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Numerical Simulation of Natural Convection Between Two Parallel Plates
by
Aishah binti Rosli
Dissertation submitted in partial fulfilment of
the requirements for the
Bachelor of Engineering (Hons)
(Chemical Engineering)
SEPTEMBER 2011
Universiti Teknologi PETRONAS
Bandar Seri Iskandar
31750 Tronoh
Perak Darul Ridzuan
CERTIFICATION OF APPROVAL
Numerical Simulation of Natural Convection between Two Parallel Plates
AppQ:;b~ =\'='
by
Aishah binti Rosli
A project dissertation submitted to the
Chemical Engineering Programme
Universiti Teknologi PETRONAS
in partial fulfilment of the requirement for the
BACHELOR OF ENGINEERING (Hons)
(CHEMICAL ENGINEERING)
(Dr. Rajashekhar Pendyala)
UNIVERSITI TEKNOLOGI PETRONAS
TRONOH, PERAK
September 2011
CERTIFICATION OF ORIGINALITY
This is to certizy that I am responsible for the work submitted in this project, that the original work is my own except as specified in the references and acknowledgements, and that the original work contained herein have not been undertaken or done by unspecified sources or persons.
AISHAH BINTI ROSLI
ABSTRACT
Free convection flow is present and significant in various engineering
circumstances, such as the design of heat exchangers, nuclear reactors, and many
chemical processes. It is also important in the application of cooling equipment. In
this project, a two-dimensional numerical model is formulated to simulate the natural
convection of air between two parallel plates using a computational fluid dynamics
(CFD) code. Several different situations are modeled to observe the effects on the
natural convection flow.
ACKNOWLEDGEMENT
First and foremost, I would like to thank my supervisor of this project, Dr. Rajashekhar Pendyala for the valuable guidance and advice. He has inspired me greatly to work on this project, and his willingness to help and motivate me has contributed greatly to this project.
I would also like to thank my family and friends for their understanding and support throughout the completion of the project. Words are inadequate in offering the thanks to all the other related parties for giving their full support and efforts in completing the project from the starting stage of the project to the end of the semester.
ii
TABLE OF CONTENTS
ABSTRACT ................................................................................................................. 1
ACKN"OWLEDGEMENT ........................................................................................ II
CHAPTER 1: INTRODUCTION ............................................................................. l
1.1 Background of Study ...................................................................... 1
1.2 Problem Statement. ......................................................................... 2
1.3 Objectives and Scope of Study ....................................................... 2
1.3.1 Objectives ............................................................................. 2
1.3.2 Scope of Study ...................................................................... 2
1.4 Significance of the Study ................................................................ 3
1.5 Feasibility ofStudy ......................................................................... 3
CHAPTER 2: LITERATURE REVIEW ................................................................. 4
2.1 Natural Convection between Parallel Plates ................................... 4
2.1.1 Natural Convection with One Plate lsothennally Heated and
the other Thennally Insulated .............................................. 4
2.1.2 Natural Convection with Constant Temperature and Mass
Diffilsion at One Boundary .................................................. 5
2.1.3 Turbulent Natural Flow with Rayleigh Number Ranging
from 105 to 107 ..................................................................... 5
2.1.4 Natural Convection with Walls Heated Asymmetrically ..... 6
2.1.5 Natural Convection Flows for Low-Prandtl Fluids .............. 6
2.1.6 Turbulent Natural Convection with Symmetric and
Asymmetric Heating ............................................................ 7
2.1.7 Laminar Flow with UWT/UWC and UHF!UMF ................. 8
2.1.8 Natural Convection with Asymmetric Heating Coupled with
Thennal Radiation ............................................................... 9
iii
CHAPTER 3: METHODOLOGY .......................................................................... lO
3.1 Research and Project Activities .................................................... 10
3.1.1 Research ............................................................................. 10
3.1.2 Simulation ofProblems ...................................................... lO
3.1.3 Comparison and Verification ............................................. lO
3.1.4 Conclusion and Report Writing ......................................... 11
CHAPTER 4: RESULTS AND DISCUSSION ..................................................... 14
4.1 Model Equations ........................................................................... 14
4.2 Mesh ............................................................................................. 15
4.2 Simulation ..................................................................................... 16
4.2.1 Case 1: Both Plates Set at Different Temperatures (Steady
State) .................................................................................. 16
4.2.2 Case 2: Right Plate Heated, Left Plate at Zero Heat Flux
(Steady State) ..................................................................... 19
4.2.3 Case 3: Left Plate Heated, Right Plate at Zero Heat Flux
(Steady State) ..................................................................... 23
4.2.4 Case 4: Right Plate Heated, Right Plate at Zero Heat Flux
(Transient) .......................................................................... 27
CHAPTERS: CONCLUSION AND RECOMMENDATIONS .......................... 35
REFERENCES ......................................................................................................... 36
APPENDIX I ............................................................................................................ 38
APPENDIX D ........................................................................................................... 39
iv
LIST OF FIGURES
Figure 1: Profiles of the mean vertical velocity component of the symmetrical flow
case, Ra = 4.0x 10 6: Dy/L = 0.98; x, y/L = 0.55; •,y/L = 0.11 . .................. 7
Figure 2: Profiles of the mean vertical velocity component of the asymmetrical flow
case, Ra = 2.0 x 10 6: Dy/L = 0.98; x, y!L = 0.55; •,y/L = 0.11 . .................. 8
Figure 3: Geometry of the Model.. ............................................................................. 15
Figure 4: Velocity Vector for Case 1 ......................................................................... 16
Figure 5: Stream Function for Case 1 ........................................................................ 17
Figure 6: Temperature Profile for Case 1 .................................................................. 17
Figure 7: Velocity XY Plot for Case 1 ....................................................................... 18
Figure 8: Temperature XY Plot for Case 1 ................................................................ 18
Figure 9: Velocity Vector for Case 2 ......................................................................... 20
