Finite-Difference Approximations to The Heat And Diffusion Equation
FINITE-DIFFERENCE APPROXIMATIONS
TO THE HEAT AND DIFFUSION EQUATION
ANDRI HANRYANSYAH
DEPARTMENT OF PHYSICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2014
THE STATEMENT ABOUT UNDERGRADUATE THESIS
AND INFORMATION RESOURCES AND WITH
COPYRIGHTS GRANTING*
Along with this I declare that this Undergraduate Thesis, which entitled
Finite-Difference Approximations To The Heat And Diffusion Equation are
really my works which directed by advisor commission and not yet proposed in any
form to any universities or colleges. Information Resources that sourced or
quoted from paper or works that is published or unpublished from another writer
are already mentioned in the text and imprinted in the bibliography at the end of
this Undergraduate Thesis. Along with this I grant my copyrights for my
Undergraduate Thesis to Bogor Agricultural University.
Bogor, July 2014
Andri Hanryansyah
NIM G74090039
ABSTRACT
ANDRI HANRYANSYAH. Finite Difference Approximations to the Heat and
Diffusion Equation. Supervised by AGUS KARTONO.
This research review two models of Heat Convection-Diffusion Equation, the
Zero-Order model and First-Order model that are derived from advection equation
and combined with heat source-sink (S) variable. The Zero-Order model’s S value
has linear characteristic as it makes the system temperature changed linearly. The
First Zero model’s S value has exponential characteristic as it makes system
temperature changed exponentially. The incompressible flow condition is used in
this research. Both models, then solved using two Finite Difference Method,
Forward Time Centered Space (FTCS) and Crank-Nicolson (CN). The convection
velocity below 0.3 Mach will not affect the heat transfer process become convection
significantly. The incompressible condition makes the convection velocity would
affect the heat transfer process if has a value above 0.3 Mach. The unstability result
will be appearing in some simulation condition, especially in FTCS method. It is
because the FTCS method is an explicit method which, unconditionally unstable
and very sensitive to iteration value, step size, and some additions constant in the
equation.
Keywords: advection, convection, Crank-Nicolson, diffusion, Finite Difference
ABSTRAK
ANDRI HANRYANSYAH. Aproksimasi Beda Hingga pada Persamaan Panas dan
Difusi. Dibimbing oleh AGUS KARTONO
Penelitian ini mengkaji dua model persamaan difusi konveksi panas, ZeroOrder model dan First-Order model yang diturunkan dari persamaan adveksi dan
dikombinasikan dengan variabel sumber atau penyusutan panas (S). Nilai S pada
Zero-Order model tidak bergantung suhu , sehingga memiliki karakteristik linear
yang menjadikan suhu sistem berubah secara konstan untuk setiap selang waktu.
Nilai S pada First-Order model bergantung terhadap suhu, sehingga suhu sistem
berubah secara eksponensial untuk setiap selang waktu. Kondisi aliran yang bersifat
incompressible diasumsikan pada penelitian ini. Kedua model tersebut diselesaikan
dengan menggunakan dua metoda Beda Hingga. Forward Time Centered Space
(FTCS) and Crank-Nicolson (CN).
Kondisi aliran yang incompressible
menyebabkan kecepatan konveksi dengan nilai dibawah 0.3 Mach tidak akan
mempengaruhi proses transfer panas menjadi konveksi secara signifikan.
Kecepatan konveksi akan mempengaruhi proses transfer panas jika memiliki nilai
diatas 0.3 Mach. Hasil yang tidak stabil akan muncul dalam beberapa kondisi
simulasi, khususnya pada metoda FTCS. Hal ini dikarenakan metoda FTCS
merupakan metoda eksplisit yang secara tanpa syarat tidak stabil dan sangat sensitif
terhadap nilai iterasi, ukuran step, dan beberapa nilai konstanta tambahan di dalam
persamaan.
Kata kunci: adveksi, Beda-Hingga, Crank-Nicolson, difusi, konveksi
FINITE-DIFFERENCE APPROXIMATIONS
TO THE HEAT AND DIFFUSION EQUATION
Undergraduate Thesis
As one of the requirements to get
Bachelor of Sciences Degree
At
Department of Physics
DEPARTMENT OF PHYSICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2014
ANDRI HANRYANSYAH
Project Title: Finite-Difference Approximations to The Heat And Diffusion
Equation
Name
: Andri Hanryansyah
NIM
: G74090039
Approved by
Dr Agus Kartono
Supervisor
Known by
Dr Akhiruddin Maddu
Chief of Department of Physics
Date of graduation :
ACKNOWLEDGEMENTS
The author would like to acknowledge His countless thanks to the Most
Gracious and the Most Merciful, ALLAH subhanawata’ala who always gives His
all the best of this life and there is no question about it. Shalawat and Salaam to the
Prophet Muhammad SAW and his family. This Undergraduate Thesis is submitted
to satisfy one of the requirements in accomplishing the Undergraduate Degree at
the Department Physics, Faculty of Mathematics and Natural Sciences in Bogor
Agricultural University.
The author would wish to take his chance to express his thanks and sincere
gratitude to the chase:
1. My Beloved Parents, Rachmat Jehan and Cucu Aisyah, who always give me
support.
2. My lovely brothers, Indra Nugraha Ramdhani and Alm. Muhammad Fauzan
Nurrahman.
3. Dr. Agus Kartono, my supervisor, who has given his best guidance, advice
and support to write and finish this Undergraduate Thesis.
4. Dr. Akhiruddin Maddu, Chief of Physics Department, Bogor Agricultural
University, Who has given guidance and inspiring thought.
5. Dr. Husin Alatas, who has been my Academic Advisor from 3rd semester
till 7th semester and inspired me to always give my best.
6. Tony Sumaryada Ph.D , my Undergraduate Thesis editor, who has given
critique and advice to my handwriting.
7. All lecturers in Physics Department for their invaluable teaching.
8. Mr. Firman, who has given help for the administration affairs.
9. Arbainah, who always given her best support and helped the writer to finish
this Undergraduate Thesis.
10. Esha Ardhie, Iman Noor, and Caesar Riyadi, my best friends who always
supported me to finish this Undergraduate Thesis.
11. Mr Adi Sunaryo, who has helped the writer to complete this Undergraduate
Thesis.
12. All Physics 46, 47, and 48, who have supported the writer.
13. Anyone that can’t be mentioned directly or indirectly who has helped the
writer in completing this Undergraduate Thesis. The writer does appreciate
any opinions, and suggestions for the improvement of this
Undergraduate Thesis.
Bogor, July 2014
Andri Hanryansyah
TABLE OF CONTENTS
TABLE OF CONTENTS ....................................................................................... vi
TABLE OF FIGURE ............................................................................................. vii
TABLE OF APPENDIX ...................................................................................... viii
INTRODUCTION ................................................................................................... 1
Backgrounds Problem .......................................................................................... 1
Formulation of the Problem ................................................................................. 1
Research Objective .............................................................................................. 1
The Benefits of Research ..................................................................................... 2
The Scope of Research......................................................................................... 2
BACKGROUND THEORY .................................................................................... 3
Heat Diffusion Equation ...................................................................................... 3
Convection-Diffusion Equation ........................................................................... 3
Dirichlet Boundary Condition ............................................................................. 4
Finite Difference Method ..................................................................................... 4
Forward Time Centered Space Scheme ............................................................... 5
Crank Nicolson Method ....................................................................................... 6
METHODS .............................................................................................................. 9
Time and Place of Research ................................................................................. 9
Research Tools ..................................................................................................... 9
Research Method ................................................................................................. 9
Research Flowchart............................................................................................ 10
RESULT AND DISCUSSION .............................................................................. 11
Heat Convection-Diffusion Simulation Using FTCS ........................................ 11
Heat Convection-Diffusion Simulation Using Crank Nicolson ........................ 13
Heat Source-Sink Value..................................................................................... 17
CONCLUSION ..................................................................................................... 18
BIBLIOGRAPHY ................................................................................................. 19
APPENDIX ........................................................................................................... 20
CURRICULUM VITAE ....................................................................................... 28
TABLE OF FIGURE
Figure 1: Mesh on a semi infinite strip used for solution in one dimensional
equation problem.3 .................................................................................................. 5
Figure 2: Four nodes that represent the FTCS calculation process ........................ 6
Figure 3: Six nodes that represent the Crank-Nicolson calculation step
process. .................................................................................................................... 7
Figure 4 : Research Flowchart .............................................................................. 10
Figure 5: A shifted function which represent the advection process11 ................. 11
Figure 6: A smoother function, which represents the diffusion process11 ............ 11
Figure 7 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with Zero-Order model (v = 10 m/s & nx = 40) ....................................... 12
Figure 8 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with First-Order model (v = 10 m/s & nx = 40) ....................................... 12
Figure 9 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 10 m/s & nx = 40). ...................... 14
Figure 10 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with First-Order model (v = 10 m/s & nx = 40). ............ 14
Figure 11 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with Zero-Order model (v = 10 m/s & nx = 400)........... 15
Figure 12 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with First-Order model (v = 10 m/s & nx = 400). .......... 15
Figure 13 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with Zero-Order model (v = 150 m/s & nx = 400)......... 16
Figure 14 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with First-Order model (v = 150 m/s & nx = 400). ........ 16
Figure 16: Temperature at the central node using FTCS for different of times
with various of k constant, & model are shown. ................................................... 17
Figure 17 : Temperature at the central node using Crank Nicolson for
different of times with various of k constant, & model are shown. ...................... 18
TABLE OF APPENDIX
Appendix 1. Example of GUI on Heat Convection-Diffusion MATLAB
Simulation Program Main Menu..................................................... 20
Appendix 2. Example of GUI on Heat Convection-Diffusion MATLAB
Simulation Program Input Menu with Simulation Result. .............. 20
Appendix 3. Zero-Order and First-Order model formulation from FTCS. .......... 21
Appendix 4. Zero-Order and First-Order model formulation from Crank
Nicolson. ......................................................................................... 22
Appendix 5. The 3-D Contour plot shows an unstable result of the
temperature distribution at various of time using FTCS ZeroOrder (left) and First-Order model (right) with v = 10 m/s, nt =
40, nx = 400, k = 0.2, & α = 0.1 ...................................................... 23
Appendix 6. The 3-D Contour plot shows an unstable result of the
temperature distribution at various of time using FTCS ZeroOrder (left) and First-Order model (right) with v = 150 m/s, nt
= 40, nx = 40, k = 0.2, & α = 0.1 ..................................................... 24
Appendix 7. The 2-D Contour plot shows The Temperature Distribution at t
= 0 until 10 seconds using Zero-Order model (left) and FirstOrder model (right) with Crank Nicolson method (v = 10 m/s
& nx = 400) ..................................................................................... 24
Appendix 8. The 2-D Contour plot shows The Temperature Distribution at t
= 0 until 10 seconds using Zero-Order model (left) and FirstOrder model (right) with Crank Nicolson method (v = 150 m/s
& nx = 400) ..................................................................................... 24
Appendix 9. Table of Temperature Dsitribution at t = 0 and t = 10 s, using
Crank-Nicolson Zero-Order Model with various convection
velocity value and k = -0.2 compared to Temperature
Distribution using Crank-Nicoloson and Exact method without
v and k value. ................................................................................... 25
Appendix 10. Table of Temperature Dsitribution at t = 0 and t = 10 s, using
Crank-Nicolson First-Order Model with various convection
velocity value and k = -0.2 compared to Temperature
Distribution using Crank-Nicoloson and Exact method without
v and k value. .................................................................................. 26
Appendix 11. Table of Temperature Distribution at t = 10 s, using CN &
FTCS with Zero-Order and First-Order model compared with
Temperature Distribution at t = 10 s, using CN and FTCS with
Ordinary Heat Diffusion model . .................................................... 27
1
INTRODUCTION
Backgrounds Problem
The heat-transfer equations play a major role in the modeling of some
physical phenomena. These equations are used when attempting to describe heat
transport processes and show the heat distribution on it.1 The heat equations are the
example of Partial Differential Equations, further the heat equations are belonging
to Parabolic Equations that are part of Quasilinear Equations, which is one of the
type Partial Differential Equations.2 Some techniques have been developed to find
an appropriate simulation method of it as same with natural process.1 Numerical
results show that the complete reliability of the proposed algorithms.3
The main concern of this research is to simulate the heat transfer on one
dimensional case with the Dirichlet boundary problem. And then, the influence of
the convection velocity and heat source-sink value which, is based from mass
diffusion equations are given in this simulation and combined with finite-difference
methods.1 Some additions from mass advection diffusion equation, hopefully could
make a better model for heat transfer.
