Connjjuugga at tee aHHeeat lTTrrannssffeerr ... · Connjjuugga at tee aHHeeat lTTrrannssffeerr AAnnaalyyssiiss wwiitthhiinn aa B otttto omm dHHeeaatteed CNNon n--cconnvveennttiioonnaall

CCoonnjjuuggaattee HHeeaatt TTrraannssffeerr AAnnaallyyssiiss wwiitthhiinn aa

BBoottttoomm HHeeaatteedd NNoonn--ccoonnvveennttiioonnaall CCyylliinnddrriiccaall

EEnncclloossuurree

AAssiiff HHuussssaaiinn MMaalliikk

0044--UUEETT//PPhhDD--MMEE--0066

Department of Mechanical Engineering

Faculty of Mechanical & Aeronautical Engineering

UUnniivveerrssiittyy ooff EEnnggiinneeeerriinngg && TTeecchhnnoollooggyy

TTaaxxiillaa –– PPaakkiissttaann

JJuullyy 22001122

ii

CCoonnjjuuggaattee HHeeaatt TTrraannssffeerr AAnnaallyyssiiss wwiitthhiinn aa

BBoottttoomm HHeeaatteedd NNoonn--ccoonnvveennttiioonnaall CCyylliinnddrriiccaall

EEnncclloossuurree

AAuutthhoorr

AAssiiff HHuussssaaiinn MMaalliikk

0044--UUEETT//PPhhDD--MMEE--0066

A thesis submitted in partial fulfilment of the requirement for the degree of

PhD Mechanical Engineering

Thesis Supervisor

Prof. Dr. Muhammad Saleem Iqbal Alvi

Thesis Supervisor‟s Signature:______________________________

___________________________________ ____________________________________

External Examiner‟s Signature External Examiner‟s Signature

PPrrooff.. DDrr.. GGhhuullaamm YYaasseeeenn CChhoohhaann PPrrooff.. DDrr.. IIjjaazz AAhhmmaadd CChhoouuddhhrryy

Department of Mechanical Engineering

UUnniivveerrssiittyy ooff EEnnggiinneeeerriinngg && TTeecchhnnoollooggyy TTaaxxiillaa -- PPaakkiissttaann

JJuullyy -- 22001122

iii

Supervisor

Prof. Muhammad Saleem Iqbal Alvi

Members of Research Monitoring Committee

Prof. Dr Shahab Khushnood

Prof. Dr Fathi M. Mahfouz

Dr Muhammad Khalid Khan Ghauri

Foreign Research Evaluation Experts

Prof. Dr. Brian Norton, Ireland

Asstt. Prof. Dr. Martin Tango, Canada

iv

Foreign Expert Evaluation Report

This thesis presents important research on conjugate heat transfer in cylindrical

enclosures. Both experimental and theoretical work are carried out very competently

with appropriate attention to detail. Good model validation is evident.

The study is grounded in a very thorough and up-to-date appreciation of the

extensive relevant previous research work on this topic. That sure foundation

certainly informs the excellent work presented but the review could be more critical

of methods and their validity and be referred to more significantly in the discussion

of the novelty of the results. The use of air as a heat transfer fluid inevitably means

that axial conduction in the metal walls dominates key aspects of system behavior.

The conclusions of this thesis reflect that.

The thesis makes a significant contribution to research in conjugate heat transfer

problems. It is recommended for the award of PhD.

Brian Norton

v

Foreign Expert Evaluation Report

This work is unique because it systematically explored the total heat transfer in

cylindrical enclosures, an application that is vital in many industrial product and

equipment. The experimental apparatus/instrumentation is well designed and the

procedure was well laid out. The numerical simulation was well articulated and is

compatible to established technique in this field.

Through robust experimental study and CFD simulation, this research work clearly

demonstrated:

a. Understanding of the thermal behavior of the enclosures of inner cylinders of

different materials with significant difference in thermal conductivities. The CFD

simulations were also performed and validated by the experimental results.

b. Understanding (using CFD simulations) of the heat transfer and buoyancy effects

within the bottom heated vertical concentric cylinder enclosure geometries using

streamlines, thermal lines and velocity vectors.

Martin Tango

vi

Dedicated

To My Sweetest

Amara

vii

Acknowledgement

All praises to Almighty Allah, the most Merciful, Compassionate, Gracious and

Beneficent, Who has created this world and is the entire source of knowledge and

wisdom endowed to mankind.

I am greatly thankful to my supervisor, Dr Muhammad Saleem Iqbal Alvi and review

monitoring members Dr Shahab Khushnood, Dr Fathi Mahfouz and Dr Khalid Khan

Ghauri for their supervision, keen interest, technical advises and support during the

research, experimentation, publications and preparation of thesis. I am really

grateful to Dr Ajmal Shah for sincere guidance, technical support during the

research, CFD simulation in Fluent software and publications. He helped me in

carrying out some of the required modifications to the CFD simulations as well as in

thesis writing.

I am thankful to Engineer Nazir Ahmed Mirza who gave his sincere support to

conduct Ph D research. I am grateful to Engineer Rafaqat Ali Mughal who gave spare

time in the office, morally supported at each and every step during Ph D research

work, allowed technical staff for help in experiments. I am also thankful to Engineer

Farakh Iqbal, Scientist Abdul Qavi Qazi and their technical staff who fabricated data

acquisition system. I am also thankful to Muhammad Asif and Muhammad Ramzan

who physically helped to conduct experiments.

I would like to express my dearest feelings and respect towards my parents for their

endless prayers and support under which I always feel secure. I am greatly indebted

to my wife for her support and constant encouragement and for inspiring me when I

needed it. I am also grateful to my cousin Dr Muhammad Yousaf Awan who gave me

his sincere advises during research work. At the end, I am grateful to all those who

consistently wished me glittering on the skies of success. May ALLAH bless them

with healthy and long lives. With my deepest gratitude,

Asif Hussain Malik

viii

Research Work Publications

Asif Hussain Malik, M.S.I. Alvi, Shahab Khushnood, F.M. Mahfouz, M.K.K. Ghauri,

Ajmal Shah, Experimental study of conjugate heat transfer within a bottom heated

vertical concentric cylindrical enclosure, International Journal of Heat and Mass

transfer, 55 (4), 2012, p. 1154-1163


Ajmal Shah, Numerical Study of Conjugate Heat Transfer within a Bottom Heated

Cylindrical Enclosure, Proceedings of 9th International Bhurban Conference on

Appplied Science and Technology (IBCAST), Islamabad, Pakistan, 9th – 12th January,

2012 held at National Centre of Physics, Islamabad, Pakistan and published by IEEE

publishers.

Asif Hussain Malik, Shahab Khushnood, Experimental Study of Flow-induced

Vibration of Prismatic Bodies in Parallel Flow, presented in “National Workshop on

Vibration Analysis” arranged by Pakistan Atomic Energy Commission in National

Centre of Physics, Islamabad on 16th – 17th April, 2012


Ajmal Shah, Effects of Wall Material on Heat Transfer within a Bottom Heated

Vertical Concentric Cylindrical Enclosure, International Journal of Heat and Fluid

Flow, Under Review

ix

Table of Contents

ACKNOWLEDGEMENT ......................................................................................................................... VII

RESEARCH WORK PUBLICATIONS .......................................................................................................... VIII

TABLE OF CONTENTS ........................................................................................................................... IX

LIST OF FIGURES ................................................................................................................................ XII

LIST OF TABLES ................................................................................................................................ XVI

NOMENCLATURE ............................................................................................................................ XVIII

ABSTRACT......................................................................................................................................XXIII

CHAPTER-1: INTRODUCTION ...................................................................................................... 1

1.1 HEAT TRANSFER IN ENCLOSURES .................................................................................................... 1

1.2 MODES OF HEAT TRANSFER .......................................................................................................... 2

1.2.1 Conduction heat transfer ............................................................................................. 2

1.2.2 Convection heat transfer .............................................................................................. 3

1.2.3 Radiation heat transfer ................................................................................................ 4

1.2.4 Conjugate heat transfer ............................................................................................... 5

1.3 PROBLEM DEFINITION ................................................................................................................. 5

1.4 RESEARCH OBJECTIVES ................................................................................................................ 7

1.5 THESIS ORGANIZATION ................................................................................................................ 8

CHAPTER-2: LITERATURE SURVEY ............................................................................................. 11

2.1 HISTORY OF HEAT TRANSFER ....................................................................................................... 11

2.2 CLASSIFICATION BASED ON HEAT SOURCE ....................................................................................... 12

2.2.1 Enclosures with internal heat source .......................................................................... 13

2.2.2 Enclosures with lateral wall heat source ..................................................................... 14

2.2.3 Enclosure with bottom wall heat source ..................................................................... 16

2.3 CLASSIFICATION BASED ON ENCLOSURE GEOMETRY ........................................................................... 20

2.3.1 Cylindrical enclosures ................................................................................................. 20

2.3.2 Rectangular enclosures .............................................................................................. 22

2.3.3 Square enclosures ...................................................................................................... 25

2.3.4 Other enclosures ........................................................................................................ 26

CHAPTER-3: MATHEMATICAL FORMULATION .......................................................................... 29

3.1 CONSERVATION OF MASS........................................................................................................... 29

3.2 CONSERVATION OF MOMENTUM ................................................................................................. 30

3.3 CONSERVATION OF ENERGY ........................................................................................................ 31

x

3.3.1 Viscous dissipation term ............................................................................................ 32

3.3.2 Energy source due to chemical reaction...................................................................... 33

3.3.3 Energy equation in solid regions ................................................................................. 33

3.4 NATURAL CONVECTION AND BUOYANCY EFFECTS ............................................................................. 34

3.4.1 Boussinesq Model ...................................................................................................... 35

3.5 RADIATION HEAT TRANSFER........................................................................................................ 36

3.6 BOUNDARY CONDITIONS ........................................................................................................... 38

CHAPTER-4: EXPERIMENTAL SYSTEM AND DATA ............................................................................. 41

4.1 CYLINDRICAL ENCLOSURE ........................................................................................................... 43

4.1.1 Bottom disc ............................................................................................................... 44

4.1.2 Inner cylinder ............................................................................................................. 46

4.1.3 Outer Cylinder ........................................................................................................... 47

4.1.4 Enclosure’s centerline ................................................................................................ 49

4.2 ELECTRIC HEATER .................................................................................................................... 49

4.3 DATA ACQUISITION SYSTEM ........................................................................................................ 49

4.4 TEMPERATURE CONTROL SYSTEM................................................................................................. 51

4.5 TEMPERATURE SENSORS ............................................................................................................ 52

4.5.1 PT-100 temperature sensors ...................................................................................... 52

4.5.2 N-type Thermocouple ................................................................................................ 53

4.6 CONVECTION HEAT TRANSFER COEFFICIENT .................................................................................... 54

4.7 UNCERTAINTY ANALYSIS ............................................................................................................ 56

CHAPTER-5: NUMERICAL ANALYSIS .......................................................................................... 57

5.1 INTRODUCTION TO THE CFD SIMULATIONS..................................................................................... 57

5.2 HEAT TRANSFER TO THE ENCLOSURE ............................................................................................. 57

5.2.1 Geometry and meshing .............................................................................................. 58

5.2.2 Boundary conditions .................................................................................................. 61

5.2.3 CFD models applied ................................................................................................... 62

5.3 GRID INDEPENDENCE STUDY ....................................................................................................... 64

CHAPTER-6: RESULTS AND DISCUSSION ................................................................................... 66

6.1 INTRODUCTION ....................................................................................................................... 66

6.2 AXIAL THERMAL BEHAVIOR ......................................................................................................... 67

6.2.1 Axis of the enclosure .................................................................................................. 67

6.2.2 Inner cylinder ............................................................................................................. 69

6.3 RADIAL THERMAL BEHAVIOR ....................................................................................................... 71

xi

6.3.1 Enclosure with aluminum inner cylinder ..................................................................... 72

6.3.2 Enclosure with mild steel inner cylinder ...................................................................... 74

6.3.3 Enclosure with stainless steel inner cylinder................................................................ 75

6.3.4 Wall thickness effects on heat transfer mechanism .................................................... 77

6.4 NON-DIMENSIONAL RESULTS ...................................................................................................... 78

6.4.1 Nusselt number ......................................................................................................... 79

6.4.2 Rayleigh number ........................................................................................................ 81

6.5 THE CFD SIMULATION RESULTS ................................................................................................... 82

6.5.1 Validation of the CFD simulation ................................................................................ 83

6.5.2 Contours of streamlines at 353 K ................................................................................ 86


the buoyancy effects are stronger while using inner cylinder of aluminum as compared to other inner

cylinders due to its high thermal conductivity. .......................................................................... 91


CHAPTER-7: CONCLUSIONS AND FUTURE RECOMMENDATIONS .............................................. 94

7.1 CONCLUSIONS ........................................................................................................................ 94

7.2 FUTURE RECOMMENDATIONS ..................................................................................................... 96

REFERENCES ..................................................................................................................................... 97

APPENDIX-A ................................................................................................................................... 107

A-1: CONVECTION HEAT TRANSFER COEFFICIENT .................................................................................... 107

A-2: EXPERIMENTAL TEMPERATURE DATA ............................................................................................. 107

APPENDIX-B ................................................................................................................................... 114

xii

List of Figures

Figure 1-1: Gas centrifuge machine ............................................................. 3

Figure 1-2: Schematic diagram of vertical concentric cylindrical enclosure ..... 6

Figure 3-1: Boundary conditions of vertical concentric cylindrical enclosure 38

Figure 4-1: Experimental Apparatus .......................................................... 42

Figure 4-2: Cross-sectional view of concentric cylindrical enclosure ............. 43

Figure 4-3: Various bottom discs used in the experiments .......................... 44

Figure 4-4: Schematic diagram of vertical concentric cylindrical enclosure showing

geometric specifications. .......................................................................... 45

Figure 4-5: Bottom disc of aluminum with the thermocouple clamp ............ 46

Figure 4-6: Various inner cylinders used in the experiments ........................ 47

Figure 4-7: Two outer cylinders O1 and O2 of mild steel used in the experiments 48

Figure 4-8: Data Acquisition System (DAS) ................................................ 50

Figure 4-9: Schematic diagram of temperature control system .................... 50

Figure 4-10: Schematic diagram of the temperature measuring system ....... 51

Figure 4-11: Temperature sensors mounted on the outer cylindrical wall ..... 52

Figure 5-1: The enclosure geometry .......................................................... 58

Figure 5-2: Meshing of enclosure geometries of outer cylinder, O1 & O2 outer cylinder

............................................................................................................... 59

Figure 5-3: Meshing of enclosure bottom geometry of outer cylinder, O1.& O2.60

xiii

Figure 5-4: Meshing of enclosure top geometry of outer cylinder, O1.& O2. .. 60

Figure 5-5: Meshing of enclosure geometry showing annular and width gaps61

Figure 6-1: Axial temperature distribution along the axis of the enclosure with inner

cylinder of (a, b) aluminum, (c, d) mild steel and (e, f) stainless steel. ........ 68

Figure 6-2: Axial temperature along the inner surface of inner cylinder of aluminum

(a, b), mild steel (c, d) and stainless steel (e, f) ......................................... 70

Figure 6-3: Radial temperature distribution with aluminum inner cylinder .... 73

Figure 6-4: Radial temperatures with mild steel inner cylinder .................... 74

Figure 6-5: Radial temperature with stainless steel inner cylinder ................ 76

Figure 6-6: Nusselt number along inner cylinder wall ................................. 80

Figure 6-7: Local Nusselt number with Rayleigh number ............................. 82

Figure 6-8: Comparison of experimental and the CFD results of enclosure with

aluminum inner cylinder ........................................................................... 84

Figure 6-9: Comparison of experimental and the CFD results of enclosure with mild

steel inner cylinder ................................................................................... 85

Figure 6-10: Comparison of experimental and the CFD results of enclosure with

stainless steel inner cylinder ..................................................................... 86

Figure 6-11: Streamlines of enclosure for configuration O1 a) aluminum inner

cylinder, c). mild steel inner cylinder, e) stainless steel inner cylinder, for

configuration O2 b) aluminum inner cylinder, mild steel inner cylinder, f), stainless

steel inner cylinder at 353 K. .................................................................... 88



xiv







Figure B-1: Thermal lines of enclosure for configuration O1, a) aluminum inner


configuration O2, b) aluminum inner cylinder, d). mild steel inner cylinder, f),

stainless steel inner cylinder at 353 K. ..................................................... 114

Figure B-2: Thermal lines of enclosure for configuration O1 a) aluminum inner


configuration O2 b) aluminum inner cylinder, d). mild steel inner cylinder, f),


Figure B-3: Thermal lines of enclosure for configuration O1 a) aluminum inner




Figure B-4: Velocity vectors of enclosure for configuration O1 a) aluminum inner








xv





xvi

List of Tables

Table 4-1: Geometric configurations of cylindrical enclosure ....................... 45

Table 4-2: Temperature Sensors Distribution in the Enclosure .................... 53

Table 5-1: Properties of Fluid (air) ............................................................ 64

Table 5-2: Properties of Materials.............................................................. 64

Table 5-3: Grid independence study .......................................................... 64

Table A-1: Convective heat transfer coefficient of air within the concentric cylindrical

enclosure............................................................................................... 107

Table A-2: Bottom disc temperature data with outer cylinder O1 ............... 108

Table A-3: Bottom disc temperature data with outer cylinder O2 ............... 108

Table A-4: Experimental temperature data on inner surface of inner cylinder with

outer cylinder O1 .................................................................................... 109

Table A-5: Experimental temperature data on inner surface of inner cylinder with

outer cylinder O2 .................................................................................... 109

Table A-6: Experimental temperature data on outer surface of inner cylinder with

outer cylinder O1 .................................................................................... 110

Table A-7: Experimental temperature data on outer surface of inner cylinder with

outer cylinder O2 .................................................................................... 110

Table A-8: Experimental temperature data on inner surface of outer cylinder with

outer cylinder O1 .................................................................................... 111

Table A-9: Experimental temperature data on inner surface of outer cylinder with

outer cylinder O2 .................................................................................... 111

xvii

Table A-10: Experimental temperature data on outer surface of outer cylinder with

outer cylinder O1 .................................................................................... 112

Table A-11: Experimental temperature data on outer surface of outer cylinder with

outer cylinder O2 .................................................................................... 112