Figure 10: Stream Function for Case 2 ...................................................................... 20
Figure 11: Temperature Profile for Case 2 ................................................................ 21
Figure 12: Velocity XY Plot for Case 2 ..................................................................... 21
Figure 13: Temperature XY Plot for Case 2 .............................................................. 22
Figure 14: Velocity Vector for Case 3 ....................................................................... 24
Figure 15: Stream Function for Case 3 ...................................................................... 24
Figure 16: Temperature Profile for Case 3 ................................................................ 25
Figure 17: Velocity XY Plot for Case 3 ..................................................................... 25
Figure 18: Temperature XY Plot for Case 3 .............................................................. 26
Figure 19: Velocity Vector at the Start of the Simulation ......................................... 28
Figure 20: Velocity Vector After Time has Passed ................................................... 28
Figure 21: Stream Function at the Start of the Simulation ........................................ 29
Figure 22: Stream Function After Time has Passed .................................................. 29
Figure 23: Temperature Profile at the Start of the Simulation ................................... 30
Figure 24: Temperature Profile After Time has Passed ............................................. 30
Figure 25: Velocity XY Plot at the Start of the Simulation ....................................... 31
Figure 26: Velocity XY Plot After Time has Passed ................................................. 31
Figure 27: Temperature XY Plot at the Start of the Simulation ................................ 32
Figure 28: Temperature XY Plot After Time has Passed .......................................... 32
v
LIST OFT ABLES
Table 1: Gantt Chart and Key Milestones for FYP I ................................................. 12
Table 2: Gantt Chart and Key Milestones for FYP II ................................................ 13
vi
CHAPTER I
INTRODUCTION
1.1 Background of Study
Natural convection, also known as free convection, is caused by density
changes that are caused by a heating process that results in the movement of the
fluid. The heating process causes the density of the fluid close to the heat-transfer
surface to decrease, while cooling causes the density of the fluid to increase, which
then causes the fluid to move in free convection due to the buoyancy forces that is
imposed on the fluid. These forces, which give rise to the free-convection currents
are called body forces, and would not be present if some external force field did not
act upon the fluid (Holman, 2001 ).
In free convection, at the heated body, which is the wall of the plate in this
case, the velocity is zero according to the no-slip boundary condition. The velocity
then increases very quickly in a thin boundary layer adjacent to the body until it
reaches a maximum value and then, far away from the body, it becomes zero again.
At first, the boundary-layer development is linear, but, depending on the properties
of the fluid and the difference in temperature between the two plates, turbulent
eddies are formed and transition to a turbulent boundary layer begins at a distance
away from the leading edge (Pitts, 1998).
The differential equation of motion for the boundary layer should be obtained
in order to analyze the heat-transfer problem. Similar to forced-convection problems,
some natural convection problems require for the relations for heat transfer in
different situations to be obtained from experimental measurements. Usually, these
circumstances involve those in which it is difficult to predict temperature and
velocity profiles analytically (Holman, 2001 ).
1
1.2 Problem Statement
Many studies have been done to observe the transient behavior of natural
convection between vertical parallel plates in various conditions, some done
experimentally while some provides analytical solutions for the problems. Numerical
solutions, on the other hand, can be used to solve complex problems, unlike
analytical solutions, which are only available for simple problems.
This project is aimed to solve some natural convection problems using
Computational Fluid Dynamics (CFD) to generate numerical solutions to the
problem and compare these solutions with the analytical solutions and experimental
findings available in the literature.
1.3 Objectives and Scope of Study
1.3.1 Objectives
• To identify various boundary conditions in which natural convection
between vertical parallel plates can occur.
• Develop a simulation method using CFD to obtain numerical solutions
for each condition.
1.3.2 Scope of Study
There are many situations in which natural convection can occur and
depending on the situation, the behavior of the flow will vary.
Therefore, the scope of study will be limited to:
• Flow between vertical parallel plates
• Flow of incompressible fluid
2
1.4 Significance of the Study
Transient natural convection flow between vertical parallel surfaces is applied
in technological processes such as the early stages of melting as well as in heating of
insulating air gaps by inserting heat at the furnaces start-up (Jha, 2010). The effects
of both thennal and mass buoyancy forces on free convection flows are also
significant in many engineering situations; in the design of heat exchangers, nuclear
reactors, solar energy collectors thenno protection systems and other chemical
processes (Mameni, 2008). These flows are also applied in the cooling of electric and
electronic equipment, nuclear reactor fuel elements and home ventilation (Badr,
2006). Free convection is a preferable heat transfer technique because of its
simplicity, reliability and cost effectiveness (Yilmaz, 2007).
1.5 Feasibility of Study
Throughout this study there are several phases that will be done throughout
completing the project:
I. Research based on literature review on natural convection from multiple
references and sources. This phase involved doing researches within the limit
of scopes of the project in order to built strong foundation on the theoretical
part before proceeding with the next phases of the project.
II. Identifying related data used for the simulation process. This phase involved
collecting all the required data needed before proceeding with the simulation.
Ill. Testing, comparing and modifying collected data. This data that are previously
collected from various trusted sources and references will be used as
comparison with the data obtained from the simulation.
3
CHAPTER2
LITERATURE REVIEW
2.1 Natural Convection between Parallel Plates
Free convection between vertical walls has been studied extensively
due to its importance in various engineering applications. Studies have been
done on this problem in various conditions.
2.1.1 Natural Convection with One Plate Isothermally Heated and the other Thermally Insulated
Jha and Ajibade (2010) (Jha, 2010) studied the natural
convection flow between two infinite vertical parallel plates with one
plate isothermally heated and the other one thermally insulated. From
the study, solutions were obtained for velocity and temperature fields
in the form of convergent infinite series for two cases; Prandtl number
i= 1 and Prandtl number = 1.