Formulation of the Problem
1.
2.
3.
4.
What is the effect of the convection velocity on heat-transfer process?
What is the effect of heat source-sink value on heat-transfer process?
What is the effect of the incompressible flow condition on heat-transfer
process?
What is the effect of the used by Zero and First-Order Model on heat-transfer
simulation result?
Research Objective
1.
2.
3.
4.
5.
Investigate and analyze the effect of convection velocity on the heat-transfer
process.
Investigate and analyze the effects of heat source-sink value on the heattransfer process.
Investigate and analyze the effects of the incompressible flow condition on
heat-transfer process.
Compare and analyze the Heat-transfer simulation result from the FTCS
method and Crank Nicholson method.
Compare and analyze the Heat-transfer Simulation Result from Zero-Order and
First-Order Model.
2
The Benefits of Research
This research delivers benefits for the field of physics modeling, especially
in Heat Transfer modeling. This Research is expected to help researchers and
people who desire to learn more about heat transfer phenomena.
The Scope of Research
The Simulation result obtained via two modified heat equation. First, heat
equation based on the zero-order diffusion equation. Second, heat equation based
on the first-order diffusion equation. Both equations were solved numerically using
two Finite Difference Methods, Forward Time Centered Space method and Crank
Nicolson method. There are some assumptions on this simulation, such as the
incompressible flow condition are preassumed, there are only two ways of
convection velocity (from right and from left), the diffusion coefficient is
considered as a constant, and The Dirichlet Boundary Condition is used as boundary
condition.
3
BACKGROUND THEORY
Heat Diffusion Equation
The Heat Diffusion Equation belongs to Parabolic Equations that are part
of Quasilinear Equations, which are one of types Partial Differential Equations.2
The equations are usually stated as follows:
∂T
∂t
=α
∂2 T
(1)
∂x2
The � constant known as thermal diffusivity which is a measure of how fast
a material can carry heat away from a higher to lower temperature.4
Convection-Diffusion Equation
Convection-diffusion equation plays a pivotal role in the modeling of some
physics simulation where heat or energy is transformed inside a physical system
due to two processes: convection and diffusion.5 The convection means the
movement of molecules or particles within heat. The diffusion describes the spread
of particles through random motion from regions of higher to lower concentration.
Besides, this equation can be described as advection-diffusion of quantities such as
heat, mass, energy, etc.5 Many people widely used this in analyzing the spread of
solute in a liquid flowing through a tube, long range transport of pollutants in the
atmosphere, flow in porous media and many more. 6,7,8
Many journals, scientific papers, and books stated that the convectiondiffusion equation depending on the context, the same equation can be called as the
advection-diffusion equation. 9 The general equation for convection–diffusion
usually stated : 8, 9,10
∂T
+∇∙ v⃗T =∇∙ α∇T +S
(2)
∂t
In the one-dimensional case, the convection-diffusion equation become
∂T
∂t
∂
∂
∂
∂x
∂x
∂x
+ ∙ v⃗T = ∙ α
T +S
(3)
In a common situation, the diffusion coefficient is considered as a constant
and the volume is incompressible, which means in any pressure condition the
volume is constant. Therefore, this condition makes the velocity won’t affect the
heat transfer process become convective unless the velocity has a value greater than
0.3 Mach.11 Then, the formula simplifies to: 11
∂T
∂t
+v⃗
∂T
∂x
=α
∂2 T
∂x2
+S
(4)
Where T as a temperature variable, which are functions of x and t variable, v as
velocity variable, t as time variable, x as space variable, α as thermal diffusivity
constant, and S as Heat Source or Sink value. If the high temperature inside the
system is increased, means that the reaction inside of it is endotherm (heating
4
phenomena). Otherwise, if the heat inside the system is decreased, means that the
reaction inside of it is exothermic (cooling phenomena).
The Convection-Diffusion with Zero-Order Source Sink Terms. The formulas
are as follows:
∂T
∂t
+v⃗
∂T
∂x
=α
∂2 T
∂x2
+k0
(5)
Where k0 is a Source/Sink value for Zero-Order model.
The Convection-Diffusion with First-Order Source Sink Terms, with the
formula are as follows:
∂T
∂t
+v⃗
∂T
∂x
=α
∂2 T
∂x2
+k1 T
(6)
Where k1 is a Source/Sink value for the First - Order model.
The difference between both formulas are The Heat Source/Sink from ZeroOrder model are not depending on the temperature and the other hand the Heat
Source/Sink from First - Order model are dependent with the temperature.
Dirichlet Boundary Condition
In engineering applications, a Dirichlet boundary condition may also be
referred to as a fixed boundary condition. 12 In this case, we assumed that the
equation has some boundary terms are:
∂T
∂t
+ v⃗
∂T
∂x
=α
∂2 T
∂x2
T x,0 = f x
T 0,t = T(L,t)
+S
0 ≤ x ≤ L, t > 0
(7)
(Initial condition)
(8)
(Boundary condition) (9)
Finite Difference Method
Finite-Difference Methods (FDM) is numerical methods for approximating
solutions of differential equations using finite difference equations. The idea of this
method is to divide the whole interval t0 ≤ t ≤ tmax into M segments of width Δt =
(tmax – t0)/M and approximate the first & second derivatives in the differential
equations for each grid point by the central difference formulas. However, in order
for this system of equations to be solved easily, it should be linear, implying that its
coefficients may not contain any term of x.12,13 The finite difference method obtains
an approximate solution for T (x, t) at a finite set of x and t. The codes developed
in this research based on some literature, the discrete x is uniformly spaced in
interval of 0 ≤ x ≤ L such as. 3
� =
−
∆�,
= , ,…
(10)
5
Where N is the total number of spatial nodes,3
∆� =
(11)
−
Similarly, the t discrete is uniformly spaced in 0 (t0) ≤ t ≤ tmax :3
� =
−
∆�,
= , ,…
(12)
Where M is the total number of time steps and Δt is the size of a time steps3
∆� =
����
−
(13)
The solution domain is depicted in Picture 1.3
Figure 1: Mesh on a semi infinite strip used for solution in one dimensional
equation problem.3
As shown in the Figure 1, the black squares with indicated by a red dash border is
the location of the known initial value T x,0 = f x . The open squares which
indicated by a blue dash border is the location of the known boundary terms
T 0,t =T(L,t). The open circles show the location of the inside points where the
finite difference approximation is calculated.3
Forward Time Centered Space Scheme
The FTCS (Forward-Time Central-Space) method is a finite difference
method used for numerically solving parabolic partial differential equations.14 It is
a first-order method in time, explicit in time, and is conditionally stable when
applied to the heat equation. When practiced as a method for advection equations,
or more generally hyperbolic partial differential equation, it is unstable unless the
artificial viscosity is included. The abbreviation FTCS was first used by Patrick
Roache.15
FTCS method is usually used in Heat Transfer or Fluid Dynamics
Computation. Figure 2 shows four nodes in the grid mesh schematically represent
the FTCS calculation process.
6
Figure 2: Four nodes that represent the FTCS calculation process
From the picture above and discrete approximations, which are based on finite
�� ��
� �
difference methods. The formula for T, , , and
can be obtained as listed
��
�� ��
below :
T=T(x,t)
(14)
��
��
��
� �
��
��
=
=
=
� �,�+ −� �,�
∆�
(15)
� �+ ,� −� �− ,�
∆�
(16)
� �+ ,� − � �,� +� �− ,�
∆�
(17)
From those formulations the Heat Convection-diffusion can be obtained :
Zero-Order model (5) becomes
� �,�+
�∆�
∆�
=
�∆�
∆�
(� �+
,�
− � �,� + � �−
,�
First-Order model (6) becomes
(� �+
,�
− � �,� + � �−
,�
)−
⃗⃗∆�
�
∆�
)−
⃗⃗∆�
�
∆�
� �,�+
(� �+
,�
(� �+
=
− � �−
,�
,�
− � �−
,�
) + � �,� +
) + � �,� +
Crank Nicolson Method
� �,� ∆�
∆�
(18)
(19)
The Crank–Nicolson method is a finite difference method used for
numerically solving the heat equation and similar partial differential equations. 16 It
is a second-order method in time, it is implicit in time and it is numerically stable.
The method was developed by John Crank and Phyllis Nicolson in the middle of
the 20th century.16,17 However, the approximate solutions can still contain
(decaying) spurious oscillations if the ratio of time step Δt times the thermal
diffusivity to the square of space step, Δx2, is large. This method utilizes the T
values in six points as shown in Figure 3.
7
Figure 3: Six nodes that represent the Crank-Nicolson calculation step process.
