ANALYTICAL SOLUTION AND NUMERICAL SIMULATION

OF ADVECTION-DIFFUSION EQUATION RELATED TO

FUM IGATION PROBLEM

MOHD AZUAN BIN MOHAMAD NOR

UNIVERSITI TEKNOLOGI MALAYSIA

ANALYTICAL SOLUTION AND NUMERICAL SIMULATION

OF ADVECTION-DIFFUSION EQUATION RELATED TO

FUMIGATION PROBLEM

MOHD AZUAN BIN MOHAMAD NOR

A dissertation submitted in partial fulfilment of the

requirements for the award of the degree of

Master of Science

Faculty of Science

Universiti Teknologi Malaysia

APRIL 2017

iii

To my beloved Ma and Abah, for your love and care,

to my siblings, Gee, Wani, Kak Elly and Abang Li for your support and concern,

and,

to my very best friend, Bella for your motivation, intelligence and guidance in life.

iv

ACKNOW LEDGEM ENT

All praise belongs to Allah S.W.T., the Lord of the world. Thankful for the

strength He has given to me to complete my master’s degree. This dissertation has

been completed with support from many people whom I wish to acknowledge their

contribution. I wish to express my sincere appreciation to my supervisor, Dr. Zaiton

Mat Isa for her continuous guidance and support. Special thanks to Assoc. Prof.

Sharidan Shafie for his invaluable sharing sessions. I am thankful to all my family

members especially my parents for their patience, support and understanding, and my

siblings for their concern, support and motivation. I am grateful to my wonderful

friends, especially Bella, Farid and Dayad for their time, continuous support and

motivation.

Once again, thanks to everyone who has directly or indirectly involved for

supporting me to complete this dissertation and my study.

v

ABSTRACT

The dissertation reports the analytical approach and numerical simulation of a

transport phenomenon which is governed by the advection-diffusion equation. The

main objective is to solve this governing equation by both analytical and numerical

methods. One-dimensional advection-diffusion equation is solved by using Laplace

Transformation method. Then, the solution is interpreted in two-dimensional graph

of solute mass concentration over the distance. The graph is plotted using a

mathematical tool i.e. MATLAB software. On the other hand, the numerical

simulation is performed by using Computational Fluid Dynamics (CFD) software,

ANSYS Fluent. In ANSYS Fluent, the solver discretizes the transport equation using

Finite Volume Method where the computational flow domain is divided into control

volumes. The transport model is partly adapted based on a phenomenon of solute

mass concentration distribution in mixture with air in the flow domain of the

cylindrical silo during fumigation. Based on this applications, the initial and

boundary conditions are defined similar to the ones in analytical approach. From this

simulation, data is obtained in form of graphical distribution profile of solute mass

which can be compared against the works from analytical approach, hence, the

verification is achieved. The verification demonstrates the accuracy of numerical

solution on this transport phenomena, thus, it can be used for various experimental

applications.

vi

ABSTRAK

Disertasi ini melaporkan pendekatan analitik dan simulasi berangka terhadap

fenomena angkutan yang diwakili oleh persamaan aliran-resapan. Objektif utama

adalah untuk menyelesaikan persamaan ini dengan kedua-dua kaedah analitik dan

berangka. Persamaan aliran-resapan berdimensi satu diselesaikan dengan

menggunakan kaedah Transformasi Laplace. Kemudian, penyelesaian ditafsirkan

dalam bentuk graf dua dimensi di mana kepekatan jisim bahan terlarut dicatat

mengikut jarak. Graf diplot dengan menggunakan perisian MATLAB. Sebaliknya,

simulasi berangka dilakukan dengan menggunakan perisian Pengiraan Dinamik

Bendalir (CFD), iaitu ANSYS Fluent. Dalam ANSYS Fluent, teknik pendiskretan

pada persamaan angkutan adalah menggunakan Kaedah Isipadu Terhingga di mana

domain aliran pengiraan dibahagikan kepada isipadu terkawal. Model angkutan

direka berdasarkan fenomena taburan kepekatan jisim bahan terlarut dalam campuran

dengan udara di dalam domain aliran silo silinder semasa pengasapan. Berdasarkan

aplikasi ini, syarat awal dan sempadan ditakrifkan sama seperti yang ditakrifkan di

dalam pendekatan analitik. Dari simulasi ini, data diperolehi dalam bentuk grafik

profil taburan jisim bahan terlarut di mana ianya boleh dibandingkan dengan kerja-

kerja dari pendekatan analitik, dengan itu, pengesahan dicapai. Pengesahan

menunjukkan ketepatan penyelesaian berangka pada fenomena angkutan, dengan itu,

ia boleh digunakan untuk pelbagai aplikasi eksperimen.