Case I: Pr i= 1
u(y, t) co
= (Pr ~ 1) I ({F1 (b3; 1.0) - F1 (b4 ; 1.0) n=O
- ( -l)n[F1(b3; Pr) + F1(b4; Pr)]} co
+ 2 I ( -1)n[F1 (b1; 1.0)- F1 (b2; 1.0)]) m=O
4
Case II: Pr = 1
co
T(y,t)= I(-1)n[erfc(2~)+erfcC~)] n=O
The functional a~, a2, a3, !4, b~, ~. b], b4, F~, F2 and F3 used in
the above equations are defined in Appendix II.
2.1.2 Natural Convection with Constant Temperature and Mass
Diffusion at One Boundary
Marneni (2008) has used the Laplace-transform technique to
solve the transient natural convection flow between two vertical
parallel plates with one boundary having constant temperature and
mass diffusion. Velocity profiles are obtained from the study for two
cases; Schmidt number =f. 1 and Schmidt number = 1. In addition, the
effect of different parameters such as the buoyancy ratio, Schmidt
number, Prandtl number and time are studied.
2.1.3 Turbulent Natural Flow with Rayleigh Number Ranging from 105
to 107
Badr, Habib, Anwar, Ben-Mansour and Said (2006) investigated
buoyancy driven turbulent natural convection in vertical parallel plate
channels that covers Rayleigh numbers ranging from 105 to 107 and
focuses on the effect of the channel geometry and the Nusselt number.
This problem was investigated for two cases; isothermal and isoflux
heating conditions. From this study, two new correlations were
obtained for the average Nusselt number in terms of the Rayleigh
number and channel aspect ratio.
5
2.1.4 Natural Convection with Walls Heated Asymmetrically
Natural convection in a vertical parallel plate channel with
asymmetric heating was studied experimentally and numerically by
Yilmaz and Frazer (2007) using a laser Doppler anemometer (LDA)
in the experiment. The investigation was done with one wall
maintained at uniform temperature and the opposing wall made of
glass. Three different LRN k-e turbulence models were used in the
numerical calculations, and correlation equations were developed for
average heat transfer and induced flow rate using the numerical
results.
Other than Yilmaz et al. (2007), Singh and Paul (2006) have
also studied natural convection between two parallel plates with
asymmetric heating, but focuses on the effect of the Prandtl number
and buoyancy force distribution parameter. This study used the
Laplace transform method to find the solutions for the velocity and
temperature fields. From this study, it has been found that the
symmetric/asymmetric nature of the flow formation can be obtained
by giving a suitable value to the buoyancy force distribution
parameter.
2.1.5 Natural Convection Flows for Low-Prandtl Fluids
While many of the studies in the literature deals with air and
water as the fluid, Campo, Manca and Morrone (2006) dealt with the
natural convection of metallic fluids, with Prandtl number of less than
1 between vertical parallel plates with isoflux heating. Correlation
equations for the induced flow rate, maximum wall temperatures, and
average Nusselt numbers were obtained as functions of the main
thermal Grashof or Rayleigh numbers and geometrical parameters.
6
2.1.6 Turbulent Natural Convection with Symmetric and Asymmetric
Heating
Habib, Said, Ahmed and Asghar (2002) investigated the
velocity characteristics of free convection in both symmetrically and
asymmetrically heated vertical channels. This study, done
experimentally, showed that for the symmetrical flow case, the
measurements indicate high velocity gradient at the shear layer close
to the hot wall and a region of reversed flow at the center of the
channel close to the channel exit as illustrated in Figure 2.1 below.
For asymmetrical flow, the results showed a large vortex with flow
upward the hot wall and down the cold wall with a wider boundary
layer close to the hot wall compared to the cold wall boundary layer
as presented in Figure 2.2.
20 v
18
cmts 16
14
12
10
a
6 Bortcm
4
2 Middle Hot woll
0 ~r "-·--·
/ /
Top
·2
-4 0 0.25 05
x/b
Figure 1: Profiles of the mean vertical velocity component of the symmetrical flow case, Ra = 4.0x 10 6:0 y/L = 0.98; x, y/L = 0.55; •,
y/L = 0.11.
7
12 -v 10
'~ cm/s 8 I ' ~'
6 kf b\ : ..... \~
4 ~ ,.. Bottom
. ·-·-· 2 -·-X •••
0 '~Middle ·-. .,, ~
·--x··x: ~J(....._ ', x, .
-2 ~·· ' '~-. -4 J(.
Top \' \ -6
-8 Hot H
-10
0 0.25 0.5 0.75 LO
wlb Figure 2: Profiles of the mean vertical velocity component of the
asymmetrical flow case, Ra = 2.0 x 10 6: Dy/L = 0.98; x, y/L = 0.55;
•, y/L = 0.11.
Another study on natural convection in an asymmetrically
heated vertical parallel plate channel was done by Fedorov and
Viskanta (1997). Scaling relations for the Reynolds and average
Nusselt numbers have been developed in terms of relevant
dimensionless parameters in this study.
2.1.7 Laminar Flow with UWTIUWC and UHFIUMF
Earlier, Lee (1999) studied natural convection heat and mass
transfer between vertical parallel plates with both unheated entry and
unheated exit. Both boundary conditions of uniform wall temperature/
uniform wall concentration (UWTIUWC) and uniform heat flux/
uniform mass flux (UHFIUMF) were considered in this study.
Analytical solutions of the dimensionless volume flow rate, the
average Nusselt number and the average Sherwood number were
8
derived under fully developed conditions for the boundary conditions
considered.
2.1.8 Natural Convection with Asymmetric Heating Coupled with
Thermal Radiation
Cheng and MUller (1997) also studied natural convection with
asynunetric heating, but with thennal radiation considered as well.
Based on the experimental and numerical results of this study, a semi
empirical correlation was developed to describe the heat transfer of
turbulent natural convection coupled with thennal radiation in a
channel with one-sided heated wall.
9
CHAPTER3
METHODOLOGY
3.1 Research and Project Activities
3.1.1 Research
Comprehensive literature study will be done to identify
situations and conditions in which natural convection can occur.