Table A-12: Experimental temperature data on axis with outer cylinder O1 113

Table A-13: Experimental temperature data on axis with outer cylinder O2 113

xviii

Nomenclature

𝐴 = Heat transfer area

𝐴1 = Aspect ratio of enclosure configuration O1

𝐴2 = Aspect ratio of enclosure configuration O2

𝐴𝑑 = Surface area of bottom disc

𝐴𝑜 = Surface area of outer cylinder

𝐴𝑠 = Surface Area

𝐵𝑟 = Brinkman number

𝐶 = Linear anisotropic phase function coefficient

𝐷 = Difference between radius of outer and inner cylinders of enclosure

𝐷𝑖 = Diameter of inner cylinder

𝐷𝑜 = Diameter of outer cylinder

𝐸𝑏 = Energy per unit area per unit time

𝐹 = External body forces

𝐹𝑟 = Body force along radial direction

𝐹𝑧 = Body force along axial direction

𝐺 = Incident radiation flux

𝐺𝑟 = Grashof number

𝐼 = Unit tensor

xix

𝐽𝑗 = Diffusion flux of species j

𝐾 = Thermal conductivity

𝐾𝑒𝑓𝑓 = Effective conductivity

𝐾𝑟 = Radiative conductivity

𝐾1 = Thermal conductivity ratio of aluminum with air

𝐾2 = Thermal conductivity ratio of mild steel with air

𝐾3 = Thermal conductivity ratio of stainless steel with air

𝐿 = Height of enclosure wall

𝑁𝑢 = Local Nusselt number

𝑂1 = Outer cylinder diameter = 256 mm

𝑂2 = Outer cylinder diameter = 300 mm

𝑄𝑐 = Conduction heat transfer

𝑄 𝑐 = Rate of heat conduction

𝑄𝑐𝑜𝑛𝑣 = Convection heat transfer

𝑄𝑖 = Heat entering the enclosure

𝑄𝑜 = Heat leaving enclosure

𝑄𝑟 = Radiation heat transfer

𝑅𝑗 = Volume rate of creation of species j

𝑅𝑎 = Rayleigh number

𝑅𝑒 = Reynolds number

𝑅𝑅1 = Radius ratio of enclosure configuration O1

𝑅𝑅2 = Radius ratio of enclosure configuration O2

𝑆𝑕 = Energy source terms

xx

𝑆𝑚 = Mass source term within differential volume

𝑇 = Variable temperature

𝑇𝑎 = Ambient temperature

𝑇𝑎𝑣 = Average temperature of outer wall of outer cylinder

𝑇𝑏𝑐 = Bottom disc central temperature

𝑇𝑐 = Cold temperature within enclosure

𝑇𝑑 = Average temperature of upper surface of bottom disc

𝑇𝑔 = Gas temperature

𝑇𝑕 = Hot surface temperature

𝑇𝑜 = Operating temperature

𝑇𝑤 = Wall temperature

∆𝑇 = Temperature gradient

𝑌𝑗 = Mass fraction of species, 𝑗

𝑍 = non-dimensional axial component of enclosure

𝑎 = Absorbing coefficient

𝑑 = Diameter of bottom disc

𝑑𝑕𝑠 = Non-dimensional diameter of heat source

𝑔 = Acceleration due to gravity

𝑕 = Sensible enthalpy

𝑕𝑒 = Natural convection heat transfer coefficient of enclosure

𝑕0 = Natural convection heat transfer coefficient of outer cylinder

𝑕𝑗𝑜 = Enthalpy of formation of species j

𝑝 = Static pressure

xxi

𝑛 = Refractive index

𝑞𝑐 = Specific conduction heat flux

𝑞𝑟 = Specific radiation heat flux

𝑞𝑖𝑛 = Specific heat input

𝑞𝑜𝑢𝑡 = Specific heat output

𝑟 = Radial coordinate

𝑟𝑖 = Refractive index

𝑟1 = Radius of bottom disc

𝑟2 = Radius of outer cylinder

𝑡 = Time

𝑣 = Velocity of fluid

𝑣𝑟 = Radial velocity of fluid

𝑣𝑧 = Axial velocity of fluid

∆𝑥 = Thickness of layer

𝑧 = Axial coordinate

𝑧1 = Inner cylinder height

𝑧2 = Outer cylinder diameter

𝛼 = Thermal diffusivity of fluid

𝛽 = Coefficient of thermal expansion

𝜌 = Thermal diffusivity

𝜌𝑜 = Constant density of fluid in a closed domain

𝜏 = Stress tensor

𝜇 = Dynamic viscosity

xxii

𝜖 = Emmisivity

𝜃 = Non-dimensional temperature

𝜍 = Stefan-Boltzman constant

𝜍𝑠 = Scattering coefficient

𝝉𝑒𝑓𝑓 = Effective stress tensor

𝛤 = Radiation parameter

𝛹 = Slip coefficient

𝜃 = Non-dimensional temperature

xxiii

Abstract

In this research work an experimental as well as numerical study of conjugate heat

transfer within an air filled bottom-heated vertical concentric cylindrical enclosure is

presented. Such enclosures have a broad spectrum of engineering applications like

in centrifuge machines, solar collectors, heat exchangers, storage tanks, internal

combustion engines, compressors, flow forming machines, modern cold rolling

machines, crude oil refinery centrifuge machines, brewery machines, blenders,

filtration equipments, moisture separators, chemical process plants, dryers,

compressed air systems, distillation columns and plants, pressure vessels, rotary

shakers etc. Eighteen different experiments are performed by varying the bottom

disc central temperature between 353-433K, using three different inner cylinders

(aluminum, mild steel and stainless steel) and two different diameter outer cylinders.

Such enclosures are generally used in centrifuge machines which are employed for

segregation of chemicals in different process industries. In such machines heat is

generated at the bottom of the enclosure due to electric motor losses, which affects

the process taking place within the enclosure. Uniform temperature is desired in

such enclosures for segregation of chemicals. To achieve the desired conditions, a

thorough heat transfer analysis of the enclosure is required.

In this study, the experimental temperature data within the enclosure was collected

at three different bottom disc central temperatures (353, 393 and 433 K) to

investigate the heat transfer behavior within the enclosure at different temperatures.

Three different inner cylinders were used to study the effect of inner cylinder

thermal conductivity on the heat transfer mechanism. Two outer cylinders of mild

steel were used having different diameters to study the effect of the outer cylinder

diameter on the heat transfer within the enclosure. The streamlines, thermal lines

and velocity vectors of the enclosure are also computed numerically by simulating

the experimental setup in 2-D axisymmetric domain.

xxiv

It was observed that due to high thermal conductivity of the inner cylinder (in case

of aluminum inner cylinder) the heat transfer in the radial as well as axial direction

was enhanced and a more uniform temperature was achieved within the enclosure.

Similarly thermal response of the enclosure with mild steel inner cylinder was

uniform as compared to the enclosure geometry with stainless steel inner cylinder.

The effect of outer cylinder diameter on the heat transfer within the inner cylinder of

aluminum material was minimum due to its high thermal conductivity as compared

to mild steel and stainless steel inner cylinders. The CFD simulations are performed

to study the effects of heat transfer and buoyancy forces on the air filled enclosure.

The CFD results are validated by the experimental results. The heat balance of the

enclosure was carried out. It was observed that the heat transfer coefficients for the

enclosure were within the range of 8-29 W.m-2K-1 for different experiments

performed. Non-dimensional analysis of the enclosure was carried out using Nusselt

and Rayleigh numbers to generalize the results.

CHAPTER-1: INTRODUCTION

1.1 Heat transfer in enclosures

There are numerous enclosure geometries with highly diverse applications in

the field of engineering including the annulus between cylinders, spherical

annulus and closed cylindrical enclosures. Such enclosures are in use since

the history of the mankind till today. Different geometrical configurations of

enclosures are in use, depending upon the specific application. More common

configurations are cylindrical, square, rectangular, cubical, trapezoidal etc.

The heat transfer in the enclosures is complicated when the buoyancy forces

overcome the fluid resistance and initiates natural convection currents. The

buoyancy forces become important at higher temperatures. Fluid adjacent to

the hotter surface of the enclosure rises up, pushing the cooler fluid to move

down making a rotational motion within the enclosures enhancing heat

transfer. The heat transfer through the enclosure depends on whether the

hotter plate is at the top or at the bottom. When hotter plate is at the top, no

convection current develops in the enclosure, because the lighter fluid is

always on the top of the heavier fluid. In this case heat is transferred through

pure conduction. When the hotter plate is at the bottom, it heats the fluid

near the bottom surface. So the lighter fluid rises up and pushes the heavier

fluids down resulting in vortex formation.

The study of heat transfer in the enclosures has been under investigation for

the last fifty years. Most of the research work has been performed by heating

the enclosures from the sides symmetrically and differentially and a few

researchers also worked on the enclosures heated from the bottom as well.

The heat transfer takes place in the fluid-filled enclosures in many practical

situations and is of interest for researchers in many fields of process industry.

It has a broad spectrum of engineering applications like in centrifuge

2

machines, building machines, solar collectors, heat exchangers, materials

processing, storage tanks, furnace designs, nuclear designs, internal

combustion engines, compressors, flow forming machines, modern cold

rolling machines, crude oil refinery centrifuge machines, brewery machinery,

all types of blenders, filtration equipments, moisture separators, chemical

process plants & machineries, all types of dryers, compressed air systems,

distillation columns and plants, pressure vessels, rotary shakers, etc.

For cooling such enclosures also involve heat transfer mechanism like cooling

the lateral surfaces of the centrifuge machines by using chillers. In many

applications natural convection is the only feasible mode of cooling the heat

source with the principal advantage of its reliability, where convection

currents are naturally generated without the need to prime movers such as

pumps or fans. Therefore, natural convection in enclosures is the most

important area in heat transfer research and gaining much importance

because of practical significance in engineering and technology. A typical

enclosure of gas centrifuge machine is shown in Figure 1.1

1.2 Modes of heat transfer

Heat is a form of energy that can be transferred from one system to another

as a result of temperature difference. The science that deals with the

determination of the rate of such energy transfers is called heat transfer. The

transfer of energy as heat is always from the higher temperature medium to

the lower one. Heat transfer stops when the two media reach the same

temperature. Heat can be transferred in three modes: conduction, convection

and radiation.

1.2.1 Conduction heat transfer

Conduction is the transfer of energy from the more energetic particles of a

substance to the adjacent less energetic ones. It mostly takes place in solids.

3

Figure ‎1-1: Gas centrifuge machine

According to the Fourier‟s law of heat conduction the rate of heat conduction

𝑄 𝑐 through a planer layer is directly proportional to the temperature

difference across the layer ∆𝑇 and the heat transfer area 𝐴, but inversely

proportional to the thickness of the layer ∆𝑥. i.e.,

𝑄 = −𝑘𝐴𝑑𝑇

𝑑𝑥 (1.1)

Where, 𝑘 is thermal conductivity of the material.

1.2.2 Convection heat transfer

Convection is a form of energy transfer due to fluid motion. It is more

common in liquids and gasses. The flow of fluid along a hot solid surface also

removes heat from the hot surface by convection. With the fast moving fluid,

4

convection heat transfer increases. If the fluid is forced to flow over the

surface by external means such as fan, pump or the wind, it is called forced

convection. But, if the fluid motion is caused by buoyancy forces that are

induced by the density differences due to the variation of temperature in the

fluid, it is called natural convection. The buoyancy forces become important

when the Rayleigh number becomes greater than 1708.

According to Newton‟s law of cooling rate of convection heat transfer 𝑄 𝑐𝑜𝑛𝑣

through the fluid in motion is directly proportional to temperature

gradient and heat transfer area 𝐴𝑠. This law can be expressed as given

below;

𝑄 𝑐𝑜𝑛𝑣 = 𝑕𝑜𝐴𝑠(𝑇𝑕 − 𝑇𝑎) (1.2)

Where 𝑕𝑜 is the convection heat transfer coefficient, 𝐴𝑠 is the surface area

through which convection heat transfer takes place, 𝑇𝑕 is the hot surface

temperature and 𝑇𝑎 is the ambient temperature of the fluid.

1.2.3 Radiation heat transfer

Radiation is the energy emitted by the matter in the form of electromagnetic

waves as a result of the changes in the electronic configurations of the atoms

or molecules. Radiation is the fastest mode of heat transfer. The form of

radiation emitted by bodies due to their temperature is called thermal

radiation.

The Stefan-Botlzmann law states that total energy radiated per unit surface

area of a black body per unit time is directly proportional to the fourth power

of the black body‟s temperature. Mathematically, it can be written as;

𝐸𝑏 = 𝜍𝑇4 (1.3)

5

Where 𝜍 is Stefan-Boltzmann constant (5.67x10-8Js-1m-2K-4), 𝐸𝑏 is energy

per unit area per unit time in J.s-1.m-2, in S.I. units. 𝑇 is the temperature in

Kelvin. The grey body does not absorb or emit the full amount of radiation

flux. Normally, it radiates small amount of radiation flux due to its emissivity,

𝜖. Therefore, total energy radiated per unit surface area per unit time

becomes;

𝐸𝑏 = 𝜖𝜍𝑇4 (1.4)

Where 𝜖 is emissivity of grey body. For a perfect black body, 𝜖 = 1, but

normally, emissivity depends on the wavelength.

Radiation heat transfer from a surface at temperature 𝑇 to its surrounding at

a temperature 𝑇𝑎 is determined from the equation given below;

𝑄 𝑟 = 𝜀𝜍𝐴𝑠(𝑇4 − 𝑇𝑎

4) (1.5)

Jozef Stefan deduced this law in 1879 on the basis of experimental

measurements, while Ludwig Boltzmann derived this law in 1884 from

theoretical considerations.

1.2.4 Conjugate heat transfer

The conjugate heat transfer is the combination of all three or any two modes

of heat transfer taking place simultaneously. It is a complex field in the joint

solid to solid, fluid to fluid, solid to fluid and fluid to solid heat transfer.

1.3 Problem definition

The study of conjugate heat transfer mechanism in enclosures is important

because the chemical processes occurring in such enclosures are strongly

dependent on temperature. One example is the segregation of chemicals. The

6

segregation of chemicals is carried out in the centrifuge machines used in

sugar industry, paper industry, nuclear industry and many other process

industries. The enclosure used in such machines is generally a double

concentric cylindrical enclosure as shown in Schematic diagram of vertical

concentric cylindrical enclosure

The inner cylinder rotates to segregate the chemicals. In such machines the

heat losses in electrical motor, which rotate the inner cylinder, affects the

segregation process being taking place within the enclosure.

Figure ‎1-2: Schematic diagram of vertical concentric cylindrical enclosure

The heat transfer mechanism is complicated because of the fact that there

are two enclosures within a single concentric cylindrical enclosure. One is the

inner enclosure within the inner cylinder and the other is outer enclosure in

between the inner and outer cylinders. In author‟s opinion the inner cylinder

material and outer cylinder diameter also affect the heat transfer mechanism

in such enclosures. The inner cylinder material and outer cylinder diameter

7

effects on the heat transfer in such enclosures have been least studied

previously.

Thus an experimental setup is required to investigate the heat transfer

phenomena in bottom heated concentric cylindrical enclosures using different

inner and outer cylinders. Numerical analysis are also required to study the

stream lines, thermal lines, velocity vectors etc. in the enclosure and

understand the heat transfer mechanism and its effects on the fluid flow.

1.4 Research objectives

The conjugate heat transfer phenomena in vertical concentric cylindrical

enclosure are studied experimentally and computed numerically. The heat

losses of electric motor, which enter the enclosure from the bottom, are

simulated by supplying heat from the bottom by an electric heater. The

temperature range of 353-433 K is required within the inner cylinder of the

enclosure to segregate the chemicals. The main objectives of this research

work are listed below.

To vary the temperature of the bottom disc from 353-433 K to investigate

the process of heat transfer within vertical concentric cylindrical

enclosures and improve the performance of such enclosures by

controlling temperature in centrifuge machines being used in the process

industries.

To use inner cylinders of different materials to study the effect of

different materials on the conjugate heat transfer within vertical

concentric cylindrical enclosures for a temperature range of 353-433 K.

To use outer cylinders of different diameters to study the effect of the

gap between the two cylinders on the heat transfer mechanism within

8

vertical concentric cylindrical enclosures for a temperature range of 353-

433 K

To determine the heat transfer coefficient of such complex enclosures.

To perform numerical simulation of vertical concentric enclosures and

validate the simulations by comparison with the experimental data.

To obtain streamlines, thermal lines and velocity vectors effects from CFD

results to understand the mechanism of conjugate heat transfer in such

enclosures.

To study the effect of wall‟s heat conduction on the thermal conditions

within the inner enclosure with respect to segregation of chemicals in

centrifuge machines.

To investigate the effects of buoyancy forces on conjugate heat transfer

within the enclosure and radius ratios on the thermal behavior of the

concentric cylindrical enclosure.

To provide valuable knowledge for design of such enclosures under

different operating temperatures, using different materials of inner

cylinder and different diameters of outer cylinder.

1.5 Thesis organization

Chapter 1 - Introduction

This chapter deals with the introduction of all modes of heat transfer in the

enclosures including conjugate heat transfer in general and gives detail about

the conjugate heat transfer in the vertical concentric cylindrical enclosures. A

brief review of the complexities involved in this enclosure, along with an

indication towards the areas for further research and main objectives of the

9

research have been defined. Finally an organization chart of the thesis has

been given.

Chapter 2 – Literature survey

In order to depict a complete picture of the issues related to all types of

enclosures along with their heat sources and enclosure geometries, a

comprehensive literature survey is carried out. The experimental and

computational work of the past researchers on different models used related

to heat transfer in the enclosures are discussed. The main emphasis has been

given on the work related to the heat transfer in the concentric cylinders and

enclosures.

Chapter 3 – Mathematical formulation

The phenomena occurring in the conjugate heat transfer analysis of bottom

heat concentric enclosures is highly complex because it is axisymmetric,

incompressible, laminar flow and involves the conservation of mass,

momentum and energy equations. Viscous dissipation, Boussinesq model,

buoyancy effects, natural convection, conduction and radiation heat transfer

have been explained. The boundary conditions applied in the experimental

model are written in detail.

Chapter 4 – Experimental system and data

This chapter explains the specifications of different enclosure geometries used

during experimentation. The enclosure geometries of different outer diameter

cylinders are explained. Three different cylinders of aluminum, mild steel and

stainless steel are used within enclosure configurations (O1, O2). The various

instruments used to measure different parameters are also described. The

10

convection heat transfer coefficients of the enclosure of eighteen experiments

have been calculated using heat balance method.

Chapter 5 – Numerical Analysis

This chapter includes the details of the CFD simulation of the experimental

work performed. Heat transfer mechanism using different parameters of CFD

simulations are discussed. Geometry and meshing are made by using Gambit.

Boundary conditions used in Fluent 6.3 are written in detail. The CFD

simulation models are discussed in detail. Finally grid independence study is

made.

Chapter 6 – Results and discussion

In this chapter the experimental results are plotted to analyze axial and radial

thermal behavior of bottom heated vertical cylindrical enclosure. Non-

dimensional results are discussed using Nusselt and Rayleigh numbers. The

CFD simulation results are discussed using streamlines within the enclosure

geometry.

Chapter 7 – Conclusions and future recommendations

In this chapter the conclusions of this research work are given. The work

done so far is in two main directions: (i) Experiments of different enclosure

configurations have been performed to study the heat transfer along with

non-dimensional analysis and (ii) The CFD simulations have opened wide

gates for simulating such transfer of heat along with its buoyancy effects for

the future researchers. Finally recommendations for future research are

suggested.

11

CHAPTER-2: LITERATURE SURVEY

2.1 History of heat transfer

The ancients viewed heat as that related to fire. Heraclitus around 500 BC

concluded that fire, earth and water were three main components of nature.

Among these components fire was the key component that controlled and

modified the others. Heraclitus concluded that all things were an exchange for

fire. In 11th century AD, Abu Rayhan Biruni explained that movement and

friction were the causes of heat. In the 13th century, Abd Allah Baydawi

described two possible causes of heat generation. He told that natural heat

was the heat of a fiery broken atom. Heat might arise through change of

motion [1].