From the picture above and discrete approximations which are based on finite
difference methods. The formula for T,
below :
��
� �
��
��
=
=
��
,
�� ��
, and
� �,�+ +� �,�
�=
��
�� ��
=
� �
��
can be obtained as listed
(20)
� �,�+ −� �,�
∆�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
4∆�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
∆�
(
)
(
)
(
)
From those formulations the Heat Convection-Diffusion can be obtained:
Zero-Order model (5) becomes
�
= − ⃗⃗⃗⃗⃗⃗ −
4∆�
=
=
�
∆�
+ ⃗⃗⃗⃗⃗⃗ (� �−
�
4∆�
,�
)+
∆�
+
�
= ⃗⃗⃗⃗⃗⃗ −
∆�
−
4∆�
�
∆�
First-Order model (6) becomes
�
=
�
∆�
+ ⃗⃗⃗⃗⃗⃗ (� �−
�
4∆�
,�
)+
4∆�
∆�
+
�
∆�
�
= ⃗⃗⃗⃗⃗⃗ −
∆�
−
4∆�
�
∆�
∆�
+
,
i=2, 3,4,5,... N-1
�
∆�
�
∆�
(� �,� ) +
= − ⃗⃗⃗⃗⃗⃗ −
=
�
�
∆�
�
∆�
,
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
,�
)++
i= 2 3,4,5,.. N-1
)
(
)
(
)
( 9)
�
(� �,� ) +
(
(28)
−
∆�
(24)
�
∆�
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
,�
)
(
)
(
)
8
Using Dirichlet Boundary Condition in equation (7) (8) and (9)
= ,
= ,
= ,
= ,
= � ,� =
= � ,� =
The initial temperature distribution on this research project is considered by
T x, 0 = f x = Sin πx⁄L
(
)
The derivation of Zero-Order and First-Order model with FTCS and CN method
can be seen in Appendix 2 and 3.
9
METHODS
Time and Place of Research
This research was held from March 2013 until January 2014 and takes the
places at the Computation Physics Laboratory, Department of Physics, Faculty of
Mathematics and Natural Sciences, Bogor Agricultural University.
Research Tools
The main tools in this research are DELL Inspiron N4030 Notebook with
specification Intel Core-i3 @2. 53 GHz Processors, 2 GB of RAM, Windows Blue
(8.1) Operating System and some software that already installed, such as: Microsoft
Office 2010, GIMP, Sublime 2 Text Editor, and MATLAB™ 2012b.
Research Method
Literature Study
Literature study was done in order to understand the mechanism of heat
transfer and the effect of convection-diffusion and the numerical methods to solve
partial differential equation. Moreover, this literature study would make easier the
process of result analysis of the program and correlate it with the heat transfer and
the diffusion equation.
Formulating the Equation
Some parameters in heat convection-diffusion such as, convection velocity
and Heat Source-Sink, are written into the equation in mathematical form and added
into the ordinary heat diffusion equation. After the heat convection-diffusion with
some parameter has been written in a new equation. Then, the equation is
discretized using the Finite Difference Method. The Dirichlet Boundary Condition
(Fixed-Boundary) was used as initial and boundary condition.
Simulation Using MATLABTM Software
The value of T(x,0) was defined using equation (32) as an initial condition.
Then, The Dirchlet Boundary Condition is considered as the boundary condition in
this simulation. After the numerical equations are obtained, initial and boundary
condition was considered, those requirements are translated into a pseudo code.
After that, the pseudo codes from the Heat Convection-Diffusion are translated into
MATLAB syntax which are used in MATLAB Software to solve the equation in
the region T(2,t+1) ~ T(L-1, t+1) (t= 1, 2,3... M), after that the result are shown in a 3D
contour or 2-D contour graph in order to simulate the temperature distribution on
the domain of x and shows the Heat Convection-Diffusion Phenomena.
10
Research Flowchart
Start
Formulating heat convectiondiffusion with Source-Sink Value
Consider initial and
boundary condition
Output : The Heat Convection-Diffusion
Equation with Source- Sink Value, initial and
boundary condition.
Formulating numerical equation based on heat
convection diffusion with source-sink value, initial
condition and boundary condition using FDM.
Output : The Heat Convection-Diffusion
Numerical Equation with Source- Sink Value,
initial and boundary condition based on FDM.
Translating the Heat Convection Diffusion numerical
equation into programable MATLAB syntax.
Create and designing GUI for Heat
Convection Diffusion MATLAB Program.
Output : A Heat Convection-Diffusion MATLAB Program
with graphic result that display temperature distribution on 1 Dimension
and can be inputted with some customizing paramater.
Inputing some parameter value to MATLAB program,
analyzing the graphic result using Ms. Excel and compare
the graph result with heat convection diffusion theory.
Output : An Analyzing and review result in form of table of data and
graphical from the Heat Convection Diffusion MATLAB Program which
are processed using Microsoft Excel 2010
Finish
Figure 4 : Research Flowchart
11
RESULT AND DISCUSSION
The advection (convection) diffusion equation are underlies the law of mass
conservation, it is also called a mass balance equation. Then, it is easily seen that
the initial function (initial temperature distribution) at (x,0) is merely shifted in time
and velocity v won’t change the function shape as shown in Fig. 5
Figure 5 : A shifted function which represent the advection process11
Fig. 5 shows that the function shifted from the initial function position. 11 The
positive value of v used on the Fig. 5, which shows the shift’s direction from left to
right and if v use a negative value, the shift’s direction will move from right to left.
Consider if the diffusion equations are the only part of the equation with α
value is greater than 0, with α denoting the heat conductivity or thermal
diffusivity.4,11 The graphic result will mainly interpret the heat convection-diffusion
equation as the result of molecular diffusion caused by Brownian motion of
particles.11
Figure 6: A smoother function, which represents the diffusion process11
Heat Convection-Diffusion Simulation Using FTCS
Generally the length of mesh L = 10 m is considered. The following
parameter input are introduced: thermal diffusivity α = 0.1 (m2/s), mesh iterations
nx = 40, time iterations nt = 40, tmax = 4 s, k constant = -0.2 and convection velocity
v = 10 m/s. The minus sign at k constant means that the system is in the cooling
process. The initial temperature distribution in the domain at t = 0 s is of the form
(32).
Figure. 7 show the temperature distributions in the domain of x for different
of times using FTCS Method with Zero-Order model. Figure. 8 show temperature
distributions in the domain of x for different of times using FTCS Methods with
First-Order model. The different values of times are shown in both of the figures.
Both figures show the convection phenomena which, shown by the temperature
distributions are shifting on each time. The maximum temperature on each
temperature distribution that indicated by a red sign. The convection phenomena
can be seen from the maximum temperatures move in the convection velocity
direction.
12
Figure 7 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with Zero-Order model (v = 10 m/s & nx = 40)
Figure 8 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with First-Order model (v = 10 m/s & nx = 40)
The diffusion phenomena are shown by the changes of temperature
distribution shape. The Shape of temperature distributions is changed become
smaller on each time which, means that the temperature on the domain spread out
to both sideways as the result of heat diffusion caused by motion of particles that
triggered by the maximum temperature on the domain.
13
Even though, those all results are not consistent with the theory that
considers an incompressible flow. It means in any pressure variations the volumes
are constant and in mathematically the v or convection will not affect until the v
value is greater than 0.3 Mach (110 m/s). Both figures result shows that convection
velocity = 10 m/s is enough to affect the heat-transfer process become convection
process. It is opposite with the theory which stated a low v value which has a value
less than 0.3 Mach. So, it won't affect the heat-transfer process become convection
process.11 A huge value of iteration should be inputted on simulation, in order to
make the simulation result become more accurate. Other parameters such as nt, tmax,
L, α, and v are inputted with the same value at first simulation.
Unfortunately the FTCS can't afford a huge iteration value or very small step
size value. It’s because the FTCS is considered as explicit numerical method.4 The
unstability calculation result from the FTCS method (both for Zero-Order and FirstOrder) are shown in Appendix 6. The FTCS method is also could become unstable
if some parameter in simulation has been inputted with unusual value, for example,
Appendix 7 shows an unstable result graphic of FTCS calculation which, has been
inputted with v = 150 m/s.
Heat Convection-Diffusion Simulation Using Crank Nicolson
The heat convection-diffusion simulation terms for CN are same as FTCS
does. The length of mesh L = 10 m is considered. The following parameter input
are introduced: thermal diffusivity α = 0.1 (m2/s), mesh iterations nx = 40, time
iterations nt = 40, tmax = 4 s, k constant = -0.2 and convection velocity v = 10 m/s.
The minus sign at k constant means that the system is in the cooling process. The
initial temperature distribution in the domain at t = 0 s is of the form (32).
Figure. 9 show the temperature distributions in the domain of x for different
of times using CN Method with Zero-Order model. Figure. 10 show temperature
distributions in the domain of x for different of times using CN Methods with FirstOrder model. The different values of times are shown in both of the figures. Both
figures show the convection phenomena which, shown by the temperature
distributions are shifting on each time. The maximum temperature on each
temperature distribution that indicated by a red sign. The convection phenomena
can be seen from the maximum temperatures move in the convection velocity
direction.
Even though, Figure 9 and 10 are not consistent with the theory that
considers an incompressible flow as same with the FTCS simulation on Figure 7
and 8. A huge value of iteration should be inputted on simulation, in order to make
the simulation result become more accurate. In this CN simulation method, the
iteration of mesh nx = 400 is considered. Meanwhile, other parameters such as nt,
tmax, L, α, and v are inputted with the same value as first simulation does.
14
Figure 9 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 10 m/s & nx = 40).
Figure 10 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with First-Order model (v = 10 m/s & nx = 40).
Both CN simulation results with nx = 400 (Zero-Order and First-Order),
shows a graphic which, is relevant to the incompressible flow as shown in Figure
11 and 12. The convection velocity with a value below 0.3 Mach (10 m/s) is not
enough to make the heat transfer process become convection process. It’s very
different with previous CN simulation result with the convection velocity value
below 0.3 Mach is enough to make the heat transfer process become convection.
15
Figure 11 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 10 m/s & nx = 400).
Figure 12 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with First-Order model (v = 10 m/s & nx = 400).
The convection velocity would affect the heat transfer process if the
convection velocity value is above 0.3 Mach. Then, in order to prove that. In this
simulation, the convection velocity with a value above 0.3 Mach is considered, v =
150 m/s. Both graphic results show a relevant result with the incompressible flow
terms. The convection velocity with a value above 0.3 Mach would affect the heat
transfer process become convection process. The Convection phenomena will
16
clearly observe when the results from Figure 13 and 14 are presented in 2-D contour
or table as shown in Appendix 8, 9, and 10. The Appendix 9 show comparison of
CN Zero-Model temperature distribution in t = 4 s with various of convection
velocity, as the temperature distributions are flattened before 10 seconds. The
Appendix 10 show comparison of CN First-Model temperature distribution at t =
10 s with various of convection velocity.
Figure 13 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 150 m/s & nx = 400).
Figure 14 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with First-Order model (v = 150 m/s & nx = 400).
17
Heat Source-Sink Value
From the previous section, the result has shown that The Zero-Order’s
model (both for FTCS and CN) temperature distributions are faster to being
flattened than The First-Order’s model result. The result differences between both
models are caused by heat source-sink value that used in both methods.The ZeroOrder model uses a Heat Source-Sink value, which causes the temperature of the
system changed constantly at each time. The k constant on Zero-Order model are
not depends with temperature on the system. The Heat Source-Sink value on the
First-Order model is not constant. It depends with the temperature on the system.