vii

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOW LEDGEM ENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF SYMBOLS / ABBREVIATIONS xiv

LIST OF APPENDICES xvi

1 INTRODUCTION

1.1 Background of the Study 1

1.2 Statement of the Problem 3

1.3 Objectives of the Study 3

1.4 Scopes of the Study 4

TABLE OF CONTENTS

1.5 Significance of the Study

1.6 Dissertation Outline

4

5

viii

2 LITERATURE REVIEW

2.1 Introduction 7

2.2 Transport Phenomena 7

2.2.1 Transport Phenomena in Grain Fumigation 9

2.3 Conservation of Mass 10

2.4 Fick’s Law 13

2.5 The Advection-Diffusion Equation 14

2.6 Numerical Simulation in Computational Fluid 15

Dynamics (CFD)

3 ANALYTICAL APPROACH

3.1 Introduction 17

3.2 Derivation of Advection-Diffusion Equation 18

3.3 Transport Model 24

3.3.1 Initial and Boundary Conditions 25

3.4 Analytical Approach by Using Laplace 27

Transformation Method

4 ANSYS FLUENT SIMULATION

4.1 Introduction 41

4.2 Flow Domain Geometry of Transport Model 41

4.3 Fluid Flow Analysis Using Fluent Solver 42

4.4 Designing of Flow Domain Geometry Using 43

Design Modeler (DM)

4.5 Meshing of the Flow Domain Using Meshing Mode 49

in ANSYS Fluent

4.6 Solution Setup Using ANSYS Fluent Solver 54

Component

4.7 Solution in ANSYS Fluent Solver Component 64

4.8 Result Analysis in CFD-Post Component 69

5 RESULTS AND DISCUSSION

5.1 Introduction 71

5.2 Verification of the Analytical Solution 72

5.3 Verification of Numerical Simulation Against 76

Analytical Solution

5.4 Concentration Distribution in Grain Fumigation 80

6 SUMMARY AND CONCLUSION

6.1 Introduction 81

6.2 Summary of the Study 81

6.3 Conclusion of the Study 82

6.4 Recommendations 83

REFERENCES 84

ix

Appendices A - C 88 - 96

x

LIST OF TABLES

TABLE TITLE PAGE

4.1 Properties of phosphine gas (Mat Isa, 2014) 58

4.2 Values used for inlet boundary condition 61

4.3 Values used for outlet boundary condition 63

5.1 Value of parameters used for verification of the analytical

solution

74

5.2 Data value obtained from works by van Genuchten and

A lves(1982)

