the model equations.

The up - wind scheme in the area of finite difference method approximates the convective

term x

u∂∂ω through the stream up ward differentiation. The horizontal second order

approximation of the convective term is given as shown below

45

0,243

0,2

43

< −+−

≈

> +−

≈

∂∂

uwhenx

u

uwhenx

ux

u

EEEPP

WWWPP

P

δωωω

δωωωω

(4.8)

For the term y

V∂∂ω due to non uniform grid lines in the y direction the problem becomes

more complex.

( ) ( )( )

( ) ( )( ) 0,

2

0,2

22

22

<

+−+++−

≈

>

+++−+

≈

∂∂

vwhenyyyy

yyyyyyV

vwhenyyyy

yyyyyyVy

V

NNNNNN

NNNNNNNPNNNNNp

SSSSSS

SSSSSSsPSSSSSp

P

δδδδωδωδδωδδδ

δδδδωδωδδωδδδω

(4.9)

Substituting equation (4.4 to 4.9) in to equation (4.3) and rearranging all the terms, one can

obtain the following equation: Appendix B: 1

baaaaaaaaaa

mP

mmSSSS

mNNNN

mWWWW

mEEEE

mSS

mNN

mWW

mEE

mPP

+++

++++++=−− 11ωωω

ωωωωωωω(4.10)

The discretisation coefficients in the above equation are:

(4.11a)

(4.11b)

(4.11c)

(4.11d)

( ) )11.4()0,max(56

)11.4()0,max(53

)11.4()0,max(53

gVyyy

ya

fux

a

eux

a

PNNNNN

NNN

PWW

PEE

−+

−=

−=

−−=

δδδδ

δ

δ

( ) ( ) ( )( ) ( ) ( )NSS

PSSS

SSSS

NSNP

NNN

NNNN

PW

PE

yyyV

yyyya

yyyV

yyyya

uxx

a

uxx

a

δδδν

δδδδ

δδδν

δδδδ

δδν

δδν

++

+=

++−

+=

+=

−+=

20,max56

20,max56

)0,max(512

)0,max(512

2

2

46

( ) )0,max(56

PSSSSS

SSS V

yyyy

aδδδ

δ+

−= (4.11h)

)11.4(

)11.4(12

)11.4(1

21

1

kx

gb

jZ

aaaaaaaaaa

it

a

we

o

P

mSSNNWWEESNWEP

mP

−−=

+++++++++=

=

−

−

δρρ

ρ

νδ

Unsteady state problems can be discretised in many different ways. Some of the well

known schemes are explicit and fully implicit schemes.

The explicit scheme essentially assumes that the old value 1−mφ prevails through out the

entire time step except at the time tt ∆+ . Explicit scheme is conditionally stable. It has a

great restriction on the time step size. The time step depends on the mesh size ( )( )2xt ∆∝∆

It becomes very expensive to improve spatial accuracy because the maximum possible time

step needs to reduce as the square of x∆ . Consequently this method is not recommended for

general transient problems [10].

The fully implicit scheme assumes that the new value mφ , prevails through out the entire

time step. In fully implicit scheme discretisation both side of the discretisation equation

contain the transport properties ( mφ ) at the new time step and system of algebraic

equations must be solved at the new time step. The time marching procedure starts with a

give initial distribution of the transport properties ( 1−mφ ). The system of equations

developed is solved after selecting time step t∆ . Next the solution mφ is assigned to

1−mφ and the procedure is repeated to progress the solution by a further time step. All the

coefficients of the discretised equations are positive; this makes the implicit scheme

47

unconditionally stable for any time step size. Since the accuracy of the scheme is only first

order in time, small time step are needed to ensure the accuracy of the result. On the ground

of its superior stability the fully implicit scheme is recommended for general purpose CFD

computations [9, 10].

4.2.3 The poison equation

It is used second order central differencing scheme to discretised the Poisson equation. The

discretisation of the Poisson equation (3.20) will result in

( ) ( )

yyy

xSPNNPNEPW

p δδψψδψψ

δψψψω −−−

+−−

= 2

2 (4.12)

PSSNNWWEEPP aaaaa ωψψψψψ −+++= (4.13)

The discretised form of the Poison equation has the following coefficients.

2

1x

aa WE δ== (4.14a)

yy

aN

N δδ1= (4.14b)

yy

aS

S δδ1= (4.14c)

SNWEP aaaaa +++= (4.14d)

4.2.4 The velocity field

The rule used to calculate the velocity component at the grid point is derived from the

definition of the stream function. The course of the stream function is under gone square

interpolation between the three points psnpew and ψψψψψψ ,,,, .

48

To calculate the value at the point “p” for the horizontal velocity component

( ) ( ) ( ) ( )PNNSS

NPN

NSN

SP yyy

yyyy

yu ψψ

δδδδψψ

δδδδ

−+

+−+

≈ (4.15a)

And for the vertical velocity component

x

V WEp δ

ψψ2−

≈ (4.15b)

4.2.5 Mass and heat transport equation in the bulk phases

The equation of mass and heat transport equations are discretised with the help of finite

volume method. The line of the grid is divided in to non overlapping rectangular control

volume. In this method the transport equation is integrated over the given control volume.

In the center of the control volume P, the transport value concentration and temperature are

calculated.

Figure 4. 3 Control Volumes for calculation of temperature and concentration distributions

49

To get the discretised equation therefore we need the equations in the form of conservative

equations as shown below.

Vkkkk

kkkk rM

yC

xC

Dy

CV

xC

ut

Cν+

∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

2

2

2

2

(4.16)

The first term of the above equation represents the time change term and is zero for steady

state problems. The finite volume integration of equation (4.16) over a control volume must

be augmented with a further integration over the finite time step tδ .

Each term in this equation are evaluated and simplified separately. The parts are then

reassembled in to a discrete equation relating at node P to the φ value at nodes ,, WE N

and S .

The temporal mass change in the control volume is approximated with first order

(backward) differencing scheme over the control volume.

yxtCC

dxdytC m

PmP

n

s

e

w

m δδδ

1−−≈

∂∂

∫ ∫ (4.17)

The convective term in equation (4.16) can be integrated once with out approximation.

( ) ( ) ( ) ( )

( ) ( )ssnnwwee

ssnnwwee

n

s

e

w

n

s

e

w

CFCFCFCF

xCvCvyCuCudxdyy

VCdxdyx

uC

−+−≈

−+−≈∂

∂+∂

∂∫ ∫∫ ∫ δδ

(4.18)

To evaluate the right hand side of the preceding expression the values of we CC , , nC

sCand need to be estimated. In the finite volume method, the values are stored only at the

nodes SandNEWP ,,,, . The method for determining an interface value (say, eC ) from the

50

nodal values (say, PE CandC , ) has important consequence for the accuracy of the

numerical model of equation (4.16).

A straight forward method for estimating for the general property eφ in terms of nodal

values PE andφφ is linear interpolation (called central differencing scheme). But it has a

disadvantage of not identifying the flow direction [9]. A better estimation is the up-wind

differencing scheme. Because it takes in to account the flow direction when determining the

value at the cell face: the convective value of eφ at a cell face is taken to be equal to the

value at the upstream node. Based on the above explanation the cell face values are given

as shown below.

0

0

<=

>=

eEe

epe

FifCCFifCC

The value of wC can be define similarly. The conditional statement can be more compactly

written if we define a new operator. We shall define max (a, b) to denote the greater of A

and B, then the up – wind differencing scheme implies

( ) ( )( ) ( )

( ) ( )( ) ( ) )19.4(0,max0,max

)19.4(,0,max0,max19.4(0,max0,max

)19.4(,0,max0,max

dFCFCCFcFCFCCFbFCFCCFaFCFCCF

sPsSss

nNnPnn

wPwWww

eEePee

−−=−−=−−=

−−=

Therefore equation (4.18) becomes

( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( )

)20.4(

0,max0,max0,max0,max

0,max0,max0,max0,max

−+−−−+−++

−−−

=−+−

sPsS

nNnP

wPwW

eEeP

ssnnwwee

FCFCFCFC

FCFCFCFC

CFCFCFCF

For the reaction term

51

dxdyrMdxdyrM Vkk

n

s

e

w Vkk νν∫ ∫ ≈ (4.21)

The third term in equation (4.18) expresses the balance of transport by diffusion in to a

control volume. The integral can be evaluated exactly with out approximation.

( )ax

CCyDx

CCyD

yxCDy

xCDdxdy

xCD

x

WPPE

we

n

s

e

w

22.4

−−

−≈

∂∂−

∂∂≈

∂∂

∂∂

∫ ∫

δδ

δδ

δδ

−−

−≈

∂∂−

∂∂≈

∂∂

∂∂

∫ ∫

s

SP

n

PN

sn

n

s

e

w

yCC

xDy

CCxD

xyCDx

yCDdydx

yCD

y

δδ

δδ

δδ (4.22b)

Therefore adding equation (4.22a) and (4.22b)

Psn

Ss

Nn

WE Cy

xDy

xDxyD

xyDC

yxDC

yxDC

xyDC

xyD

+++−+++

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ (4.22c)

The four diffusive fluxes are replaced by finite difference approximation.

Substituting equation (4.17 to 4.22) in equation (4.16) and rearranging all terms gives as

shown in Appendix B: 2, one can get the following equation.

bCaCaCaCaCaCa mP

mmSS

mNN

mWW

mee

mPP ++++= −− 11 (4.23)

Equation (4.23) applies to each internal node in computational nodes.

The coefficients of the discretised equations are given below

( )

( )

( ) ( )cFy

xDa

bFxyDa

aFxyDa

nN

N

wW

eE

24.40,max

)24.4(0,max

)24.4(0,max

−+=

+=

−+=

δδ

δδδδ

52

( ) ( )

( )( )fb

et

yxa

dFy

xDa

m

SS

S

24.40

24.4

24.40,max

1

=

=

+=

−

δδδ

δδ

( )NSEWmPNWEP FFFFaaaaa −+−−+++= −1 (4.24g)

Where ( )nSew FandFFF ,, represents the volumetric flow rate through the face of the

control volume. They are calculated using the definition of the stream function.

sene

ne

se

ne

see dy

yudyF

ψψ

ψ

−=∂∂== ∫∫

(4.25a)

swnwWF ψψ −= (4.25b)

senw

ne

nw

ne

nwn dx

xVdxF

ψψ

ψ

−=∂∂−== ∫∫ (4.25c)

seswsF ψψ −= (4.25d)

Applying the Continuity equation

( ) ( ) ( ) ( ) ( ) 0≡−−−+−−−=−+− nenwseswseneswnwNSEW FFFF ψψψψψψψψ (4.26)

Therefore equation (4.25) becomes

1−++++= mPSNWEP aaaaaa (4.27)

The up - wind differencing scheme used to approximate the transport value at the control

volume face over sizes the convective term over the conductive term for small stream

velocities. To improve this several methods have been developed. Some of them are hybrid

differencing, QUICK and Power law scheme [10].