Many studies provide analytical solutions, which are only available
for simple problems, while numerical solutions can be used to solve
more complex situations. These analytical solutions and experimental
results are to be used later for comparison with the numerical
solutions obtained in this project to estimate the deviation between the
solutions. Preparing literature review is vital at the very early phase of
the project since it can create a good foundation and improves
understanding towards completing the project.
3.1.2 Simulation of Problems
All required data are collected from several sources during the
early phase of the project. After identifying and choosing the
conditions for the problems, a simulation using computational fluid
dynamics (CFD) code, FLUENT will be done for each problem to
obtain numerical solutions for the problems and to get related profiles
such as velocity and temperature profiles.
3.1.3 Comparison and Verification
Outputs gained from the simulation run are analysed and
compared to those output data presented in several sources collected
before from the previous phase of the project. The input data are
modified accordingly with respect to the objective of the project that
is to study the effect of operating conditions on the efficiency of the
process. After the numerical solutions and the appropriate profiles are
10
obtained, these solutions are then compared to the analytical solutions
found in the literature as well as experimental results if it is available.
3.1.4 Conclusion and Report Writing
A conclusion is justified based on the output data collected
from the simulation runs. All these data and analysis involved
throughout the completion of the project are summarized into a
complete documentation of final report thesis writing.
11
3.2 Gantt Chart and Key Milestones
Table 1: Gantt Chart and Key Milestones for FYP I
No. DetaWWeek 1 2 3 4 5 6 7 8 9 10 ll 12 13 14
1 Selection of Project Topic: Simulation of Natural Convection Between Parallel Plates Using CFD ~
2 Preliminary Research Work CQ
3 Submission of Extended Proposal • ~ Cll ~
4 Proposal Defence s ~
5 Project Work Continues (I)
I
6 Submission of Interim Draft Report to Supervisor "0 Jl
;
~ 7 Submission of Final Report to Coordinator
~- - ~
r-. D Process
• Key Milestone
12
Table 2: Gantt Chart and Key Milestones for FYP II
No. DetaiVWeek 1 2 3 4 s 6 7 8 9 10 11 12 13 14 15 1 Project Work Continues
2 Submission of Progress Report
~ • 3 Project Work Continues CQ • 4 Pre-SED EX .... 0 ..... fll
Jl 5 Submission of Draft Report 0 e 0
t 6 Submission of Dissertation (Soft Bound) en I
I
7 Submission of Technical Paper "'0
~ 8 Oral Presentation Jl 9 Submi_~Si<:!n _of pi"Qj_e_~t Dissertation (Hard Bound} •
D Process
• Key Milestone
13
CHAPTER4
RESULT AND DISCUSSION
4.1 Model Equations
Consider a vertical, parallel-plate channel shown schematically in Figure 3.
Air enters the channel at temperature T0 • In this case, the thermal buoyancy force is
considered to be driving the fluid motion. The flow is considered incompressible and
radiation heat transfer is neglected. The Navier-Stokes equations of motion for two
dimensional incompressible flow in a vertical channel can be written as:
Conservation of mass (continuity)
iJ(pu) + iJ(pv) = O iJx iJy
Conservation of x-momentum
~ (puu) + ~ (puv) = ilp + a [CJJ. + J1. ) au] iJx iJy - iJx iJx t iJx
a [ au] 2 ak +- (p. + fl.t)- - -p-ay ay 3 ax
Conservation of y-momentum
:x (puv) + :/PVV) = - :~ + :J (p. + Jl.t) ::1 a [ av] 2 ak +- (p. + fl.t)- - -p-
ay ay 3 ay
+ (p- Po)B
14
Conservation of energy equation
!... ( uT) + !... (pvT) = !... [(.!:. + .!!:!.) ar] + !... [(.!:. + .!!:!.) ar] ax P oy ax cp Prt ax oy cp Prt oy
4.2 Mesh
A mesh, or grid, to model the problem has been formulated in order to
simulate the problem using ANSYS Workbench 12.1. The geometry of the model is
as shown in Figure 3.
Outlet
t
10m
y~ X
Inlet
Figure 3: Geometry of the Model
15
4.2 Simulation
Different boundary conditions are imposed on the plates, and simulations
were done in steady state and transient conditions. At steady state, the boundary
conditions used were: both plates heated at different temperatures, the left plate
heated while the right one is set at zero heat flux, and the right plate heated while the
left one is set at zero heat flux. For transient condition simulation, one plate was
isothermally heated while the other was adiabatic. For all the conditions, the fluid is
set as air, and the inlet temperature at 300 K.
4.2.1 Case 1: Both Plates Set at Different Temperatures (Steady State)
In this case, the temperature of the left plate is set at 1000 K
while the temperature of the right plate is set at 2000 K. The profiles
obtained from the simulation are as shown below.