Around 1600, Francis Bacon concluded that heat itself was motion and

nothing else. In the mid-17th century, Robert Hooke declared that heat was

nothing but a brisk and violent agitation of the elements of a body. In 1761,

Joseph Black formulated a theory of latent heat and demonstrated that

different bodies have their own specific heats. James Watt invented the Watt

engine. Thomas Newcomen and James Watt invented steam engine. In 1797

Sir Benjamin Thompson used friction to convert work to heat. In 1783,

Lavoisier showed the role of oxygen in burning and proposed the caloric

theory. In 1824 Sadi Carnot declared that production of motive power was

due to the transfer from warm to cold body. In 1738, Daniel Bernoulli

developed the kinetic energy of gases and proposed that gases containing

molecules moved in all directions and as a result pressure developed. Internal

energy of a substance was the arithmetic sum of the kinetic of each molecule,

and heat transfer takes place from more energetic regions to less energetic

ones [2].

http://en.wikipedia.org/wiki/Fire

http://en.wikipedia.org/wiki/Heraclitus

http://en.wikipedia.org/wiki/Motion_(physics)

http://en.wikipedia.org/wiki/Friction

http://en.wikipedia.org/wiki/Francis_Bacon

http://en.wikipedia.org/wiki/Robert_Hooke

http://en.wikipedia.org/wiki/Joseph_Black

http://en.wikipedia.org/wiki/Latent_heat

http://en.wikipedia.org/wiki/James_Watt

http://en.wikipedia.org/wiki/Steam_engine

12

Joule and Mayer presented that heat and work were the same types of

energy leading to the principle of the conservation of energy mentioned by H.

Helmholtz in 1847. In 1850 Clausius declared that caloric theory could be

incorporated with kinetic theory with the condition that the energy

conservation was utilized and stated the First Law of Thermodynamics. In

1851 William Thomson declared that heat was not a substance, but an active

mechanical effect. Mechanical work and heat must have equivalence. In the

modern research, heat is a type of energy that is transferred from one body

to the other due to temperature difference or generated by friction etc. In

1701 Newton presented law of cooling known as Newton‟s law of cooling to

explain the convection heat transfer. In 1822 Fourier expressed Fourier‟s law

of heat conduction. In 1879 Joseph Stefan experimentally determined and in

1884 Ludwig Boltzmann theoretically verified the law of radiation emitted by

black bodies called Stefan-Boltzmann law [3].

A lot of research work on different modes of heat transfer has been carried

out by the past researchers. The heat transfer mechanism in the enclosures is

important because of its applications in the field of engineering. The past

researchers have performed experimental work and computed numerical work

on the heat transfer in the enclosures which may be classified with respect to

the heating source and the enclosure geometry. Their detail is given below:

2.2 Classification based on heat source

The past researchers have studied heat transfer in enclosures using different

locations of heat source like bottom heated enclosures, sidewall heated

enclosures, heat source within the enclosures etc. Depending upon the type

and location of heating source, they have simulated various practical

applications of transfer of heat in the enclosures. Based on the location of

heating source, their research work can be classified into the following

categories.

http://en.wikipedia.org/wiki/Conservation_of_energy

http://en.wikipedia.org/wiki/First_Law_of_Thermodynamics

13

2.2.1 Enclosures with internal heat source

Previously, various experimental and numerical analyses of fluid-filled

enclosures, being heated from within the enclosures, have been carried out.

Such types of enclosures have their engineering applications in thermal

energy storage systems, buildings, furnaces, nuclear reactors and cooling of

electronic equipments etc. The buoyancy flow analysis in the enclosures

being heated internally is useful specifically for process industries. Some of

the past research work, related to enclosures with internal heat source, have

been discussed below.

Liaqat and Baytas [4] numerically examined the conjugate transfer of heat in

the square enclosure containing uniform volumetric heat sources. They

compared their results with non-conjugate analysis. They analyzed the

problem using control volume approach keeping the outsides of the walls at

constant temperature. Natural connective flow fields with radioactive

tritium gas as constant, uniformly distributed, internal heat source was

experimentally studied and numerically computed by Kee et al. [5]. They

analyzed the spherical and cylindrical enclosures bounded with isothermal

walls.

Blair et al. [6] experimentally studied heat transfer response for a heat

source within the enclosure through the toroid centered within a cylinder

filled with water. They analyzed the effect of changing the height of the coil

within the enclosure. Kuznetsov and Sheremet [7] numerically examined

the conjugate transfer of heat in a rectangular enclosure with the heat

generating core as a heat source. Such type of enclosure is used in nuclear

reactors and thermal storage tanks. They have described the effects of

Grashof number, thermal conductivity of solid walls and size of heat source

on the thermal behavior of the enclosure.

14

Arnas and Ebadian [8] presented convective heat transfer between two

concentric circular pipes. The walls of pipes were heated and/or cooled

independently and subjected to uniform heat generation. They simulated the

nuclear accident in which the contaminated coolant shall generate energy.

They graphically presented the ratio of average Nusselt number with heat

generation to that without heat generation. Heat generation affects the

Nusselt number in the case of equal wall temperatures of the two pipes.

Khalilollahi and Sammakia [9] numerically analyzed buoyancy-induced flow

produced by isothermal flat vertical surface surrounded in a large rectangular

enclosure. The walls of this enclosure are assumed adiabatic. Teertstra et al.

[10] experimentally described the measurements of natural convection for an

isothermally heated sphere centrally located in an isothermally cooled

spherical enclosure. The electronic equipments are protected from

environmental contaminants i.e. dust or moisture. The circuits are placed in

the closed enclosures. Malik, A. H. et al. [11] experimentally studied

conjugate heat transfer within a bottom heated vertical concentric cylindrical

enclosure. Top and bottom walls except the bottom disc are assumed to be

adiabatic. The inner cylinder is heated from the bottom disc and act as an

internal heat source for the outer cylinder making a outer sub-enclosure.

2.2.2 Enclosures with lateral wall heat source

Enclosures with lateral wall heat source are important due to their vast

engineering applications. Such enclosures have their engineering applications

in clinical blood oxygenators, gas centrifuges, barrel reactors, heat

exchangers, compact electronic packaging, nuclear reactors, shipping

containers for spent fuel, solar collectors, electronic packaging, thermal

storage tanks etc. Because of such a vast application domain the past

15

research work include both experimental and numerical study of such

enclosures. Some of the past research work has been mentioned below.

Rahman et al. [12] numerically studied mixed convection heat transfer within

a vented square enclosure. Its heat conducting horizontal solid circular

cylinder is placed in the centre of the enclosure to simulate electronic cooling

and ventilation of buildings. The top, bottom, and left vertical walls of the

enclosure are kept adiabatic, while its right vertical wall is kept at uniform

temperature. They studied the effects of cylinder on the flow and heat

transfer within the enclosure by comparing it with the flow and heat transfer

without that cylinder. Ball et al. [13] presented experimental results of

convection flows and heat transfer produced within the annular gap

between the concentric vertical cylinders to simulate the cooling of rotating

machinery. They used rotating heated inner cylinder, stationary cooled

outer cylinder and horizontal endplates forming an enclosure. They

investigated mixed convection flows and heat transfer within vertical annulus.

They studied the effects of both buoyancy and centrifugal forces on the

stability of fluid flow.

Lipkea and Springer [14] performed an experimental investigation of heat

transfer through gases placed within vertical concentric cylinders. In a modified

hot wire type thermal conductivity cell these cylinders have different

temperatures. They have evaluated the end effects along with the overall heat

transfer between concentric cylinders and concluded that the end effects take

active role only in the corner regions. Glakpe et al. [15] investigated natural

convection heat transfer in the annular space formed by a square rod

enclosed within a cylinder and two horizontal surfaces using finite difference

procedure. The inner rod has constant heat flux and enclosing cylinder is

isothermal and horizontal surfaces are assumed to be adiabatic.

Sankar et al. [16] numerically investigated the effect of surface tension on

natural convection in a vertical cylindrical annular enclosure. Inner and outer

16

cylinders of the enclosure are at different uniform temperatures, while

horizontal top and bottom walls are thermally insulated. Numerical results

indicated multi-cellular flows even in smaller aspect ratio cavities induced by the

thermo-capillary forces. Keyhani et al. [17] measured natural convection heat

transfer in a vertical annulus whose inner cylinder is at constant surface heat

flux and the outer cylinder is at constant temperature. They concluded that

energy transferred by thermal radiation varied with Rayleigh number and

working fluid. With air as a working fluid up to 50 percent of heat transfer

takes place by radiation, while for helium radiation is up to 30 percent of heat

transfer rate.

Sarr et al. [18] numerically investigated natural convection heat transfer in an

enclosure. The enclosure has with constant wall heat flux on the inner

cylinder and uniform temperature on the outer cylinder, while assuming other

walls to be adiabatic. They compared heat transfer of different fluids such as

air, ammonia-liquid and carbon dioxide-liquid.

2.2.3 Enclosure with bottom wall heat source

Enclosures with bottom wall heat source are important due to their vast

engineering applications. Such enclosures are generally employed in solar

systems, cooling of electronic equipments, geophysics, meteorology, cooling

low powered laptop computers, energy storage, fire control, TV, safety of

nuclear reactors, furnaces, monitors etc. The past researchers have analyzed

such enclosures both experimentally and numerically. Some of their

contributions are mentioned here.

Buell et al. [19] studied the effect of lateral wall conductivity on the stability

of a fluid-filled cylinder heated from bottom. The authors examined natural

convection in a fluid-filled cylinder. Its top and bottom surfaces are rigid and

perfectly conducting ones, while its sidewall has an arbitrary thermal

conductivity and assumed to be insulated outside the lateral walls. Vargas et

17

al. [20] experimentally and numerically studied natural convection in small

aspect ratio cylindrical enclosure. The enclosure has laterally insulated wall

while top and bottom surfaces are at constant but different temperatures.

They studied natural convection vertically using PIV technique.

Zhao et al. [21] numerically investigated natural convection in a bottom

heated rectangular enclosure, polluted with symmetrically placed finite

thermal and pollutant sources. The top wall of the enclosure is at lower

temperature, while vertical and bottom walls, except the discrete bottom

strips, assumed to be insulated. They simulated oceanography and geology in

this investigation. Dagtekin et al. [22] numerically analyzed natural convection

heat transfer and fluid flow of two hot partitions located at the bottom of a

square enclosure. They insulated the right and bottom walls and maintained left

and top walls at a uniform temperature to simulate the electronic components in

the laboratory. They studied the heat transfer and fluid flow effects of location

and heights of the partitions.

Natural convection in rectangular enclosure is numerically studied by [23]. They

used a discrete flush-mounted rectangular heat source that heats the bottom

while rest of the bottom surface being insulated. The enclosure is cooled from

the top surface and considered either adiabatic or constant temperature on the

sidewalls. They simulated the problem of cooling electronic equipments.

Kuznetsov and Sheremet [24] numerically investigated heat transfer in a vertical

rectangular enclosure with local heat and containment sources at the bottom

with constant temperature and concentration respectively. Top, bottom and right

side walls are assumed adiabatic from outside, while left wall is assumed to

exchange heat with environment. They simulated the combined effect of

environment and local heat source in the design of micro-electronic equipment.

They studied the effects of Grashof number, buoyancy ratio and transient factor

on flow modes, heat and mass transfer.

18

Maki et al. [25] measured the average heat transfer rates for natural convection

of air in a cylindrical enclosure. It is heated from bottom and cooled from top in

a bore space of an inclined super-conducting magnet keeping vertical walls to be

adiabatic. The observed that average heat transfer rates varies with the angle of

inclination and the average Nusselt number is increased for the large positive net

acceleration. The conduction becomes dominant by heating the top and cooling

the bottom of enclosure.

Matthias [26] described finite difference model within soil under the cylindrical

enclosure. They measured trace gas between soil and atmosphere while

mounting the enclosure on soil and determined changes in the concentration of

trace gas within the enclosed space. Akamatsu et al. [27] performed

numerical analysis to study the Kelvin force effects on a water-flow

vertical cylindrical enclosure. It is heated at the bottom and cooled at the

top in a vertical magnetic field. The authors of [28] numerically analyzed

natural convection heat transfer in a prismatic enclosure. They studied the

flow structure sensitivity to governing parameters, Rayleigh number and

enclosure aspect ratio.

The authors of [29] analyzed natural convection heat transfer in the square

enclosure. The top and bottom of the enclosure are assumed adiabatic and

side walls are at lower isothermal temperature. The heat source of higher

isothermal temperature is placed at the bottom of the inner square enclosure.

They simulated the cooling of electronic equipments. The influence of a small

heat source positioned in the bottom of a square enclosure is experimentally

investigated by [30]. The bottom wall, except heated section and the top wall

are assumed adiabatic and the vertical walls are assumed at uniform

temperature. They observed a symmetrical distribution of local Nusselt

number.

Aydin et al [31] numerically investigated the natural convection heat transfer

in a rectangular enclosure. They heated the bottom and symmetric cooled the

19

sides. The bottom surface, except the heated section and the top surface are

assumed to be adiabatic. They simulated the cooling of electronic

equipments. Brito et al. [32] numerically analyzed natural convection in the

air-filled inner square enclosure. The enclosure is heated by the heat source

from the bottom by insulating the top. The vertical surfaces of enclosure are

considred as isothermal at low temperature. Rayleigh number and

dimensionless heat source length were main parameters of interest.

Saha et al. [33] numerically presented natural convection in a rectangular

enclosure using a finite element method. They assumed the top and bottom

walls as adiabatic, except central heated part of the bottom and maintained

two vertical walls at constant low temperature. The electronic component is

treated as heat source placed on the flat surface. They performed parametric

study of Grashof number, dimensionless heat source length, inclination angle

with horizontal axis and aspect ratio of the enclosure. Al-Bahi et al. [34]

numerically investigated natural convection of inclined rectangular enclosure.

They heated the bottom by the heater. Its side walls are isothermal heat

sinks while the top and bottom walls except the bottom heated section are

assumed to be insulated. At 0° inclination (upright geometry), the flow

formed two cells pattern with a mushroom like isotherms. At 180° inclination

(inverted geometry) the flow formed circulation cells within the top section of

the enclosure.

Aswatha et al. [35] numerically studied natural convection in the rectangular

enclosures whose bottom wall was subjected to uniform/sinusoidal/linearly

varying temperatures. Malik, A. H. et al. [11] experimentally studied

conjugate heat transfer within a bottom heated vertical concentric cylindrical

enclosure. Top and bottom walls except the bottom disc are assumed to be

adiabatic. The inner cylinder is heated from bottom disc making the inner

sub-enclosure, while the inner cylinder act as a heat source for the outer

cylinder making outer sub-enclosure.

20

The heat transfer mechanism is complicated because such enclosures contain

two enclosures within a single concentric cylindrical enclosure. One is the

inner enclosure within the inner cylinder and the other is outer enclosure in

between the inner and outer cylinders. The inner enclosure is heated from the

bottom while for the outer enclosure the inner cylinder acts as a heat source.

This situation makes the problem more diverse in the sense that the inner

enclosure falls in the category of enclosure with bottom heat source and the

outer enclosure in the category of enclosure with centrally heated source.

2.3 Classification based on enclosure geometry

The study of heat transfer in the enclosures has been under investigation for

the last fifty years. There are numerous research papers on the study of heat

transfer in the enclosure geometries including their cylindrical, rectangular,

square, cubical, trapezoidal and prismatic configurations. The past

researchers performed experimental and numerical analyses of heat transfer

in different enclosure geometries to study different parameters of interest and

get insight of fluid behavior within those enclosures to understand heat

transfer characteristics. It has a broad spectrum of engineering applications

like in centrifuge machines, building machines, solar collectors, heat

exchangers, materials processing, storage tanks, furnace designs, nuclear

designs, I. C. engines, compressors, flow forming machines, modern cold

rolling mills etc. The past researchers have worked on the heat transfer in the

enclosure geometries, which are further subdivided into cylindrical enclosures,

rectangular enclosures, square enclosures and others enclosures given as

under:

2.3.1 Cylindrical enclosures

The research work on the heat transfer in cylindrical enclosures has been

performed experimentally and numerically by the past researchers. Their work

21

has engineering applications in centrifuge machines, compressors, heat

exchangers, flow forming machines, modern cold rolling machines etc. In

such types of enclosures the research work performed in the past is given

below:

Amara et al. [36] studied natural convection occurred in a vertical cylinder

opened from two opposite ends. The cylinder is heated at a periodical lateral

heat flux. Lai et al. [37] numerically studied natural convection in a enclosure

of vertical cylindrical geometry. They heated the cylindrical surface at

constant heat flux, with top and bottom surfaces being adiabatic. This

research has given insight into the behavior of the system at the boundary

between laminar (1010 ≥ 𝑅𝑎 ≥ 1013) and turbulent (5 × 1013 ≥ 𝑅𝑎 ≥

1015) flow.

Numerical simulation of natural convection is proposed by [38] in a vertical

cylinder. They analyzed the effects of velocity, temperature, aspect ratio,

Nusselt, Rayleigh and Prandtl numbers at steady-state condition and

investigated the fluid transient behavior. Sharma et al. [39] numerically

investigated the results of conjugate natural convection heat transfer heated

by a volumetric energy generating source within a cylindrical enclosure. The

heat conducting body is placed in the enclosure between isothermal lateral

walls.

The authors of [6], [20], [40] and [41] investigated natural convection and

conduction in cylindrical enclosures. Ho et al. [42] numerically investigated

the buoyancy-driven fluid convection heat transfer for air surrounding the two

horizontal, differentially heated cylinders within an adiabatic circular enclosure.

They studied the effects of gap width between thermally active cylinders on

natural convection fluid flow and heat transfer and inclination angle of an

enclosure due to gravity.

22

Sparrow et al. [43] experimentally studied heat transfer is for a rotating, re-

circulating, forced convection flow in a cylindrical enclosure. They determined

local heat transfer coefficients along the heated parallel disk system. Smith et

al. [44] analyzed radiation and convection transfer in fossil-fueled energy

cylindrical units. They calculated axial and radial temperature profiles of gas

with wall heat flux/wall temperatures for different values of the system

parameters.

Ben Salah et al. [45] solved radiation transfer problems by the finite-volume

method in an axisymmetric cylindrical enclosure with absorbing, emitting, and

scattering gray medium. Natural convection in the vertical storage tanks is

presented by [46]. They derived a correlation for the Nusselt number associated to

the cooling processes. The authors of [47] laterally heated cylindrical enclosure at a

uniform heat flux and studied the Rayleigh number effects on the temperature.

Keyhani et al. [48] experimentally examined natural convection heat transfer in two

vertical rod bundles within the cylindrical enclosure for a wide range of Rayleigh

numbers.

2.3.2 Rectangular enclosures

The past researchers have performed experimentally and computed

numerically the research work on the heat transfer in rectangular enclosures.

Their work has engineering applications in migration of contaminants in the

buildings and microelectronic equipments etc. Some of the past researchers

have used such types of enclosures as given below:

Niu et al. [49] experimentally investigated mixed convection heat transfer in a

large enclosure. They studied the effects of geometric factors and aspect

ratio of enclosure. Prud`homme et al. [50] analyzed linear stability of natural

convection of vertical enclosure by heating it at uniform heat flux at four

walls. Trevisan and Bejan [51] analytically studied and numerically computed

natural convection in the closed cavities caused by combined temperature and

23

concentration buoyancy effects with uniform heat and mass fluxes along the

tall vertical cavities. Natural convection is numerically investigated by [21] in a

bottom heated rectangular enclosure with top at lower temperature and vertical

and bottom walls, except heated discrete strips assumed to insulated.