53

The hybrid differencing scheme of Spalding (1972) is based on a combination of central

and upwind differencing schemes [9]. The central differencing scheme which is accurate to

second order, is employed for small peclet number ( )2<eP and the upwind scheme, which

is accurate to first order but accounts for transportiveness, is employed for large peclet

numbers

The hybrid differencing scheme uses piecewise formulae based on the local peclet number

to evaluate at the face of the control volume. The scheme is fully conservative and since the

coefficients are all positive it is unconditionally bounded. It satisfies the transportiveness

requirement by using an upwind formulation for large values of peclet number.

A better approximation to the exact solution is given by the power law scheme which is

described in Patankar (1979) [10]. Although some what more complicated than the hybrid

scheme, the Power law expressions are not particularly expensive to compute, and they

provide an extremely good representation of the exponential behavior.

The Quadratic upstream interpolation for convective kinetics (QUICK) scheme of Leonard

(1979) uses a three - point upstream-weighted Quadratic interpolation for cell, face values

[10].

Generally each scheme has its own advantage and disadvantage. The hybrid scheme

produces physically realistic solutions and is highly stable when compared with the higher

order schemes: Hybrid differencing and Power law scheme. Its disadvantage is that the

accuracy in terms of Taylor series truncation error is only first order while third order for

QUICK scheme.

54

Based on the types of the problem one can use one of the scheme mentioned above.

However, for general CFD code hybrid differencing scheme is widely used. Thus, define

the peclet numberDuLPe= , therefore the face values of the peclet becomes

( )

( )

( )

)28.4(

28.4

b28..4

28.4

dxDyFPe

cxDyFPe

yDxFPe

ayDxFPe

sss

nnn

ww

ee

δδδδδδδδ

=

=

=

=

Pe is the peclet number, the dimensionless parameter that describes the relative strength of

convection to diffusion.

Multiplying the discretisation coefficient by the weighting factor ( )PeA

( ) ( ) ( )

( ) ( ) ( )boFwPeAxyDa

aoFePeAxyDa

wW

eE

29.4,max

29.4,max

+=

−+=

δδ

δδ

( ) ( ) ( )

( ) ( ) ( )doFsPeAy

xDa

coFnPeAy

xDa

sS

S

nN

N

29.4,max

29.4,max

+=

−+=

δδ

δδ

The various differencing schemes have different function of ( )PeA . The expression ( )PeA

for various differencing scheme are summarized in table (4.1) [9].

55

Table 4:1 the function ( )PeA for different schemes

Scheme Formula for ( )PeA

Central difference Pe5.01−

Up – wind 1

Hybrid ( )( )PeMax 5.01,0 −

Power law ( )( )51.01,0 PeMax −

Exponential (exact)

( )1exp −

=PePe

PeA

One can get the above equations after solving the convective - conduction equation in the y

direction.

( )30.42

2

dxCdD

dxCu =∂

The best formulation is given by the power low method. It can be applied with out

modification for general transport equation in two or three dimensions. The method

switches between the two peclet number ranges 10≤ep and 10>ep . For the power law

the function is given by

( ) ( )31.41011,0max

5

−= PePeA ne

It is more accurate than the upwind and hybrid differencing schemes.

56

The heat transport equation is discretised as shown below with the same analogy as the

mass transport equation

( )( )32.4

21

2

2

2

2

dxdyTT

ZUHr

c

yT

xTa

dydxyTV

xTu

tT n

s

e

ws

orR

o

n

s

e

w∫ ∫∫ ∫

−−∆

−

∂∂+

∂∂

=

∂∂+

∂∂+

∂∂

ρ

The time function can be discretised

yxtTTdxdy

tT m

Pm

Pn

s

e

w

m

δδδ

1−−≈∂∂

∫ ∫ (4.33)

The convective term is discretised as follows

( ) ( ) ( ) ( )

( ) ( ) ( )34.4ssnnwwee

ssnnwwee

n

s

e

w

n

s

e

w

TFTFTFTF

xTVTVyTuTudxdyy

VTdxdyx

uT

−+−=

−+−≈∂

∂+∂

∂∫ ∫∫ ∫ δδ

Using the up-wind differencing scheme

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )dFTFTTF

cFTFTTFbFTFTTFaFTFTTF

sPsSss

nNnPnn

wPwWww

eEePee

35.40,max0,max35.40,max0,max35.40,max0,max35.40,max0,max

−−=−−=−−=

−−=

( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )

( )36.40,max0,max

0,max0,max0,max0,max0,max0,max

−+−−−+−+−−−

=−+−

sPsS

nNnPwP

wWeEeP

ssnnwwee

FTFTFTFTFT

FTFTFTTFTFTFTF

57

Integrating the diffusion term with in the control volume gives as show below

( )

PSN

SS

NN

WE

S

SP

N

PNWPPE

snwe

n

s

e

w

n

s

e

w

Ty

xay

xaxya

xyaT

yxaT

yxaT

xyaT

xya

yTT

xay

TTxa

xTT

yaxTT

ya

xyTax

yTay

xTay

xTadydx

yTa

ydxdy

xTa

x

+++−+++=

−−

−+

−−

−=

∂∂−

∂∂+

∂∂−

∂∂=

∂∂

∂∂+

∂∂

∂∂

∫ ∫∫ ∫

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδδδ

37.4

The discrete contribution of the source term is obtained by assuming that the source term

has a uniform value through out the control volume.

( ) ( )38.422

121

−

+∆=−+∆ ∫ ∫∫ ∫

PPo

oS

Po

o

vHpo

n

s

e

wS

Po

ovH

n

s

e

w po Tc

yxUyTx

cU

dxdyHrc

dxdyTTc

UdxdyHr

cρ

δδδδρ

ρρρ

Substituting equation (4.33 to 4. 38) in equation (4.32) and rearranging all the terms as

shown in appendix B: 3, one can get the following equation

bTaTaTaTaTaTa mP

mmSS

mNN

mWW

mEE

mPP ++++= −− 11

(4.39)

Where the coefficients of the discretisation equations are

( ) ( ) ( )

( ) ( ) ( )bFPeAxyaa

aFPeAxyaa

wwW

eeE

40.40,max

40.40,max

+=

−+=

δδ

δδ

( ) ( ) ( )

( ) ( ) ( )dFPeAy

xaa

cFPeAy

xaa

ssS

S

nnN

N

40.40,max

40.40,max

+=

−+=

δδδδ

58

( )

( )

( )gTZc

yxUb

fZc

yxUaaaaaa

et

yxa

SPo

o

Po

omSNWEP

m

40.42

40.42

40.42

1

1

ρδδ

ρδδ

δδδ

=

+++++=

=

−

−

The peclet number for the heat transport equation becomes

( )

( )

( )cxayF

Pe

byaxF

Pe

ayaxF

Pe

Nnn

ww

ee

41.4

41.4

41.4

δδδδ

δδ

=

=

=

( )dxayFPe Ss

s 41.4δδ

=

4.2.6 Mass transfer equation at the interface

The control volume is separated in two smallest half at the interface as shown in figure

(4.4). In the upper phase concentration is represented by andC p* the lower phase pC . The

concentration in the lower half is determined by the equilibrium relation as follows.

59

Figure 4. 4 Control volume for mass and heat transport equations at the interface

( )( )( )cNCC

bNCCaCNC

EE

PP

WW

42.442.4

42.4

*

*

*

=

=

=

N is the distribution coefficient of the transported component in phase equilibrium. In a

real system N depends on concentration and temperature. Both dependence of distribution

coefficient is neglected here. For initial concentration distribution N is calculated based

specific equilibrium data. And it is kept constant for the rest of the simulation.

It is important to consider the total flux J that is made up of the convection flux ( )φρu and

the diffusion flux

∂∂Γ−

xφ , thus,

( )x

uJ∂∂Γ−= φφρ (4.43)

60

In this case the control volume is divided in to two parts. The upper part contains the

horizontal flux **ee JandJ , and the lower part contains the horizontal flux of ew JandJ

SNWWEEtotl JJJJJJJ +++++= ** (4.44)

Using equation (3.48)

( )45.40

02

2

2

2

=++∂∂

=−

∂∂

+∂∂

−∂∂+

∂∂

+∂∂+

∂∂

+∂∂

bJt

C

rVMyC

xC

DyvC

yC

VxuC

xC

ut

C

totalk

kkkk

kkk

kkk ν

Vkk

kkkk

kk

ktotal

rMbyC

xC

DyvC

yC

vxuC

xC

uJwhere

ν−=

∂∂

+∂∂

−∂∂+

∂∂

+∂∂+

∂∂

=− 2

2

2

2

:

The temporal mass change is discretised in the same fashion as the bulk phase

( )t

CNt

Ct

Ct

C kkk

∂∂+=

∂∂+

∂∂

=∂∂ **

21

21 (4.46)

yxtCCNdxdy

tC m

PmP

n

s

e

w

m

δδδ

1*

21 −−+≈

∂∂

∫ ∫ (4.47)

The fluxes at the boundary are given as follows

(4.48a)

With the same analogy as above the other flux can formulated as follows

( )

( ) ( )0,max0,max2

2

22

*****

*

***

***

eEePPEe

e

e

eeee

e

eeee

FCFCxCCyD

J

xCyD

CFJ

xCyDyCuJ

−−+

−−=

∂∂−=

∂∂−=

δδ

δ

δδ

61

( ) ( )( )0,max0,max2

*****

eEePPEe

e FCFCNxCCN

yDJ −−+

−−=

δδ

(4.48b)

( ) ( )0,max0,max2

*****

*wPwW

WPww FCFC

xCCyD

J −−+

−−=

δδ

(4.48c)

( ) ( )[ ]0,max0,max2

*****

wPwWWPw

w FCFCNxCC

NyD

J −−+

−−=

δδ

(4.48d)

( ) ( )[ ]0,max0,max**

nnnPn

Pnnn FCFC

yCC

xDJ −−+

−−=

δδ (4.48e)

( ) ( )[ ]0,max0,max **

sPsss

sPss FNCFC

yCNC

xDJ −−+

−−=

δδ (4.48f)