75

5.3 Value of parameters used in analytical solution and

numerical simulation

77

FIGURE

1.1

2.1

2.2

2.3

2.4

3.1

3.2

3.3

4.1

4.2

4.3

xi

LIST OF FIGURES

TITLE PAGE

Example of transport phenomena involving diffusion 2

process

Graphical symbols of the mass conservation equation for a 11

control volume

Schematic of a control volume for the equation of mass 12

conservation

Schematic of solute mass diffuses from the location with 13

higher concentration to the location with lower

concentration

Stage of the discretization process (Moukalled et al., 2016) 16

Schematic of a control volume for one-dimensional 18

problem

An example of cylindrical silo in grain cultivation industry 24

Rectangular as flow domain geometry of the transport 26

model and its boundary conditions

Flow domain geometry of the transport model, adapted 42

from vertical cross-sectional area of cylindrical silo

Fluent Solver (ANSYS Fluent) as fluid flow analysis tool 43

Design Modeler (DM) is used for designing the flow 44

domain geometry

4.4 Selecting the analysis type of 2D and initiating new

xii

44

4.5

geometry to open Design Modeler (DM) component

browser

XY plane selection and sketching of a rectangular to 45

4.6

represent the flow domain for one-dimensional simulation

General dimensions sketching tool used to define the value 45

4.7

of vertical length V1 and horizontal length H2

Selection of sketch for surface generating 46

4.8 Generating of surface based on the selected sketch 47

4.9 Defining the bottom line of geometry as “Inlet” boundary 47

4.10 Generating the “Inlet” boundary 48

4.11 Inlet and outlet boundaries and wall of the flow domain are 48

4.12

defined

Geometry meshing using Meshing Mode component 49

4.13 Initiating the Meshing Mode component in ANSYS Fluent 50

4.14

Solver

Generating of mesh in Meshing Mode component 50

4.15 Default meshing and its number of cell elements 51

4.16 Inserting the face sizing and local refinement options 51

4.17 Setting of the face sizing option 52

4.18 Updating mesh after face sizing set up 52

4.19 Local refinement and its geometry selection 53

4.20 Updating mesh after local refinement set up 53

4.21 Solution setup in ANSYS Fluent Solver component 54

4.22 General dialog box in Solution setup 55

4.23 Mesh checking and report quality function 56

4.24 Defining chemical species transport as the transport model 56

4.25 Local list of phosphine gas creation in Materials setup 57

4.26 Mixture-template of phosphine gas and air 59

4.27 Cell zone conditions set up 60

4.28 The mixture-template is selected in Fluid dialog box 61

4.29 Setting of values used for inlet boundary condition 62

4.30 Setting of values used for outlet boundary condition 63

4.31 Reference Values dialog box 64

4.32 Solution Methods and Solution Controls dialog boxes 65

4.33 Residual Monitors dialog box 66

4.34 Solution Initialization dialog box 67

4.35 Run Calculation dialog box 68

4.36 Example of calculation being run 68

4.37 Vertical line in the flow domain geometry to represent the 70

movement of the solute gas in y-direction

4.38 Example of the concentration distribution profile of solute 70

mass within the flow domain over the distance in y-

direction at given time

5.1 Comparison between works by van Genuchten and Alves 76

(1982) and analytical solution for this study

5.2 Distribution profiles of Phosphine Mass Fraction at time 78

t=900 s

5.3 Distribution profiles of Phosphine Mass Fraction at time 78

t=1800 s.

5.4 Distribution profiles of Phosphine Mass Fraction at time 79

t=2700 s.

5.5 Distribution profiles of Phosphine Mass Fraction at time 79

t=3600 s.

5.6 Concentration distribution of Phosphine Mass Fraction 80

xiii

xiv

p Density

c Solute mass concentration

t Time

St Time interval

x, y, z Cartesian coordinates

Sx, Sy, Sz Space volume

qx Diffusive mass flux in x-direction

Jx Total mass flux in x-direction

d m cv Net change of mass within the Control Volume (CV) during time

^ interval

%min Total mass entering the CV during time interval

'Z'Mout Total mass leaving the CV during time interval

D Solute dispersion coefficient

u = ( u , v , w ) Velocity components

m Mass flow rate

V Del operator

A Input concentration

ro Infinity

LIST OF SYMBOLS / ABBREVIATIONS

xv

K Diffusion dependent variable

L Laplace Transformation

L -1 Inverse Laplace Transformation

/C Laplace transformed diffusion dependent variable

s Laplace transformed time

m 1, m 2 Roots of the characteristic equation

c1, c2 Arbitrary constants

exp Exponential

erfc Error function

CFD Computational Fluid Dynamics

FDM Finite Difference Method

FEM Finite Element Method

FVM Finite Volume Method

xvi

APPENDIX

A

B1

B2

C

LIST OF APPENDICES

TITLE

Table of Laplace Transformation (van Genuchten and

Alves, 1982)

MATLAB Programming Codes for Plotting for

Analytical Solution and works by van Genuchten and

A lves(1982)

Data from works by van Genuchten and Alves (1982)

MATLAB Programming Codes for Plotting for

Analytical Solution and Numerical Simulation

PAGE

88

94

95

96

CHAPTER 1

INTRODUCTION

1.1 Background of the Study

In our daily life, there are so many phenomena occurring in and within the

surrounding environment. These phenomena are taking place right in front of our eyes

and in fact, within human body. From movement of surrounding fluids to blood flow

inside the blood veins, these are among many examples of transport phenomena.

Through phenomena of breathing for example, air is inhaled into the lung.

Within air, there is oxygen gas as one of the components. The molecules of oxygen

are carried by air and moves around throughout the respiratory system until they reach

the lung. Then, the oxygen molecules are carried by blood and moves around within

the blood vessel system all the way through human body. This is happening right

within human body in daily life, and indeed, every second of human life. This transport

phenomena is known as advection and diffusion transport process (Quarteroni et al.,

2001). Advection denotes the oxygen gas is carried by air which is caused by pressure

gradient when breathing (Beard and Bassingthwaighte, 2001). On the other hand, the

2

natural tendency of oxygen molecules move around within air in human body system

can be described by diffusion process (Beard and Bassingthwaighte, 2001).