Natural convection is experimentally studied and numerically computed by

[52] in a cavity. They symmetrically heated both sides of the cavity with a

uniform heat flux assuming bottom wall as adiabatic and top wall as left open

to atmosphere. Flow pattern in the cavity is different from differentially

heated enclosures. Joly et al. [53] investigated the Soret effect on natural

convection in a binary liquid-filled vertical enclosure. They kept two vertical

walls of enclosure at constant heat fluxes and assumed two horizontal walls

as adiabatic. The buoyancy force effects oppose each other at equal intensity.

Sigey et al. [54] studied natural convection heat transfer in a rectangular

enclosure mounting electric heater below the window keeping other walls as

insulated using finite difference method. Chen et al. [55] numerically

investigated natural convection in rectangular enclosures in which side, top

and bottom walls are at uniform temperatures such that 𝑇𝑠 > 𝑇𝑡 > 𝑇𝑏 . They

calculated Rayleigh numbers and aspect ratios. They modeled the top wall as a

rigid surface or moving boundary.

The authors of [56] and [57] investigated all modes of heat transfer in the

enclosure. They examined effects of heat sources on the enclosure walls to

predict thermal behavior of heated enclosures. They determined the impact of

each interaction on convective heat transfer from the heat source surface.

Khalilollahi et al. [58] numerically analyzed natural convection heat transfer

by isothermal vertical surface enclosed in a long rectangular enclosure by

assuming the enclosure walls to be adiabatic. The heat is transferred in

different regimes. Initially, the temperature distribution and heat transfer

coefficient follow one dimensional conduction solution.

24

Bouali et al. [59] have numerically analyzed the effects of radiation on heat

transfer and flow structures in an inclined rectangular enclosure containing a

centered conducting body along with its inclination effects. Ganguli et al. [60]

performed CFD simulations to predict the variation in natural convection heat

transfer coefficient for various vertical enclosures with a range of heights, gap

widths and temperature differences. They compared simulations with the

experimental and numerical studies.

Guimaraes et al. [61] performed numerical analysis of forced convection in

an enclosure with a tube bank of 18 stationary cylinders is . One wall is hot,

while others are insulated inducing the flow by one fan near the top wall.

They studied the effect of temperature and velocity distributions on the

Nusselt number for different Reynolds numbers. Khanafer and Vafai [62]

reduced the effective boundary conditions at the open side of structures to a

closed-ended domain to save the CPU and memory usage. They obtained

boundary conditions for both temperature and flow fields covering a

comprehensive range of controlling parameters.

Wilkes et al. [63] studied natural convection heat transfer in a rectangular

enclosure. They assumed one vertical wall hotter than the other, while the

top and bottom walls are alternatively assumed perfect insulation and linear

variation. Saha et al. [34] numerically investigated natural convection in a

inclined rectangular enclosure with adiabatic sidewalls and isothermally heat

sink top wall. Oosthuizen et al. [64] numerically studied natural convection in

rectangular enclosure. They mounted a heated isothermal rectangular

element on the centre of its one vertical wall with the top cooled wall keeping

remaining walls as adiabatic. They studied the effects of convection heat

transfer on the Nusselt number and aspect ratios.

25

2.3.3 Square enclosures

The past researchers performed experimental and numerical analysis of the

research work on the heat transfer in square enclosures. Their work has

diverse engineering applications in cooling of electronic equipments, building

designs etc. Some researchers have used such enclosures as given below:

Corvaro and Paroncini [29] experimentally and numerically analyzed the

natural convection heat transfer in a partially divided square enclosure, while

Dagtekin and Oztop [65] numerically analyzed the natural convection heat

transfer of two heated partitions within the square enclosure. Xia et al. [66]

numerically studied natural convection heat transfer in a square enclosure.

The enclosure vertical walls are differential heating with the insulated top

and bottom walls. In the hot vertical wall the temperature varies with time,

while the cold vertical wall is at a constant temperature. They checked the

flow field and heat transfer to achieve desirable goals.

Kim and Viskanta [67] experimentally presented and numerically computed

the buoyancy induced flow in an enclosure. The conduction heat transfer

taken place in the enclosure walls simultaneously stabilized and

destabilized natural convection flow depending on the location of the

enclosure. Kwak et al. [68] numerically investigated natural convection of

an incompressible fluid in a square enclosure. Aktas et al. [69] numerically

investigated the effects of thermo-acoustic wave motion on natural convection

in a square enclosure. Paroncini et al. [30] experimentally investigated the

effect of heat source in the bottom of the square enclosure. Refai Ahmed and

Yovanovich [70] used finite difference technique of the Marker and Cell

method to obtain solutions of natural convection heat transfer from discrete

heat sources in a square enclosure.

Aminossadati and Ghasemi [71] numerically investigated natural convection in

a square enclosure at different angles of inclination. They kept two adjacent

26

walls of the enclosure as insulated and the other two at different

temperatures. They studied the effects of differential heating of the enclosure

walls and inclination angle on convection flow. Habibollah et al. [72]

numerically studied natural convection in a partitioned square cavity. The

enclosure vertical left and right walls hot and cold respectively, while other

walls and partition are assumed to be adiabatic. They determined the effects

of Rayleigh number, partition height and location, volume fractions of nano-

particles.

2.3.4 Other enclosures

The research work on the heat transfer in cubical, spherical, trapezoidal, and

prismatic types of enclosures has been performed experimentally and

computed numerically by the past researchers. Their work has diverse

engineering applications in electrical packaging, nuclear reactor safety,

electronic cooling, furnaces fire control, energy storage etc. Previously,

researchers have performed experimental and numerical analysis using such

enclosures as given below:

Warrington and Crupper [73] experimentally investigated natural

convection heat transfer from a fixed array of four isothermal, heated

cylinders to an isothermal, cooled cubical enclosure for both a horizontal

and vertical position of the array to determine the effect of the position of

the tubes within the enclosure. Tagawa and Ozoe [74] numerically

investigated natural convection of liquid metal in a cubical enclosure under a

static external magnetic field horizontally and parallel to the heated vertical

wall.

Newport et al. [75] experimentally investigated heat transfer between a

horizontal isothermal cylinders present at the center of an isothermal cubical

enclosure.

27

Bohn and Anderson [76] experimentally studied heat transfer between parallel

and perpendicular vertical walls in the enclosure and discussed many aspects of

natural convection flows. They carried out the testing in a cubical enclosure

with an adiabatic top and bottom and isothermal sides. Kee et al. [77]

experimentally characterized and numerically computed natural

connective flow fields for the fluids having constant, uniformly distributed

internal heat sources with the sphere or cylinder having isothermal walls.

Natarajan et al. [78] numerically studied the visualization of natural

convective heat transfer within a trapezoidal cavity by differentially heating

the lateral walls.

In the past research work, the researchers used heat source at the bottom of

the different types of enclosures, but the least attention related to the bottom

heated concentric cylindrical enclosure has been given by the past

researchers. For this reason the present research work is unique in its

research to get insight into the inner cylinder as well as between outer and

inner cylinders of the enclosure by heating from the bottom. The concentric

cylindrical enclosure studied in this research work actually contains two

enclosures, one enclosure within the inner cylinder, while the other enclosure

between the inner and outer cylinders. Inner enclosure is heated from the

bottom while for the outer enclosure the inner cylinder acts as an internal

heat source.

In this research work, the conjugate heat transfer within a bottom heated

concentric cylindrical enclosure has been investigated experimentally and

computed numerically. The combined natural convection in the air,

conduction in the solid and radiation between different walls are investigated.

In spite of the above mentioned research, the effects of bottom disc

temperature, inner cylinder materials and outer cylinder diameters on the

non-conventional cylindrical enclosure has been least studied todate to the

best of authors‟ knowledge. Experimental model has been designed to

28

conduct the experiments using two different configurations with three

different materials of the inner cylinders and discs and three different bottom

disc temperatures (353, 393, 433 K) at the upper central location of the disc.

This study provides a tool to adjust the cooling system according to the

requirements, gives insight of different configurations and inner cylinder

materials and promotes understanding of the conjugate heat transfer. This

gives an open guidance for the future and leaves open fields for studying on

different materials and geometries.

29

CHAPTER-3: MATHEMATICAL FORMULATION

The heat transfer taking place in the fluid-filled enclosures is complicated due

to the presence of various modes of heat transfer mechanisms. The

convection heat transfer takes place in the fluid filled enclosure. The

conduction heat transfer takes place within the solid parts of the enclosure

geometry and radiation heat transfer may take place at the hot surfaces. At

high temperatures the buoyancy forces come into play, which creates vortices

in the fluid inside the enclosure and thus further complicates the mechanism

of heat transfer. Due to a broad spectrum of engineering applications in

centrifuge machines, flow forming machines, heat exchangers, nuclear

reactors, modern cold rolling machines etc., these enclosures are highly

diverse in their shape. Therefore, proper mathematical modeling of the heat

transfer mechanism in such enclosures is required to address the above

mentioned complexities.

Mathematical formulation of the conjugate heat transfer mechanism in the

fluid-filled enclosures includes the conservation of mass, momentum and

energy equations to solve the flow domain. To deal with the conduction

within the solids and radiations from hot surfaces their mathematical

formulation must be added. In the subsequent articles a brief introduction of

the equations necessary for solving conjugate heat transfer processes is

given. These equations have been mentioned in detail in [79-82].

3.1 Conservation of mass

In fluid mechanics the conservation of mass or continuity equation in its

differential form can be written as under;

30

𝜕𝜌

𝜕𝑡+ 𝛻. 𝜌𝒗 = 𝑆𝑚 (3.1)

Equation 3.1 is a general form of the mass conservation equation, valid for

both compressible and incompressible flows. The term 𝑆𝑚 stands for the

mass source term within the differential volume. In 2-D cylindrical coordinate

system (r, z) the continuity equation (Eq. 3.1) can be written as under;

𝜕𝜌

𝜕𝑡+

𝜕

𝜕𝑧 𝜌𝑣𝑧 +

𝜕

𝜕𝑟 𝜌𝑣𝑟 +

𝜌𝑣𝑟

𝑟= 𝑆𝑚 (3.2)

Where, 𝑟 and 𝑧 are the radial and axial coordinates, while, 𝑣𝑟 and 𝑣𝑧 are

the radial and axial velocities of the fluid respectively and 𝜌 is the density of

fluid.

3.2 Conservation of momentum

Conservation of momentum equation states that the algebraic sum of all the

forces acting on a small fluid volume is equal to zero. It is a vector equation

and in an inertial reference frame the differential form of momentum equation

is given as under;

𝜕(𝜌𝑣)

𝜕𝑡+ ∇. 𝜌𝑣𝑣 = −∇𝑝 + ∇. 𝝉 + 𝜌𝒈 + 𝑭 (3.3)

Where, p, 𝝉, 𝒈 and 𝑭 are static pressure, stress tensor, acceleration due to

gravity and external body forces, respectively. 𝑭 includes other model-

dependent source terms such as porous medium and user-defined sources.

The stress tensor 𝝉 is given by;

𝝉 = 𝜇[ 𝛻𝒗 + 𝛻𝒗𝑇 −2

3𝛻. 𝒗𝑰] (3.4)

Where 𝜇 is the dynamic viscosity, 𝑰 is the unit tensor. The second term on

the right side of Eq. 3.4 is the effect of volume dilatation.

31

In 2-D cylindrical coordinate system (r, z) the two components of momentum

equation are given below;

𝜕

𝜕𝑡 𝜌𝑣𝑧 +

1

𝑟

𝜕

𝜕𝑧 𝑟𝜌𝑣𝑧𝑣𝑧 +

1

𝑟

𝜕

𝜕𝑟 𝑟𝜌𝑣𝑟𝑣𝑧 =

−𝜕𝑝

𝜕𝑧+

1

𝑟

𝜕

𝜕𝑧 𝑟𝜇 2

𝜕𝑣𝑧

𝜕𝑧−

2

3 ∇. 𝒗 +

1

𝑟

𝜕

𝜕𝑟 𝑟𝜇{

𝜕𝑣𝑧

𝜕𝑟+

𝜕𝑣𝑟

𝜕𝑧} + 𝐹𝑧

(3.5)

and

𝜕

𝜕𝑡 𝜌𝑣𝑟 +

1

𝑟

𝜕

𝜕𝑧 𝑟𝜌𝑣𝑧𝑣𝑟 +

1

𝑟

𝜕

𝜕𝑟 𝑟𝜌𝑣𝑟𝑣𝑟

= −𝜕𝑝

𝜕𝑟+

1

𝑟

𝜕

𝜕𝑧 𝑟𝜇 2

𝜕𝑣𝑟

𝜕𝑧+

𝜕𝑣𝑧

𝜕𝑟

+1

𝑟

𝜕

𝜕𝑟 𝑟𝜇 2

𝜕𝑣𝑟

𝜕𝑟−

2

3 ∇. 𝒗 − 2𝜇

𝑣𝑟

𝑟2

+2

3

𝜇

𝑟 ∇. 𝒗 𝜌

𝑣𝑧2

𝑟+ 𝐹𝑟

(3.6)

3.3 Conservation of energy

In order to incorporate the effect of different modes of heat transfer, the

energy equation in conjugate heat transfer process must be modified. The

general differential form of energy conservation equation is given as under;

32

𝜕

𝜕𝑡 𝜌𝐸 + 𝛻. 𝒗 𝜌𝐸 + 𝑝 = 𝛻. (𝑘𝑒𝑓𝑓 𝛻𝑇 − 𝑕𝑗𝑗 𝑱𝑗 +

𝝉𝑒𝑓𝑓 . 𝒗 ) + 𝑆𝑕 (3.7)

Where, 𝑘𝑒𝑓𝑓 is the effective conductivity including the effect of turbulent

thermal conductivity and 𝑱𝑗 is the diffusion flux of species, 𝑗. 𝝉𝑒𝑓𝑓 is the

effective stress tensor.

The first three terms on the right side of Eq. 3.7 give energy transfer due to

conduction, species diffusion and viscous dissipation, respectively. 𝑆𝑕 is the

energy source term including radiation source, chemical reaction source etc.

𝐸 = 𝑕 −𝑝

𝜌+

𝑣2

2 (3.8)

Where, 𝐸 is the energy transferred during the chemical reaction, 𝑕 is

sensible enthalpy. For ideal fluid 𝑕 is given by;

𝑕 = 𝑌𝑗𝑗 𝑕𝑗 (3.9)

For incompressible flows sensible enthalpy 𝑕 is given by;

𝑕 = 𝑌𝑗𝑗 𝑕𝑗 +𝑝

𝜌 (3.10)

Where 𝑌𝑗 is the mass fraction of species 𝑗 and

𝑕𝑗 = 𝑐𝑝,𝑗𝑇

𝑇𝑎𝑑𝑇 (3.11)

3.3.1 Viscous dissipation term

The viscous dissipation term is the third term on the right hand side of the

energy conservation equation (Eq. 3.7). It expresses the heat energy

generated by viscous shear in the fluid flow. Viscous dissipation term

becomes dominant when the Brinkman number, 𝐵𝑟 is close to unity, where

33

𝐵𝑟 =𝜇𝑣2

𝑘∆𝑇 (3.12)

Where, ∆𝑇 stands for temperature gradient in the system, 𝜇 is the dynamic

viscosity of fluid, 𝑣 is the velocity of flow, and 𝑘 is thermal conductivity of

fluid. In general, for compressible flows 𝐵𝑟 ≥ 1.

3.3.2 Energy source due to chemical reaction

Energy is either released to the surrounding or added to the system due to

the chemical reaction within the system. The chemical reaction is the

exothermic one when energy is released to the surroundings in the form of

heat but the endothermic one when the energy is added to the system.

Energy addition or rejection due to chemical reaction is incorporated in the

energy equation through source term 𝑆𝑕 and is given by;

𝑆𝑕 ,𝑟𝑥𝑛 = − 𝑕𝑗

0

𝑀𝑗𝑗 𝑅𝑗 (3.13)

Where, 𝑕𝑗0 and 𝑅𝑗 are the enthalpy of development of species 𝑗 and

volumetric rate of development of species 𝑗, respectively.

3.3.3 Energy equation in solid regions

In solid regions the energy transport equation used is in the following form;

𝜕

𝜕𝑡 𝜌𝑕 + 𝛻. 𝒗𝜌𝑕 = 𝛻. 𝑘𝛻𝑇 + 𝑆𝑕 (3.14)

The convection energy transfer because of the motion in rotation or

translation of the solids is characterized by the second term on the left hand

side of Eq. 3.14. The velocity field 𝒗 is calculated from the motion particular

to the solid zone. While, the terms on the right hand side of Eq. 3.14

34

correspond to the conduction heat fluxes and volumetric heat sources within

the solid, respectively.

3.4 Natural convection and buoyancy effects

When an object is partially or fully immersed in a fluid the upward force

exerted on the object is called buoyancy force. The value of this buoyancy

force is equal to the weight of the fluid displaced by the object.

The fluid density varies with temperature by the addition of heat to a fluid

and flow is induced by the gravitational force acting on the density variations

called natural convection flows. This is simply the thermo-siphoning that

circulates the liquid without any mechanical pump either in an open loop or

closed-loop circuit. By heating the liquid in the loop it expands and becomes

less dense and more buoyant than the cooler liquid at the bottom of the loop.

Due to convection the hotter liquid moves upwards in the system replaced by

the cooler one through the gravitational force. In the mixed convection flows

the buoyancy forces can be designated by the ratio of Grashof and Reynolds

numbers as shown below;

𝐺𝑟

𝑅𝑒 2=

𝑔𝛽∆𝑇𝐿

𝑣2 (3.15)

There are strong buoyancy effects on the fluid flow when the above number

from Eq. 3.14 approaches or exceeds unity. However, when this number is

very small, buoyancy forces are insignificant and can be neglected in the

mathematical modeling. The strength of the buoyancy-induced flow in the

pure natural convection is measured by the Rayleigh number;

𝑅𝑎 =𝑔𝜌𝛽 ∆𝑇𝐿3

𝜇𝛼 (3.16)

35

Where, 𝛽 is the coefficient of thermal expansion and 𝛼 is the thermal

diffusivity.

𝛼 =𝑘

𝜌𝑐𝑝 (3.17)

A buoyancy-induced laminar flow is assumed at Ra<108 and transition from

laminar to turbulence is assumed in the range of 108<Ra<1010.

Two different approaches are used to incorporate the natural convection

effects in the mathematical modeling of heat transfer in a closed domain.

Performing transient calculations in which the density is calculated initially

from initial values of pressure and temperature. This approach is

employed when there are large temperature differences in the flow

domain.

Performing steady state calculations using the Boussinesq approximation.

In this approach, the density is initially assumed constant and latter on

calculated through Boussinesq approximation. This technique is valid only

when the temperature gradients are small in the flow domain.

In natural convection flows the Boussinesq approximation is generally

employed to capture the buoyancy effects. It is explained below.

3.4.1 Boussinesq Model

The Boussinesq approximation is used in the buoyancy driven flows. This

approximation states that the density gradients are so small at low

temperatures that these are neglected, except in the buoyancy where

gravitational forces act. In most of the flows this approximation makes the

mathematics and physics much easier. In Boussinesq approximation the

density of the fluid is assumed constant in all governing equations, except in

36

the buoyancy term used in the momentum equation. Let 𝜌𝑜 stands for the

constant density of the fluid in a closed domain, 𝑇𝑜 is the operating

temperature and 𝛽 is the coefficient of thermal expansion, then according to

the Boussinesq approximation the density of the fluid is given by;

𝜌 = 𝜌𝑜[1 − 𝛽 𝑇 − 𝑇𝑜 ] (3.18)

The above equation is incorporated in the buoyancy term to eliminate 𝜌.