The Heat Source-Sink value will change if the temperature value is changes.
The simulation has been run with a non - convection term. So, it makes
easier to analyze. The midpoint temperature in this simulation is used as a tracking
point for heat source sink phenomena. The k constant on First-Order would make a
heat sink value that decreased each time step if the systems are in cooling process
as shown by the blue curve in Figure 16 and 17. It is because the k constant in the
First - Order model used as multiplied factor to produce heat sink value. The heat
sink value becomes smaller each time step and become saturated. This cooling
curve has a same trends result with the cooling curve based on newton laws of
cooling.16 Meanwhile, the k constant would produce a bigger heat source value each
time step if the systems are in the heating process as shown on Figure 16 and 17
the temperature increased exponentially.
Figure 15: Temperature at the central node using FTCS for different of times
with various of k constant, & model are shown.
18
Figure 16 : Temperature at the central node using Crank Nicolson for different
of times with various of k constant, & model are shown.
CONCLUSION
A Heat Convection-Diffusion Equation are derived from the heat diffusion
equation, advection equation and combined with heat source-sink (S) value. The S
value is a constant which defines a reaction inside the system are exothermic or
endothermic. The incompressible flow is used as the condition in this simulation.
The Equation categorized into two models. First, the Zero-Order model with the S
value also defined as k constant. Second, the First-Order model with the S value in
the equation depends on system temperature (T) value. Both models, then solved
using two Finite Difference Method, Forward Time Centered Space (FTCS) and
Crank-Nicolson (CN). The Zero-Order model’s S value has linear characteristic as
S of the model makes system temperature changed linearly. The First Zero model’s
S value has exponential characteristic as S on the model makes system temperature
changed exponentially. The incompressible flow condition makes the convection
velocity with a value below 0.3 Mach will not affect the heat transfer process
become convection significantly. The convection velocity would affect the heat
transfer process if has a value above 0.3 Mach. The unstability will be appear in
some simulation, especially in FTCS method. It is because the FTCS method is an
explicit method which, unconditionally unstable and very sensitive to iteration
value, step size, and some additions constant in the equation.
19
BIBLIOGRAPHY
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
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Introduction to Scientific Computing. Berlin: Springer, 2007.
Matthews, John H; Fink, Kurtis D.;. Numerical Methods Using MATLAB.
London: Pearson, 2004.
Recktenwald, Gerald W.;. Finite-Difference Approximations to the Heat
Equation. Portland-Oregon: Portland State University, 2004.
Lienhard IV, John H.; Lienhard V, John H. A Heat Transfer Textbook
Third Edition. Massachussets: Phlogiston Press, 2003.
Mohammadi, Reza. "Exponential B-Spline Solution of ConvectionDiffusion Equations." (Applied Mathematics and Computation) 2013.
Dehgan, M. "Weighted Finite Difference Techniques for the OneDimensional Advection Diffusion Equation." (Applied Mathematics and
Computation) 147, no. 2 (2004): 307-319.
R. C., Mittal and R. K., Jain. "Redefined Cubic B-Splines Collocation
Method for Solving Convection-Diffusion Equations." (Applied
Mathematical Modelling) 2012.
M. K. , Kadalbajoo and P. , Arora. "Taylor-Galerkin B-Spline Finite
Element for the One-Dimensional Advection-Diffusion Method for the
One-Dimensional Advection-Diffusion Equation." (Numerical Methods
for Partial Differential Equations) 26, no. 5 (2010).
Bejan A. Convection Heat Transfer. Wiley, 2013.
Salkuyeh, Davod Khojasteh. "On The Finite Difference Approximation To
The Convection-Diffusion Equation." (Elsevier) 2006.
Hundsdorfer , Willem; Verwer, Jan;. Numerical Solution of TimeDependent Advection-Diffusion- Reaction Equations. Edited by La Jolla
R.Bank, La Jolla R.L. Graham, Wurzburg J.Stoer, Kent R.Varga, &
Tubingen H.Yserentant. New York: Springer, 2003.
A., Cheng; D. T., Cheng;. "Heritage and Early History of The Boundary
Element Method." Engineering Analysis with Boundary Elements
(Elsevier), no. 29 (February 2005): 268-302.
Kaw, Autar, and E. Eric Kalu. Numerical Methods with Applications.
Florida: http://www.autarkaw.com, 2008.
Tannehill, John C.; Anderson, Dale A.; Pletcher, Richard H.;.
Computational Fluid Mechanics and Heat Transfer. 3rd. London: Taylor
& Francis, 2011.
Roache, Patrick;. Fundamentals of Computational Fluid Dynamics.
London: Hermosa Pub, 1998.
Cebeci, Tuncer. Convective Heat Transfer. 2nd Rev. California: Horizon
Publishing, 2002.
Yang, Won Y.; Cao, Wenwu; Chung, Tae-Sang; Morris, John;. Applied
Numerical Methods Using MATLAB. New Jersey: John Wiley and Sons,
2005.
20
APPENDIX
Appendix 1. Example of GUI on Heat Convection-Diffusion MATLAB
Simulation Program Main Menu.
Appendix 2. Example of GUI on Heat Convection-Diffusion MATLAB Simulation
Program Input Menu with Simulation Result.
21
Appendix 3. Zero-Order and First-Order model formulation from FTCS.
The formula for T,
�� ��
,
�� ��
, and
� �
��
��
��
��
� �
��
��
=
of FTCS are listed below:
T=T(x,t)
=
=
� �,�+ −� �,�
∆�
� �+ ,� −� �− ,�
∆�
� �+ ,� − � �,� +� �− ,�
∆�
From those formulations the Heat Convection-Diffusion can be obtained :
Zero-Order model (5) becomes
� �,�+ −� �,�
∆�
+ �⃗
� �+ ,� −� �− ,�
∆�
=�
� �+ ,� − � �,� +� �− ,�
∆�
+
And simplifies it becomes
� �,�+
=
�∆�
∆�
(� �+
,�
− � �,� + � �−
,�
)−
⃗⃗∆�
�
∆�
(� �+
,�
− � �−
,�
) + � �,� +
∆�
First-Order model (6) becomes
� �,�+ −� �,�
∆�
+ �⃗
� �+ ,� −� �− ,�
∆�
And simplifies it becomes
�∆�
∆�
(� �+
� �,� ∆�
,�
=�
� �+ ,� − � �,� +� �− ,�
∆�
� �,�+
− � �,� + � �−
,�
)−
=
⃗⃗∆�
�
∆�
(� �+
,�
+
− � �−
� �,�
,�
) + � �,� +
22
Appendix 4. Zero-Order and First-Order model formulation from Crank Nicolson.
The formula for T,
�� ��
,
�� ��
, and
� �
��
�=
��
��
� �
��
��
=
=
Of Crank Nicolson are listed below:
��
=
� �,�+ +� �,�
� �,�+ −� �,�
∆�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
4∆�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
∆�
From those formulations the Heat Convection-Diffusion can be obtained:
Zero-Order model (5) becomes
� �,�+ −� �,�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
+ �⃗
∆�
4∆�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
�
∆�
=
+
Simplifies it and the equation becomes
− ⃗⃗⃗⃗⃗⃗ −
�
4∆�
�
�
+
∆�
�
∆�
�
⃗⃗⃗⃗⃗⃗
4∆�
(� �−
(� �−
,�
,�+
)+
)+
∆�
−
∆�
+
�
∆�
First-Order model (6) becomes
�
∆�
�
(� �,�+ ) + ⃗⃗⃗⃗⃗⃗ −
(� �,� ) +
�
∆�
4∆�
�
⃗⃗⃗⃗⃗⃗
−
4∆�
�
∆�
(� �+
(� �+
,�
)+
,�+
)=
� �,�+ −� �,�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
+ �⃗
=
∆�
4∆�
� �,�+ +� �,�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
+
∆�
Simplifies it and the equation becomes
�
�
(�
)+
+
− ⃗⃗⃗⃗⃗⃗ −
�
4∆�
∆�
�
∆�
∆�
(� �+
,�+
�− ,�+
)=
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
,�
)
�
∆�
+
∆�
�
⃗⃗⃗⃗⃗⃗
4∆�
�
∆�
(� �−
−
,�
)+
�
(� �,�+ ) + ⃗⃗⃗⃗⃗⃗ −
∆�
−
�
∆�
+
4∆�
(� �,� ) +
The Zero-Order and First-Order model can be represented in matrix format which
later will be formulated using tridiagonal matrix.3
23
Where the coefficients of the interior nodes are:
Zero-Order model :
�
�
= − ⃗⃗⃗⃗⃗ −
,
∆�
= (
=
�
∆�
+ ⃗⃗⃗⃗⃗⃗ (� �−
�
4∆�
First-Order model :
,�
)+
= (
∆�
−
∆�
�
⃗⃗⃗⃗⃗⃗⃗
�
�
)
∆�
+
∆�
�
−
= − ⃗⃗⃗⃗⃗ −
�
∆�
= (
=(
�
∆�
+
�
⃗⃗⃗⃗⃗⃗⃗
∆�
) (� �− ,� ) + (
∆�
= (
∆�
−
)
∆�
�
(� �,� ) +
∆�
+
�
⃗⃗⃗⃗⃗⃗⃗
∆�
�
∆�
i=2,3,4,5,...N-1
∆�
�
∆�
−
+
∆�
�
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
�
−
∆�
)
)
) (� �,� ) + (
Using Dirichlet Boundary Condition in equation (7) (8) & (9)
= ,
= ,
= ,
= ,
)+
i=2,3,4,5,...N-1
,
∆�
,�
= � ,� =
= � ,� =
�
∆�
−
�
⃗⃗⃗⃗⃗⃗⃗
) (� �+ ,� )
∆�
Appendix 5. The 3-D Contour plot shows an unstable result of the temperature
distribution at various of time using FTCS Zero-Order (left) and First-Order model
(right) with v = 10 m/s, nt = 40, nx = 400, k = 0.2, & α = 0.1
24
Appendix 6. The 3-D Contour plot shows an unstable result of the temperature
distribution at various of time using FTCS Zero-Order (left) and First-Order model
(right) with v = 150 m/s, nt = 40, nx = 40, k = 0.2, & α = 0.1
Appendix 7. The 2-D Contour plot shows The Temperature Distribution at t = 0
until 10 seconds using Zero-Order model (left) and First-Order model (right) with
Crank Nicolson method (v = 10 m/s & nx = 400)
Appendix 8. The 2-D Contour plot shows The Temperature Distribution at t = 0
until 10 seconds using Zero-Order model (left) and First-Order model (right) with
Crank Nicolson method (v = 150 m/s & nx = 400)
Appendix 9. Table of Temperature Dsitribution at t = 0 and t = 10 s, using Crank-Nicolson Zero-Order Model with various convection
velocity value and k = -0.2 compared to Temperature Distribution using Crank-Nicoloson and Exact method without v and k value.