Another example of transport phenomena also can be found in fabric

manufacturing industry. In order to get the desired colour to dye the fabric, the

concentrated colour dye is poured into a solvent. The mixing reaction between the

liquid dye and the solvent involves diffusion process. At some point, the dye will stop

spreading when its concentration is the same within the solvent or in other words, there

is no more concentration gradient. These processes are illustrated in Figure 1.1.

\

Figure 1.1 Example of transport phenomena involving diffusion process.

There are many other example of real applications involving the advection and

diffusion transport process. The ability to understand, predict and control such

phenomena is essential. This ability offers substantial economic benefits and

contributes for a better life to human being. Thus, through this work, an alternative

approach of mathematical modelling is applied to study this advection and diffusion

transport process.

3

1.2 Statem ent of the Problem

In the subject of transport phenomena, the transport process is full of

uncertainty when it comes to real life examples in many industrial applications such

as reactor design, multiphase flows in furnaces and heat exchangers, design of better

inkjet printers, and grain fumigation in the silo. However, the conventional engineering

methods through field experiments to investigate these uncertainties involve a very

high cost. Thus, an alternative way is being considered in wide range of research area.

Mathematical modelling is one of such alternative to experimental method.

Among many applications, the transport phenomena of solute mass distribution

is an example of advection-diffusion transport process which can be studied by

mathematical modelling, either by analytical or numerical approach. Hence, this work

of solute mass transport process analysis is initiated to contribute to the subject of the

transport phenomena. The one-dimensional advection-diffusion equation of solute

mass transport is considered to be solved by both analytical approach and numerical

simulation.

1.3 Objectives of the Study

The following are list of objectives defined for this study:

i. To solve the one-dimensional advection-diffusion equation by analytical

approach.

4

ii. To perform numerical simulation for the one-dimensional advection-diffusion

equation by using ANSYS Fluent software.

iii. To verify results of the numerical simulation against the analytical solution for

one-dimensional problem of the advection-diffusion equation.

1.4 Scopes of the Study

The transport phenomena of phosphine gas particularly in grain fumigation

application is considered to be studied. This transport process consists of both

advection and diffusion reactions. Only movement of phosphine gas or its

concentration (in terms of mass fraction) distribution in one-dimensional problem is

considered. The transport model is partly adapted from grain fumigation application

in the silo which it is assumed free from the grain and fully filled with air. This model

is solved by both analytical and numerical works. The computational programming

tool in MATLAB software is used to demonstrate the result of analytical work. On the

other hand, ANSYS Fluent software is used to perform the numerical simulation.

Then, the results from both works are compared for verification.

1.5 Significance of the Study

Advection and diffusion processes are indeed among interesting transport

phenomena which occur in surrounding of human life. Fumigation application in a silo

is among these significant phenomena in which there are many uncertainties about

how the fumigant gas (i.e. phosphine gas) is distributed in air within the flow domain

of silo. Thus, a study need to be developed in order to understand the behaviour of

phosphine gas distribution throughout the silo. A mathematical modelling is applied

5

as an alternative approach to conventional engineering method which is normally very

costly. This study is expected to contribute to the answer of those uncertainties so that

the outcomes can be useful in the wide range of related applications in various

industries. In addition, this study is expected to encourage the academic community to

explore further on the mathematical approach which can be applied in various

phenomena of transport and fluid dynamics.

1.6 Dissertation Outline

This dissertation is structured in six chapters. Chapter 1 which consists of six

sections, introduces the study frameworks. First section describes a brief background

of this study. The second section explains the statement of problem, followed by a list

of the objectives for this study. Scopes of the study are presented in the following

section and significance of the study is described in the second last section. Lastly, in

this section, the contents of this dissertation is briefly explained.

In Chapter 2, the literature review from previous and current researches are

discussed. The advection-diffusion equation is introduced as a governing equation in

transport phenomena and its principle of mass conservation and Fick’s Law will be

discussed in details. Subsequently, a brief background of the numerical simulation is

discussed.

The analytical approach of this study will be discussed in Chapter 3. It consists

of the derivation of advection-diffusion equation and finding its analytical solution by

defining the initial and boundary conditions based on the transport model of grain

fumigation. Then, the numerical simulation is discussed for which the step by step of

transport model simulation procedures including the geometry design of flow domain,

6

meshing, setup and solution procedures of ANSYS Fluent software are shown in

Chapter 4.

In Chapter 5, the results of study are presented, compared and the verification

is achieved. The results include the findings from both analytical and numerical works

for which the comparison between both approaches can be performed. Finally,

together with the summary of this study, the conclusion and few possible suggestions

for further study are recommended in Chapter 6.

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