1.57•03
1.49..03
1.42e-03
1.34e-03
126•03
1.1S.03
1.10e-03
1.02•03
9.46•04
8.67•04
7.89•04
7.11e-04
6.33&-04
5.54•04 4 .76&-04
3.98•04
3.20..04
2.41e-04
1.63•04
8.48..05
6.57•06
Velocity Vectors Colored By Velocity Magnitude (m/s) Nov 13, 2011 ANSYS FLUENT 12.1 (2d, pbns,lam)
Figure 4: Velocity Vector for Case 1
16
4.34.01
4.13.01
3.91•01
3.69•01
3.48.01
3.26•01
3.04•01
2.82•01
2.61•01
2.39•01
2.17•01
1.95•01
1.74•01
1.52•01
1.30•01
1.09•01
8.69•02
6.52•02
4.34.02
2.17•02
O.OOt~O
Contours of Stream Function (kg/s) Nov 13, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 5: Stream Function for Case 1
2.00.-+03
1 .91t~3
1 .83t~3
1 .74t~3
1 .66t~3
1.57e-+03
1 .49t~3
1 .401~3
1.32t~3
1.23t~3
1 .15~3
1 .06.~3
9.77.~2
8.91t~2
8.06~2
7 .21t~2
6.351~2
5.50t~2
4.65e~2
3.79t~2
2.94.~2
Contours of Static Temperature (k) Nov 13, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 6: Temperature Profile for Case 1
17
I• 'E4
1.20&-03
1.00e-03
8.00&-04
6.00&-04
4.00e-04
y 2.00&-04
Veloci~ (m/s O.OOe+OO
~2.008-04
-4.008-04
-6.008-04 • •
-8.008-04 • 0 0.1 0.2 0.3
YVelocity
•
•
• • • •
•
0.4 0.5 0.6 0.7 0.8 0.9
Position ( m)
Nov 13, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 7: Velocity XY Plot for Case 1
2.00&+03
1.808+03
1.60&+03
1.40e+03
Static 1.20o+03 Temperature
(k) 1.000+03
8.008+02
6.00&+02 • •
••
•
• • • • • • • 4.008+02 -+--~~-~-~-~-~~-~-~~
static Temperature
0 ~ ~ ~ M M ~ U M M
Position (m)
Dec 18,2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 8: Temperature XY Plot for Case 1
18
From the velocity vector, it can be seen that the velocity of air
is higher near the right plate, which is heated at a higher temperature,
and that the velocity of air increases as it goes up the plate from the
inlet to the outlet. The recirculatory patterns observed in the stream
function are due to the natural convection between the plates. The
temperature profile predictably shows a higher temperature near the
right plate, which is heated at 2000 K as opposed to the left plate,
which is heated at 1000 K. The temperature profile also shows a
decrease in the temperature of air around the middle of the plates as it
approaches the outlet. This may be caused by the reversed flow that
occurs between the plates.
The velocity XY plot shows the velocity of air in around the
middle of the plates, and it can be observed that for x = 0.55 to x = 1,
reverse flow does not occur whereas it occurs for x = 0 to x = 0.55,
due to the lower temperature of the left plate. The magnitude of the
upward flow increases near the right plate while the magnitude of the
downward flow decreases near the left wall as time increases. Finally
at steady state, the upward flow was formed near the right wall and
the downward flow near the left wall. Exactly at the right and left
plates, the velocity is zero, which is true according to the no-slip
boundary condition. The temperature XY plot at the same position
shows a decrease of temperature as it moves away from the left plate,
then starts to increase at approximately x = 0.45 as it approaches the
right plate.
4.2.2 Case 2: Right Plate Heated, Left Plate at Zero Heat Flux (Steady
State)
In this case, the temperature ofthe right plate is set at 1000 K
while the left plate is set at zero heat flux. The profiles obtained from
the simulation are as shown below.
19
1.17e-03
1.11&-03
1.05e-03
9.93..o4
9.34e-04
8.76&-04
8.18..o4
7.5~
7.01&-04
6.42&-04
5.84e-04
5.26&-04
4 .67&-04
4 .09&-04
3.50&-04
2.92&-04
2.34&-04
1.75e-04
1.17&-04
5.85•05
7.78.0S
Velocity Vectors Colored By Velocity Ma!Jlllude (m/s) Dec 06. 2011 ANSYS FLUENT 12.1 (2d. pbns, lam)
Figure 9: Velocity Vector for Case 2
2.31e-01
2.20.01
2.0S.01
1.97&-01
1.85&-01
1.74e-01
1.62&-01
1.50&-01
1.39&-01
1.27e-01
1.16&-01
1.04e-01
9.26&-02
8.10.02
6.946-02
5.79&-02
4.63&-02
3.47e-02
2.31e-02
1.16&-02
0.00.+00
Contours of stream Foocbon (kg/s) Dec 06, 2011 ANSYS FLUENT 12 1 (2d. pbns, lam)
Figure 10: Stream Function for Case 2
20
1.001'+03
9.651-+02
9.301-+02
8.951-+02
8.601-+02
8.251-+02
7.901-+02
7.551-+02
7 .201-+02
6.85...C2
6.501-+02
6.15...C2
5.801-+02
5.451-+02
5.101-+02
4.751-+02
4.401-+02
4.051-+02
3.701-+02
3.35...C2
3.00...C2
Contours of static Temperature (k) Dec06, 2011 ANSYS FLUENT 12.1 (2d, pbns,lam)
Figure 11: Temperature Profile for Case 2
I • Y=~
y Veloci~
(m/s
YVelocity
8.00.04
6.00e-04
4 .001-04
2.00e-04
0.00...00 •
• -2.0o.o4 • -• .... 00.04
0 0.1 0.2 0.3
• • •
•
•
, • • ,
• • • •
0.4 0.5 0.6 0.7 0.8 0.9
Position (m)
Dec06, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 12: Velocity XY Plot for Case 2
21
I• ~sto 1.00e+03
9.00e+02
8.00e+02
7.00e+02
Static •
• • •
•
Temperature (k)
6.00e+02
5.00e+02
4.00e+02
3.00e+02
0
Static Temperature
..
• • • •
• • -• •
0.1 0.2 0.3 0.4 0.5 0 6 0.7 0.8 0.9
Position (m)
Dec 18, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 13: Temperature XY Plot for Case 2
From the velocity vector, it can be seen that the velocity of air
is higher near the heated right plate, and that the velocity of air
increases as it goes up the plate from the inlet to the outlet. The stream
function shows the mass flow rate near the inlet to be high and
concentrated at the heated plate, while as the flow goes toward the
outlet it moves toward the adiabatic left plate as well, even as it
increases near the heated right plate. The temperature profile
predictably shows a higher temperature near the right plate, which is
heated while near the left plate, the temperature of air remains at 300
K., which is the initial temperature of air when it enters the channel.
The temperature profile also shows that near the heated plate, as the
air approaches the outlet, the region of heated air near the plate gets
larger, further from the plate.
The velocity XY plot shows the velocity of air in around the
middle of the plates, and it can be observed that for x = 0.7 to x = 1,
22
reverse flow does not occur whereas it occurs for x = 0 to x = 0. 7, as a
result of more cooling of the air near the left plate. The magnitude of
the upward flow increases near the right plate while the magnitude of
the downward flow decreases near the left wall as time increases.