25
26
Appendix 10. Table of Temperature Distribution at t = 0 and t = 10 s, using Crank-Nicolson First-Order Model with various convection
velocity
TO THE HEAT AND DIFFUSION EQUATION
ANDRI HANRYANSYAH
DEPARTMENT OF PHYSICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2014
THE STATEMENT ABOUT UNDERGRADUATE THESIS
AND INFORMATION RESOURCES AND WITH
COPYRIGHTS GRANTING*
Along with this I declare that this Undergraduate Thesis, which entitled
Finite-Difference Approximations To The Heat And Diffusion Equation are
really my works which directed by advisor commission and not yet proposed in any
form to any universities or colleges. Information Resources that sourced or
quoted from paper or works that is published or unpublished from another writer
are already mentioned in the text and imprinted in the bibliography at the end of
this Undergraduate Thesis. Along with this I grant my copyrights for my
Undergraduate Thesis to Bogor Agricultural University.
Bogor, July 2014
Andri Hanryansyah
NIM G74090039
ABSTRACT
ANDRI HANRYANSYAH. Finite Difference Approximations to the Heat and
Diffusion Equation. Supervised by AGUS KARTONO.
This research review two models of Heat Convection-Diffusion Equation, the
Zero-Order model and First-Order model that are derived from advection equation
and combined with heat source-sink (S) variable. The Zero-Order model’s S value
has linear characteristic as it makes the system temperature changed linearly. The
First Zero model’s S value has exponential characteristic as it makes system
temperature changed exponentially. The incompressible flow condition is used in
this research. Both models, then solved using two Finite Difference Method,
Forward Time Centered Space (FTCS) and Crank-Nicolson (CN). The convection
velocity below 0.3 Mach will not affect the heat transfer process become convection
significantly. The incompressible condition makes the convection velocity would
affect the heat transfer process if has a value above 0.3 Mach. The unstability result
will be appearing in some simulation condition, especially in FTCS method. It is
because the FTCS method is an explicit method which, unconditionally unstable
and very sensitive to iteration value, step size, and some additions constant in the
equation.
Keywords: advection, convection, Crank-Nicolson, diffusion, Finite Difference
ABSTRAK
ANDRI HANRYANSYAH. Aproksimasi Beda Hingga pada Persamaan Panas dan
Difusi. Dibimbing oleh AGUS KARTONO
Penelitian ini mengkaji dua model persamaan difusi konveksi panas, ZeroOrder model dan First-Order model yang diturunkan dari persamaan adveksi dan
dikombinasikan dengan variabel sumber atau penyusutan panas (S). Nilai S pada
Zero-Order model tidak bergantung suhu , sehingga memiliki karakteristik linear
yang menjadikan suhu sistem berubah secara konstan untuk setiap selang waktu.
Nilai S pada First-Order model bergantung terhadap suhu, sehingga suhu sistem
berubah secara eksponensial untuk setiap selang waktu. Kondisi aliran yang bersifat
incompressible diasumsikan pada penelitian ini. Kedua model tersebut diselesaikan
dengan menggunakan dua metoda Beda Hingga. Forward Time Centered Space
(FTCS) and Crank-Nicolson (CN).
Kondisi aliran yang incompressible
menyebabkan kecepatan konveksi dengan nilai dibawah 0.3 Mach tidak akan
mempengaruhi proses transfer panas menjadi konveksi secara signifikan.
Kecepatan konveksi akan mempengaruhi proses transfer panas jika memiliki nilai
diatas 0.3 Mach. Hasil yang tidak stabil akan muncul dalam beberapa kondisi
simulasi, khususnya pada metoda FTCS. Hal ini dikarenakan metoda FTCS
merupakan metoda eksplisit yang secara tanpa syarat tidak stabil dan sangat sensitif
terhadap nilai iterasi, ukuran step, dan beberapa nilai konstanta tambahan di dalam
persamaan.
Kata kunci: adveksi, Beda-Hingga, Crank-Nicolson, difusi, konveksi
FINITE-DIFFERENCE APPROXIMATIONS
TO THE HEAT AND DIFFUSION EQUATION
Undergraduate Thesis
As one of the requirements to get
Bachelor of Sciences Degree
At
Department of Physics
DEPARTMENT OF PHYSICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2014
ANDRI HANRYANSYAH
Project Title: Finite-Difference Approximations to The Heat And Diffusion
Equation
Name
: Andri Hanryansyah
NIM
: G74090039
Approved by
Dr Agus Kartono
Supervisor
Known by
Dr Akhiruddin Maddu
Chief of Department of Physics
Date of graduation :
ACKNOWLEDGEMENTS
The author would like to acknowledge His countless thanks to the Most
Gracious and the Most Merciful, ALLAH subhanawata’ala who always gives His
all the best of this life and there is no question about it. Shalawat and Salaam to the
Prophet Muhammad SAW and his family. This Undergraduate Thesis is submitted
to satisfy one of the requirements in accomplishing the Undergraduate Degree at
the Department Physics, Faculty of Mathematics and Natural Sciences in Bogor
Agricultural University.
The author would wish to take his chance to express his thanks and sincere
gratitude to the chase:
1. My Beloved Parents, Rachmat Jehan and Cucu Aisyah, who always give me
support.
2. My lovely brothers, Indra Nugraha Ramdhani and Alm. Muhammad Fauzan
Nurrahman.
3. Dr. Agus Kartono, my supervisor, who has given his best guidance, advice
and support to write and finish this Undergraduate Thesis.
4. Dr. Akhiruddin Maddu, Chief of Physics Department, Bogor Agricultural
University, Who has given guidance and inspiring thought.
5. Dr. Husin Alatas, who has been my Academic Advisor from 3rd semester
till 7th semester and inspired me to always give my best.
6. Tony Sumaryada Ph.D , my Undergraduate Thesis editor, who has given
critique and advice to my handwriting.
7. All lecturers in Physics Department for their invaluable teaching.
8. Mr. Firman, who has given help for the administration affairs.
9. Arbainah, who always given her best support and helped the writer to finish
this Undergraduate Thesis.
10. Esha Ardhie, Iman Noor, and Caesar Riyadi, my best friends who always
supported me to finish this Undergraduate Thesis.
11. Mr Adi Sunaryo, who has helped the writer to complete this Undergraduate
Thesis.
12. All Physics 46, 47, and 48, who have supported the writer.
13. Anyone that can’t be mentioned directly or indirectly who has helped the
writer in completing this Undergraduate Thesis. The writer does appreciate
any opinions, and suggestions for the improvement of this
Undergraduate Thesis.
Bogor, July 2014
Andri Hanryansyah
TABLE OF CONTENTS
TABLE OF CONTENTS ....................................................................................... vi
TABLE OF FIGURE ............................................................................................. vii
TABLE OF APPENDIX ...................................................................................... viii
INTRODUCTION ................................................................................................... 1
Backgrounds Problem .......................................................................................... 1
Formulation of the Problem ................................................................................. 1
Research Objective .............................................................................................. 1
The Benefits of Research ..................................................................................... 2
The Scope of Research......................................................................................... 2
BACKGROUND THEORY .................................................................................... 3
Heat Diffusion Equation ...................................................................................... 3
Convection-Diffusion Equation ........................................................................... 3
Dirichlet Boundary Condition ............................................................................. 4
Finite Difference Method ..................................................................................... 4
Forward Time Centered Space Scheme ............................................................... 5
Crank Nicolson Method ....................................................................................... 6
METHODS .............................................................................................................. 9
Time and Place of Research ................................................................................. 9
Research Tools ..................................................................................................... 9
Research Method ................................................................................................. 9
Research Flowchart............................................................................................ 10
RESULT AND DISCUSSION .............................................................................. 11
Heat Convection-Diffusion Simulation Using FTCS ........................................ 11
Heat Convection-Diffusion Simulation Using Crank Nicolson ........................ 13
Heat Source-Sink Value..................................................................................... 17
CONCLUSION ..................................................................................................... 18
BIBLIOGRAPHY ................................................................................................. 19
APPENDIX ........................................................................................................... 20
CURRICULUM VITAE ....................................................................................... 28
TABLE OF FIGURE
Figure 1: Mesh on a semi infinite strip used for solution in one dimensional
equation problem.3 .................................................................................................. 5
Figure 2: Four nodes that represent the FTCS calculation process ........................ 6
Figure 3: Six nodes that represent the Crank-Nicolson calculation step
process. .................................................................................................................... 7
Figure 4 : Research Flowchart .............................................................................. 10
Figure 5: A shifted function which represent the advection process11 ................. 11
Figure 6: A smoother function, which represents the diffusion process11 ............ 11
Figure 7 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with Zero-Order model (v = 10 m/s & nx = 40) ....................................... 12
Figure 8 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with First-Order model (v = 10 m/s & nx = 40) ....................................... 12
Figure 9 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 10 m/s & nx = 40). ...................... 14
Figure 10 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with First-Order model (v = 10 m/s & nx = 40). ............ 14
Figure 11 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with Zero-Order model (v = 10 m/s & nx = 400)........... 15
Figure 12 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with First-Order model (v = 10 m/s & nx = 400). .......... 15
Figure 13 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with Zero-Order model (v = 150 m/s & nx = 400)......... 16
Figure 14 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using
Crank Nicolson method with First-Order model (v = 150 m/s & nx = 400). ........ 16
Figure 16: Temperature at the central node using FTCS for different of times
with various of k constant, & model are shown. ................................................... 17
Figure 17 : Temperature at the central node using Crank Nicolson for
different of times with various of k constant, & model are shown. ...................... 18
TABLE OF APPENDIX
Appendix 1. Example of GUI on Heat Convection-Diffusion MATLAB
Simulation Program Main Menu..................................................... 20
Appendix 2. Example of GUI on Heat Convection-Diffusion MATLAB
Simulation Program Input Menu with Simulation Result. .............. 20
Appendix 3. Zero-Order and First-Order model formulation from FTCS. .......... 21
Appendix 4. Zero-Order and First-Order model formulation from Crank
Nicolson. ......................................................................................... 22
Appendix 5. The 3-D Contour plot shows an unstable result of the
temperature distribution at various of time using FTCS ZeroOrder (left) and First-Order model (right) with v = 10 m/s, nt =
40, nx = 400, k = 0.2, & α = 0.1 ...................................................... 23
Appendix 6. The 3-D Contour plot shows an unstable result of the
temperature distribution at various of time using FTCS ZeroOrder (left) and First-Order model (right) with v = 150 m/s, nt
= 40, nx = 40, k = 0.2, & α = 0.1 ..................................................... 24
Appendix 7. The 2-D Contour plot shows The Temperature Distribution at t
= 0 until 10 seconds using Zero-Order model (left) and FirstOrder model (right) with Crank Nicolson method (v = 10 m/s
& nx = 400) ..................................................................................... 24
Appendix 8. The 2-D Contour plot shows The Temperature Distribution at t
= 0 until 10 seconds using Zero-Order model (left) and FirstOrder model (right) with Crank Nicolson method (v = 150 m/s
& nx = 400) ..................................................................................... 24
Appendix 9. Table of Temperature Dsitribution at t = 0 and t = 10 s, using
Crank-Nicolson Zero-Order Model with various convection
velocity value and k = -0.2 compared to Temperature
Distribution using Crank-Nicoloson and Exact method without
v and k value. ................................................................................... 25
Appendix 10. Table of Temperature Dsitribution at t = 0 and t = 10 s, using
Crank-Nicolson First-Order Model with various convection
velocity value and k = -0.2 compared to Temperature
Distribution using Crank-Nicoloson and Exact method without
v and k value. .................................................................................. 26
Appendix 11. Table of Temperature Distribution at t = 10 s, using CN &
FTCS with Zero-Order and First-Order model compared with
Temperature Distribution at t = 10 s, using CN and FTCS with
Ordinary Heat Diffusion model . .................................................... 27
1
INTRODUCTION
Backgrounds Problem
The heat-transfer equations play a major role in the modeling of some
physical phenomena. These equations are used when attempting to describe heat
transport processes and show the heat distribution on it.1 The heat equations are the
example of Partial Differential Equations, further the heat equations are belonging
to Parabolic Equations that are part of Quasilinear Equations, which is one of the
type Partial Differential Equations.2 Some techniques have been developed to find
an appropriate simulation method of it as same with natural process.1 Numerical
results show that the complete reliability of the proposed algorithms.3
The main concern of this research is to simulate the heat transfer on one
dimensional case with the Dirichlet boundary problem. And then, the influence of
the convection velocity and heat source-sink value which, is based from mass
diffusion equations are given in this simulation and combined with finite-difference
methods.1 Some additions from mass advection diffusion equation, hopefully could
make a better model for heat transfer.