Substituting equation (4.47 and 4.48) in equation (4.45) and rearranging all the terms as

shown in appendix B: 4, one can get the following equation

1*1*** −−++++= mP

mSSNNWWEEPP CaCaCaCaCaCa (4.49)

The coefficients of the discretisation equations are

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) )50.4(0,max0,max22

50.40,max0,max22

***

***

bFFNPeAxyDPeNA

xyDa

aFFNPeAxyDPeNA

xyDa

wwwww

eeeeE

+++=

−+−++=

δδ

δδ

δδ

δδ

( ) ( ) ( )cFPeNAy

xDa nnN

N 50.40,max2

*

−+=δδ

( ) ( )

( ) ( )( )fkCb

et

yxNa

dFPeAyxDa

k

m

nssS

S

510.4

50.42

1

)50.4(0,max2

1

=

+=

+∆=

−

δδδ

δ

The value of the peclet number at the control volume face is given by

62

( )

( )byD

xFPe

ayDxF

Pe

ee

ee

51.42

51.42

**

δδ

δδ

=

=

( )cyDxF

Pe ww 51.4

2δδ

=

( )

( )

( )fxDyF

Pe

exDyF

Pe

dyD

xFPe

Sss

Nnn

ww

51.4

51.4

51.42 *

*

δδδδδδ

=

=

=

The volume flow rates are given by:-

( )aF nee 52.4int* ψψ −=

( )( )( )dF

cFbF

sww

nww

see

52.452.4

52.4

int

int*

int

ψψψψψψ

−=−=

−=

Therefore the coefficient Pa becomes

( ) ( )sewnewmm

SNWEP FFFNFFFaaNaaaaa −−−−−−+++++= −− **21 ( 4.53)

The last term in equation (4.53) represents the volume flow rate balance of the two half

control volumes. Through the interface there is no convective exchange, so each of the

balances in the corresponding phases fulfills the continuity equation. Applying continuity

equation

( )aFFF new 54.40** =−−

( )bFFF sew 54.40=−−

63

Therefore equation (4.53) becomes

( )55.41−++++= mSNWEP aNaaaaa

At the interface the transport equation is coupled with the amount of substance generated

by the chemical reaction. Hence we should calculate the amount of substance generated at

the interface ( )intJ . For the upper phase it is expressed as follows:-

( )56.4*int

**1

new

mP

mP JJJJyx

tCC

−+−=− −

δδδ

Solve for intJ

( )57.4**1

int new

mP

mP JJJyx

tCC

J ++−−

=−

δδδ

4.2.7 Heat transfer equation at the interface

With the same approach applied to mass transfer, we can formulate the discretised equation

for the heat transfer at the interface. The only difference is the presence of the source

term ( )PQ . According to the explanation at the end of chapter (3.3.2) the reaction term at

the interface should be approximated through a mass flux dependent heat source.

( )58.4inthJlQ TP ∆−=

hT∆ is the specific amount of enthalpy releasing during the mass transfer process from the

donating to the receiving phase. It is negative for exothermic and positive for endothermic

reaction. Under the assumption that the solvent of both phases are immiscible, it is possible

to calculate the value with the mixing enthalpies of the transfer component in both phases.

The factor l shows the direction of mass transport. It has the value one for mass transport to

the upper phase and negative one to the lower phase.

64

The heat balance at the control volume face of figure (4.4) from equation (3.61) can be

discretised as follows

( )TTZU

rHyT

yxT

yTVc

xTu

tTc S

ovRpopo −+∆−

∂∂

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ 2

2

2

λλρρ (4.59)

With the same analogy as that of mass transfer, the heat fluxes for each control volume

face are given below

( )

( )

( ) ( )[ ] ( )aFTFTcxTTyJ

xTTyTFcJ

xTyyTucJ

eeePPoPe

e

PeeePoe

eeePoe

60.40,max0,max2

2

22

*********

*

********

******

−−+

−−=

−−=

∂−=

ρδ

δλ

δδλρ

δδλδρ

Since the system at the interface are in equilibrium, unlike the mass transfer the heat transfer has

equal interface temperature. One can use the temperature of the lower phase for this particular work.

With the same formulation as above the remaining flux are given below

( ) ( ) ( )[ ] ( )bFTFeTcPeAxTTyJ eePPoe

Pee 60.40,max0,max

2−−+

−−= ρ

δλδ

( ) ( ) ( )[ ] ( )

( ) ( ) ( )[ ] ( )dFTFTcPeAy

TTxJ

cFTFTcPeAy

TTxJ

sPssPosS

sps

nnnPPonN

Pnn

60.40,max0,max

60.40,max0,max ****

−−+

−−=

−−+

−−=

ρδ

λδ

ρδ

λδ

The integral value of the time function in the control volume is given by

( )61.42

1**

yxtTTcc

dxdytTc

mP

mPpopo

n

s

e

w

mpo δδ

δρρ

ρ−−+

≈∂∂

∫ ∫

The heat lost to the surrounding is given by

65

( ) ( ) ( )62.422

yxTTZU

dxdyTTZU

So

n

s

e

wu

o δδ−≈−∫ ∫

Therefore equation (3.61) has the form

( ) ( )63.42

2 int

1**

yxTTZU

hJlJJJJyxtTTcc m

PSo

Tnsew

mP

mPpopo δδδδ

δρρ

−+∆−−+−=−+ −

Substituting equation (4.60to 62) in equation (4.51) and rearranging all the terms as shown

in appendix B: 5, one can get the following equation

( )64.411 bTaTaTaTaTaTa mP

mmSS

mNN

mWW

mEE

mPP +++++= −−

Where the coefficients are given as follows

( ) ( )[ ] ( ) ( ) ( )

( ) ( )[ ] ( ) ( ) ( )bFcFcPeAPeAx

ya

aFcFcPeAPeAx

ya

wpowpowwW

epoepoeeE

65.40,max0,max2

65.40,max0,max2

*****

*****

ρρλλδδ

ρρλλδδ

+++=

−+−++=

( ) ( ) ( )

( ) ( ) ( )

( )fyxt

cca

dPeAy

xFca

cPeAy

xFca

popomp

sS

spoS

nN

npoN

65.42

65.40,max

65.40,max

**1

****

δδδρρ

δλδρ

δδλρ

+=

+=

+−=

−

( )

( )ghJlyTxZU

b

fyxZU

aaaaaa

TSo

omSNWEp

65.42

65.42

int

1

∆−=

+++++= −

δδ

δδ

4.2.8 The vorticity equation at the interface

The horizontal grid lines are distributed over the calculation area in such away that the two

lines are forming the horizontal boundaries of the concentration - temperature control

volume at the interface. The interface itself is in between these lines. For the calculation of

66

the velocity field it is necessary to solve the vorticity transport equation at the interface to

get the two vorticity fields *ωωand . Therefore an additional grid point is implemented at

the interface. The vorticity transport equation has to be discretised at the new grid point,

figure 4.5. The temperature and concentration values at the grid points are available which

allows us to calculate the x - differential of density and interfacial tension with out

interpolation.

Around the interface the vorticity transport equation is discretised for two phase’s for the

horizontal grid points,

Figure 4. 5 ωψ − grid points at the interface

From the correlation for tension balance equation (3.34), we get the vorticity force for the

lower part of the interface.

( )

( )bx

ax

WWW

WWWWWW

66.41

66.41

**

**

ωηησ

ηω

ωηησ

ηω

+∂∂=

+∂∂=

( )

( )

( )ex

dx

cx

EEEEEE

EEE

PPP

66.41

66.41

66.41

**

**

**

ωηησ

ηω

ωηησ

ηω

ωηησ

ηω

+∂∂=

+∂∂=

+∂∂=

67

The main equation for vorticity transport at the interface as derived in chapter three is given

by equation (3.37)

xg

Zyyxxu

t yoo ∂∂−−

∂∂

∂∂+

∂∂=

∂∂+

∂∂ ρωηωηωηωρωρ 22

2 125

6 (4.67)

Each terms of the vorticity transport equation (4.67) are discretised with the help of

equation (4.4), (4.5), (4.6), (4.7) and (4.9).The discrete value of the lower phase

( PEEEWWW ωωωωω ,,,, ) are replaced with the above relations, equation (4.66).

( )68.422

12112

21

21

21

21

56

21

21

***

2

2

*2*

2

2**

**

∂∂+

∂∂−

+−

∂∂

∂∂+

∂∂+

∂∂=

∂∂+

∂∂+

∂∂+

∂∂

xxg

Z

yyxxxxu

tt

y

oooo

ρρωηηω

ωηωηωηωρωρωρωρ

Each terms of equation (4.68) is discretised one by one using finite difference and finally the

result is reassembled.

The time function is discretised as follows

tttt

mP

mP

P

mP

mP

ompoo δ

ωωρδωωρωρωρ

1***

1**

21

21

21

21 −− −+−=

∂∂+

∂∂ (4.69a)

Substituting the discrete value of the lower phase by equation (4.66)

( )69.4143

21

21 *

*1*

**

**

∂−

+−−

+=

∂∂+

∂∂ −

xtttttwem

Pom

Po

Pooo

ooσσ

ηω

δρω

δρω

ηηρρ

δρωρωρ

For the convection term we have only the x-component.

68

( )

( )bx

u

xu

xu

ax

u

xu

xu

EEEPoP

WWWPoPoP

EEEPoP

WWWPoPoP

70.42

3106

23

106

106

70.4243

106

243

106

106

*****

*****

*

δωωωρ

δωωωρωρ

δωωωρ

δωωωρωρ

−+

−+=

∂∂

++− +−

=

∂∂

Substituting for the lower part of the interface by equation (4.66), equation (4.70) becomes

as shown in appendix B: 6.

The diffusion term is discretised by finite difference

( )axx

wxx

EPWEPW 72.42

212

21

21

21

2

****

22

*2*

2

2

+−+

+−=

∂∂+

∂∂

δωωωη

δωωηωηωη

ncorrelatiothebypartslowertheforngSubstituti . For x - component

( )b

xxxx

xxxxx eee

Ewe

Pwww

W

72.4

211

22

1

21

21

*2

*

2

*2

**

2

*

2

*2*

2

2

∂−

++∂−

−−∂−

+=

∂∂+

∂∂

σσδχ

ωδησσ

δ

ωδησσ

δχω

δη

ωηωη

ncorrelatiothebypartslowertheforngSubstituti for y - component

( )

( )byyxyyyyyyyy

ayyyyyyyyyy

Ss

we

SP

SnN

n

Ss

PS

Pn

Nn

73.41

73.4

***

***

ωδδ

ησσδδ

ωδδ

ηδδ

ηωδδ

η

ωδδ

ηωδδ

ηωδδ

ηωδδ

ηωη

+∂−

⋅−

−−=

+−−=

∂∂

∂∂

The source term is discretised at the grid point p.

