numerical simulation of natural convection between two

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)


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)


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)


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)


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)



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.


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)


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)


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)


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)


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)



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.


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

numerical simulation of natural convection between two

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