Formulation of the Problem
1.
2.
3.
4.
What is the effect of the convection velocity on heat-transfer process?
What is the effect of heat source-sink value on heat-transfer process?
What is the effect of the incompressible flow condition on heat-transfer
process?
What is the effect of the used by Zero and First-Order Model on heat-transfer
simulation result?
Research Objective
1.
2.
3.
4.
5.
Investigate and analyze the effect of convection velocity on the heat-transfer
process.
Investigate and analyze the effects of heat source-sink value on the heattransfer process.
Investigate and analyze the effects of the incompressible flow condition on
heat-transfer process.
Compare and analyze the Heat-transfer simulation result from the FTCS
method and Crank Nicholson method.
Compare and analyze the Heat-transfer Simulation Result from Zero-Order and
First-Order Model.
2
The Benefits of Research
This research delivers benefits for the field of physics modeling, especially
in Heat Transfer modeling. This Research is expected to help researchers and
people who desire to learn more about heat transfer phenomena.
The Scope of Research
The Simulation result obtained via two modified heat equation. First, heat
equation based on the zero-order diffusion equation. Second, heat equation based
on the first-order diffusion equation. Both equations were solved numerically using
two Finite Difference Methods, Forward Time Centered Space method and Crank
Nicolson method. There are some assumptions on this simulation, such as the
incompressible flow condition are preassumed, there are only two ways of
convection velocity (from right and from left), the diffusion coefficient is
considered as a constant, and The Dirichlet Boundary Condition is used as boundary
condition.
3
BACKGROUND THEORY
Heat Diffusion Equation
The Heat Diffusion Equation belongs to Parabolic Equations that are part
of Quasilinear Equations, which are one of types Partial Differential Equations.2
The equations are usually stated as follows:
∂T
∂t
=α
∂2 T
(1)
∂x2
The � constant known as thermal diffusivity which is a measure of how fast
a material can carry heat away from a higher to lower temperature.4
Convection-Diffusion Equation
Convection-diffusion equation plays a pivotal role in the modeling of some
physics simulation where heat or energy is transformed inside a physical system
due to two processes: convection and diffusion.5 The convection means the
movement of molecules or particles within heat. The diffusion describes the spread
of particles through random motion from regions of higher to lower concentration.
Besides, this equation can be described as advection-diffusion of quantities such as
heat, mass, energy, etc.5 Many people widely used this in analyzing the spread of
solute in a liquid flowing through a tube, long range transport of pollutants in the
atmosphere, flow in porous media and many more. 6,7,8
Many journals, scientific papers, and books stated that the convectiondiffusion equation depending on the context, the same equation can be called as the
advection-diffusion equation. 9 The general equation for convection–diffusion
usually stated : 8, 9,10
∂T
+∇∙ v⃗T =∇∙ α∇T +S
(2)
∂t
In the one-dimensional case, the convection-diffusion equation become
∂T
∂t
∂
∂
∂
∂x
∂x
∂x
+ ∙ v⃗T = ∙ α
T +S
(3)
In a common situation, the diffusion coefficient is considered as a constant
and the volume is incompressible, which means in any pressure condition the
volume is constant. Therefore, this condition makes the velocity won’t affect the
heat transfer process become convective unless the velocity has a value greater than
0.3 Mach.11 Then, the formula simplifies to: 11
∂T
∂t
+v⃗
∂T
∂x
=α
∂2 T
∂x2
+S
(4)
Where T as a temperature variable, which are functions of x and t variable, v as
velocity variable, t as time variable, x as space variable, α as thermal diffusivity
constant, and S as Heat Source or Sink value. If the high temperature inside the
system is increased, means that the reaction inside of it is endotherm (heating
4
phenomena). Otherwise, if the heat inside the system is decreased, means that the
reaction inside of it is exothermic (cooling phenomena).
The Convection-Diffusion with Zero-Order Source Sink Terms. The formulas
are as follows:
∂T
∂t
+v⃗
∂T
∂x
=α
∂2 T
∂x2
+k0
(5)
Where k0 is a Source/Sink value for Zero-Order model.
The Convection-Diffusion with First-Order Source Sink Terms, with the
formula are as follows:
∂T
∂t
+v⃗
∂T
∂x
=α
∂2 T
∂x2
+k1 T
(6)
Where k1 is a Source/Sink value for the First - Order model.
The difference between both formulas are The Heat Source/Sink from ZeroOrder model are not depending on the temperature and the other hand the Heat
Source/Sink from First - Order model are dependent with the temperature.
Dirichlet Boundary Condition
In engineering applications, a Dirichlet boundary condition may also be
referred to as a fixed boundary condition. 12 In this case, we assumed that the
equation has some boundary terms are:
∂T
∂t
+ v⃗
∂T
∂x
=α
∂2 T
∂x2
T x,0 = f x
T 0,t = T(L,t)
+S
0 ≤ x ≤ L, t > 0
(7)
(Initial condition)
(8)
(Boundary condition) (9)
Finite Difference Method
Finite-Difference Methods (FDM) is numerical methods for approximating
solutions of differential equations using finite difference equations. The idea of this
method is to divide the whole interval t0 ≤ t ≤ tmax into M segments of width Δt =
(tmax – t0)/M and approximate the first & second derivatives in the differential
equations for each grid point by the central difference formulas. However, in order
for this system of equations to be solved easily, it should be linear, implying that its
coefficients may not contain any term of x.12,13 The finite difference method obtains
an approximate solution for T (x, t) at a finite set of x and t. The codes developed
in this research based on some literature, the discrete x is uniformly spaced in
interval of 0 ≤ x ≤ L such as. 3
� =
−
∆�,
= , ,…
(10)
5
Where N is the total number of spatial nodes,3
∆� =
(11)
−
Similarly, the t discrete is uniformly spaced in 0 (t0) ≤ t ≤ tmax :3
� =
−
∆�,
= , ,…
(12)
Where M is the total number of time steps and Δt is the size of a time steps3
∆� =
����
−
(13)
The solution domain is depicted in Picture 1.3
Figure 1: Mesh on a semi infinite strip used for solution in one dimensional
equation problem.3
As shown in the Figure 1, the black squares with indicated by a red dash border is
the location of the known initial value T x,0 = f x . The open squares which
indicated by a blue dash border is the location of the known boundary terms
T 0,t =T(L,t). The open circles show the location of the inside points where the
finite difference approximation is calculated.3
Forward Time Centered Space Scheme
The FTCS (Forward-Time Central-Space) method is a finite difference
method used for numerically solving parabolic partial differential equations.14 It is
a first-order method in time, explicit in time, and is conditionally stable when
applied to the heat equation. When practiced as a method for advection equations,
or more generally hyperbolic partial differential equation, it is unstable unless the
artificial viscosity is included. The abbreviation FTCS was first used by Patrick
Roache.15
FTCS method is usually used in Heat Transfer or Fluid Dynamics
Computation. Figure 2 shows four nodes in the grid mesh schematically represent
the FTCS calculation process.
6
Figure 2: Four nodes that represent the FTCS calculation process
From the picture above and discrete approximations, which are based on finite
�� ��
� �
difference methods. The formula for T, , , and
can be obtained as listed
��
�� ��
below :
T=T(x,t)
(14)
��
��
��
� �
��
��
=
=
=
� �,�+ −� �,�
∆�
(15)
� �+ ,� −� �− ,�
∆�
(16)
� �+ ,� − � �,� +� �− ,�
∆�
(17)
From those formulations the Heat Convection-diffusion can be obtained :
Zero-Order model (5) becomes
� �,�+
�∆�
∆�
=
�∆�
∆�
(� �+
,�
− � �,� + � �−
,�
First-Order model (6) becomes
(� �+
,�
− � �,� + � �−
,�
)−
⃗⃗∆�
�
∆�
)−
⃗⃗∆�
�
∆�
� �,�+
(� �+
,�
(� �+
=
− � �−
,�
,�
− � �−
,�
) + � �,� +
) + � �,� +
Crank Nicolson Method
� �,� ∆�
∆�
(18)
(19)
The Crank–Nicolson method is a finite difference method used for
numerically solving the heat equation and similar partial differential equations. 16 It
is a second-order method in time, it is implicit in time and it is numerically stable.
The method was developed by John Crank and Phyllis Nicolson in the middle of
the 20th century.16,17 However, the approximate solutions can still contain
(decaying) spurious oscillations if the ratio of time step Δt times the thermal
diffusivity to the square of space step, Δx2, is large. This method utilizes the T
values in six points as shown in Figure 3.
7
Figure 3: Six nodes that represent the Crank-Nicolson calculation step process.