Finally at steady state, the upward flow was formed near the right wall
and the downward flow near the left wall. Exactly at the right and left
plates, the velocity is zero, which is true according to the no-slip
boundary condition.
The temperature XY plot at the same position shows an
increase of temperature as it approaches the right plate, while close to
the left plate, the temperature of air is 300 K, the initial temperature of
air at the inlet. However, it can be observed from the graph that at y =
7, the temperature of the air increases faster as it approaches the right
plate compared to air at y = 5 andy= 10, even though at y = 10, the
air is at the outlet. Theoretically, at y = I 0, the temperature of air
should increase faster as it approaches the heated right plate compared
to both at y = 5 and y = 7. This may be due to the reversed flow that
exists between the two plates.
4.2.3 Case 3: Left Plate Heated, Right Plate at Zero Heat Flux (Steady
State)
In this case, the temperature of the left plate is set at 1000 K
while the right plate is set at zero heat flux. The profiles obtained from
the simulation are as shown below.
23
1 09•03
1.D4.o3
9.81•04
9.26•04
8.72•04
8.17•04
7 63•04
7.08.04
6.54.04
5.99•04
5.45•04
4.90.04
4.36•04
3.81•04
3.27.0..
2.73•04
218•04 1.64.0..
1.09.0..
5.47•05
1.80.07
Velocity Vectors Colored By Velocity Magn~tude (m/s) Nov 13, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 14: Velocity Vector for Case 3
1.9J.-01
1 .8~1
17...01
1.64..o1
1.54.01
1.4S.01
1.3S.01
1.2»01
1.1&.01
1.0&.01
9.6»02
8.68.02
7.72.02
6.75•02
5.7S.02 4.82•02
3.8&.02
2.8S.02
1.9*02 9.6S.03
0.00.+00
Contours of Stream Function (kg/s) Nov 13.2011 ANSYS FLUENT 121 (2d, pbns, lam)
Figure 15: Stream Function for Case 3
24
1 .00e+03
9.65e+02
9.308+02
8.95e+02
8.60e+02
8.258+02
7.90.+02
7.55e+02
7.20.+02
6.85e+02
6.508-+02
6.158-+02
5.80e+02
5.o45e+02
5.10e+02
U5e+02
o4.o40e+02
4 .058+02
3.70e+02
3.358+02
3.00e+02
Contours of Static Temperature (k) Nov 13.2011 ANSYS FLUENT 12.1 (2d. pbns. lam)
Figure 16: Temperature Profile for Case 3
I • v=4
6.~
5.00e-04 •
4.~
3.~
2.~ y
Veloci~ 1.00.04
(m/s O.OOe+OO
• •
-1.00.04
-2.~
-3.~
0 0.1 0.2 0.3
YVelocity
•
• •
• • •• •
0.4 0.5 0.6 0.7 0.8 0.9
Position (m)
Nov 13, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 17: Velocity XY Plot for Case 3
25
I• '€4
1.00e+03
9.00e+02
8.00e+02
7.00e+02
Static Temperature 6.00e+02
(k) 5.00e+02
4.00e+02
Static Temperature
•
• • 0 0.1 u 0~ u u u u u u
Position ( m)
Dec 18,2011 ANSYS FLUENT 12.1 (2d, pbns, lam)
Figure 18: Temperature XY Plot for Case 3
From the velocity vector, it can be seen that the velocity of air
is higher near the heated left plate, and that the velocity of air
increases as it goes up the plate from the inlet to the outlet The stream
function shows the mass flow rate near the inlet to be high and
concentrated at the heated plate, while as the flow goes toward the
outlet it moves toward the adiabatic right plate as well, even as it
increases near the heated left plate.
The temperature profile predictably shows a higher
temperature near the left plate, which is heated while near the right
plate, the temperature of air remains at 300 K, which is the initial
temperature of air when it enters the charmel. The temperature profile
also shows that near the heated plate, as the air approaches the outlet,
the region of heated air near the plate gets larger, further from the
plate. However, it can be observed from the graph that at the outlet,
the region of heated air is smaller than that neat the outlet
Theoretically, the outlet, the region of heated air should be larger
26
compared to further inside the plates. This may be due to the reversed
flow that exists between the two plates.
The velocity XY plot shows the velocity of air in around the
middle of the plates, and it can be observed that for x = 0 to x = 0.3,
reverse flow does not occur whereas it occurs for x = 0.3 to x = l, as a
result of more cooling of the air near the right plate. The magnitude of
the upward flow increases near the left plate while the magnitude of
the downward flow decreases near the right wall as time increases.
Finally at steady state, the upward flow was formed near the left wall
and the downward flow near the right wall. Exactly at the right and
left plates, the velocity is zero, which is true according to the no-slip
boundary condition. The temperature XY plot at the same position
shows an increase of temperature as it approaches the left plate, while
close to the right plate, the temperature of air is 300 K, the initial
temperature of air at the inlet.
4.2.4 Case 4: Right Plate Heated, Left Plate at Zero Heat Flux
(Transient)
In this case, the transient simulation was done with the
temperature of the right plate set at 1000 K while the left plate set at
zero heat flux. The profiles obtained from the simulation are as shown
below.