3.5 Radiation heat transfer

Fourth term of the energy equation (Eq. 3.7) (𝑺𝒉) also includes the heat

source due to radiation. In order to calculate the radiation heat transfer

different models are used which are given as under;

Discrete transfer radiation model

Rosseland radiation model

P-1 radiation model

Surface to surface radiation model

Discrete ordinate radiation model

The details of the mathematical equations of only Rosseland radiation model

are mentioned here. All these models are explained in detail in [79].

According to the P-1 radiation model the radiative heat flux in a gray medium

can be approximated by the equation given below;

𝑞𝑟 = −𝛤∇𝐺 (3.19)

Where, the parameter 𝛤 is given by;

𝛤 =1

(3 𝑎+𝜍𝑠 −𝐶𝜍𝑠 ) (3.20)

37

Where 𝑎 is the absorbing coefficient, 𝜍𝑠 is the scattering coefficient, 𝐺 is the

incident radiation flux, 𝐶 is the linear anisotropic phase function coefficient

In the Rosseland radiation model the assumption is made that the intensity is

the black body intensity at the gas temperature. 𝐺 is given as under;

𝐺 = 4𝜍𝑛2𝑇4 (3.21)

Where, 𝑛 is the refractive index. So putting the value of 𝐺 in the radiative

heat flux equation, the equation becomes;

𝑞𝑟 = −16𝜍𝛤𝑛2T3∇𝑇 (3.22)

Both the specific radiative (𝑞𝑟 ) and specific conduction heat fluxes (𝑞𝑐) have

the same form, therefore, these can be written as,

𝑞 = 𝑞𝑐 + 𝑞𝑟 = −(𝑘 + 𝑘𝑟)∇𝑇 (3.23)

𝑘𝑟 = −16𝜍𝛤𝑛2T3 (3.24)

Where, 𝑘 is the thermal conductivity, 𝑘𝑟 is the radiative conductivity. The

radiative heat flux at the wall boundary is given by;

𝑞𝑟 ,𝑤 = −𝜍(𝑇𝑤

4−𝑇𝑔4)

𝛹 (3.25)

Where, 𝑇𝑤 is the wall temperature, 𝑇𝑔 is the temperature of the gas at the

wall and 𝛹 is the slip coefficient.

The Rosseland radiation model is faster than P-1 model and has less memory

requirement as compared to other radiation models. It is able to deal with

axisymmetric geometries and contain the effect of scattering.

38

3.6 Boundary Conditions

Appropriate boundary conditions are required as closure laws to solve the

Navier-Stokes equations.

Figure ‎3-1: Boundary conditions of vertical concentric cylindrical enclosure

Boundary conditions are derived from the known parameters available at

different physical boundaries of the domain in question. In this research work

the experimental model used is the vertical concentric cylindrical enclosure as

shown in Figure 3.1.

39

The top and bottom walls of the enclosure are insulated (adiabatic), except

the bottom disc, which act as a heat source for the enclosure. No slip

boundary conditions are enforced at these walls of the enclosure. The outer

side of the outer cylinder wall is exposed to natural convection at ambient

conditions. The mathematical formulation of these boundary conditions is

given below;

Bottom disc wall boundary

Heat source is applied at the bottom of the disc of radius, 𝑟1. N-type

thermocouple probe is mounted at the upper central location of the bottom

disc and connected to the temperature controller. The temperatures are

specified for different experiments at 𝑧 = 0 and known. Its boundary

conditions are given below;

At 𝑧 = 0; 0 < 𝑟 < 𝑟1;

𝑇 is known;

𝑣𝑟 = 𝑣𝑧 = 0 (No slip condition);

Bottom insulated wall boundary

Bottom surface of enclosure is assumed to be insulated except the bottom

disc heated by the heat source. Neither heat is entered into the bottom

insulated wall nor does it come out of that wall. Its boundary conditions are

given below;

At 𝑧 = 0; 𝑟1 < 𝑟 < 𝑟2;


𝑞𝑖𝑛 = 𝑞𝑜𝑢𝑡 = 0 for an adiabatic wall.

40

Side wall boundary

At the side wall of enclosure the natural convection heat transfer coefficient

𝑕𝑜 is taken as a boundary of enclosure. Heat entering from the hot bottom

disc after passing through the walls of inner cylinder comes out of the wall of

outer cylinder of enclosure. Its boundary conditions are given below;

At 𝑟 = 𝑟2; 0 𝑧 < 𝑧2;

Symmetry boundary

In 2-D axisymmetric geometry the axis is used as a symmetry boundary and

half of the geometry is taken for the analysis as shown in Figure 3.1 as used

by the past researchers ([39], [58], [83], [84] and [85]). At the axis as a

boundary conditions all the gradients of temperature and velocity are zero. Its

boundary conditions are given below;

At 𝑟 = 0; 0 𝑧 < 𝑧2;

𝜕𝑇

𝜕𝑟= 0 ;

𝜕𝑣𝑧

𝜕𝑟= 0;

𝜕𝑣𝑟

𝜕𝑟= 0;

Top insulated wall boundary

Top wall of enclosure is taken as insulated or adiabatic one. Neither heat is

entered into the bottom insulated wall nor does it come out of that wall. Its

boundary conditions are given below;

At 𝑧 = 𝑧2, ; 0 𝑟 < 𝑟2;


𝑞𝑖𝑛 = 𝑞𝑜𝑢𝑡 = 0 for an adiabatic wall.

41

CHAPTER-4: EXPERIMENTAL SYSTEM AND DATA

Experiments are always performed to depict the real scenario and solve the

problems faced by individuals, educational, social and industrial organizations

etc. The experiments diagnose the core reason of the problems faced. From

the pre-historical period to till now the experiments are the most authentic

source of resolving problems. Performing experiments on the actual physical

set up is generally uneconomical. Therefore, experiments are mostly

performed on the experimental models to limit the expenditures. In the

experiments the instrumentations are generally required in order to record

data. All the required instrumentations like thermocouples, pressure gauges,

flow-meters, temperature controllers, heaters, data acquisition systems,

personal computers etc are arranged to collect the experimental data.

Experiments are generally conducted in the controlled environment to avoid

complexities.

In the present research work the heat transfer mechanism in vertical

concentric cylindrical enclosure has been studied experimentally and

computed numerically, with particular focus on centrifuge machines.

Centrifuge machines are generally used in the process industries for

segregation of chemicals. In these machines the heat is generated due to the

electrical losses at the bottom of the enclosure and rotation of inner cylinder,

which affect the process of segregation within the enclosure. That is, the heat

transfer mechanism plays a vital role in the processes occurring in such

enclosures. In this research, an experimental setup of vertical concentric

cylindrical enclosure is designed in which such losses were simulated by

heating the bottom disc by an electric heater. From these experiments the

data is generated to study conjugate heat transfer mechanism in such

enclosures. The heat transfer mechanism is complicated because such

enclosures contain two enclosures, one enclosure within the inner cylinder,

while the other enclosure in between the inner and outer cylinders. The inner

42

enclosure was heated from the bottom while for the outer enclosure the inner

cylinder acted as a heat source. This situation made the problem more

diverse in the sense that the inner enclosure falls in the category of enclosure

with bottom heat source and the outer enclosure in the category of enclosure

with centrally heated source.

Figure ‎4-1: Experimental Apparatus

The experimental model is shown in Figure 4-1, which consists of two vertical

concentric cylinders [11]. With this experimental model total eighteen

experiments have been performed by varying the bottom disc temperature,

inner cylinder material and outer cylinder diameter. The bottom disc is heated

with an electric heater to simulate the actual scenario occurring in such

43

enclosures. The temperature at the upper central location of the bottom disc

is varied from 353-433 K.

Figure ‎4-2: Cross-sectional view of concentric cylindrical enclosure

The temperature data is collected at different locations with the help of PT-

100 temperature sensors shown in Figure 4.2. The steady state is assumed

when the temperatures inside the enclosure remained unchanged for fifteen

minutes as assumed by [49]. Different parts/systems of the experimental

setup are discussed below in detail.

4.1 Cylindrical enclosure

The enclosure studied, consists of two concentric cylinders filled with air at

ambient conditions. There is a circular disc at the bottom central location to

act as a heat source for the enclosure. The rest of the bottom portion and the

whole of the top are covered with mild steel sheets and insulated with

44

ceramic wool and teflon layers. The cylinders are placed in the vertical

direction as shown in Figure 4.2. The outer cylinder is longer than inner

cylinder allowing a gap between the inner cylinder and the top cover. This

gap connects the inner enclosure with the outer one. The details of different

parts of the enclosure are given below.

4.1.1 Bottom disc

The heat generated due to losses in electrical motor, used for rotating the

inner cylinder, in the centrifuge machines affects the chemical process of

segregation within the enclosure.

Figure ‎4-3: Various bottom discs used in the experiments

In order to simulate these heat losses occurring in centrifuge machine and

study their effect on the heat transfer in the enclosure an experimental model

has been designed. In this model the bottom disc is heated with the help of

an electric heater. Three different bottom discs are used in these experiments

as shown in Figure 4.3. The central disc temperature is controlled with the

help of a temperature controller. The temperature is measured at eleven

different locations along the diameter of the disc. At the disc central point the

temperature has been varied within the range of 353-433K in different

experiments.

45

Table ‎4-1: Geometric configurations of cylindrical enclosure

Parts of

geometry Material

Diameter

(m)

Thickness

(m)

Height

(m)

Bottom disc Aluminum, Mild

steel, Stainless steel 0.138 0.004 -

Inner cylinder Aluminum, Mild

steel, Stainless steel 0.146 0.005 1.48

Outer cylinder Mild steel O1=0.256,

O2=0.300 0.01 1.54

Figure ‎4-4: Schematic diagram of vertical concentric cylindrical enclosure

showing geometric specifications.

46

Figure ‎4-5: Bottom disc of aluminum with the thermocouple clamp

Disc of the same material as the inner cylinder is used in each experiment.

The bottom disc made of aluminum and showing the clamp for holding

thermocouple probe is shown in Figure 4.5. The dimensions of different

bottom discs are same and given in Table 4.1 and shown in Figure 4.4.

Bottom disc temperatures observed and measured in eighteen different

experiments are tabulated in appendix A.

4.1.2 Inner cylinder

Thermal behavior inside the inner cylinder as well as its material is very

important and performs pivotal role in the centrifuge machines. In the ideal

conditions, a uniform temperature is required in such enclosures. Generally,

during the process in the inner cylinder of the centrifuge machines the

temperature rises up to 473 K. Three inner cylinders made of aluminum, mild

steel and stainless steel are used in this research work as shown in Figure

4.6. The experiments are performed using inner cylinders made of different

materials to investigate their effect on the heat transfer mechanism in the

enclosure. The experiments are performed within a temperature range of

353-433 K required to segregate the chemicals.

47

Figure ‎4-6: Various inner cylinders used in the experiments

The three inner cylinders used have same dimensions as tabulated in Table

4.1 and shown in Figure 4.4. The temperature data is measured at the inner

as well as outer surface of the inner cylinder walls for eighteen different

experiments and tabulated in appendix A.

4.1.3 Outer Cylinder

Two outer cylinders of mild steel, named as O1 and O2, have been used in

these experiments as shown in Figure 4.7. Geometric specifications of the

outer cylinders (O1, O2) are given in Table 4.1 and shown in Figure 4.4. The

48

outer cylinders are made of mild steel and have different diameters. In these

experiments the effect of outer cylinder diameter is studied on the conjugate

heat transfer occurring within the enclosure. The temperature data on inner

and outer surface of outer cylinder walls is measured experimentally and

tabulated in appendix A.

Figure ‎4-7: Two outer cylinders O1 and O2 of mild steel used in the

experiments

49

4.1.4 Enclosure’s centerline

The axis of the geometry has key importance in such types of enclosures. As

the vertically placed enclosure, under study, has axisymmetric geometry, the

temperature and velocity gradients across the axis line are believed to be

zero. The heat transfer and flow vortices are assumed symmetric about the

axis of the geometry and therefore 2-D axisymmetric geometry is considered

in the numerical simulation. Experimental temperature data is recorded at the

axis of the enclosure for eighteen different experiments and tabulated in

appendix A.

4.2 Electric Heater

A coil type electric heater of 500 W is placed below the bottom disc in a

bucket inside the foundation box. The heater is capable of maintaining

temperatures up to 1116 K. A contactor (solid state relay), temperature

controller and N-type thermocouple are used to control the temperature at

the central location of upper surface of bottom disc. The temperature control

system kept the temperature of the disc at a desired value.

4.3 Data acquisition system

Data acquisition system (DAS) converts the analog waveforms into digital

values for further processing. Its sensors convert physical parameters into the

electrical signals, while the signal conditioning circuits convert sensor signals

into such a form which can be converted to the digital values. DASs are

guided by softwares such as BASIC, C, Fortran, Java, Pascal etc.

In this research work DAS microcontroller AT mega 168 based system,

interfacing with the computer through serial port and having ADC sampling

rate of 62.5 KHz, is used. It accepts up to 64-channels inputs with 60 second

resolution and is shown in the Figure 4.8. This DAS is connected to PC

50

through serial interface. This device provides pre-amplification, scaling and

automatic multiplexing of thermocouple input signals and has been used

previously by [13], [52], [67], [86] and [87].

Figure ‎4-8: Data Acquisition System (DAS)

Figure ‎4-9: Schematic diagram of temperature control system

The schematic diagram of the temperature measuring system is shown in

Figure 4.10. The DAS temperature monitor receives signal from the attached

thermocouples and send the results to the personal computer attached. The

temperature readings are recorded when a steady state condition is achieved.

51

Figure ‎4-10: Schematic diagram of the temperature measuring system

4.4 Temperature control system

The temperature of the central point of the bottom disc is controlled through

a controller-contactor system. The N-type thermocouple senses the

temperature at the upper central point of the bottom disc. The thermocouple

sends data to the temperature controller. The temperature controller

compares the temperature of the disc with the desired set point temperature.

When the disc temperature reaches a set point value, the temperature

controller conveys a message of ON or OFF to the contactor. The contactor is

solid state relay, which switches ON or OFF the heater according to the signal

received from temperature controller as shown in Figure 4.9.

The temperature controller CAL series 9900 with supply voltage of 230 volts

and frequency of 50~60 Hz, power consumption capacity of 5 VA,

temperature range of -200-1800°C was used. It can receive and measure the

signal from J, K, R, S, T, E, L, N and PT-100 type thermocouples. Its

calibration accuracy is ±0.25% and control is ±0.15% typically.

A solid state relay is an electronic switching device. Its coil supply is 230 volts

through relay. Its maximum contactor rated voltage is 500 volts and

temperature error is ±0.5%.

52

4.5 Temperature sensors

The temperature is measured experimentally at different locations in the

experimental setup. The exact location of temperature sensors is shown in

Figure 4.2. Two different type of temperature sensors are used in this

research work. They are discussed below.

4.5.1 PT-100 temperature sensors

PT-100 temperature sensors are platinum resistance thermometers (PRTs)

that give high accuracy over the temperature range (-50 ~ +500°C).

Figure ‎4-11: Temperature sensors mounted on the outer cylindrical wall

These sensors have the resistance of 100 ohms at 0°C and 138.4 ohms at

100°C. There is approximately linear relationship between temperature and

resistance in a small temperature range (0 ~ 100°C), the error at 50°C is

0.4°C.

In this work platinum resistance flat film standard element PT-100

temperature sensors (stock No. 362-9799, class-A) with two wires, lead

length of 10 mm, width of 2 mm and thickness of 1.4 mm have been used.

53

Such temperature sensors are used for the precise temperature measurement

with an accuracy of ±1.45°C. In Figure 4.11 the experimental model is shown

in which the PT-100 temperature sensors on the outer wall are shown.

Table ‎4-2: Temperature Sensors Distribution in the Enclosure

Inner Cylinder Outer Cylinder Axis Disc Ambient Temp.

Inner side Outer side Inner side Outer side

14 14 14 14 14 11 1

In total 81 PT-100 temperature sensors are used to monitor the temperature

at different locations of the enclosure. The distribution of these temperature

sensors is shown in Figure 4.2 and Table 4.2. First row of the temperature

sensors is placed at an axial distance of 5 mm from the bottom disc. The

successive rows are then 116 mm apart except the last row. In the last row

the sensors are placed at the end of inner cylinders and at the same location

on the outer cylinders.

N-type thermocouple probe is mounted at the upper central point of the

bottom disc. While 5 PT-100 temperature sensors are placed in a line on both

sides of the central point, each 14 mm apart from its neighbors. One

thermocouple is placed in the open atmosphere to measure the ambient

temperature. The thermocouples are connected to A T mega based data

acquisition system of 64 channels.

4.5.2 N-type Thermocouple

N-type thermocouple is mineral insulated temperature sensor which is

suitable for use at temperature range -230 ~ 1300 °C, due to its stability and

ability to resist high temperature oxidation. This thermocouple is more

popular due to higher sensitivity than K-type. Its probe material is of nickel-

chromium-silicon/nickel-silicon with probe diameter of 3 mm and probe length

54

of 300 mm and its lead length is 1 m. Its accuracy is ±1.5 °C for -40~375 °C

and sensitivity is 39 µV°C-1 at 900 °C.

The N-type thermocouple is used to sense the temperature at the upper

central point of the bottom disc and send the data to the temperature

controller. The temperature controller after comparing the temperature of the

disc with the desired set point temperature conveys a message of ON or OFF

to the contactor. Ultimately, the contactor switches ON or OFF the heater

after receiving signal from the temperature controller.

4.6 Convection heat transfer coefficient

The convection heat transfer coefficient is not a fluid property. It is an

experimentally determined parameter. Its value depend on all the variables

affecting the convection such as surface geometry, the nature of fluid motion,

properties of the fluid and the bulk fluid velocity as mentioned by Cengel [88]

The case when the geometry is simple and heat transfer is known, the

convection heat transfer coefficient is easy to be calculated. But, calculating

the convection heat transfer coefficient for vertical concentric cylindrical

enclosure is a difficult task.

The heat input to the enclosure is discrete and unknown in this case. The

temperature controller controls and maintains the required upper central

temperature of the bottom disc. Therefore, its convection heat transfer

coefficient is calculated by performing a heat balance at the enclosure

geometry under steady state conditions. In this enclosure the heat entered

through the bottom disc and after passing through the enclosure left it

through the wall of the outer cylinder. Under steady state, the heat entering

the enclosure must be equal to the heat leaving it. The heat is lost from the

enclosure to the surrounding by natural convection. The ambient temperature

and pressure are 299 K and 93.4 KPa respectively. The natural convection

55

heat transfer coefficient of air at the outer cylinder is taken as 10 W.m-2.K-1

by trial and error method as mentioned by Sukhatme [89]. The heat leaving

the enclosure (𝑄𝑜) can be estimated by using the Newton‟s law of cooling;

𝑄𝑜 = 𝑕𝑜𝐴𝑜(𝑇𝑎𝑣 − 𝑇𝑎) (4.1)

𝐴𝑜 = 𝜋𝐷𝑜𝑧2 (4.2)

Where, 𝐴𝑜 is the surface area of outer cylinder, 𝑕𝑜 is the natural convection

heat transfer coefficient, 𝐷𝑜 is diameter of outer cylinder, 𝑧2 is the height of

outer cylinder, 𝑇𝑎𝑣 is the average temperature of the outer cylinder, 𝑇𝑎 is the

ambient temperature.