( )

( )75.422

74.412621

2112

***

**22

**2

−+

−=

∂∂+

∂∂

+∂−

⋅=

+

xxg

xxg

ZxZZ

weweyy

Pwe

δρρ

δρρρρ

ωησσωηωη

69

Substituting equation (69, 71, 72, 73, 74 & 75 in equation (4.68) and rearranging all the

terms one can get

( )76.41*1*1211

*****

baaaaaaaaaa

mp

mmp

mmp

m

mSS

mNN

mWWWW

mEEEE

mWW

mEE

mPP

+++

++++++=−−−−−− ωωω

ωωωωωωω

The coefficients of the discretised equations are

( )auxx

a P

oo

E 77.4)0,max(56

**

2

*

−+

+=δ

ηηρρ

δη

( )

( )cyy

a

buxx

a

NN

P

oo

W

77.4

77.4)0,max(56

*

**

2

*

δδη

δηηρρ

δη

=

++=

( )

( )

( )fux

a

eux

a

dyy

a

P

oo

WW

P

oo

EE

SS

77.4)0,max(103

77.4)0,max(103

77.4

**

**

δηηρρ

δηηρρ

δδη

+−=

−+

−=

=

( ) ( )iZ

aaaaaaaaa

ht

a

gt

a

mmWWEESNWEP

om

om

77.412

)77.4(

)77.4(

2

*1

*1*

*

*1*

1

ηηη

ηη

δρδρ

++++++++=

=

=

−−

−

−

70

( )

( ) ( )0,max10

30,max

103

77.4)0,max(56

21)0,max(

56

21

109611

43

2

22

22

**

pwwwwwo

peeeeeo

powww

poeee

po

S

owewewey

uxx

uxx

juxxx

uxxx

uxZyyxtxxx

gb

δσσ

ηδρ

δσσ

ηδρ

ηδρ

δδσσ

ηδρ

δδσσ

ηδρ

δδδηδρ

δσσ

δρρ

δρρ

−⋅−−

−⋅−

+

−+

−+

−+

++++

−−

−+

−−=

The Poisson equation (3.20) does not have to be solved at the interface, because the normal

velocity is zero at the interface. The interface can be represented by a stream line which is

given by any constant stream function values intψ and get the boundary conditions at the

interface.

4.3 Discretisation of boundary and initial conditions

Once the physical principle involved in the problem have been stated in the form of partial

differential equations, the next step is to lay down the boundary and / or initial condition

applying to the specific physical problem at hand [9]. The formulations of the boundary

conditions require a clear concept of the physical problem to be solved, in order to have

some assurance that a sensible solution exists.

4.3.1 Horizontal boundary

For the stream function and the transport properties TandC,ω , we have to formulate

the condition at the boundaries. Like the phase boundary it is also possible to describe the

horizontal boundaries because the normal velocity component is zero. At the point of the

boarder the value of the stream function is constant.

=== int* ψψψ HRHR Constant (4.78)

The vorticity force is not as such easy to fix at the boundaries. If we use the Poisson

equation, it can happen stability problem and the result it self unreliable. But for free spaces

71

or surface far from the interface of the stream, we can assume that it is free of rotation or

spinning. In case of capillary flow, the horizontal boundaries are given by

HRHR ωω =* (4.79)

For mass transfer, there is no mass flux at the horizontal boundaries.

0=∂∂=

∂∂ ∗

HRHR yC

yC (4.80a)

Besides the normal velocity component is zero, so that there is no convective transport

through the boundaries. To release this condition in the discretisation equations, we set the

coefficients of the boundary nodes to zero. For example, the coefficient ( Na ) of the

upper boundary is zero. The other coefficients are the same as the bulk phase.

For heat transfer, the heat flux through the wall at the horizontal boundaries can be

neglected and taken as adiabatic boundaries as follows.

0=∂∂=

∂∂ ∗

HRHR yT

yT (4.80b)

4.3.2 Interface Boundaries

At the interface we already have the transport equation except the stream function. Under

the condition that 0=V , the values of the stream function along the interface is to be

constant. Hence the Poisson equation at the interface does not have to be solved.

4.3.3 Vertical boundaries

At the vertical boundaries symmetrical boundary conditions are assumed. Symmetrical

boundary conditions are very easy to formulate. A grid point at the left boundaries shall

72

have 0=i and right boundary Mi = . The distance to each of these boundaries shall be x∆21 .

For the vorticity property this symmetrical pattern can be described

( )( )( )( )d

cb

a

jjM

jjM

jMj

jMj

81.481.481.4

81.4

,1,2

,0,1

,1,2

,,1

ωωωω

ωωωω

=

=

=

=

+

+

−−

−

and for the stream function as follows

( )( )b

a

jjM

jmj

82.482.4

,0,1

,,1

ψψψψ=

=

+

−

The boundary points ( ) ( )jMi ,&,0 can be treated like any other point in the middle of the

calculation field because the neighbor points have the necessary wandψ values.

The half control volume for concentration and temperature distribution at the boundary of

the calculation field is united to the complete control volume.

( )( )bTT

aCC

jjM

jjM

83.4

83.4

,0,1

,0,1

=

=

+

+

( )( )bTT

aCC

jMj

jMj

84.4

84.4

,,1

,,1

=

=

−

−

The discretisation of the concentration and temperature equations has to be solved for the

control volume in the range of Mi<=<=0 .

73

4.3.4 Initial conditions

For the start of the simulation initial conditions are required which already fulfill the

differential equation. Initially it is assumed that both phases are at rest. Thus initial

distributions of the stream function and vorticity are given by

( ) ( )( )86.40)0,,(

85.4tan0,,=

=yx

tconsyxωψ

The initial distribution of the concentration and temperature are the result of a one

dimensional diffusion equation. The mass transport equation for one dimension is given by

2

2

yCD

tC

∂∂=

∂∂ (4.87)

The following boundary conditions are considered.

1. At 0=t in both phases , one can have the bulk concentration *∞∞ CandC ( )0≠y

2. At the interface 0=y , for all 0≤t the continuity of mass transfer is granted. i.e.

y

CDyCD

∂∂=

∂∂ ∗

* (4.88)

3. The transport resistance of the interface is neglected. At the interface the phases are

in equilibrium ( )*intint NCC = .Applying the above conditions in equation (4.87) one

can obtain the following solutions.

0≤yfor

( )aDty

DtNCC

Dty

DtCC

yC 89.4

4exp

4exp

2int

2int

−−−=

−−−=

∂∂ ∗

∞∞

ππ

74

0≥yfor

( )btD

ytD

NCCDD

tDy

DtCC

yC 89.4

4exp

4exp

2int

*

2*int ⋅

−−−=

−−−=

∂∂

∗∗

∗∞

∗

∗∞

ππ

Integrating equation (4.89) for the lower phase gives

( ) ( )DtyerfNCCNCtyC

2, *

int*int −−= ∞ (4.90a)

and for the upper phase

( ) ( )tD

yerfNCCDDCtyC

*

*int

*int

2, −

∗−= ∞

∗ (4.90b)

The error function is defined as:-

dte

xerfxerfcx

t∫

−−=

−=

0

221

1

π (4.91)

Away from the interface the initial distribution of both phases is taken to be the bulk

concentration of the diffusing component. Applying the above condition and equation

(4.90) one can obtain a correlation for the interface concentration *intC of the diffusing

component.

( ) ( )92.4*intint NCC

DDCC −∗

−= ∞∗∗

∞

After rearranging it will be

( )93.41

*

int

NDD

CDDC

C

∗+

∗−

=∞∞

∗

75

Equation (4.93) is implicit in ∗intC because the distribution coefficient itself depends on ∗

intC .

The interface concentration has determined through iteration of equation (4.93) using an

equation for ( )CfN = .

To obtain the initial temperature distribution, the heat conduction equation in the capillary

space should be solved

( ) ( )aTTAyTa

tT

94.402

2

−−∂∂=

∂∂

( )bZc

UA

p

o 94.42

0ρ=

In deriving equation (4.94) the convective, reaction and the second derivative of differential

equation in x – direction are neglected and the surrounding and the initial temperature of

the system are the same.

)95.4(0,0, ≠== ytTT o

To obtain the initial temperature distribution from equation (4.94), let us find the solution

for the following equation.

( )96.42

2

yTa

tT

∂∂=

∂∂

This equation describes the one dimensional heat conduction equation in y – direction.

Using equation (4.89a) gives for the mass flow rate expressed in concentration of the lower

phase.

( ) ( )97.4int0int CCt

DtCDm y −=∂∂−= ∞= π

According to the explanation in section (4.2.7), the heat flux can be related as follows

76

( )

( ) ( )bCCDhlB

Where

at

Bmhlq

T

T

98.4

98.4

int

intint

−∆−=

=∆−=

∞π

�

The heat generated at the interface ( )intq� is distributed in to two streams *intq� and intq� for

the upper and lower phases. The heat flow rate of the lower phase is given by

( )at

Bfq 99.4=�

and the upper phase

( ) ( )bt

Bfq 99.41* −=D

The weighting factor (f) ranges between zero and one. The boundary condition at the

interface is therefore

( )

( ) ( )byyT

tBf

ayyT

tBf

100.40,1

100.40,

** +=∂∂=−

−=∂∂=

λ

λ

at the bulk phase

∞±== yTT o , (4.100c)

Under consideration of these boundary conditions one can obtain the result of equation

(4.96) with the help of Laplace transformation for the two phases as follows

( ) ( )ayTat

yerfcaBftyT o 101.40,2

, ≤+=λπ

( ) ( ) ( )byTta

yerfcaBftyT o 101.40,2

1,**

** ≥+−=

λπ

77

Through multiplication of the first term on the right side byAte−

, equation (4.101) is

modified in such way that the heat loss to the surrounding through the wall is considered.