27
1.50e-11 1.4-4e-11 1.388-11 1.32e-11 1.26e-11 1.20e-11 1.14e-11 1.08e-11 1.02e-11 9.62e-12 9.02e-12 8.42e-12 7.83e-12 7.23e-12 6.63e-12 6.03e-12 5.43e-12 4.84e-12 4.24e-12 3.64e-12 3.04e-12 2.44e-12 1.85e-12 1 .. 25e-12 6.50e-13 5.17e-14
Velocity Vectors Colored By Velocity Magnttude (m/s) (Time=2.0000e-01) Dec 27, 2011 ANSYS FLUENT 12.1 (2d. pbns. lam. transient)
Figure 19: Velocity Vector at the Start of the Simulation
9.72e-05 9.33e-05 8.94e-05 8.568-05 8.17e-05 7.788-05 7.40..05 7.01e-05 6.62e-05 6.23e-05 5.85e-05 5.46e-05 5.07e-05 4.68e-05 4.30e-05 3.91e-05 3.52e-05 3.14e-05 2.75e-05 2.36e-05 1.97e-05 1.59•05 1.20e-05 8.13e-06 4.26e-06 3.90•07
Velocity Vectors Colored By Velocity Magnitude (mls) (Time=1 .0000e+03) Dec 27. 2011 ANSYS FLUENT 12.1 (2d, pbns, lam. transient)
Figure 20: Velocity Vector After Time has Passed
28
8.546-09 8~09
7.866-09 7.52e-09 7.1S.09 6.83e-09 6.496-09 6.156-09 5.816-09 5.476-09 5.13e-09 4 .78e-09 4 .44e-09 4 .10..09 3.766-09 3.42e-09 3.08e-09 2.73e-09 2.396-09 2.056-09 1.71e-09 1.376-09 1.03e-09 6.836-10 3.42e-10 0.00..00
Cortours of stream Ft.netion (kgls) (Time=8.()()()()e.()1) Dec 27, 2011 ANSYS FLUENT 12.1 (2d, pbns,lam, transler'i)
Figure 21: Stream Function at the Start of the Simulation
6.63e-03 6.36e-03 6.10e-03 5.836-03 5.576-03 5.306-03 5.04•03 4 .776-03 4.!516-03 4.24e-03 3.98e-03 3.716-03 3.456-03 3.186-03 2.92e-03 2.65e-03 2.39e-03 2.126-03 1.86e-03 1.596-03 1.336-03 1.066-03 7.956-04 5.30..04 2.656-04 0.00..00
ContOli'S of Stream Function (kgls) (Time=1.0000e+03) Dec 27, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam, translert)
Figure 22: Stream Function After Time has Passed
29
1.00e+03 9.72e+02 9.44e-+02 9.168+02 8.88e+02 8.60e+02 8.32e+02 8.04e+02 7.768+02 7.48e+02 7.204H02 6.92...02 6.648+02 6.368+02 6.08e+02 5.80e+02 5.52...02 5.24e+02 4 .968+02 4 .68e+02 4.40e+02 4 .12e+02 3.84e+02 3.568+02 3.28e+02 3.00e+02
ContOli'S of Static Temperature (k) (Time=2.0000e-01) Dec 27, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam, transiet't)
Figure 23: Temperature Profile at the Start of the Simulation
1.00e+03 9.72•+02 9.44e+02 9.168+02 8.88e+02 8.60e..o2 8.32e..o2 8.04e+02 7.768+02 7.48e+02 7.20e+02 6.92e+02 6.64e+02 6.368+02 6.08e+02 15.80e..o2 15.152e+02 15.2441-+02 4.96e..o2 4 .68e+02 4.40.-+02 4 .12e+02 3.84e+02 3.568+02 3.28e+02 3.00.+02
ContOli'S of Static Temperat\J'e (k) (Tlme=1 .0000e+03) Dec 27, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam, transieti)
Figure 24: Temperature Profile After Time has Passed
30
2.00e-08
Velocity o.ooe-+Oo Magnitude
(m/s)
···-·····-·-··· ..
-2.00e-08 +---.---r---.----.--r---y---,---,.----.---. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Position (m)
Velocity Magnitude (Time=2.0000e-01) Dec 27,2011 ANSYS FLUENT 12.1 (2d, pbns, lam, transient)
Figure 25: Velocity XY Plot at the Start of the Simulation
6.00e-05
5.00..05
4.00e-05
Velocity 3.00e-05
Magnitude (m/s)
2.00e-05
1.00e-05 • • • -.
O.OOe+OO • • • • •
0 0.1 0.2 0.3
Velocity Magnitude (Time=1.0000e+03)
• • • -• •
0.4 0.5 0.6
Position (m)
• • • . -•
0.1
•
•
0.8
•
' 0.9
• •
•
Dec 27. 2011 ANSYS FLUENT 12.1 (2d, pbns. lam. transient)
Figure 26: Velocity XY Plot After Time has Passed
31
1.00&+03
9.00e+02
8.00e+02
7.00&+02
Static 6.ooe+02
Temperature (k) 5.00e+02
4 .00e+02
3.00&+02 ... -..... -. • • • • 2.00&+02 -+--..------.---.---.----,.---,---..---.----r-~
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Position (m)
Static Temperature (Time=2.0000e-01 ) Dec 27, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam, transient)
Figure 27: Temperature XY Plot at the Start of the Simulation
1.00e+03
9.00&+02
8.00&+02
7.00e+02
Static 6.00&+02
Temperature (k) 5.00&+02
4 .00e+02
3.00&+02 1
2.00e+02 0
• . ··-·····-·-·· 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9
Position ( m)
•
static Temperature (Time=1 .0000e+03) Dec 27, 2011 ANSYS FLUENT 12.1 (2d, pbns, lam, transient)
Figure 28: Temperature XY Plot After Time has Passed
32
The stream function contours show that the flow rate of air
increases as the simulation goes on, and that the mass flow rate near
the heated plate is always higher compared to the flow rate near the
adiabatic plate. This agrees with the other simulations done in the
steady state condition. The stream function shows that the mass flow
rate of air increases with time. The temperature profile, on the other
hand, does not show any change in the temperature of air, even as
time passes.
The XY plots show the value of the parameters at different
positions along the y-axis: (1) black at y=0.1 near the inlet, (2) red at
y=S in the middle and (3) green at y=9.9 near the outlet.
The velocity XY plot in the at the beginning shows the same
velocity, at 0 m/s, and then, as time passes, the velocity of the air
increases, with the air close to the heated right plate at approximately
x = 0.95 relatively higher than the velocity of air away from the
heated plate. Near the entrance of the plate, the velocity increases
rapidly from the cold left wall to the heated right wall, while away
from the inlet, the velocity of air that is not close to the heated plate is
approximately uniform. This is because at the inlet the temperature
gradient of air near the cold wall and the right wall is great, while this
gradient decreases as air moves toward the outlet. There is a slight
decrease in velocity near the heated right plate, due to the reverse
flow. Even as the velocity of air increases with time, exactly at the
plates, the velocity is zero, which adheres to the no-slip boundary
condition.