The heat entering the enclosure (𝑄𝑖) must be equal to the heat lost 𝑄𝑜 from

the enclosure. i.e.;

𝑄𝑜 = 𝑄𝑖 = 𝑕𝑒𝐴𝑑((𝑇𝑑 − 𝑇𝑐) (4.3)

𝑕𝑒 =𝑄𝑜

𝐴𝑑((𝑇𝑑−𝑇𝑐) (4.4)

𝐴𝑑 =𝜋𝑑2

4 (4.5)

where, 𝐴𝑑 is surface area of the bottom disc, 𝑕𝑒 is the heat transfer

coefficient of enclosure, 𝑑 is diameter of the bottom disc, 𝑇𝑑 is the average

temperature of the bottom disc, 𝑇𝑐 is the cold temperature within the

enclosure.

The convection heat transfer coefficient of the air-filled enclosure calculated

from this heat balance is varying from 8-29 W.m-2.K-1 for the experiments

performed in this research work (Table A-1), which lies in the range

mentioned by Cengel [88] using the similar materials. The values of heat

56

transfer coefficient in the enclosure, calculated for different experiments, are

tabulated (Table A-1) in appendix A.

4.7 Uncertainty analysis

Experimental measurements always contain some uncertainties in spite of

using high precision measuring instruments. Such uncertainties come from

the measuring instrument, the items being measured, the environment, the

operator and many other sources. The researchers generally estimate such

uncertainties by using statistical method presented by Moffat [90]. In this

study the same method is used to estimate the uncertainties in experimental

data.

The N-type thermocouple used for measuring the temperature at the center

of bottom disc has an accuracy of ±0.75%, whereas, PT-100 temperature

sensor has an accuracy of ±0.3%. The temperature controller „CAL Series

9900‟ and contactor (solid state rely) have accuracies of ±0.25% and ±0.5%,

respectively. The uncertainty in the reading of N-type thermocouple is

estimated to be less than 1.5%.and that of PT-100 temperature sensor it is

1.1%, including the effect of the observed experimental data scatter.

57

CHAPTER-5: NUMERICAL ANALYSIS

5.1 Introduction to the CFD simulations

Due to rapid development in the past two-three decades in the field of computer

hardware and software the computational techniques are now used worldwide to

study thermal hydraulics problems. In this research work the conjugate heat transfer

phenomena in vertical concentric cylindrical enclosure is studied experimentally and

simulated numerically. The numerical analysis provided a thorough insight of the

flow and heat transfer phenomena that took place in the enclosure. Due to

symmetry of the enclosure the CFD simulations are carried out in 2-D axisymmetric

domain.

FLUENT 6.3 software is used to compare and validate numerical results with the

experimental ones and get insight of the flow and heat transfer mechanism within

the enclosure as used by the past researchers ([20], ([29], [91] and [92]). The CFD

simulations are performed to achieve following objectives;

To study the heat, mass and momentum transfer mechanisms within the enclosure.

To get insight of the enclosure through thermal lines and streamlines.

To study different heat transfer parameters in the enclosure.

To study thermal behavior of the enclosure along the axial as well as radial

direction.

5.2 Heat transfer to the enclosure

The experimental analysis of conjugate heat transfer within a bottom heated vertical

concentric cylindrical enclosure has been numerically simulated. The CFD simulations

results have been compared with the experimental results. Half of the 2-D

axisymmetric geometry of the enclosure being simulated has been shown in the

58

Figure 5.1. The CFD simulations have been conducted using the steps given as

under;

Figure ‎5-1: The enclosure geometry

5.2.1 Geometry and meshing

In this research work the geometry considered for analysis is a 2-D axisymmetric

one due to which the axis is used as a symmetry boundary. The half of the geometry

is taken for the analysis due to its axisymmetric geometry as used by the past

researchers ([39], [58], [83], [84] and [85]). The top and bottom walls of the

59

enclosure, except the bottom disc, are taken as an adiabatic with no heat flux and

no heat generation. No slip condition is considered for all walls of the enclosure. The

walls within the enclosure are thermally coupled. The convection heat transfer

coefficient, h is selected as a boundary condition at the outer surface of the outer

vertical cylinder by trial and error method. Gambit 2.3.16 has been used as a

preprocessor to construct the geometry and mesh. The geometry is generated and

meshed in the Gambit software as shown in the Figure 5.2. Square mesh is used to

mesh the geometry. Along the axis of the geometry there were 1541 grid points,

while at the top and bottom of the geometry there are 211 grid points.

Figure ‎5-2: Meshing of enclosure geometries of outer cylinder, O1 & O2 outer cylinder

Due to change in diameters of the radial grid are 139 and 151 for outer cylinder, O1

& O2 respectively.

60

Figure ‎5-3: Meshing of enclosure bottom geometry of outer cylinder, O1.& O2.

Figure ‎5-4: Meshing of enclosure top geometry of outer cylinder, O1.& O2.

61

Meshing of enclosure bottom and top geometries are shown in the Figures 5.3 and

5.4. the grid size of 1 mm is shown in these geometries. The enclosure geometry

section shown in the Figure 5.5 indicates the annular gaps between axis and inner

cylinder and between outer and inner cylinders, while the gap width between inner

cylinder and top cover plate.

Figure ‎5-5: Meshing of enclosure geometry showing annular and width gaps

5.2.2 Boundary conditions

Proper boundary conditions of the CFD simulations must be applied to get exact

converged solution. After generating and meshing the geometry on the Gambit the

boundary conditions are applied. The axis is taken as symmetry boundary. Top and

bottom walls except the bottom disc are assumed as insulated. The inner and outer

cylinders walls are considered as conducting walls. The two mild-steel outer

62

cylinders of different diameters and three inner cylinders of aluminum, mild-steel

and stainless-steel having similar dimensions are used in different configurations. Air

is used as a fluid. The outer surface of the simulated enclosure is far away from heat

source, it is assumed that the pressure and temperature at the outer surface of the

enclosure are equal to the ambient pressure and temperature respectively to allow

free movement across this surface. There is an ambient temperature at the top of

the vertical enclosure, so the enclosure is assumed as an infinite enclosure.

5.2.3 CFD models applied

The boundary conditions are applied on the Gambit and the geometry is created and

meshed. The meshed geometry file is exported to the Fluent. Different models are

applied in the Fluent and the CFD simulations are carried out. The following settings

are made in the Fluent before starting the simulations.

Two dimensional steady state analysis is carried out.

The Rosseland radiation model is selected due to its utility only with the

pressure-based solver.

Air is treated as a stationary, incompressible and laminar fluid.

The FLUENT have used the pressure-based solver specifically developed for

incompressible flows [79].

The second order scheme is used for pressure, while the second order upwind

discretization schemes are used for momentum and energy [7].

The flow and energy equations are solved by the FLUENT [79].

The user defined function has incorporated the experimental temperature results

of bottom disc and coupled with the FLUENT

63

In the FLUENT the Green-Gauss cell-based gradient scheme is used for the

structured meshes, therefore, this scheme is selected for this research work.

Boussinesq approximation is used assuming the density as a constant, except in

the buoyancy term of vertical momentum equation.

SIMPLE algorithm is used to solve the pressure-velocity coupling. This algorithm

is used by the previous researchers ([23], [39], [40], [59], [71], [93] and [94]).

For the pressure, density, body forces and momentum the default values given in

the FLUENT are used in the under-relaxation factors.

The continuity, x-velocity, y-velocity and energy residuals have used the absolute

convergence criteria of 1E-04, 1E-04, 1E-04 and 1E-06 respectively.

The convergence criteria are different for different cases as shown in the

previous studies by [21], [87] and [95]. Iterations took place till the convergence

is achieved to the required solution in the range of 765 ~ 4968 iterations.

Two different configurations with three different materials of inner cylinders and

discs are used in the analysis. In total 18 experiments are performed at the

specific temperatures of 353 K, 393 K and 433 K on the upper central location of

bottom discs.

The convection heat transfer coefficient is taken as a boundary condition on the

outer surface of the enclosure.

After simulating the above mentioned models, the simulations are started. The

above mentioned models involve computations to be performed at each cell. The

mesh selected is square and total numbers of cells of both configurations (O1 and

O2) are 213960 and 247400 respectively.

The properties of fluid and materials [88] used in these simulations are given in the

Tables 5.1 and 5.2 respectively.

64

Table ‎5-1: Properties of Fluid (air)

Properties Air

Density, kg/m3 1.184

Specific heat capacity, J/kg.K 1007

Thermal conductivity, W/m.K 0. 02551

Dynamic viscosity, Pa.sec 1.8949E-5

Absorption coefficient, 1/m 0.49919, constant

Scattering phase function Isotropic

Scattering coefficient, 1/m 0

Thermal Expansion Coefficient, 1/K 0.0033445

Table ‎5-2: Properties of Materials

Properties Aluminum Mild Steel Stainless Steel

Density, kg/m3 2739 7833 8238

Specific heat capacity, J/kg.K 896 502 468

Thermal conductivity, W/m.K 222 45.3 13.4

5.3 Grid independence study

Table ‎5-3: Grid independence study

Grid of cell

size =0 5

mm

Grid of cell

size =1 mm

Grid of cell

size = 1.5

mm

Grid of cell

size = 2

mm

Number of cells 855840 213960 96135 54260

Percentage error 1.91 1.94 2.9 4.2

65

The grid independence study is carried out by taking the grids of cell sizes 2 mm,

1.5 mm, 1mm and 0.5 mm. There is a prominent percentage error of cell sizes 2

mm, 1.5 mm and 1mm, while cell sizes 1mm and 0.5 mm have almost the same

percentage error with the experimental results of [11] .Therefore, cell size of 1 mm

is taken while meshing the enclosure under research.

66

CHAPTER-6: RESULTS AND DISCUSSION

6.1 Introduction

The study of heat transfer within the cylindrical enclosure is important because of its

numerous applications in the process industry. During the last couple of decades a

lot of research work has been done related to heat transfer in cylindrical enclosures.

The past researchers [36-48] have thoroughly studied the heat transfer mechanism

in cylindrical enclosures. In this research work the phenomena of conjugate heat

transfer within vertical concentric cylindrical enclosure is investigated experimentally

and computed numerically. The unique feature of this study is that the heat transfer

within the enclosure is studied using inner cylinders made of different materials. The

effect of inner cylinder material on heat transfer in such enclosures has not been

studied previously. The heat source is placed at the central bottom location of the

enclosure with a proper temperature control mechanism. The temperature of the

source central location is varied in the range of 353-433 K. This research work also

includes the study of effect of outer cylinder diameter on the heat transfer within the

enclosure. The numerical analysis is carried out using Fluent code. Numerical

simulation results of vertical concentric cylindrical enclosure are validated by

comparing with the experimental data. Streamlines, thermal lines and velocity

vectors are obtained from CFD simulation which helped to understand the conjugate

heat transfer mechanism within the enclosure.

Thermal conductivity of inner cylinder material is thought to play a key role in

affecting thermal behavior within the enclosure. For the material analysis three inner

cylinders made of aluminum, mild steel and stainless steel are used. Thermal

conductivity of aluminum is about 17 times greater than that of stainless steel and

about 5 times greater than mild steel.

As mentioned earlier that a uniform temperature is desired within the inner

enclosure for segregation of chemicals in centrifuge machines. Therefore, to know

67

the thermal conditions within the enclosure, the study of axial and radial thermal

behavior of the enclosure is important. To generalize the study related to vertical

concentric cylindrical enclosure, non-dimensional analysis must be done. Similarly to

understand the heat transfer mechanism and buoyancy effects within the enclosure,

the study of CFD results is required. In the subsequent articles the above mentioned

areas and parameters related to cylindrical enclosures have been studied.

6.2 Axial thermal behavior

The temperature distribution along the axis and inner cylinder wall are very

important to study the thermal behavior within the enclosure. Amara et al. [36]

have drawn both axial and radial temperature profiles of vertical cylindrical enclosure

by heating it with a periodical lateral heat flux density. In this research work the

axial temperature along the axis, inner cylinder and outer cylinder of the enclosure

have been measured experimentally and computed numerically. The experimental

temperature data is given in appendix B. The experimental and CFD results of axial

temperature along the axis and inner cylinder of the enclosure are shown in Figures

6.1 and 6.2 respectively. The axial temperature distribution at axis of the geometry

and inner cylinder are discussed in detail in the following articles.

6.2.1 Axis of the enclosure

The analysis of thermal behavior along the axis of the enclosure is performed

experimentally and computed numerically. The thermal behavior at the axis of

enclosure gives true picture of heat transfer mechanism within the enclosure. The

segregation of chemicals in centrifuge machines takes place around the axis of the

geometry and therefore strongly depends on the thermal behavior along the axis.

Temperature at the axis is studied by varying bottom disc central temperature

between 353-433 K, using inner cylinders of different materials and outer cylinders

of different diameters. The axial temperature distribution along the axis of vertical

concentric cylindrical enclosure under different experimental conditions is shown in

68

Figure 6.1. In this research work inner cylinders of aluminum, mild steel and

stainless steel materials have been used and their effects on temperature along the

axis of enclosure have been discussed.

Figure ‎6-1: Axial temperature distribution along the axis of the enclosure with inner

cylinder of (a, b) aluminum, (c, d) mild steel and (e, f) stainless steel.

Six different experiments are performed using inner cylinder made of aluminum. The

same experiments are repeated using inner cylinders of mild steel and stainless

steel. These experiments are performed at different temperatures of the bottom disc

central location (353, 393, and 433 K) and using two outer cylinders (O1 and O2) of

different diameters. These experiments are also numerically simulated in 2-D

axisymmetric domain using Fluent 6.3 software. The experimental results of axial

temperature are compared with the CFD results and shown in Figure 6.1. The

69

experimental and numerical results of axial temperature match closely with each

other for these eighteen different cases and therefore validate the numerical

simulations. The aspect ratio and radius ratio of the enclosure geometry with outer

cylinder O1 are 6.016 and 1.73, while with outer cylinder O2 these values are 5.133

and 2.055 respectively. Figure 6.1 shows that the axial temperature decreases along

the height of the enclosure for all the experiments performed due to the presence of

heat source at the bottom of the enclosure. Figure 6.1 (a-b) shows that the axial

temperature distribution is almost independent of the outer cylinder diameter.

However, Figure 6.1 (c-f) shows that the axial temperature distribution is a function

of the outer cylinder diameter especially at higher temperatures of the bottom disc.

This response might be due to high thermal conductivity of aluminum as compared

to mild steel and stainless steel.

6.2.2 Inner cylinder

The axial temperature distribution along the inner side of inner cylinder is another

important parameter to study the heat transfer and thermal behavior of vertical

concentric cylindrical enclosures. The temperature distribution along the axis and

inner cylinder tells the whole story of heat transfer in such enclosures. The effect of

thermal conductivity of inner cylinder material on the heat transfer mechanism in

vertical concentric cylindrical enclosure have not yet been studied by the past

researchers to the best of the authors‟ knowledge. In centrifuge machines, uniform

temperature within the inner cylindrical enclosure is ideally required for the

segregation of different chemicals. Design engineers give special attention to design

the centrifuge geometry for attaining ideal thermal conditions required within the

inner cylinder. The experimental and the CFD results for axial temperature along the

inner side of inner cylinder are shown in Figure 6.2. Figure 6.2 (a-b) shows the axial

temperature distribution along the inner cylinder, using inner cylinder made of

aluminum material, under different temperatures of the bottom disc and using two

different outer cylinders O1 and O2. Figure 6.2 (c-d) and Figure 6.2 (e-f) show the

same results using inner cylinders of mild steel and stainless steel respectively.

70

Figure ‎6-2: Axial temperature along the inner surface of inner cylinder of aluminum

(a, b), mild steel (c, d) and stainless steel (e, f)

In Figure 6.2 (a-b) it is observed that using inner cylinder of aluminum the axial

temperature distribution along the inner cylinder is independent of the outer cylinder

diameter. However, in Figure 6.2 (c-d) and Figure 6.2 (e-f) it is observed that using

inner cylinders of mild steel and stainless steel the axial temperature along the inner

cylinder depends on the diameter of the outer cylinder. This behavior of the

enclosure can be explained as follow. Due to higher thermal conductivity of

aluminum the heat is transferred easily to the outer enclosure irrespective of the

diameter of the outer cylinder. However, in the case of mild steel and stainless steel

inner cylinders the thermal conductivities of the inner cylinder are low and therefore

the axial temperature along the inner cylinder is a function of the diameter of the

71

outer cylinder as well. In other words, it can be stated that in case of aluminum

inner cylinder the radial heat transfer is controlled by the thermal conductivity of the

inner cylinder material. Similarly in case of mild steel and stainless steel inner

cylinders the radial heat transfer is controlled by thermal conductivity as well as the

diameter of the outer cylinder.

Due to high thermal conductivity of aluminum the heat conduction in the axial

direction is also more as compared to mild steel and stainless steel, therefore, a

more uniform axial temperature is observed in Figure 6.2 (a-b) as compared to

Figure 6.2 (c-f). The spread in temperature distribution for various bottom disc

temperatures is more in Figure 6.2 (c-f) as compared to Figure 6.2 (a-b). The reason

might be due to higher thermal conductivity of aluminum as compared to mild steel

and stainless steel. Figure 6.2 indicates that a more uniform axial temperature is

obtained using inner cylinder of aluminum as compared to mild steel and stainless

steel inner cylinders, under different experimental conditions of bottom disc and

outer cylinder. The close agreement between the experimental and the CFD results

of axial temperature along inner cylinder support the numerical simulation of vertical

concentric cylindrical enclosure.

6.3 Radial thermal behavior

Under steady state conditions the heat is transferred mainly in the radial direction in

the vertical concentric cylindrical enclosure, heated at the bottom and insulated at

the top and bottom. To understand the thermal behavior of such enclosures the

thermal distribution in radial direction must be studied. The heat is transferred from

the hot bottom disc to the inner enclosure in the axial as well as radial direction

through radiation and convection. The convection heat transfer takes place within

the enclosure as well as at the outer lateral walls of the enclosure, while conduction

heat transfer takes place in the walls of inner and outer cylinders. Thermal

conductivity of the material is the main parameter affecting the transfer of heat

72

through the walls of inner and outer cylinders. The effect of annular gap (between

the inner and outer cylinders) of the enclosure on the radial heat transfer

mechanism is also investigated.

Along the radial direction the effects of different materials (aluminum, mild steel and

stainless steel) of inner cylinder, diameter of the outer cylinder and the temperature

of the bottom disc in the range of 353-433 K have been studied. The radial thermal

behavior is studied at three different axial locations 5, 700 and 1396 mm, measured

from the bottom of the geometry. The radial temperature distributions of the

enclosure geometries using inner cylinders of aluminum, mild steel and stainless

steel materials have been explained in detail as given below.

6.3.1 Enclosure with aluminum inner cylinder

Using aluminum inner cylinder the radial temperature distribution is obtained

experimentally as shown in Figure 6.3. These results are obtained at three different

axial heights (5, 700 and 1396 mm) for different operating conditions of the bottom

disc and outer cylinder. At an axial location of 5 mm the temperature distribution for

the two different outer cylinders (O1, O2) has been shown in Figure 6.3 (a-b). Radial

thermal response at an axial height of 5 mm (Figure 6.3 (a-b)) is almost same for

the two different outer cylinders (O1, O2). This uniformity in thermal behavior of both

configurations might be due to high thermal conductivity of aluminum inner cylinder.