Assuming that the coefficient which describes the intensity of the heat flow through the

wall be constant, one can obtain

( ) ( ) ( )

( ) ( ) ( ) ( )byTta

yerfcAtaBftyT

and

ayTat

yerfcAtaBftyT

o

o

102.40,2

1,

102.40,2

exp,

**

** ≥+−−=

≤+−=

λπ

λπ

The average value for A is

( )ZccU

Apopo

o**

4ρρ +

= (4.103)

Equation (4.102a) and (4.102b) are the required result for the initial equation (4.94a).The

temperature profile in both phases is correlated through the condition *TT = which allows

calculating the weight factor f. With y = 0, equate equation (4.102a) with equation

(4.102b), the result will be

1

1

*

*

+=

aa

f

λλ (4.104)

If one uses a certain time ot in equation (4.90) and (4.102) for the start of the simulation,

then initial distribution of concentration and temperature can be obtained. Depending on

ot different concentration gradients at the interface are resulting.

78

4.3.5 Algorithm

The Algorithm for the solution of transient problems combined mass, heat and momentum

transport equations has described below.

4.3.6 Solution of the discretised equation

In the previous section the governing equations of fluid flow, mass and heat transfer have

been discretised using finite volume and finite difference methods. These result in system

of linear algebraic equations which needs to be solved. The complexity and size of the set

of equations depends on the dimensionality of the problem, the number of grid points and

the discretisation practice [9]. There are two families of solution techniques for linear

algebraic equations: direct method and indirect or iterative methods. The equation system

for the calculation of the distribution ( )yxm ,φ based on the discretisation coefficients is

solved numerically through the inner iteration process.

79

Time iteration

Outer iteration

No

No

Yes

No Yes

Figure 4. 6 Algorithms for the simulation of mass and heat transport discretisation equations using implicit scheme.

Start

Properties, geometry and discretising Data

0:=t Mesh generation, Guess the initial values of TC ,,ω

ttt δ+=: { }TC,,ωφ∈∀ mm φφ =− :1 Estimated value for ψφ &m distribution

Calculate the coefficient of mass transfer according to Eqn. (4.29), (4.24e), (4.27), (4.50) Inner Iteration: )49.4(),23.4(.),,( EqnyxC m and )57.4(.EqnJ in Calculate the coefficient of heat transfer according to Eqn. (4.40), (4.65) Inner Iteration: )64.4(),39.4(.),,( EqnyxT m ),(),,( yxyx σρ Calculate the coefficient vorticity transport equation according to Eqn. (4.11), (4.77) Inner Iteration: )76.4(),10.4(.),,( Eqnyxmω Inner Iteration: )13.4(),( Eqnyxψ , Coefficients Eqn (4.14)

Convergence { },, mmm TCωφ ∈∀

Print out the out put: ),( yxT m ),(),,(),,( yxCyxyx mmm ψω

?endtt == End

80

Iterative methods are based on the repeated application of a relative simple algorithm

leading to eventual convergence after some- time a large number of repetitions. The main

advantage of iterative solution methods are that only non zero coefficients of the equation

need to be stored in core memory.

Jacobean and Gauss - seidel iterative methods are easy to implement in simple computer

programs, but they can be slow to converge when the system of equations are large. Hence

they are not suitable for general CFD procedures. Thomas (1949) developed a technique for

rapid solving tri - diagonal systems that is now called the Thomas Algorithm or the Tri -

diagonal Matrix Algorithm (TDMA) [10]. The tri-diagonal matrix algorithm is actually a

direct method for one dimensional equation, but it can be applied iteratively, in a line-by-

line fashion, to solve multi-dimensional problems and is widely used in CFD program.

Generally the discretisation equations are rearranged so that the equation system along the

horizontal and vertical grid lines have the following form. For instance for the calculation

of the vorticity transport equation m

ji ,ω , where ( )Mi ,,.........2,1,0= is shown blow.

The general vorticity transport equation in the discretised form is rearranged along the

horizontal grid line.

dCBwADE

baaaaaaaaaaa

mEE

mE

mP

mW

mWW

mP

mmP

mmSSSS

mNNNN

mSS

mNN

mEEEE

mEE

mPP

mWW

mWWWW

=−−+−

+++

+++=−−+−−−−−

ωωωω

ωωωωωωωωωωω

)105.4(211

81

=

−

−

−

−

−

−

−−−−

−−−−−

−−−−−

jM

jM

jM

jM

ji

j

j

j

j

mjM

mjM

mjM

mjM

mji

mj

mj

mj

mj

MMM

MMMM

MMMMM

MMMMM

iiiii

OOO

dddd

d

dddd

ADEBADECBADE

CBADE

CBADE

CBADECBADE

CBADCBA

,

,1

,2

,3

,

,3

,2

,1

,0

,

,1

,2

,3

,

,3

,2

,1

,0

1111

22222

33333

33333

22222

1111

.

.

.

.

.

.

.

.

.

.

.

.

0....................................00..............................0

0........................0000

.........

.........

.............00........0

.........

.........

.........0............................................000................................................00.....................................................00.......................................................0

ω

ω

ω

ω

ω

ω

ω

ω

ω

(4.106)

The element of the coefficient matrix is given below

( )( )( )( )( )eaE

daDcaCbaBaaA

jiWWi

jiWi

jiEEi

jiEi

jipi

107.4107.4

107.4

107.4

107.4

,,

,,

,,

,,

,,

−=

−=

−=

−=

=

And the element of the resultant vector

( )faa

aaaabd

mji

jimmji

jim

mjijiSS

mjijiS

mjijiNN

mjijiNi

i 107.42

,,,21

,,,1

2,,,1,,,2,,,1,.,

++

+++−=

−−−−

−−++

ωω

ωωωω

The elements ( )MMi ,1,1,0 −= of the resultant vector contains the boundary conditions

)113.4(: ,2,0,,1,0, gaadd mjjWW

mjjWoo −− ++= ωω

82

( )( )

)107.4(:

107.4:

107.4:

1,,,1,,

,1,1,11

,2,1,11

jaadd

iadd

hadd

mMjMEE

mjMjMEMM

mjMjMEEMM

mjjWW

++

+−−−

−

++=

+=

+=

ωω

ω

ω

The coefficient matrix of the vorticity equation has penta - diagonal shape because of the

number of neighbor points considered per coordination directions. At the discretisation

point of the Poisson, mass and heat equations the immediate neighbors have been involved.

Therefore tri-diagonal matrixes are resulted. The discretised equation of the Poisson, mass

and heat equations have the following forms

dBACbaaaaaaa

mN

mP

mS

mP

mmEE

mPP

mWW

mNN

mP

mmSS

=++

++++=−+− −−−

φφφφφφφφφφ )108.4(211

The matrix is shown below for general function φ

−−−

−−−

mm

mmm

mmm

ii

OO

ACBAC

BAC

BAC

BACBAC

BA

i

0.......000

000......0.......000..0..............0.....000......00.......0

111

222

222

111

−

−

mmj

mmj

mmj

mij

mj

mj

mj

,

1,

2,

,

2,

1,

0,

.

.

.

.

φ

φ

φ

φ

φ

φ

φ

=

−

−

mmj

mmj

mmj

mij

mj

mj

mj

d

d

d

d

d

d

d

,

1,

2,

,

2,

1,

0,

.

.

.

.

(4.109)

83

The coefficients are

( )( )( ))110.4

110.4

110.4

,

,

,

caCbaBaaA

ijS

ijN

ijp

−=

−=

=

The resultant vector is given by

baaad mP

mmWW

mEE +++= −− 11φφφ (4.111)

To solve the penta-diagonal coefficient matrix in an effective way it is possible to use

Guess elimination.

In practice, the iterative process is terminated when some arbitrary convergence criterion is

satisfied [9]. An appropriate convergence criterion depends on the nature of the problem

and on the objective of the computation, A common procedure is to examine the most

significant quantities given by the solution (such as the maximum velocity, total shear

force, a certain pressure drop or overall heat flux) and to require that the iterations be

continued only until the relative change in these quantities between two successive

iterations is greater than a certain small number often the relative change in the grid-point

values of all the dependent variables is used to formulate the convergence criterion . This

type of criterion can sometimes be misleading when heavy under relaxation is used the

change in the dependent variables between successive iterations is intentionally slowed

down; this may create an illusion of convergence although the computed solution may be

far from being converged. A more meaningful method of equations is satisfied by the

current values of the dependent variables.

84

For each grid point, a residual R can be calculated from

nbnbnbnb abaR φφ∑ −+= (4.112)

Obviously, when the discretisation equation is satisfied, R will be zero. A suitable

convergence criterion is to require that largest value of /R/ be less than a certain small

number.

85

5. Results and discussion

The mathematical models developed in the previous chapter have been solved using

MATLAB programming computer code. The code has been written for both one and two

dimensional diffusion equation, and two dimensional diffusion convection equations. The

results have been discussed in the following section. The model has been validated using

analytical solutions of one dimensional diffusion equations and other results from literature

[13].

The stream function vorticity method has some attractive features. The pressure term is

eliminated from the momentum equation by cross differentiation, and instead of dealing

with the continuity equation, and two momentum equation, we need to solve only two

equations to obtain the stream function and the vorticity. Although it has attractive

features, it has also short comings. The vorticity and vorticity potential vectors involve

concepts that are hard to visualize and interpret than the meaning of the velocity component

and pressure. For this reason the vorticity potential has used here only to calculate the

coefficient of the convective term in the discretisation equations.

The solutions of the discretised equations can be obtained for increasing time. Starting with

the given initial concentration and temperature distributions, the values of the concentration

and temperature distribution are obtained at the next time step. The results thus, obtained

are then used to evaluate the concentration and temperature at the end of the second time

step. Thus, the solution proceeds for increasing time until results are obtained over a

specified time or until a particular concentration and temperature level or the steady state is

attained [14].

86

The parameters used in this study have been taken from literature. Some of them are

experimental data [15] and some of them have been calculated using empirical correlations

[16]. For detail information see appendix A.

Table 5. 1 Parameter used for the simulation

Component Density

(m3kg-1) Diffusivity (m2s-1)

Thermal conductivity (w/m.k)

Heat capacity (J/kg.k)

Viscosity (Ns/m2)

Water solution 1041 91088.0 −× 0.609 4186 3109.0 −×

Cyclohexane 779 91017.1 −× 0.135 1263 3107.1 −×

In addition to the parameters given in table (5.1) the following parameters are also used.

The value of distribution coefficient (N) has taken the value 168. The heat of reaction of the

system has the value 11067 −−=∆ kgkJH [13].

As it has been discussed in section (4.1.2), the solution of numerical methods depends on

the size of the mesh. Hence the optimal grid point used for the simulation of concentration

and temperature distributions in this thesis are 51 by 51 grid points. The discussions of the

results are based on those values of the grid points.

