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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
Asif Hussain Malik, M.S.I. Alvi, Shahab Khushnood, F.M. Mahfouz, M.K.K. Ghauri,
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
Asif Hussain Malik, M.S.I. Alvi, Shahab Khushnood, F.M. Mahfouz, M.K.K. Ghauri,
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
6.5.3 Contours of streamlines at 393 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. .......................................................................... 91
6.5.4 Contours of streamlines at 433 K ................................................................................ 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
Figure 6-12: Streamlines of enclosure for configuration O1 a) aluminum inner
cylinder, c). mild steel inner cylinder, e) stainless steel inner cylinder, for
xiv
configuration O2 b) aluminum inner cylinder, mild steel inner cylinder, f), stainless
steel inner cylinder at 393 K. .................................................................... 90
Figure 6-13: 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 433 K. .................................................................... 92
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. ..................................................... 114
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. ..................................................... 115
Figure B-3: 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 433 K. ..................................................... 116
Figure B-4: Velocity vectors 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. ..................................................... 117
Figure B-5: Velocity vectors 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. ..................................................... 118
xv
Figure B-6: Velocity vectors 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 433 K. ..................................................... 119
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
𝑑𝑠 = 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].
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.
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;
𝑣𝑟 = 𝑣𝑧 = 0 (No slip condition);
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;
𝑣𝑟 = 𝑣𝑧 = 0 (No slip condition);
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
The comparison of experimental results with 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) 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
The comparison of experimental results with 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) 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.
6.5.3 Contours of streamlines at 393 K
The contours of streamlines are obtained from CFD simulations for the enclosure
configurations O1 and O2 at bottom disc central temperature of 393 K and are shown
in Figure 6.12. Figure 6.12 (a-b) shows streamlines for two configurations (O1, O2)
of aluminum inner cylinder. Similarly Figures 6.12 (c-d) and (e-f) show the
corresponding results for mild steel and stainless steel inner cylinders. It is observed
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.
The above mentioned observations regarding Figure 6.12 can be explained as given
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
Figure 6-12: 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
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.
6.5.4 Contours of streamlines at 433 K
The contours of streamlines are obtained from CFD simulations for the enclosure
configurations O1 and O2 at bottom disc central temperature of 433 K and are shown
in Figure 6.13. Figure 6.13 (a-b) shows streamlines for two configurations (O1, O2)
of aluminum inner cylinder. Similarly Figures 6.13 (c-d) and (e-f) show the
corresponding results for mild steel and stainless steel inner cylinders. It is observed
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.
The above mentioned observations regarding Figure 6.13 can be explained as given
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
Figure 6-13: 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
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
References
1. Griffiths, J.G., The Orders of Gods in Greece and Egypt (According to Herodotus.
The Journal of Hellenic Studies, 1955. 75(DOI:10.2307/629164. JSTOR 629164):
p. 21-23.
2. Baeyer, H.C.v.N.Y., Warmth Disperses and Time Passes - The History of Heat
1998( The Modern Library ISBN 0-375-75372-9).
3. Mahon, B., The Man Who Changed Everything - the Life of James Clerk Maxwell.
Hoboken, NJ: Wiley. 2003( ISBN 0-470-86171-1).
4. Liaqat, A.a.B., A.C., Conjugate Natural Convection in a Square Enclosure
Containing Volumetric Sources International Journal of Heat and Mass Transfer,
September 2001. 44(17): p. 3273-3280.
5. Kee, R.J., C.S. Landram, and J.C. Miles, Natural Convection of a Heat-
Generating Fluid Within Closed Vertical Cylinders and Spheres. Journal of Heat
Transfer, 1976. 98(1): p. 55-61.
6. Blair, N.J., Beckman, W. A. and J. W. Mitchell, J. W., Experimental Transient
Natural Convection in Enclosures. The 1994 American Solar Energy Society
Annual Conference, June 25-30, 1994: p. 280-285.
7. Kuznetsov, G.V.a.S., Mikhail A., Conjugate Natural Convection in an Enclosure
with a Heat Source of Constant Heat Transfer Rate International Journal of Heat
and Mass Transfer, January 2011. 54(1-3): p. 260-268.
8. Arnas, O.A.a.E., M.A., , Convective Heat Transfer in a Circular Annulus with
Various Wall Heat Flux Distributions and Heat Generation. Journal of Heat
Transfer, Transactions of the ASME, May 1985. 107: p. 334-337.