While comparing both configurations in Figure 6.3 (a) and Figure 6.3 (b) it is

observed that the spread in temperature at the outer cylinder in Figure 6.3 (a) is

more as compared to Figure 6.3 (b) over a range of bottom disc temperatures (353-

433K). The reason of that spread might be that the outer cylinder has a lower

thermal conductivity and the heat transfer mechanism is controlled by outer cylinder

diameter in this region.

Figure 6.3 (c-d) shows the radial thermal behavior at an axial height of 700 mm

using aluminum inner cylinder. By analyzing these figures it is observed that the

radial response at axial location of 700 mm has the same trend as that at axial

73

location of 5 mm, but the spread in temperature in this location is less prominent as

compared to axial location of 5 mm possibly due to the low temperature gradient

across the radius of the enclosure at 700 mm height.

Figure ‎6-3: Radial temperature distribution with aluminum inner cylinder

At the axial location of 1396 mm radial thermal behavior is plotted as shown in

Figure 6.3 (e, f). Due to very low temperature gradient across the radius of the

enclosure at 1396 mm height the spread in temperature is not very prominent as

compared to the radial thermal behavior at 5 and 700 mm.

74

6.3.2 Enclosure with mild steel inner cylinder

The radial temperature distribution with mild steel inner cylinder is measured

experimentally for two different configurations (O1, O2) and three different bottom

disc temperatures (353, 393 and 433 K). The radial thermal behavior is studied at

three different axial heights (5, 700 and 1396 mm). At an axial location of 5 mm the

radial temperature of the two different enclosure configuration (O1, O2) with mild

steel inner cylinder has been shown in Figure 6.4 (a-b). The radial temperature at

the axis and both sides of inner cylinder wall of mild steel are higher in Figure 6.4

(a-b) as compared to Figure 6.3 (a-b) for aluminum inner cylinder. The possible

reason might be the difference between the thermal conductivities in the two cases.

Figure ‎6-4: Radial temperatures with mild steel inner cylinder

75

However, the behavior of temperature at outer radial positions shows a greater

spread with configuration O1 (Figure 6.4 (a)) as compared to configuration O2

(Figure 6.4 (b)). The same behavior is noted with aluminum inner cylinder (Figure

6.3 (a-b)). The reason might be stronger dependence of heat transfer mechanism on

the diameter of outer cylinder. The radial temperatures of configuration, O1 (Figure

6.4 (a)) are higher than that of configuration, O2 (Figure 6.4 (b)). This might be due

increase in outer cylinder diameter from O1 to O2.

Figure 6.4 (c-d) shows the radial thermal behavior at an axial height of 700 mm

using mild steel inner cylinder. The radial temperature at the axis and both sides of

inner cylinder wall of mild steel are higher in Figure 6.4 (c-d) as compared to Figure

6.3 (c-d) for aluminum inner cylinder. The same behavior is noted with aluminum

inner cylinder (Figure 6.3(c-d)). It is observed that the radial response at axial

location of 700 mm has the same trend as compared to axial location of 5 mm, but

the spread in temperature in this location is less prominent as compared to axial

location of 5 mm possibly due to the low temperature gradient across the radius of

the enclosure at 700 mm height. The radial temperatures of configuration, O1

(Figure 6.4 (c)) are higher as compared to configuration, O2 (Figure 6.4 (d)) just by

changing outer cylinder diameter.

At the axial location of 1396 mm radial thermal behavior is plotted as shown in

Figure 6.4 (e, f). The radial temperature at the axis and both sides of inner cylinder

wall of mild steel are higher in Figure 6.4 (e-f) as compared to Figure 6.3 (e-f) for

aluminum inner cylinder. Due to very low temperature gradient across the radius of

the enclosure at 1396 mm height the spread in temperature is not so prominent as

compared to the radial thermal behavior at 5 and 700 mm.

6.3.3 Enclosure with stainless steel inner cylinder

The radial temperature distribution with stainless steel inner cylinder is measured

experimentally for two different configurations (O1, O2) and three different bottom

76

disc temperatures (353, 393 and 433 K). The radial thermal behavior is studied at

three different axial heights (5, 700 and 1396 mm). At an axial location of 5 mm the

radial temperature of the two different enclosure configuration (O1, O2) with

stainless steel inner cylinder has been shown in Figure 6.5 (a-b). The radial

temperature at the axis and both sides of inner cylinder wall of stainless steel are

higher in Figure 6.5 (a-b) as compared to Figure 6.3 (a-b) and Figure 6.4 (a-b) for

aluminum and mild steel inner cylinders respectively. The possible reason might be

due to the lowest thermal conductivity of stainless steel as compared to the mild

steel and aluminum. In this simulation it is observed that stainless steel inner

cylinder is a function of outer cylinder diameter.

Figure ‎6-5: Radial temperature with stainless steel inner cylinder

77

However, the spread in the radial temperature of enclosure configuration O1 (Figure

6.5 (a)) is more as compared to enclosure configuration O2 (Figure 6.5 (b)). This

shows that spread in the radial temperature of enclosure configuration (O1) with

stainless steel inner cylinder might be due strong dependence of heat transfer

mechanism on outer cylinder diameter. The radial temperatures of configuration O1

(Figure 6.5 (a)) are higher than that of configuration, O2 (Figure 6.5 (b)) possibly

due to increase of outer cylinder diameter.

At an axial height of 700 mm Figure 6.5 (c-d) shows the radial thermal behavior

using stainless steel inner cylinder. It is observed that the radial response at axial

location of 700 mm has the same trend as compared to axial location of 5 mm, but

the spread in temperature in this location is less prominent as compared to axial

location of 5 mm possibly due to the low radial temperature gradient at 700 mm

height. The radial temperatures of enclosure configuration, O1 (Figure 6.5 (c)) are

higher as compared to configuration, O2 (Figure 6.5 (d)) possibly due to increase in

outer cylinder.

At the axial location of 1396 mm radial thermal behavior is shown in Figure 6.5 (e,

f). Due to very low radial temperature gradient at 1396 mm height the spread in

temperature is not so prominent as compared to the radial thermal behavior at 5

and 700 mm.]

6.3.4 Wall thickness effects on heat transfer mechanism

The inner cylinders made of three different materials of aluminum, mild steel and

stainless steel are used in the vertical cylindrical enclosure. Thermal conductivity of

aluminum is 5 times greater than mild steel and 17 times greater than stainless

steel. The effects of thermal conductivities of inner cylinders of aluminum, mild steel

and stainless steel are prominent. In these cases thermal conductivities and air are

the main controlling parameters of the heat transfer mechanism within the enclosure

as shown in the Figure 6.2 (a-f). In these cases thickness of inner cylinder of all the

three materials are the same. More heat is transferred from aluminum inner cylinder

78

Figure 6.2 (a-b) as compared to the mild steel and stainless steel inner cylinders

Figure 6.2 (c-f). Similarly, more heat is transferred from mild steel inner cylinder

Figure 6.2 (c-d) as compared to stainless steel inner cylinder Figure 6.2 (e-f).

In these experiments the diameter to thickness ratios of inner cylinders are 29.2 and

outer cylinders O1 and O2 are 25.6 and 30 respectively. These all inner and outer

cylinders are thin cylinders. In the Figures 6.3 through 6.5 the temperatures of the

inner and outer surfaces of the inner cylinders are the same in the steady state

conditions. Similarly the temperatures of the inner and outer surfaces of the outer

cylinders are the same under the same conditions. In this research work wall

thicknesses of the cylinders have their negligible effect on the heat transfer

mechanism of the enclosure.

6.4 Non-dimensional results

Non-dimensional quantities are often the products or ratios of quantities with the

same dimensions. Even though a dimensionless quantity has no physical dimension

associated with it, it can still have dimensionless units. To show the quantity being

measured, often the same units in both the numerator and denominator (i.e., kg/kg,

m/m) are used.

In the field of heat transfer, Nusselt number, Biot number, Rayleigh number,

Grashof number, Prandtl number, Reynolds number etc are the non-dimensional

numbers generally used to indicate the significance of various heat transfer

parameters indicate the fact that non-dimensional study is carried out to generalize

the results, i.e., independent of a specific geometry. While studying the conjugate

heat transfer mechanism in vertical concentric cylindrical enclosures the local Nusselt

number and Rayleigh number study is important. These parameters are discussed

below.

79

6.4.1 Nusselt number

It is a dimensionless parameter that expresses the heat transfer coefficient in

convection heat transfer study. Local Nusselt number Nu can be expressed as a

derivative of the non-dimensional temperature θ with respect to the radius of the

enclosure as given below.

𝑁𝑢 𝑧 = − 𝜕𝜃

𝜕𝑧 𝑧=0

(6.1)

Where, Nu(z) is local Nusselt number, z is axial component in the enclosure and

𝜃 =𝑇−𝑇𝑐

𝑇𝑕 −𝑇𝑐 (6.2)

Where, T is the variable temperature, Th is temperature of hot surface and Tc is

temperature of cool surface.

Equation 6.1 has also been used by the past researchers [96-97] for heat transfer in

such type of enclosures. Local Nusselt numbers are plotted along the axial height of

different types of enclosures by Al-Bahi [34], Sharma et al. [39] and Kim [67]. Sarr

et al [18] have drawn Nusselt number against Grashof number in the

cylindrical enclosure showing same trends as shown in the Figure 6.7. Local

Nusselt number is calculated at the inner cylinder wall and plotted in the Figure 6.6

on a semi-log scale along the non-dimensional axial height of the enclosure. The

non-dimensional diameter dhs of the heat source i.e. bottom disc for configuration

(O1) is 0.54 and for configuration (O2) is 0.46. Six different experiments are

performed using inner cylinder made of aluminum. The same experiments are

repeated using inner cylinders of mild steel and stainless steel. These experiments

are performed at different temperatures of the bottom disc central location (353,

393 and 433 K) and using two outer cylinders (O1, O2) of different diameters.

80

Figure ‎6-6: Nusselt number along inner cylinder wall

Local Nusselt number near the heat source is high. It decreases with axial height of

the cylinder. The decrease in Nu is very steep near the bottom of the cylinder than

rest of the axial height of the cylinder as shown in Figure 6.6. In the middle section

of the inner cylinder the decrease in Nu is also steep for the enclosure configuration

O1 with non-dimensional diameter dhs of 0.54 (Figure 6.6 (a, c, e)) as compared to

the enclosure configuration O2 with non-dimensional diameter dhs of 0.46 (Figure 6.6

(b, d, f)). The effect of inner cylinder materials of two different configurations is

clearly shown in the graphs of local Nusselt number along the axial height of the

enclosure in the Figure 6.6. The same trend of Nu is observed by the past

researchers ([96-98]).

81

6.4.2 Rayleigh number

Rayleigh number is an important dimensionless parameter associated with buoyancy

driven flow (natural convection flows). It depicts the mode of heat transfer within

the fluids. This number is calculated by the relation given as under.

𝑅𝑎 =𝑔𝛽𝜌 ∆𝑇𝐷3

𝛼𝜇 (6.3)

In this study Rayleigh number Ra is calculated along the inner cylinder wall using the

following parameters., ∆T is temperature difference between the temperature at a

certain location on the inner cylinder wall and ambient temperature within the

enclosure geometry, D is difference between the radius of the outer cylinder and

inner cylinder of the enclosure, g is acceleration due to gravity, is thermal

diffusivity of air, β is coefficient of thermal expansion and µ is dynamic viscosity of

air. In such types of enclosures the past researchers [99-102] have used the same

relation for heat transfer calculations in their research.

The graphs of local Nusselt number along the Rayleigh number at the inner cylinder

wall of different materials (aluminum, mild steel, stainless steel) are shown in

Figures 6.7 for both enclosure configurations (O1, O2). The graphs of local Nusselt

number with the Rayleigh number at the inner cylinder wall of aluminum for both

geometric configurations (O1, O2) are shown in Figure 6.7 (a-b). The non-

dimensional diameter (dhs) of the heat source for configuration O1 is 0.54 and for

configuration O2 is 0.46. The local Nusselt number (Nu) increases with Rayleigh

number (Ra) for constant non-dimensional diameter (dhs)of heat source. Comparing

Figure 6.7 (a, c, e) with Figure 6.7 (b, d, f) it is observed that the Nusselt number

(Nu) increases with increasing non-dimensional diameter (dhs) of heat source while

keeping Rayleigh number Ra constant. The past researchers [96-97] have reported

similar behavior of Nusselt number variation with Rayleigh number at constant value

of non-dimensional diameter dhs of heat source and Nusselt number variation with

non-dimensional diameter dhs of heat source at constant value of Rayleigh number.

82

Figure ‎6-7: Local Nusselt number with Rayleigh number

In all the experiments performed for both the enclosure configurations (O1, O2), high

Rayleigh number Ra is achieved with stainless steel inner cylinder as compared to

mild steel, while keeping the bottom disc temperature constant as shown in Figure

6.7. Similarly, high Rayleigh number Ra is achieved with mild steel inner cylinder as

compared to aluminum inner cylinder, while keeping the bottom disc temperature

constant.

6.5 The CFD simulation results

In this research work the heat transfer mechanism in vertical concentric cylindrical

enclosure has been investigated experimentally and computed numerically. Eighteen

83

different experiments are performed in which the temperature at 82 different

locations have been measured as tabulated in chapter 4. The experimental

temperature data provided important information about the heat transfer in the

enclosure as discussed in the above articles. However, to study the heat transfer

mechanism and buoyancy effects in vertical concentric cylindrical enclosure in more

depth CFD results have been discussed in the following sections.

6.5.1 Validation of the CFD simulation

Before discussing the CFD results an effort is made to validate the CFD simulations.

The experimental data of eighteen different experiments is used to validate the CFD

results. Three different inner cylinders made of aluminum, mild steel and stainless

steel have been used in the experiments. The CFD and experimental results for the

six experiments performed with each cylinder have been compared in the following

sections.

6.5.1.1 Validation of the CFD simulation of enclosure using aluminum inner cylinder

The comparison of experimental results with the CFD simulation results of enclosure

configurations (O1, O2) with aluminum inner cylinder at different bottom disc central

temperatures (353, 393 and 433 K) have been shown in Figure 6.8.

The positive and negative errors of the CFD simulation results with the experimental

ones are 0.8% and 1.3% respectively (Figure 6.8). This shows that the CFD

simulation results are in best agreement with experimental ones. Hence, the CFD

simulation results of enclosure configurations (O1, O2) with aluminum inner cylinder

at different bottom disc central temperatures (353, 393 and 433 K) are validated

with the experimental results.

84

Figure ‎6-8: Comparison of experimental and the CFD results of enclosure with

aluminum inner cylinder

6.5.1.2 Validation of the CFD simulation of enclosure using mild steel inner cylinder


configurations (O1, O2) with mild steel inner cylinder at different bottom disc central

temperatures (353, 393 and 433 K) is shown in the Figure 6.9.

The positive and negative errors of the CFD simulation results with the experimental

ones are 1% and 1.3% respectively. This shows that the CFD simulation results are

in the best agreement with experimental results. Hence, the CFD simulation results

of enclosure configurations (O1, O2) with mild steel inner cylinder at different bottom

disc central temperatures (353, 393 and 433 K) are validated with the experimental

results.

85

Figure ‎6-9: Comparison of experimental and the CFD results of enclosure with mild

steel inner cylinder

6.5.1.3 Validation of the CFD simulation of enclosure using stainless steel inner cylinder


configurations (O1, O2) with stainless steel inner cylinder at different bottom disc

central temperatures (353, 393 and 433 K) is shown in the Figure 6.8. The positive

and negative errors of the CFD simulation results with the experimental ones are

0.48% and 1.23% respectively.

This shows that the CFD simulation results are in the best agreement with

experimental results. Hence, the CFD simulation results of enclosure configurations

(O1, O2) with stainless steel inner cylinder at different bottom disc central

temperatures (353, 393 and 433 K) are validated with the experimental results.

86

Figure ‎6-10: Comparison of experimental and the CFD results of enclosure with

stainless steel inner cylinder

6.5.2 Contours of streamlines at 353 K

Papanicolaou and Belessiotis [83] have drawn streamlines of water-filled cylindrical

enclosure laterally heated at constant heat flux while insulating its top and bottom

walls. Hamady et al. [41] have drawn streamlines by differentially heating an air-

filled cylindrical enclosure by rotating the enclosure above its longitudinal horizontal

axis. Sharma et al. [39] numerically investigated the results of conjugate natural

convection heat transfer heated by a volumetric energy generating source within a

cylindrical enclosure. The heat conducting body is placed in the enclosure between

isothermal lateral walls. Khanafer and Vafai [62] reduced the effective boundary

conditions at the open side of structures to a closed-ended domain to save the CPU

87

and memory usage. They obtained boundary conditions for both temperature and

flow fields covering a comprehensive range of controlling parameters. The

streamlines and isotherms are drawn to show the buoyancy forces and the fluid

flow.

The contours of streamlines are obtained from the CFD simulations for the enclosure

configurations O1 and O2 at bottom disc central temperature of 353 K and are shown

in Figure 6.11. Figure 6.11 (a-b) shows streamlines for two configurations (O1, O2)

of aluminum inner cylinder. Similarly Figures 6.11 (c-d) and (e-f) show the

corresponding results for mild steel and stainless steel inner cylinders. It is observed

in Figure 6.11 (a-b) that the streamlines within the inner cylinder are same for both

configurations (O1, O2) of aluminum inner cylinder. The same behavior is observed in

case of mild steel (Figure 6.11 (c-d)) and stainless steel inner cylinders (Figure 6.11

(e-f)). However, the streamlines in the inner cylinder of Figure 6.11 (a-b) shows

weak buoyancy effects as compared to Figure 6.11 (c-f). Another important

observation made regarding Figure 6.11 is that the buoyancy effects are stronger in

the outer annulus with outer cylinder configuration O2 as compared to configuration

O1 for all three inner cylinders used. The buoyancy effects in the outer annulus are

stronger in case of aluminum inner cylinder as compared to mild steel and stainless

steel for the same outer configuration. The same behavior is observed in case of

mild steel inner cylinder as compared to stainless steel.

The above mentioned observations regarding Figure 6.11 can be explained as given

below. At the bottom disc central temperature of 353 K the quantity of heat added

to the enclosure is small and therefore the effect of outer cylinder configuration on

the heat transfer from the inner cylinder to the outer annulus region is negligible.

However, due to difference of thermal conductivity buoyancy effects are relatively

weak in Figure 6.11 (a-b) as compared to Figure 6.11 (c-f). With outer cylinder

configuration O2 the buoyancy effects are stronger due to increase in volume as

compared to configuration O1 for three inner cylinders used. In the outer annulus

88

Figure ‎6-11: Streamlines of enclosure for configuration O1 a) aluminum inner cylinder, c). mild steel inner cylinder, e) stainless

steel inner cylinder, for configuration O2 b) aluminum inner cylinder, mild steel inner cylinder, f), stainless steel inner cylinder at

353 K.

89

the buoyancy effects are stronger while using inner cylinder of aluminum as

compared to other inner cylinders due to its high thermal conductivity.


The contours of streamlines are obtained from CFD simulations for the enclosure





in Figure 6.12 (a-b) that the streamlines within the inner cylinder are same for both

configurations (O1, O2) of aluminum inner cylinder. The same behavior is observed in

case of mild steel (Figure 6.12 (c-d)) and stainless steel inner cylinders (Figure 6.12

(e-f)). However, the streamlines in the inner cylinder of Figure 6.12 (a-b) shows

weak buoyancy effects as compared to Figure 6.12 (c-f). Another important

observation made regarding Figure 6.12 is that the buoyancy effects are stronger in

the outer annulus with outer cylinder configuration O2 as compared to configuration

O1 for all three inner cylinders used. The buoyancy effects in the outer annulus are

stronger in case of aluminum inner cylinder as compared to mild steel and stainless

steel for the same outer configuration. The same behavior is observed in case of

mild steel inner cylinder as compared to stainless steel.


below. At the bottom disc central temperature of 393 K the quantity of heat added

to the enclosure is small and therefore the effect of outer cylinder configuration on

the heat transfer from the inner cylinder to the outer annulus region is negligible.