87

5.1 Simulation of Concentration Profile

5.1.1 Mass transfer with out chemical reaction

The simulation result for mass transfer with out chemical reaction is described below both

for one and two dimensional diffusion, and two dimension diffusion convection equations.

The results have been presented in the subsequent sections in the form of graphs both for

analytical and numerical solutions.

The concentration profile of acetic acid for both organic and water phases as a function of

position from the interface are plotted for different times as shown in figure 5.2. Initially it

has been assumed that there is no acetic acid in the water phase. Due to molecular motion

of the diffusing component however, acetic acid transfers from the interface to the water

phase. The concentration of the diffusing component (Acetic acid) decreases from its initial

concentration ( )oC in the bulk phase of cyclohexane to the interface and starts to develop

its concentration distribution in the water phase.

The analytical and numerical solutions for one dimensional diffusion equations are

presented in figure 5.1 and figure 5.2 respectively.

88

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

y, cm

C, g

/lt1 = 5 st2 = 30 st3 = 55 st4 = 80 st5 = 105 s

Figure 5. 1 One dimensional Concentration profiles during mass transfer at the interface as function of distance from the interface, in the water phase negative, in the Cyclohexane phase positive, Analytical Solution

To validate the result of the numerical method used, analytical method has been used for

one dimensional diffusion equations. The analytical equation has been solved using Laplace

transformation of one dimensional diffusion equation. It has been solved by implementing

two point boundary conditions and one initial condition [6]. As it has been presented in

figure 5.1 the highest concentration gradient occurs near the interface of the two phases.

Away from the interface the concentration distribution attains its bulk concentration.

89

The result obtained from the numerical discretisation is presented in figure 5.2. The

numerical solutions are compared with the analytical solutions to check the validity of the

numerical method used.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

y, cm

C,

g/l

t1 = 5 s t2 = 30 st3 = 55 s t4 = 80 st5 = 105 s

Figure 5. 2 One dimensional Concentration profiles during mass transfer at the interface as a function of distance from the interface, in the water phase negative, in the Cyclohexane phase positive, Numerical solutions.

As it has been shown in figure 5.2 the result obtained from the numerical discretisation

equations is in good agreement with the analytical solution of one dimensional diffusion

equation. The comparison is presented in section 5.2.

90

Similarly, concentration distribution for two dimensional diffusion equations has been

developed and solved using numerical discretisation methods. The solutions of two

dimensional equations are validated using data from literature [13].

In the discretisation of two dimensional diffusion equations, the value of the required

parameter (temperature or concentration) takes the third dimension. In this sense, the

concentration and temperature distribution of the discretised equations are interpreted in

three dimensional cartitian coordinate system. The result of two dimensional plots for

concentration distribution is shown in figure 5.3. As it has been presented in figure 5.3, the

variation of concentration along the horizontal direction as compared to along the vertical

direction is very small. Since the major variation of concentration is along the vertical

direction emphasis has been given to the variation along the vertical direction for our

discussion. However, the variation of concentration along the horizontal direction is small;

its variation hasn’t been ignored. Instead, the average value of the concentration along the

vertical direction has been considered through out our discussion.

91

0

0.5

1

-1

-0.5

00

10

20

30

40

50

x,cmy, cm

C, g

/l

0

0.5

1

0

0.5

10

10

20

30

40

x, cmy, cmC

, g/l

a). Organic phase b). Water phase

Figure 5. 3 Two dimensional Concentration profile of the diffusing component. Taking this in to consideration the concentration variation along the horizontal direction is

small and in order to visualize the system, the coordinate system has been changed in to

one dimensional coordinate system by taking the average value of concentration along the

vertical direction.

Thus, through out this thesis the discussion is based on the average value of the

concentration along the vertical direction.

92

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

50

y, cm

C,

g/l

t1 =10 st2 = 35 st3 = 65 st4 = 85 st5 = 110 s

Figure 5. 4 Concentration profiles during mass transfer at the interface as a function of distance from the interface, in the water phase negative, in the Cyclohexane phase positive, average value along the horizontal direction for two dimension. When the numerical solutions of one dimensional diffusion equation is compared with two

dimensional diffusion equations, there is some significant difference in the result of two

dimensional discretisation equation from the analytical solution of one dimensional

diffusion equation. The concentration profile for one dimensional discretisation equations

are relatively in good agreement with the analytical solution than two dimensional

discretisation equations. In two dimensional diffusion equations, there is some significant

effect of the lateral diffusion of the diffusing component on the concentration distribution.

Due to this, the concentration distribution becomes flat in two dimensional discretisation

equations. The effect is more pronounced in the organic phase. This may be due to the

solubility difference of acetic acid in cyclohexane and water phases.

93

For pure diffusion processes, the diffusing component is dominantly transferred

perpendicular to the interface. The lateral diffusion (diffusion in the second direction) can

be neglected compare to diffusions perpendicular to the interface. This is conformed from

the result obtained for both one and two dimensional discretisation equations. I.e. for

physical mass transfer both dimension have similar concentration distribution of the

diffusing component. Of course it has been taken for two dimensional discretisation

equations, the average value of the concentration along the horizontal direction.

5.1.2 Mass transfer with chemical reaction

As it has been discussed in section (3.3.2), the reaction has been taken place at the interface

of the two fluids. Due to this the reaction term is not considered in the bulk phases of the

two liquids. Instead it has been considered only at the interface of the two liquids and used

as a source term in the discretisation of mass and heat transport equations.

Reactions that have Hatta number greater than two (Ha > 2) are considered to be fast

reactions. For the reaction considered in this study, the value of the Hatta number is 10.

Hence the reaction can be considered as fast reaction. Since the reaction that has been

considered is fast reaction and it has been taken place at the interface of the two liquids, the

diffusing component is consumed immediately as it reaches the interface. Hence, the

possibility of the diffusing component to penetrate the second phase compared to mass

transfer with out chemical reaction is very small. As a result its distribution in the second

phase is very small. The result of the distribution is shown in figure 5.5. Compared with the

distribution of mass transfer with out chemical reaction, it can be concluded that

94

concentration distribution of the diffusing component grows at slower rate to transfer to the

second phase. However, its possibility to penetrate the second phase is small, once it passes

the interface its transfer mechanism is not affected by chemical reaction. Because the

reaction has been taken place only at the interface. The transfer mechanism is simply

physical mass transfer. This is clearly shown in figure 5.5.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

y, cm

C,g

/l

t1 = 5 st2 = 30 st3 = 55 st4 = 80 st5 = 105 s

Figure 5. 5 Concentration profiles during mass transfer at the interface as a function of distance from the interface, in the water phase negative, in the Cyclohexane phase positive, average value along the horizontal direction for two dimension.

The simulation result obtained for diffusion - convection discretisation equations are some

what different from the result obtained for diffusion equations. This is due to the

incorporation of the convective terms in the discretisation equations. As it has been shown

95

in figure 5.6, the profile is not as smooth as the profile obtained from the discretisation of

two dimensional diffusion equations. The contribution of the convective term in the

transport process is small as compared to the contribution of diffusion term. For both

phases the highest concentration gradient occurs at the interface of the two fluids. This is in

good agreement with the explanation of the two - Film theory.

-1 -0.5 0 0.5 10

5

10

15

20

25

30

35t1 = 5 st2 = 30 st3 = 55 st4 = 80 st5 = 105 s

Figure 5.6 Concentration profiles during mass transfer at the interface as a function of distance from the interface, in the water phase negative, in the Cyclohexane phase positive, average value along the horizontal direction for two dimension.

As it has been shown in the above figures 5.1, 5.2, 5.4, 5.5, and 5.6 for both dimensions and

for all cases there is a common characteristic on the distribution of the diffusing

component. The concentration distribution of the diffusing component decreases from the

bulk phase to the interface for organic phase. But it increases from the bulk phase to the

interface for water phase. With respect to time, the distribution of the diffusing component

decreases from its initial concentration Co to the specified or required concentration in the

96

organic (Cyclohexane) phase and its distribution increases with time in receiving (water)

phase. Since the source of the diffusing component is cyclohexane, the concentration of the

diffusing component decreases with time as it transfers to the second (water) phase.

5.2 Simulation of temperature Profile

Similarly, the temperature distribution of the system has been discussed for both one and

two dimensional discretisation equations. The temperature distribution is obtained due to

the heat generated at the interface of the two liquids by the chemical reaction. The heat

generated has been transported to the two bulk phases by diffusion as well as convection

mode of heat transfer.

The simulation result of one dimensional diffusion / conduction/ equation shows that the

temperature gradient is higher at the interface. This temperature gradient decreases as one

goes towards the bulk Phases of the two liquids. This clearly shows that the heat source

occurs only at the interface of the two liquids.

The temperature distribution is calculated for different times. The temperature profile

increases with increasing time. This will proceed until the required temperature distribution

is obtained or the specified time is attained. This is because the heat generated by the

reaction is greater than the heat lost to the surrounding. The result of one dimensional

temperature distribution is presented in figure 5.7.

97

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 120

20.5

21

21.5

22

22.5

23

y, cm

T, 0

C

t1 = 5 st2 = 30 st3 = 55 st4 = 80 st5 = 105 s

Figure 5. 7 One dimensional temperatures profile at the interface as function of distance from the interface, in the water phase negative, in the Cyclohexane phase positive ,for Uo = 5 w m-1 k-1

As it has been shown in figure 5.7 the temperature distribution decreases from the interface

to the two bulk phases. For this particular model the temperature distribution of the phases

are nearly symmetrical to the interface, because the thermal diffusivity coefficients of the

two fluids are nearly equal. But thoroughly observing the distribution there is still a slight

difference on the growth rate of the temperature distribution on two phases. The

symmetricity disappeared as the thermal diffusivity coefficients differences of the two

liquids become higher and higher.

The temperature distribution of two dimensional discretisation equation is presented in

figure 5.8. From the temperature distribution one can infer that the temperature variation is

98

not the same for the two coordinate systems. I.e. the temperature variation along the

vertical direction is higher than the temperature variation along the horizontal direction.

00.2

0.40.6

0.81

-1-0.5

00.5

119

20

21

22

23

24

25

x, cmy, cm

T, o

C

t1 = 75 s

Figure 5. 8 Two dimensional temperature distribution of the system, for Uo = 5 w m-1 k-1

Considering that the temperature variation along the horizontal direction is small and in

order to visualize the system, the coordinate system has been changed in to one coordinate

system. Accordingly, the temperature variation along the horizontal direction has been

averaged for each raw along the vertical direction. Thus, through out this thesis the

discussion has been based on the average value of the temperature along the vertical

direction.