From the picture above and discrete approximations which are based on finite
difference methods. The formula for T,
below :
��
� �
��
��
=
=
��
,
�� ��
, and
� �,�+ +� �,�
�=
��
�� ��
=
� �
��
can be obtained as listed
(20)
� �,�+ −� �,�
∆�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
4∆�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
∆�
(
)
(
)
(
)
From those formulations the Heat Convection-Diffusion can be obtained:
Zero-Order model (5) becomes
�
= − ⃗⃗⃗⃗⃗⃗ −
4∆�
=
=
�
∆�
+ ⃗⃗⃗⃗⃗⃗ (� �−
�
4∆�
,�
)+
∆�
+
�
= ⃗⃗⃗⃗⃗⃗ −
∆�
−
4∆�
�
∆�
First-Order model (6) becomes
�
=
�
∆�
+ ⃗⃗⃗⃗⃗⃗ (� �−
�
4∆�
,�
)+
4∆�
∆�
+
�
∆�
�
= ⃗⃗⃗⃗⃗⃗ −
∆�
−
4∆�
�
∆�
∆�
+
,
i=2, 3,4,5,... N-1
�
∆�
�
∆�
(� �,� ) +
= − ⃗⃗⃗⃗⃗⃗ −
=
�
�
∆�
�
∆�
,
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
,�
)++
i= 2 3,4,5,.. N-1
)
(
)
(
)
( 9)
�
(� �,� ) +
(
(28)
−
∆�
(24)
�
∆�
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
,�
)
(
)
(
)
8
Using Dirichlet Boundary Condition in equation (7) (8) and (9)
= ,
= ,
= ,
= ,
= � ,� =
= � ,� =
The initial temperature distribution on this research project is considered by
T x, 0 = f x = Sin πx⁄L
(
)
The derivation of Zero-Order and First-Order model with FTCS and CN method
can be seen in Appendix 2 and 3.
9
METHODS
Time and Place of Research
This research was held from March 2013 until January 2014 and takes the
places at the Computation Physics Laboratory, Department of Physics, Faculty of
Mathematics and Natural Sciences, Bogor Agricultural University.
Research Tools
The main tools in this research are DELL Inspiron N4030 Notebook with
specification Intel Core-i3 @2. 53 GHz Processors, 2 GB of RAM, Windows Blue
(8.1) Operating System and some software that already installed, such as: Microsoft
Office 2010, GIMP, Sublime 2 Text Editor, and MATLAB™ 2012b.
Research Method
Literature Study
Literature study was done in order to understand the mechanism of heat
transfer and the effect of convection-diffusion and the numerical methods to solve
partial differential equation. Moreover, this literature study would make easier the
process of result analysis of the program and correlate it with the heat transfer and
the diffusion equation.
Formulating the Equation
Some parameters in heat convection-diffusion such as, convection velocity
and Heat Source-Sink, are written into the equation in mathematical form and added
into the ordinary heat diffusion equation. After the heat convection-diffusion with
some parameter has been written in a new equation. Then, the equation is
discretized using the Finite Difference Method. The Dirichlet Boundary Condition
(Fixed-Boundary) was used as initial and boundary condition.
Simulation Using MATLABTM Software
The value of T(x,0) was defined using equation (32) as an initial condition.
Then, The Dirchlet Boundary Condition is considered as the boundary condition in
this simulation. After the numerical equations are obtained, initial and boundary
condition was considered, those requirements are translated into a pseudo code.
After that, the pseudo codes from the Heat Convection-Diffusion are translated into
MATLAB syntax which are used in MATLAB Software to solve the equation in
the region T(2,t+1) ~ T(L-1, t+1) (t= 1, 2,3... M), after that the result are shown in a 3D
contour or 2-D contour graph in order to simulate the temperature distribution on
the domain of x and shows the Heat Convection-Diffusion Phenomena.
10
Research Flowchart
Start
Formulating heat convectiondiffusion with Source-Sink Value
Consider initial and
boundary condition
Output : The Heat Convection-Diffusion
Equation with Source- Sink Value, initial and
boundary condition.
Formulating numerical equation based on heat
convection diffusion with source-sink value, initial
condition and boundary condition using FDM.
Output : The Heat Convection-Diffusion
Numerical Equation with Source- Sink Value,
initial and boundary condition based on FDM.
Translating the Heat Convection Diffusion numerical
equation into programable MATLAB syntax.
Create and designing GUI for Heat
Convection Diffusion MATLAB Program.
Output : A Heat Convection-Diffusion MATLAB Program
with graphic result that display temperature distribution on 1 Dimension
and can be inputted with some customizing paramater.
Inputing some parameter value to MATLAB program,
analyzing the graphic result using Ms. Excel and compare
the graph result with heat convection diffusion theory.
Output : An Analyzing and review result in form of table of data and
graphical from the Heat Convection Diffusion MATLAB Program which
are processed using Microsoft Excel 2010
Finish
Figure 4 : Research Flowchart
11
RESULT AND DISCUSSION
The advection (convection) diffusion equation are underlies the law of mass
conservation, it is also called a mass balance equation. Then, it is easily seen that
the initial function (initial temperature distribution) at (x,0) is merely shifted in time
and velocity v won’t change the function shape as shown in Fig. 5
Figure 5 : A shifted function which represent the advection process11
Fig. 5 shows that the function shifted from the initial function position. 11 The
positive value of v used on the Fig. 5, which shows the shift’s direction from left to
right and if v use a negative value, the shift’s direction will move from right to left.
Consider if the diffusion equations are the only part of the equation with α
value is greater than 0, with α denoting the heat conductivity or thermal
diffusivity.4,11 The graphic result will mainly interpret the heat convection-diffusion
equation as the result of molecular diffusion caused by Brownian motion of
particles.11
Figure 6: A smoother function, which represents the diffusion process11
Heat Convection-Diffusion Simulation Using FTCS
Generally the length of mesh L = 10 m is considered. The following
parameter input are introduced: thermal diffusivity α = 0.1 (m2/s), mesh iterations
nx = 40, time iterations nt = 40, tmax = 4 s, k constant = -0.2 and convection velocity
v = 10 m/s. The minus sign at k constant means that the system is in the cooling
process. The initial temperature distribution in the domain at t = 0 s is of the form
(32).
Figure. 7 show the temperature distributions in the domain of x for different
of times using FTCS Method with Zero-Order model. Figure. 8 show temperature
distributions in the domain of x for different of times using FTCS Methods with
First-Order model. The different values of times are shown in both of the figures.
Both figures show the convection phenomena which, shown by the temperature
distributions are shifting on each time. The maximum temperature on each
temperature distribution that indicated by a red sign. The convection phenomena
can be seen from the maximum temperatures move in the convection velocity
direction.
12
Figure 7 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with Zero-Order model (v = 10 m/s & nx = 40)
Figure 8 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using FTCS
method with First-Order model (v = 10 m/s & nx = 40)
The diffusion phenomena are shown by the changes of temperature
distribution shape. The Shape of temperature distributions is changed become
smaller on each time which, means that the temperature on the domain spread out
to both sideways as the result of heat diffusion caused by motion of particles that
triggered by the maximum temperature on the domain.
13
Even though, those all results are not consistent with the theory that
considers an incompressible flow. It means in any pressure variations the volumes
are constant and in mathematically the v or convection will not affect until the v
value is greater than 0.3 Mach (110 m/s). Both figures result shows that convection
velocity = 10 m/s is enough to affect the heat-transfer process become convection
process. It is opposite with the theory which stated a low v value which has a value
less than 0.3 Mach. So, it won't affect the heat-transfer process become convection
process.11 A huge value of iteration should be inputted on simulation, in order to
make the simulation result become more accurate. Other parameters such as nt, tmax,
L, α, and v are inputted with the same value at first simulation.
Unfortunately the FTCS can't afford a huge iteration value or very small step
size value. It’s because the FTCS is considered as explicit numerical method.4 The
unstability calculation result from the FTCS method (both for Zero-Order and FirstOrder) are shown in Appendix 6. The FTCS method is also could become unstable
if some parameter in simulation has been inputted with unusual value, for example,
Appendix 7 shows an unstable result graphic of FTCS calculation which, has been
inputted with v = 150 m/s.
Heat Convection-Diffusion Simulation Using Crank Nicolson
The heat convection-diffusion simulation terms for CN are same as FTCS
does. The length of mesh L = 10 m is considered. The following parameter input
are introduced: thermal diffusivity α = 0.1 (m2/s), mesh iterations nx = 40, time
iterations nt = 40, tmax = 4 s, k constant = -0.2 and convection velocity v = 10 m/s.
The minus sign at k constant means that the system is in the cooling process. The
initial temperature distribution in the domain at t = 0 s is of the form (32).
Figure. 9 show the temperature distributions in the domain of x for different
of times using CN Method with Zero-Order model. Figure. 10 show temperature
distributions in the domain of x for different of times using CN Methods with FirstOrder model. The different values of times are shown in both of the figures. Both
figures show the convection phenomena which, shown by the temperature
distributions are shifting on each time. The maximum temperature on each
temperature distribution that indicated by a red sign. The convection phenomena
can be seen from the maximum temperatures move in the convection velocity
direction.
Even though, Figure 9 and 10 are not consistent with the theory that
considers an incompressible flow as same with the FTCS simulation on Figure 7
and 8. A huge value of iteration should be inputted on simulation, in order to make
the simulation result become more accurate. In this CN simulation method, the
iteration of mesh nx = 400 is considered. Meanwhile, other parameters such as nt,
tmax, L, α, and v are inputted with the same value as first simulation does.
14
Figure 9 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 10 m/s & nx = 40).
Figure 10 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with First-Order model (v = 10 m/s & nx = 40).
Both CN simulation results with nx = 400 (Zero-Order and First-Order),
shows a graphic which, is relevant to the incompressible flow as shown in Figure
11 and 12. The convection velocity with a value below 0.3 Mach (10 m/s) is not
enough to make the heat transfer process become convection process. It’s very
different with previous CN simulation result with the convection velocity value
below 0.3 Mach is enough to make the heat transfer process become convection.
15
Figure 11 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 10 m/s & nx = 400).
Figure 12 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with First-Order model (v = 10 m/s & nx = 400).
The convection velocity would affect the heat transfer process if the
convection velocity value is above 0.3 Mach. Then, in order to prove that. In this
simulation, the convection velocity with a value above 0.3 Mach is considered, v =
150 m/s. Both graphic results show a relevant result with the incompressible flow
terms. The convection velocity with a value above 0.3 Mach would affect the heat
transfer process become convection process. The Convection phenomena will
16
clearly observe when the results from Figure 13 and 14 are presented in 2-D contour
or table as shown in Appendix 8, 9, and 10. The Appendix 9 show comparison of
CN Zero-Model temperature distribution in t = 4 s with various of convection
velocity, as the temperature distributions are flattened before 10 seconds. The
Appendix 10 show comparison of CN First-Model temperature distribution at t =
10 s with various of convection velocity.