However, due to difference of thermal conductivity buoyancy effects are relatively

weak in Figure 6.12 (a-b) as compared to Figure 6.12 (c-f). With outer cylinder

configuration O2 the buoyancy effects are stronger due to increase in volume as

compared to configuration O1 for three inner cylinders used. In the outer annulus

90



393 K.

91

the buoyancy effects are stronger while using inner cylinder of aluminum as

compared to other inner cylinders due to its high thermal conductivity.


The contours of streamlines are obtained from CFD simulations for the enclosure





in Figure 6.13 (a-b) that at bottom disc temperature of 433 K the streamlines

behavior within the inner cylinder changes with outer cylinder configuration for

aluminum inner cylinder. Similarly the same behavior is observed for mild steel

(Figure 6.13 (c-d)) and stainless steel inner cylinders (Figure 6.13 (e-f)). The most

remarkable observation made regarding Figure 6.13 is that the buoyancy effects are

stronger in the outer annulus with outer cylinder configuration O2 as compared to

configuration O1 for all three inner cylinders used. The buoyancy effects in the outer

annulus are stronger in case of aluminum inner cylinder as compared to mild steel

and stainless steel for the same outer configuration. The same behavior is observed

in case of mild steel inner cylinder as compared to stainless steel.


below. At a bottom disc central temperature of 433 K the outer cylinder diameter

also affects the heat transfer mechanism within the inner cylinder due to large

quantity of heat transfer through the enclosure. With outer cylinder configuration O2

the buoyancy effects are stronger due to increase in volume as compared to

configuration O1 for three inner cylinders used. In the outer annulus the buoyancy

effects are stronger while using inner cylinder of aluminum as compared to other

inner cylinders due to its high thermal conductivity.

92



433 K.

93

The stream lines in Figures 6.11 to 6.13 also suggest that due to high thermal

conductivity of aluminum inner cylinder, the heat conduction in inner cylinder wall in

the axial direction is high as compared to inner cylinders of mild steel and stainless

steel. This behavior is more clearly visible in Figure 6.13, where due to low thermal

conductivity of mild steel and stainless steel double vortices are formed in the outer

annulus region with outer cylinder configuration O2 (Figure 6.13 (d and f)). Apart

from the above discussion there are other numerous points which can be highlighted

using the CFD simulation results. The simulation results of contours of thermal lines

and velocity vectors are given in appendix B.

94

CHAPTER-7: CONCLUSIONS AND FUTURE

RECOMMENDATIONS

7.1 Conclusions

In this research work the conjugate heat transfer mechanism within a bottom heated

non-conventional cylindrical enclosure is investigated experimentally and computed

numerically. In total eighteen different experiments are performed in a controlled

environment. The axial thermal behavior along the axis and inner cylinder of the

enclosure and radial thermal behavior at an axial height of 5, 700 and 1396 mm of

the enclosure are studied by varying bottom disc central temperature (353-433K),

inner cylinder material (aluminum, mild steel, stainless steel) and outer cylinder

diameter (O1, O2). The main focus of this study is on the thermal behavior of the

enclosure by using three inner cylinders of different materials and having large

difference in their thermal conductivities. The CFD simulations are also performed

and validated by the experimental results. The heat transfer and buoyancy effects

within the enclosure geometry are highlighted through streamlines, thermal lines

and velocity vectors obtained from CFD simulations. As a result of these experiments

and CFD simulations following points are concluded.

A more uniform axial temperature is obtained at the inner wall of aluminum

cylinder at the bottom disc central temperatures (353-433K) and outer cylinder

diameters (O1, O2) as compared to mild steel and stainless steel. The same is

true for mild steel as compared to stainless steel.

The heat transfer in the enclosure with stainless steel inner cylinder is more

sensitive to outer cylinder diameter as compared to mild steel. In this case

thermal conductivity and the outer cylinder diameter are both the controlling

parameters for the heat transfer mechanism. Aluminum is non-sensitive to the

outer cylinder diameters (O1, O2).

95

In mild steel and stainless steel inner cylinders the temperature spread near the

bottom disc in the temperature range of the bottom disc (353-433K) decreases

with increasing outer cylinder diameter.

The radial thermal distribution at 5, 700 and 1396 mm height with inner cylinder

of aluminum material indicate that the temperature at the axis and inner cylinder

are independent of outer cylinder diameter, while towards the exterior of the

enclosure it is function of outer cylinder diameter due to low thermal conductivity

of mild steel.

The decrease in Nusselt number (Nu) is very steep near the bottom of the

cylinder as compared to the rest of the cylinder height. Similarly, the decrease in

Nusselt number (Nu) is steep in the middle section of the inner cylinder for non-

dimensional diameter of heat source (dhs) of 0.54 as compared to non-

dimensional diameter of heat source (dhs) of 0.46.

The Nusselt number Nu increases with Rayleigh number (Ra) for constant non-

dimensional diameter of heat source (dhs) and increases with increasing non-

dimensional diameter of heat source (dhs) keeping Rayleigh number (Ra)

constant.

At the bottom disc temperature up to 393 K the streamlines within the inner

cylinder are almost same for both configurations (O1, O2) of three inner cylinders

of aluminum, mild steel and stainless steel being independent of outer diameter

used.

With outer cylinder configuration O2 the buoyancy effects are stronger due to

increase in volume as compared to configuration O1 for three inner cylinders

used.

In the outer annulus the buoyancy effects are stronger while using inner cylinder

of aluminum as compared to other inner cylinders due to its high thermal

conductivity.

96

7.2 Future recommendations

In order to enhance the research work further, the following recommendations are

suggested.

Air is used as a fluid within vertical concentric cylindrical enclosure in this

research work. It is commended to study thermal behavior different fluids within

such enclosures.

It is recommended to study thermal behavior different materials for inner

cylinder fluids within the vertical concentric cylindrical enclosures other than

studied in this research work.

It is recommended to study thermal behavior different outer cylinder diameters

of the vertical concentric cylindrical enclosures other than used in this research

work to get more data for further research.

The heat transfer mechanism is studied using three bottom disc temperatures of

353, 393 and 433 K. It is recommended to study such enclosures using higher

bottom disc temperatures.

It is recommended to study thermal behavior of vertical concentric cylindrical

enclosures by rotating inner cylinder while keeping outer cylinder stationary.

It is recommended to explore the contribution of effect of wall thickness such

that the thermal expansions are negligible.

It is recommended to emphasize on the significance of energy efficiency by

using the energy balance.

It is recommended to completely insulate the outer cylinder used in the

experimental apparatus and study the thermal effects.

97

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Appendix-A

A-1: Convection heat transfer coefficient

Table A-1: Convective heat transfer coefficient of air within the concentric cylindrical

enclosure

S. No Input temp (K)

Geometry

Q(W)

he

(W.m-

2K-1) Inner cylinder Outer cylinder

1 353 Aluminum O1 19.0758 25.836

2 353 Mild steel O1 13.3531 18.634

3 353 Stainless steel O1 15.2606 22.357

4 393 Aluminum O1 20.9834 16.025

5 393 Mild steel O1 27.6599 21.167 6 393 Stainless steel O1 23.8447 19.207

7 433 Aluminum O1 15.1721 7.874

8 433 Mild steel O1 34.3364 17.821


10 353 Aluminum O2 17.6935 24.645

11 353 Mild steel O2 21.011 27.892


13 393 Aluminum O2 23.2227 18.033

14 393 Mild steel O2 29.8577 22.384


16 433 Aluminum O2 37.5984 19.793

17 433 Mild steel O2 46.4454 24.346


A-2: Experimental temperature data

Experimental temperature data at various locations of the enclosure was recorded.

The temperature data was measured along the diameter of bottom disc, along the

axis of the enclosure, along the inner and outer surfaces of the inner and outer

108

cylinder walls and the ambient to the enclosure. The steady state temperature data

is tabulated in appendix A.

Table A-2: Bottom disc temperature data with outer cylinder O1

Radial distance, m 0 0.014 0.028 0.042 0.056 0.069

Tc=353K

Aluminum 353 351 350 349 348 347

Mild steel 353 351 349 348 346 343

Stainless

steel 353 348 346 344 341 340

Tc=393K

Aluminum 393 391 388 387 385 384

Mild steel 393 391 390 388 384 381

Stainless

steel 393 391 388 381 376 374

Tc=433K

Aluminum 433 431 430 429 427 425

Mild steel 433 432 430 428 427 425

Stainless

steel 433 427 424 421 413 410

Table A-3: Bottom disc temperature data with outer cylinder O2

Radial distance, m 0 0.014 0.028 0.042 0.056 0.069

Tc=353K

Aluminum 353 351 348 346 344 343

Mild steel 353 352 350 349 348 346

Stainless steel 353 350 348 346 345 344

Tc=393K

Aluminum 393 389 386 385 384 383

Mild steel 393 391 390 389 388 386

Stainless steel 393 391 386 381 377 375

Tc=433K

Aluminum 433 430 428 427 425 422

Mild steel 433 430 428 427 426 424

Stainless steel 433 431 428 420 413 407

109

Table A-4: Experimental temperature data on inner surface of inner cylinder with outer cylinder O1

Axial distance, mm 5 120 236 352 468 584 700 816 932 1048 1164 1280 1396 1480

Tc=353K

Aluminum 305 305 304 304 303 303 303 302 302 301 301 300 300 300

Mild steel 306 305 305 304 304 303 303 302 302 301 301 300 300 300

Stainless steel 307 306 305 304 303 302 302 302 301 301 300 300 300 300

Tc=393K

Aluminum 311 309 308 307 306 305 305 304 303 302 302 302 301 301

Mild steel 313 311 310 308 307 306 305 304 303 303 302 301 301 301

Stainless steel 314 311 308 307 305 305 304 303 302 302 301 301 301 300

Tc=433K

Aluminum 315 313 311 310 308 307 306 305 304 303 303 302 302 301

Mild steel 319 316 314 311 309 308 307 306 305 304 303 302 302 301

Stainless steel 319 318 315 312 309 307 306 305 304 303 303 302 302 301

Table A-5: Experimental temperature data on inner surface of inner cylinder with outer cylinder O2


Tc=353K

Aluminum 305 305 304 304 303 303 303 302 302 301 301 300 300 300

Mild steel 305 305 304 304 303 303 303 302 302 302 301 301 300 300

Stainless steel 306 305 304 303 303 302 302 302 301 301 301 300 300 300

Tc=393K

Aluminum 310 309 308 307 306 305 305 304 304 303 302 302 301 301

Mild steel 312 311 310 307 306 305 304 303 303 302 302 301 301 301

Stainless steel 310 309 308 307 305 305 304 303 302 302 301 301 301 300

Tc=433K

Aluminum 315 314 312 310 308 307 306 305 305 304 303 303 302 301

Mild steel 317 315 314 312 309 307 306 305 304 304 303 302 302 301

Stainless steel 317 316 314 312 309 307 306 305 304 303 303 302 302 301

110

Table A-6: Experimental temperature data on outer surface of inner cylinder with outer cylinder O1


Tc=353K

Aluminum 305 305 304 304 303 303 303 302 302 301 301 300 300 300

Mild steel 306 305 305 304 304 303 303 302 302 301 301 300 300 300

Stainless steel 307 306 305 304 303 302 302 302 301 301 300 300 300 300

Tc=393K

Aluminum 311 309 308 307 306 305 305 304 303 302 302 302 301 301

Mild steel 313 311 310 308 307 306 305 304 303 303 302 301 301 301

Stainless steel 314 311 308 307 305 305 304 303 302 302 301 301 301 300

Tc=433K

Aluminum 315 313 311 310 308 307 306 305 304 303 303 302 302 301

Mild steel 319 316 314 311 309 308 307 306 305 304 303 302 302 301

Stainless steel 319 318 315 312 309 307 306 305 304 303 303 302 302 301

Table A-7: Experimental temperature data on outer surface of inner cylinder with outer cylinder O2


Tc=353K

Aluminum 305 305 304 304 303 303 303 302 302 301 301 300 300 300

Mild steel 305 305 304 304 303 303 303 302 302 302 301 301 300 300

Stainless steel 306 305 304 303 303 302 302 302 301 301 301 300 300 300

Tc=393K

Aluminum 310 309 308 307 306 305 305 304 304 303 302 302 301 301

Mild steel 312 311 310 307 306 305 304 303 303 302 302 301 301 301

Stainless steel 310 309 308 307 305 305 304 303 302 302 301 301 301 300

Tc=433K

Aluminum 315 314 312 310 308 307 306 305 305 304 303 303 302 301

Mild steel 317 315 314 312 309 307 306 305 304 304 303 302 302 301

Stainless steel 317 316 314 312 309 307 306 305 304 303 303 302 302 301

111

Table A-8: Experimental temperature data on inner surface of outer cylinder with outer cylinder O1


Tc=353K

Aluminum 302 301 301 301 301 300 300 300 300 300 300 300 300 300

Mild steel 302 301 301 300 300 300 300 300 300 300 299 299 299 299

Stainless steel 302 301 301 301 301 300 300 300 300 300 299 299 299 299

Tc=393K

Aluminum 302 302 301 301 301 301 300 300 300 300 300 300 300 300

Mild steel 303 303 302 302 302 301 301 301 300 300 300 300 300 300

Stainless steel 302 302 301 301 301 301 301 301 301 300 300 300 300 300

Tc=433K

Aluminum 303 303 302 302 302 302 302 301 301 301 301 301 301 301

Mild steel 305 304 303 302 302 302 301 301 301 301 300 300 300 300

Stainless steel 304 304 303 303 302 302 301 301 301 300 300 300 300 300

Table A-9: Experimental temperature data on inner surface of outer cylinder with outer cylinder O2


Tc=353K

Aluminum 302 302 301 301 300 300 300 300 300 300 299 299 299 299

Mild steel 302 302 301 301 301 300 300 300 300 300 300 300 299 299

Stainless steel 301 300 300 300 300 299 299 299 299 299 299 299 299 299

Tc=393K

Aluminum 302 302 301 301 301 300 300 300 300 300 300 300 300 300

Mild steel 303 302 301 301 301 301 301 301 301 301 300 300 300 300

Stainless steel 303 303 302 302 301 301 301 301 301 301 301 300 300 300

Tc=433K

Aluminum 303 303 303 302 302 302 302 301 301 301 300 300 300 300

Mild steel 304 304 303 303 303 303 302 302 302 302 301 301 300 300

Stainless steel 303 303 303 302 302 302 301 301 301 301 300 300 300 300

112

Table A-10: Experimental temperature data on outer surface of outer cylinder with outer cylinder O1


Tc=353K

Aluminum 302 301 301 301 301 300 300 300 300 300 300 300 300 300

Mild steel 302 301 301 300 300 300 300 300 300 300 299 299 299 299

Stainless steel 302 301 301 301 301 300 300 300 300 300 299 299 299 299

Tc=393K

Aluminum 302 302 301 301 301 301 300 300 300 300 300 300 300 300

Mild steel 303 303 302 302 302 301 301 301 300 300 300 300 300 300

Stainless steel 302 302 301 301 301 301 301 301 301 300 300 300 300 300

Tc=433K

Aluminum 303 303 302 302 302 302 302 301 301 301 301 301 301 301

Mild steel 305 304 303 302 302 302 301 301 301 301 300 300 300 300

Stainless steel 304 304 303 303 302 302 301 301 301 300 300 300 300 300

Table A-11: Experimental temperature data on outer surface of outer cylinder with outer cylinder O2


Tc=353K

Aluminum 302 302 301 301 300 300 300 300 300 300 299 299 299 299

Mild steel 302 302 301 301 301 300 300 300 300 300 300 300 299 299

Stainless steel 301 300 300 300 300 299 299 299 299 299 299 299 299 299

Tc=393K

Aluminum 302 302 301 301 301 300 300 300 300 300 300 300 300 300

Mild steel 303 302 301 301 301 301 301 301 301 301 300 300 300 300

Stainless steel 303 303 302 302 301 301 301 301 301 301 301 300 300 300

Tc=433K

Aluminum 303 303 303 302 302 302 302 301 301 301 300 300 300 300

Mild steel 304 304 303 303 303 303 302 302 302 302 301 301 300 300

Stainless steel 303 303 303 302 302 302 301 301 301 301 300 300 300 300

113

Table A-12: Experimental temperature data on axis with outer cylinder O1


Tc=353K

Aluminum 310 307 306 304 304 303 303 302 302 302 302 301 301 301

Mild steel 311 308 306 305 304 304 303 303 302 301 301 301 301 301

Stainless steel 309 307 306 304 303 302 301 301 301 301 300 300 300 300

Tc=393K

Aluminum 321 315 312 309 308 306 305 304 303 302 302 301 301 301

Mild steel 322 317 314 313 310 309 307 306 305 304 303 302 302 302

Stainless steel 320 315 312 309 306 306 304 303 303 302 301 301 300 300

Tc=433K

Aluminum 335 326 320 316 313 311 309 307 306 305 304 304 303 302

Mild steel 337 329 322 318 314 312 309 308 306 305 304 303 303 302

Stainless steel 334 325 320 316 311 309 308 306 305 304 303 302 301 301

Table A-13: Experimental temperature data on axis with outer cylinder O2


Tc=353K

Aluminum 309 307 305 304 303 302 302 302 301 301 301 301 300 300

Mild steel 310 307 306 306 305 303 303 302 302 302 301 301 300 300

Stainless steel 309 306 305 304 304 303 302 302 301 301 301 300 300 300

Tc=393K

Aluminum 319 315 311 309 307 306 305 304 303 303 302 302 301 301

Mild steel 322 315 312 309 308 306 305 304 303 302 301 301 301 301

Stainless steel 318 313 311 309 307 305 304 303 303 302 301 301 301 300

Tc=433K

Aluminum 334 326 319 316 313 310 308 307 306 305 304 303 303 302

Mild steel 336 326 319 316 313 310 308 306 305 304 303 302 301 301

Stainless steel 334 326 319 315 313 311 308 306 305 304 303 302 302 301

114

Appendix-B

Figure B-1: Thermal lines of enclosure for configuration O1, a) aluminum inner cylinder, c). mild steel inner cylinder, e) stainless

steel inner cylinder, for configuration O2, b) aluminum inner cylinder, d). mild steel inner cylinder, f), stainless steel inner cylinder

at 353 K.

115

Figure B-2: Thermal lines of enclosure for configuration O1 a) aluminum inner cylinder, c). mild steel inner cylinder, e) stainless

steel inner cylinder, for configuration O2 b) aluminum inner cylinder, d). mild steel inner cylinder, f), stainless steel inner cylinder at

393 K.

116

Figure B-3: Thermal lines of enclosure for configuration O1 a) aluminum inner cylinder, c). mild steel inner cylinder, e) stainless


433 K.

117

Figure B-4: Velocity vectors of enclosure for configuration O1 a) aluminum inner cylinder, c). mild steel inner cylinder, e) stainless


353 K.

118



393 K.

119



433 K

120

121

122

123

124

125

126

127

128

129

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