Post on 11-Mar-2020
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.
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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
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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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