After averaging the values of the temperature distribution along the horizontal direction, for

two dimensional discretisation equations, similar result has been obtained for the

temperature distribution as that of one dimensional discretisation equation.

99

20.0

20.5

21.0

21.5

22.0

22.5

23.0

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

y , cm

T, o

Ct4 = 100 st3 = 55 st2 = 30 st1 = 5 s

Figure 5. 9 Temperatures profile at the interface as function of distance from the interface, in the water phase negative, in the Cyclohexane phase positive for kmwU o /5= , Average value of temperature along the horizontal direction for two dimensions.

As it has been shown in figure 5.9, the temperature distribution of two dimensional

discretised equations have similar trend as that of one dimensional discretised equation.

However, they have the same trend the gradient of the curves are different. The distribution

for one dimension is sharper than that of two dimensionional discretisation equations. On

the other hand, the temperature distribution of two dimensional discretisation equations is a

little bit flat compared to one dimensional discretisation equation. There are two

possibilities for the above explanation. The first one may be due to the information

(disturbance) transfer from one corner to the other corner in two dimensional discretisation

is very slow. The second one may be due to the convection heat transport of the fluid

particle. In this case, the heat that has been generated at the interface of the two liquids is

100

conveyed to the bulk phases by the convection motion of the fluid particles and in some

extent by molecular motion of the fluid particles. i.e. convective mode of heat transport is

very fast than conductive transport. Consequently the heat has been distributed to the two

bulk phases immediately it generates at the interface. Due to this the temperature

distribution in two dimensional discretisation equation becomes flat.

The maximum temperature occurs at interface of the two liquids for both one and two

dimensional discretised equations. This is because the heat that has been generated due to

chemical reaction occurs only at the interface of the two liquids. From the temperature

profile one can infer that the temperature decreases from the interface to the two bulk

phases at different rates. This is due to different in heat conductivity of the two liquids.

5.3 Validation of the simulation result

Up to now it has been discussed about the results that have been obtained from the

numerical discretisation of this work. But, whether the results are valid or not the results

should be checked with other similar works. Based on this, the results that have been

obtained in this study are compared with other results obtained from literature [13].

Moreover the numerical solutions for one dimensional diffusion equations are compared

with analytical solutions. Validation of one dimensional model is shown in figure 5.10 and

two dimensional models are shown in figure 5.11.

The comparison result of one dimension is presented in figure 5.8. It can be seen that the

numerical solution is in good agreement with the analytical solution. In validating the result

the error analysis technique has been employed. The average error calculated is two percent

(2%).

101

0

5

10

15

20

25

30

35

40

45

50

55

60

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y, cm

C, g

/l

Analytical solution phase1Numerical solution phase1Analytical solution phase2Numerical solution phase2t = 105 s

Figure 5. 10 Concentration profile as a function of distance from the interface, Comparison of numerical solution with analytical solution

The comparison of the temperature distributions is shown in figure 5.11. Some discrete

data has been taken from the literature [13]. The data that has been taken from the literature

are plotted together with the result obtained for this study on the same chart. Comparisons

of the two results show that there is some significant difference between the two results.

The deviation has been calculated using error analysis technique. The average absolute

error obtained is around two percent (1.887%).

102

20

20.5

21

21.5

22

22.5

23

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0y ,c m

T, o

C

Literature resultNumerical solution t = 75 s

Figure 5. 11 Temperature distributions as function of distance from the interface, Comparison of numerical solution with other literature [13]

From the error obtained both for concentration distribution and temperature distribution it

has been concluded that the models that have been developed in thesis simulates the

concentration and temperature distribution satisfactorily.

103

6. Conclusion and Recommendation

6.1 Conclusion

In this study a numerical simulation code has been developed that can be used to analyze

the concentration and temperature distribution of a diffusing component at the interface of

two immiscible liquids. Two dimensional capillary space was considered to develop the

mathematical models for mass, heat and momentum equations. The mathematical model

has been developed for both the bulk phases of the two liquids and at their interface. The

analysis has made using numerical discretised techniques; the Finite Volume Method

(FVM) and Finite difference method. The discretised equations are converted in to

computer code using MATLAB software. The code has been developed for both one and

two dimensional diffusion equations as well as for two dimensional convection diffusion

equations.

Based on the result obtained, the following specific conclusions can be concluded

1. The concentration profile of the diffusing component (Acetic acid) obtained from

the numerical solution is in good agreement with the analytical solution of one

dimensional diffusion equation.

2. The concentration of acetic acid decreases in the cyclohexane and increases in the

water phase with time as it is expected from mass transfer theory.

3. The concentration profile of acetic acid shows similar result for both dimensions of

conduction equations. This leads to the conclusion that the diffusing component is

dominantly transferred in the direction perpendicular to the interface.

104

4. Since the heat has been generated at the interface, the maximum temperature occurs

at the interface of the two fluids.

5. The temperature gradient of one dimensional discretisation equation is sharper than

two dimensional discretisation equations.

As a general conclusion the selected numerical discretisation technique (finite volume

method) give a good result for both concentration and temperature distributions. This

method is widely used in CFD code development, because it considers the conservation of

species in the discretisation processes. As a main conclusion interface mass transfer with

chemical reaction can be simulated with the numerical solution method presented in this

study. By comparing the model result with data obtained from other similar work, it has

been concluded that the model developed for both one and two dimensional diffusion

equations, and two dimensional diffusion convection equations simulate the concentration

and temperature distribution satisfactorily. Moreover from the comparison of the numerical

results with analytical solutions and literature result, it can be conclude that the error is in

the reasonable range.

6.2 Recommendation

As it has been discussed in chapter three, it has been considered that the reaction occurs at

the interface of the two immiscible liquids. Based on this the reaction term is considered as

a source term in the discretisation of mass and heat transport equations. Considering all the

above explanations the following points are recommended as a future work.

105

1. One can include the reaction term in the bulk phases in the discretisation of mass

and heat transport equation.

2. The model can be extended for different reaction schemes with known reaction

kinetics.

3. Experiment is required to determine the accuracy of the model developed provided

that the experimental set up is available.

4. As it has been described in section 3.3.2, the interface is assumed to be flat and

stagnant. Therefore one can extend the model for movable interface and different

shapes of the interface. For example curved interface.

5. The model developed in this study was based on the natural convection generated

by buoyancy force. Therefore one can extend the model for flow generated by

forced convection.

6. The model can be extended to three dimensional discretisation equations

106

Symbols

A Arial, 2m

C Concentration, g l-1

D Diffusion coefficient, m2s-1

F Volumetric flow rate, m-2 s-1, Constant factor for mesh size

H Height, m

∆RH Molar reaction enthalpy, J/mol

J Mass flux

K Over all mass transfer coefficients, m s-1

M Molecular mass, 1−molg

N Distribution coefficient, Molar flux, mol m-2 s-1

Pe Peclet number

T Temperature k, OC

Tb Boiling temperature k, OC

Ts Surrounding temperature k, OC

U Perimeter, m

V Volume, m3

Uo Over all heat transfer coefficient, w m-2 s-1

Z Length, m

a Temperature coefficient, m2 s-1 Coefficient of discretisation equations.

b Source term (Constant term in discretisation equation)

107

pc Heat Capacity, kJ/ Kg k

erf, ercf error function and complementary error function

g Gravity, m2 s-1

h Height., m

hT∆ Specific enthalpy due to mass transfer, kgkJ

hD Individual mass transfer Coefficient, m s-1

ioh Individual heat transfer coefficient, w m-2 k-1

l Mass transfer direction

mC Mass flow rate, kg m-2 s-1

p Pressure, Pa

q� Heat flux, w m-2

rv reaction rate per unit volume, 13 −− smkmol

S thickness, m

t Time, s

u Horizontal Velocity component, m s-1

v Vertical Velocity Component, m s-1

V Velocity Vector, ms-1

x Horizontal orthogonal coordinate, m

y Vertical orthogonal coordinate, m

z Normal orthogonal coordination, m

108

Greece symbols

δ Film thickness, m

Φ Dissipation energy, J

φ General representation of transport Variable

ω Vorticity-component vector, s-1

ψ Stream function, m2 s-1

α Volume expansion, m3 mol-1

β Thermal expansion, k-1

ε Interface shear viscosity, kg s-1

η Dynamic viscosity, Pa s

κ Interfacial dilatation viscosity, kg s-1

λ Thermal conductivity, W m-1 k-1

ν Kinematics viscosity, m2 s-1

Kν Stoichiometry coefficient for component K

ρ Density, kg m-3

σ Surface tension, N m-1

τ Viscous force /shear force/, Pa

Index

k Component index

i, j vector representation

O initial value

∞ Infinitive

109

E, .N, W, S, EE, NN, WW, SS location of nodal point (gird point)

e, n, w, s Location of control volume face

ne, nw, sw, se Edge of the control volume

P Central grid point

int. Interface

Equ. Equilibrium

Eqn Equation

∗ Upper phase

m time indices

HR Horizontal boundary

110

Reference

1. G. Astarita. Elsevier, 1976, Mass transfer with chemical reaction, Amsterdam,

London.

2. Seader.J.D. Ernest Henley.J., (1998), Separation process principles, John Wiley

and Sons, Inc.

3. Robert E. Treybal, 1981, Mass Transfer operations, McGraw-Hill Book Company-

Singapore

4. Barbara Elvars, 1992, Principles of chemical reaction engineering and plant design,

ULLMan’s Encyclopedia of Industrial chemistry, Volume B4.

5. W. D. Deekwer, 1985, Bubble Column Reactors, John Wiley and Sons, New York

6. J.M. Coulson, J.F. Richardson’s, J.R.Backhurst and J,H. Harker,1999, Coulson &

Richardson’s, Fluid flow, heat transfer and mass transfer. Chemical Engineering,

volume 1, Sixth edition,

7. D. Avnir and M.L. Kagan, 1994, The Evaluation of chemical patterns in reactive

liquids, driven by hydrodynamic instabilities, American Institute of Physics,

CHAOS,Vol. 5, No.3, PP - 589-601

8. Khalid H.Javed, John D.Thornton, Tarry J.Anderson, 1989, Surface phenomena and

Mass transfer rate in Liquid – Liquid systems: Part 2, AIChE Journal, Vol. 35,

No.7,PP-1125-1135

9. Suhas V. Patankar, 1980, Numerical Heat transfer and Fluid flow, Hemisphere

Publishing Corporation, Washington, New York, London

10. H.K Versteeg and W. Malalsekera, 1995, An introduction to computational fluid

dynamics, The finite volume method, Addison Wesley Longman Limited.