Figure 13 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with Zero-Order model (v = 150 m/s & nx = 400).
Figure 14 : Temperature Distribution at 0, 2, 4, 6, 8 and 10 seconds using Crank
Nicolson method with First-Order model (v = 150 m/s & nx = 400).
17
Heat Source-Sink Value
From the previous section, the result has shown that The Zero-Order’s
model (both for FTCS and CN) temperature distributions are faster to being
flattened than The First-Order’s model result. The result differences between both
models are caused by heat source-sink value that used in both methods.The ZeroOrder model uses a Heat Source-Sink value, which causes the temperature of the
system changed constantly at each time. The k constant on Zero-Order model are
not depends with temperature on the system. The Heat Source-Sink value on the
First-Order model is not constant. It depends with the temperature on the system.
The Heat Source-Sink value will change if the temperature value is changes.
The simulation has been run with a non - convection term. So, it makes
easier to analyze. The midpoint temperature in this simulation is used as a tracking
point for heat source sink phenomena. The k constant on First-Order would make a
heat sink value that decreased each time step if the systems are in cooling process
as shown by the blue curve in Figure 16 and 17. It is because the k constant in the
First - Order model used as multiplied factor to produce heat sink value. The heat
sink value becomes smaller each time step and become saturated. This cooling
curve has a same trends result with the cooling curve based on newton laws of
cooling.16 Meanwhile, the k constant would produce a bigger heat source value each
time step if the systems are in the heating process as shown on Figure 16 and 17
the temperature increased exponentially.
Figure 15: Temperature at the central node using FTCS for different of times
with various of k constant, & model are shown.
18
Figure 16 : Temperature at the central node using Crank Nicolson for different
of times with various of k constant, & model are shown.
CONCLUSION
A Heat Convection-Diffusion Equation are derived from the heat diffusion
equation, advection equation and combined with heat source-sink (S) value. The S
value is a constant which defines a reaction inside the system are exothermic or
endothermic. The incompressible flow is used as the condition in this simulation.
The Equation categorized into two models. First, the Zero-Order model with the S
value also defined as k constant. Second, the First-Order model with the S value in
the equation depends on system temperature (T) value. Both models, then solved
using two Finite Difference Method, Forward Time Centered Space (FTCS) and
Crank-Nicolson (CN). The Zero-Order model’s S value has linear characteristic as
S of the model makes system temperature changed linearly. The First Zero model’s
S value has exponential characteristic as S on the model makes system temperature
changed exponentially. The incompressible flow condition makes the convection
velocity with a value below 0.3 Mach will not affect the heat transfer process
become convection significantly. The convection velocity would affect the heat
transfer process if has a value above 0.3 Mach. The unstability will be appear in
some simulation, especially in FTCS method. It is because the FTCS method is an
explicit method which, unconditionally unstable and very sensitive to iteration
value, step size, and some additions constant in the equation.
19
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
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[15]
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Mohammadi, Reza. "Exponential B-Spline Solution of ConvectionDiffusion Equations." (Applied Mathematics and Computation) 2013.
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Computation) 147, no. 2 (2004): 307-319.
R. C., Mittal and R. K., Jain. "Redefined Cubic B-Splines Collocation
Method for Solving Convection-Diffusion Equations." (Applied
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M. K. , Kadalbajoo and P. , Arora. "Taylor-Galerkin B-Spline Finite
Element for the One-Dimensional Advection-Diffusion Method for the
One-Dimensional Advection-Diffusion Equation." (Numerical Methods
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Bejan A. Convection Heat Transfer. Wiley, 2013.
Salkuyeh, Davod Khojasteh. "On The Finite Difference Approximation To
The Convection-Diffusion Equation." (Elsevier) 2006.
Hundsdorfer , Willem; Verwer, Jan;. Numerical Solution of TimeDependent Advection-Diffusion- Reaction Equations. Edited by La Jolla
R.Bank, La Jolla R.L. Graham, Wurzburg J.Stoer, Kent R.Varga, &
Tubingen H.Yserentant. New York: Springer, 2003.
A., Cheng; D. T., Cheng;. "Heritage and Early History of The Boundary
Element Method." Engineering Analysis with Boundary Elements
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Kaw, Autar, and E. Eric Kalu. Numerical Methods with Applications.
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Tannehill, John C.; Anderson, Dale A.; Pletcher, Richard H.;.
Computational Fluid Mechanics and Heat Transfer. 3rd. London: Taylor
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Roache, Patrick;. Fundamentals of Computational Fluid Dynamics.
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Publishing, 2002.
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2005.
20
APPENDIX
Appendix 1. Example of GUI on Heat Convection-Diffusion MATLAB
Simulation Program Main Menu.
Appendix 2. Example of GUI on Heat Convection-Diffusion MATLAB Simulation
Program Input Menu with Simulation Result.
21
Appendix 3. Zero-Order and First-Order model formulation from FTCS.
The formula for T,
�� ��
,
�� ��
, and
� �
��
��
��
��
� �
��
��
=
of FTCS are listed below:
T=T(x,t)
=
=
� �,�+ −� �,�
∆�
� �+ ,� −� �− ,�
∆�
� �+ ,� − � �,� +� �− ,�
∆�
From those formulations the Heat Convection-Diffusion can be obtained :
Zero-Order model (5) becomes
� �,�+ −� �,�
∆�
+ �⃗
� �+ ,� −� �− ,�
∆�
=�
� �+ ,� − � �,� +� �− ,�
∆�
+
And simplifies it becomes
� �,�+
=
�∆�
∆�
(� �+
,�
− � �,� + � �−
,�
)−
⃗⃗∆�
�
∆�
(� �+
,�
− � �−
,�
) + � �,� +
∆�
First-Order model (6) becomes
� �,�+ −� �,�
∆�
+ �⃗
� �+ ,� −� �− ,�
∆�
And simplifies it becomes
�∆�
∆�
(� �+
� �,� ∆�
,�
=�
� �+ ,� − � �,� +� �− ,�
∆�
� �,�+
− � �,� + � �−
,�
)−
=
⃗⃗∆�
�
∆�
(� �+
,�
+
− � �−
� �,�
,�
) + � �,� +
22
Appendix 4. Zero-Order and First-Order model formulation from Crank Nicolson.
The formula for T,
�� ��
,
�� ��
, and
� �
��
�=
��
��
� �
��
��
=
=
Of Crank Nicolson are listed below:
��
=
� �,�+ +� �,�
� �,�+ −� �,�
∆�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
4∆�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
∆�
From those formulations the Heat Convection-Diffusion can be obtained:
Zero-Order model (5) becomes
� �,�+ −� �,�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
+ �⃗
∆�
4∆�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
�
∆�
=
+
Simplifies it and the equation becomes
− ⃗⃗⃗⃗⃗⃗ −
�
4∆�
�
�
+
∆�
�
∆�
�
⃗⃗⃗⃗⃗⃗
4∆�
(� �−
(� �−
,�
,�+
)+
)+
∆�
−
∆�
+
�
∆�
First-Order model (6) becomes
�
∆�
�
(� �,�+ ) + ⃗⃗⃗⃗⃗⃗ −
(� �,� ) +
�
∆�
4∆�
�
⃗⃗⃗⃗⃗⃗
−
4∆�
�
∆�
(� �+
(� �+
,�
)+
,�+
)=
� �,�+ −� �,�
� �+ ,� −� �− ,� +� �+ ,�+ −� �− ,�+
+ �⃗
=
∆�
4∆�
� �,�+ +� �,�
� �+ ,� − � �,� +� �− ,� +� �+ ,�+ − � �,�+ +� �− ,�+
+
∆�
Simplifies it and the equation becomes
�
�
(�
)+
+
− ⃗⃗⃗⃗⃗⃗ −
�
4∆�
∆�
�
∆�
∆�
(� �+
,�+
�− ,�+
)=
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
,�
)
�
∆�
+
∆�
�
⃗⃗⃗⃗⃗⃗
4∆�
�
∆�
(� �−
−
,�
)+
�
(� �,�+ ) + ⃗⃗⃗⃗⃗⃗ −
∆�
−
�
∆�
+
4∆�
(� �,� ) +
The Zero-Order and First-Order model can be represented in matrix format which
later will be formulated using tridiagonal matrix.3
23
Where the coefficients of the interior nodes are:
Zero-Order model :
�
�
= − ⃗⃗⃗⃗⃗ −
,
∆�
= (
=
�
∆�
+ ⃗⃗⃗⃗⃗⃗ (� �−
�
4∆�
First-Order model :
,�
)+
= (
∆�
−
∆�
�
⃗⃗⃗⃗⃗⃗⃗
�
�
)
∆�
+
∆�
�
−
= − ⃗⃗⃗⃗⃗ −
�
∆�
= (
=(
�
∆�
+
�
⃗⃗⃗⃗⃗⃗⃗
∆�
) (� �− ,� ) + (
∆�
= (
∆�
−
)
∆�
�
(� �,� ) +
∆�
+
�
⃗⃗⃗⃗⃗⃗⃗
∆�
�
∆�
i=2,3,4,5,...N-1
∆�
�
∆�
−
+
∆�
�
− ⃗⃗⃗⃗⃗⃗ (� �+
�
4∆�
�
−
∆�
)
)
) (� �,� ) + (
Using Dirichlet Boundary Condition in equation (7) (8) & (9)
= ,
= ,
= ,
= ,
)+
i=2,3,4,5,...N-1
,
∆�
,�
= � ,� =
= � ,� =
�
∆�
−
�
⃗⃗⃗⃗⃗⃗⃗
) (� �+ ,� )
∆�
Appendix 5. The 3-D Contour plot shows an unstable result of the temperature
distribution at various of time using FTCS Zero-Order (left) and First-Order model
(right) with v = 10 m/s, nt = 40, nx = 400, k = 0.2, & α = 0.1
24
Appendix 6. The 3-D Contour plot shows an unstable result of the temperature
distribution at various of time using FTCS Zero-Order (left) and First-Order model
(right) with v = 150 m/s, nt = 40, nx = 40, k = 0.2, & α = 0.1
Appendix 7. The 2-D Contour plot shows The Temperature Distribution at t = 0
until 10 seconds using Zero-Order model (left) and First-Order model (right) with
Crank Nicolson method (v = 10 m/s & nx = 400)
Appendix 8. The 2-D Contour plot shows The Temperature Distribution at t = 0
until 10 seconds using Zero-Order model (left) and First-Order model (right) with
Crank Nicolson method (v = 150 m/s & nx = 400)
Appendix 9. Table of Temperature Dsitribution at t = 0 and t = 10 s, using Crank-Nicolson Zero-Order Model with various convection
velocity value and k = -0.2 compared to Temperature Distribution using Crank-Nicoloson and Exact method without v and k value.
25
26
Appendix 10. Table of Temperature Distribution at t = 0 and t = 10 s, using Crank-Nicolson First-Order Model with various convection
velocity