Although from the temperature profile, no apparent change in
the temperature can be observed even as time passes, looking at the
temperature XY plot, it can be seen that there is a change in
temperature that happens slowly. At the beginning of the simulation,
the temperature is uniform across the plate at 300 K, which is the inlet
temperature of air, except at x = 1, due to the right plate being heated
at 1000 K. As time passes, the temperature of air close to the heated
33
right plate at approximately x = 0.95 increases slightly, even though
the temperatures further away from the heated plate remains the same.
As even more time passes, the temperature of the air further from the
heated plate starts to increase as well even as the temperature of air
near the heated right plate increases, and this goes on until steady state
is reached.
34
CHAPTERS
CONCLUSION AND RECOMMENDATIONS
A mesh for the specified geometry of two vertical parallel plates with height
of 10 m has been created for the simulation purposes of this study. Simulations have
been done for four different situations using the model for both steady state and
transient conditions, using air as the working fluid. From the profiles obtained from
the CFD code, it can be observed that the velocity of air increases near a heated
plate, but becomes exactly zero at the plate itself due to the no-slip boundary
condition. In some cases, the velocity at certain parts between the plates is negative,
due to the reversed flow that occurred between the plates. The mass flow rate is
higher close to the heated plate, as does the temperature. For velocity, mass flow rate
and temperature of the air, all increases as the air moves from the inlet to the outlet,
but in certain cases near the outlet the parameter decreases slightly due to reverse
flow. According to the results from the transient simulation, the velocity, mass flow
rate and temperature increases with time.
Further study can be done for different working fluids, instead of air. In these
simulations, radiation was not taken into account. The CFD code provides several
radiation models, which can be used so that radiation can be taken into account in the
simulation. Simulations can be done with radiation taken into account, and the most
suitable radiation model for the simulation should be determined. Other operating
conditions and geometries should be investigated to observe their effects on natural
convection.
35
REFERENCES
Badr, M. H., Habib, A. M., Anwar, S. Ben-Mansour, R. & Said, A. S. (2006).
Turbulent Natural Convection in Vertical Parallel-Plate Channels. Journal of
Heat and Mass Transfer, 43, 7 3 - 84.
Campo, A., Manca, 0. & Morrone, B. (2006). Nwnerical Investigation of The
Natural Convection Flows for Low-Prandtl Fluids in Vertical Parallel-Plates
Channels. Journal of Applied Mechanics, 73.
Cheng, X. & MUller, U. (1997). Turbulent Natural Convection Coupled with
Thermal Radiation in Large Vertical Channels with Asymmetric Heating.
International Journal of Heat Mass Transfer, 41, 1681-1692.
Fedorov G. A. & Viskanta, R. (1997). Turbulent Natural Convection Heat Transfer
in an Asymmetrically Heated, Vertical Parallel-Plate Channel. International
Journal of Heat Mass Transfer, 40, 3849- 3860.
Habib, A. M., Said, M. A. S., Ahmed, A. S. & Asghar, A. (2002). Velocity
Characteristics of Turbulent Natural Convection in Symmetrically and
Asymmetrically Heated Vertical Channels. Experimental Thermal and Fluid
Science, 26, 77- 87.
Holman, J.P. (2001). Heat Transfer: Eighth SI Metric Edition (8th Ed.). New York:
McGraw-Hill Higher Education.
Jha, K. B. & Ajibade, 0. A. (2010). Transient Natural Convection Flow Between
Vertical Parallel Plates: One Plate Isothermally Heated and The Other
Thermally Insulated. Journal of Process Mechanical Engineering, 224, 247-
252.
Lee, T. K. (1999). Natural Convection Heat and Mass Transfer in Partially Heated
Vertical Parallel Plates. International Journal of Heat and Mass Transfer, 42,
4417-4425.
Marneni, N. (2008). Transient Free Convection Flow Between Two Long Vertical
Parallel Plates with Constant Temperature and Mass Diffusion. World
Congress on Engineering 2008.
36
Pitts, R. D., & Sissom, E. L. (1998). Theory and Problems of Heat Transfer (2"d
Ed.). New York: McGraw-Hill.
Singh, K. A. & Paul, T. (2006). Transient Natural Convection Between Two Vertical
Walls Heated/ Cooled Asymmetrically. International Journal of Applied
Mechanics and Engineering, 11, 143 -154.
Wright, J. L. & Sullivan, H. F. (1194). A 2-D Numerical Model for Natural
Convection in a Vertical, Rectangular Window Cavity. ASHRAE
Transactions, Vol100, Pt. 2, pp. 1193-1206.
Yilmaz, T. & Fraser, M. S. (2007). Turbulent Natural Convection in a Vertical
Parallel-Plate Channel with Asymmetric Heating. International Journal of
Heat and Mass Transfer, 50, 2612- 2623.
37
APPENDIX I
u Horizontal velocity component of the fluid in the x-direction
v Vertical velocity component of the fluid in the y-direction
t Time component
Pr Prandtl number
T Temperature of the fluid
T0 Initial temperature
p Density
k Kinetic energy of turbulence or thermal conductivity
g Acceleration due to gravity
cp Specific heat
38
APPENDIX II
The functions below are used in the equations from a journal described in Section 2.1.1.
(a ~ ~ ( a
2 b) F2 (a; b; c)= ab erfc 2 ~~}- 2 ~"'ii"exp - 4,
2 f."' erfc('q) = ...[ii 'I exp( -x2) dx
a1 = (2n+1)ffr+2m+y
a2 = (2n + 1)ffr + (2m+ 2) - y
a3 = 2n+ 1-y
a4 = 2n+ 1 +y
39