111

11. William M. Deen, 1998, Analysis of Transport phenomena, Oxford university

press, New York.

12. Alexander G. Volkov, David W. Deamer, Darrell L. Tanelian, Vladislav S. Markin,

1998, Liquid interface in Chemistry and Biology, John Willey and Sons, Inc., New

York.

13. Alexander Grahn, 2003, Stroemungs instabilitaeten bei Stoffuebergang und

chemischer Reaktion aneder ebenen Grenzflaeche zwischen zweinicht mischbaren

Fluessigkeiten, PhD thesis, German

14. Yogash Jaluria, Kannaeth E. Torrance, 1986, Computational heat transfer, Edward

Brothers Inc., USA

15. Robert C. Reid, John M. Prausnitz, Bruce E. Poling, The properties of Gasses and

Liquids, Fourth Edition

16. R.K. Sinnott, 1999, Coulson and Richardson’s, Chemical Engineering Design,

Chemical Engineering Volume 6, third edition, Butterworth-Heinemann, USA.

17. Stanley M. Walas, 1991, Modeling with differential equations in chemical

engineering, Butterworth-Heinemann, USA.

18. John C. Berbg and Carl R.Morig, 1969, Density effects in interfacial convection,

Chemical engineering science, Vol. 24,pp 937 - 946

19. Jack Winnick, Chemical Engineering Thermodynamics, John Wiley and Sons, Inc.

112

APPENDICES

Appendix A: Prediction of physical Properties of substances

When ever possible, experimental determines Values of Physical Properties should be used.

If reliable values can not be found in the literature and if time or facilities are not available

for their determination then in order to proceed with the design the designer must restore to

estimation.

1. Thermal conductivity of liquids

The temperature dependency of Lλ is weak, and usually Lλ decreases with an increase in

temperature. Empirical Equation for thermal conductivity

2CTBTAL ++=λ

Component A B C λ[w/m.k] T[k] Temp Rang

Water -3.838e-1 5.254e-3 -6.369e-8 6.09e-1 293 273-623

Cyclohexane 2.031e-1 -2.254e-4 -2470e-8 1-35e-1 293 178-581

2. Diffusivity: most liquid-diffusion coefficient equation result from empirical modification

of the stokes Einteine equation, which predicates for the diffusion of a large spherical

molecule A through a dilute solution B. a widely used correlation for oABD is the Wilke -

Chang technique

( )

6.0

5.08104.7

AB

BoAB

TMD

νηΦ

×= −

Where oAbD = Mutual diffusion Coefficient of Solute at very low concentration in solvent B, sec

2cm

113

MB = Molecular weight of solvent B

T = Assault temperature, K

ηB = viscosity of solvent B, cp 048..1285.0 cA V=V

AV = Molar volume of solute A at its normal boiling temperature, molgcm

.3

Φ = Association factor for solvent B, dimensionless

Wilke and Chang recommended that ϕ be chosen as follows

=Φ

edsolventunassociatforMethanolforEthanolfor

waterissolventtheif

19.15.16.2

3. Liquid Heat Capacity

To determine heat capacities of organic liquids over wide temperature ranges,

corresponding states methods are normally used and the difference between the heat

capacity of liquids of in the ideate gas state is correlated with the eccentric factor of the

reduced temperature.

( ) ( ) ( ){ }[ ]14 1634.0164.1167.32.25.0 −−+−++−= rroppL TTRCC ϖ

cr T

TT = Reduced temperature

ϖ = Acentric factor

32 dTcTbTaC op +++=

=opC Ideal gas heat capacity at the same temperature to evaluate pLC

114

The values of the constants are given below.

Component H2O Water C2H4O2

Acetic Acid

C6 H12

Cyclohexane

a 32.243 4.846 -54.541

b 19.238x10-4 25.485x10-2 61.127x10-2

c 10.555x10-6 -1.753x10-4 -2.523x10-4

d 3.596x10-9 49.488x10-4 13.214x10-9

Mw (g/mol) 18.015 60.052 84.162

Tb (OC) 100 117.9 82.9/80.7

Tc (OC) 647.3 592.7 553.4

Pc (bar) 220.5 57.9 40.7

Ρ (kg/m3) 998 1049 779

VA 658.25 600.94 653.62

VB 283.13 306.21 290.84

Vc (cm3/mol) 57 172 308

ω 0.344 0.447 0.212

4. Viscosity Of liquids Experimental Value

Empirical Equation for Viscosity

CTBAL +

+=ηln

Where A, B and C are constants, T is in 0 C

Compound A B C T Rang, 0C η,cp,(T,0C) Ref

Water -2.471e1 4.209e3 4.577e2 0-370 0.90 (25) 224

Acetic Acid -4.515 1.384e3 - 15-120 1.30 (18) 212

Cyclohexane -4.398 1.380e3 -1.55e3 7-280 0.88 (25) 224

115

Appendix B: Mathematical manipulation of the Discretisation Equations 1. Discretisation equation for vorticity equation

( )

( )( )( ) ( )

( )( ) ( )

( )

( )

( ) ( ) ( )( ) ( ) ( )

( )

( )

( )

−

−+−+

−+

−−−

−

++−

+

+

++

+

+

+

+

−+

=

++

++

+−+

+

+++

+++

+−

++

−

1.

1)0,max(

)0,max(

)0,max(53)0,max(

53

20,max

20,max

)0,max(512

)0,max(512

122

2

0,max2

56

0,max2

56

0,max59

0,max591

1

2

2

2

22

Bx

g

wt

Vwyyy

y

Vwyyy

y

uwx

uwx

wyyy

Vyy

yy

wyyy

Vyy

yy

wuxx

wuxx

w

Zyyy

yyy

Vyyy

yy

yxx

Vyyy

yy

ux

uxt

we

o

P

mPPNN

NNNNN

N

PSSSSSSS

S

PEEPWW

NNSN

PNNN

NNN

SNSS

PSSS

SSS

WP

EP

P

SSSS

NNNN

PNNNN

NNN

N

PSSSS

SS

P

P

δρρ

ρ

δδδδδ

δδδδ

δδ

δδδν

δδδδ

δδδν

δδδδδδ

νδδ

ν

νδδδ

νδδδ

νδδδ

δδ

δδν

δν

δδδδδ

δ

δδ

2. Discretisation equation for mass transport equation

( )

( ) ( )( )

( ) ( )

( ) ( )

( )

+

++

−+

+

++

−+

=

+−++−

++++

+++

− 2.

0,max0,max

0,max0,max

0,max0,max0,max

0,max

1 BCtyx

CFy

xDCFy

xD

CFxyDCF

xyD

C

KMFFF

Fy

xDy

xDxyD

xyD

tyx

mP

Sss

Nnn

WwEe

P

kks

nw

esn

δδδ

δδ

δδ

δδ

δδ

ν

δδ

δδ

δδ

δδ

δδδ

116

3. Discretisation equation for Heat transport equation

( ) ( )( ) ( )

( ) ( )

( ) ( )

( )

+

+

++

−+

+

++

−+

=

+−+++−+

+++++

− 3.2

0,max0,max

0,max0,max

20,max0,max

0,max0,max

1 BTc

yxUT

tyx

TFy

xaTFy

xa

TFxyaTF

xya

T

ZcyxU

FFFF

yxa

yxa

xya

xya

tyx

SPo

omP

SsS

NnN

WwEe

P

Po

o

sn

we

NS

ρδδ

δδδ

δδ

δδ

δδ

δδ

ρδδ

δδ

δδ

δδ

δδ

δδδ

4. Discretisation equation for mass transport equation at the interface

( ) ( )

( )

( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

+

+

++

−+

+

+++

+

−+−++

=

−+++

+−+

+−+

++

++++

− 4..2

1

0,max0,max

0,max0,max22

0,max0,max22

0,max0,max

0,max

0,max2

0,max2

0,max22

1

1

*

**

**

*

BCt

yxN

CFyxDCF

yxD

CFFNxyDN

xyD

CFFNxyDN

xyD

C

FyxDF

yxD

FyxD

FNNyxD

FxyD

FNNxyD

tyxN

mP

SsNn

Www

Eee

P

sn

w

wn

e

e

δδδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδδδδδδδ

δδ

δδδ

117

5. Discretisation equation for Heat transport equation

( )( )

( ) ( )( )( ) ( )

( ) ( )

( ) ( )

( )

( )( )

( ) ( )( )( )

( ) ( )( ) ( )

( ) ( )

( )

∆−

++

+

+

+

+−

+

++

+

+

−+−+

+

=

+−

++

++−

+−+

+

++

+

++

+

−

+

+

5.2

2

0,max

0,max

0,max0,max

2

0,max0,max

2

20,max

0,max

0,max

0,max2

0,max0,max

22

int

1**

****

***

**

***

**

**

****

**

***

**

**

BhJlyTxZU

yTxt

cc

TPeAy

xFc

TPeAy

xFc

T

FcFc

PeA

PeA

xy

T

FcFc

PeA

PeA

xy

T

yxZUFc

PeAy

xFc

PeAy

xFc

FcPeA

PeA

xy

FcFc

PeA

PeA

xy

tyxcc

TSo

mp

popo

mSs

Sspo

mNn

Nnpo

mW

wpowpo

w

w

mE

epoepo

e

e

mP

ospo

sS

npo

nN

wpo

wpow

w

epoepo

e

epopo

δδ

δδδρρ

δλδρ

δδλρ

ρρ

λ

λ

δδ

ρρ

λ

λ

δδ

δδρ

δλδρ

δδλρ

ρλ

λ

δδ

ρρ

λ

λ

δδ

δδδρρ

6. Discretisation equation for Vorticity transport equation at the interface.

( )

( )

( )

( )

( )6.

0,max56

103

0,max56

56

0,max56

103

0,max56

56

109

109

106

106

**

*

**

*

**

*

**

*

**

*

* B

uxxx

uxxx

uxxx

uxxx

uxx

ux

xu

xu

Peeeeeo

EEoo

Peeeo

Eoo

Pwwwwo

WWoo

Pwwwo

Woo

Pweo

PPoo

oP

oP

−

∂−

+

+

+−

∂−

+

+

−

∂−

+

+

+

∂−

+

+

−

∂−

+

+

=

∂∂

+

∂∂

σσηδ

ρωηηρρ

δ

σσηδ

ρωηηρρ

δ

σσηδ

ρωηηρρ

δ

σσηδ

ρωηηρρ

δ

σσηδ

ρωηηρρ

δ

ωρ

ωρ