9. Khalilollahi, S., Unsteady Natural Convection Generated By A Heated Surface
Within An Enclosure. Numerical Heat Transfer, 1986. 9: p. 715-730.
10. Teertstra, P.M., Yovanovich, M. Michael, and Culham, J. Richard, Natural
Convection Measurements for a Concentric Spherical Enclosure. Journal of Heat
Transfer, Transactions of the ASME, June 2006. 128: p. 580-587.
11. Malik, A.H., M. S. I. Alvi, et al. , Experimental Study of Conjugate Heat Transfer
within a Bottom Heated Vertical Concentric Cylindrical Enclosure International
Journal of Heat and Mass Transfer, 2012. 55(4): p. 1154-1163.
98
12. Rahman, M.M., Alim, M. A., Saha, Sumon and Chowdhury, M. K., Mixed
Convection in a Vented Square Cavity with a Heat Conducting Horizontal Solid
Circular Cylinder. Journal of Naval Architecture and Marine Engineering,
December 2008. 2: p. 37-46.
13. Ball, K.S., Farouk, B. and Dixit, V.C., An Experimental Study of Heat Transfer in
a Vertical Annulus with a Rotating Inner Cylinder. International Journal of Heat
and Mass Transfer, 1989. 32(8): p. 1517-1527.
14. Lipkea, W.H. and G.S. Springer, Heat transfer through gases contained between
two vertical cylinders at different temperatures. International Journal of Heat and
Mass Transfer, 1968. 11(9): p. 1341-1350.
15. Glakpe, E.K., Watkins, C.B. and Kurien, B.J., Effect of Radiation and Specified
Heat Flux on Natural Convection in a Vertical Region with a Rectangular Inner
Boundary, . AIAA and ASME, Joint Thermodynamics and Heat Transfer
Conferences, 4th Boston, M.A. , June 2-4, 1986: p. 10 pages.
16. Sankar, M.a.V., M., Numerical Investigation of Combined Buoyancy and Surface
Tension Driven Convection in an Axisymmetric Cylindrical Annulus Nonlinear
Analysis: Modeling and Control, 2007. 12(4): p. 541-552.
17. Keyhani, M., F.A. Kulacki, and R.N. Christensen, Free Convection in a Vertical
Annulus With Constant Heat Flux on the Inner Wall. Journal of Heat Transfer,
1983. 105(3): p. 454-459.
18. Sarr, J., Moow, C., Chehouani, H., Zeghmati, B., Benet, S., and Daguenet, M.,
Study of Natural Convection in an Enclosure Bounded by Two Concentric
Cylinders and Two Diametric Planes Journal of Heat Transfer, Transactions of
the ASME, February 1995. 117: p. 130-137.
19. Buell, J.C.a.C., I., The Effect of Wall Conduction on the Stability of a Fluid in a
Right Circular Cylinder Heated from Below. Journal of Heat Transfer,
Transactions of the ASME, May 1983. 105: p. 255-260.
20. Vargas, M., Sierra, F. Z., Ramos, E., and Avramenko, A. A., Steady Natural
Convection in a Cylindrical Cavity, . International Communication of Heat and
Mass Transfer, 2002. 29(2): p. 213-221.
21. Zhao, F.-Y., Liu, Di and Tang, Guang-Fa, Natural Convection in an Enclosure
with Localized Heating and Salting from Below International Journal of Heat
and Mass Transfer, 2008. 51: p. 2889-2904.
99
22. Dagtekin, I.a.O., H.F., Natural Convection Heat Transfer by Heated Partitions
within Enclosure. International Communication Heat and Mass Transfer, 2001.
28(6): p. 823-834.
23. Sezai, I.a.M.A.A., Natural Convection Heat Transfer from a Discrete Heat Source
on the Bottom of a Horizontal Enclosure International Journal of Heat and Mass
Transfer, 2000. 43: p. 2257-2266.
24. Kuznetsov, C.V.a.S., M. A., Conjugate Heat Transfer in an Enclosure under the
Condition of Internal Mass Transfer and in the Presence of the Local Heat Source,
, . International Journal of Heat and Mass Transfer, 2009. 52: p. 1-8.
25. Maki, S., Tagawa, Toshio, and Ozoe, Hiroyuki, Average Heat Transfer Rates
Measured and Numerically Analyzed for Combined Convection of Air in an
Inclined Cylindrical Enclosure due to Both Magnetic and Gravitational Fields
Experimental Thermal and Fluid Science, 2003. 27: p. 891-899.
26. Matthias, A.D., and Peralta Hernandez, A. R., Modeling Temperatures in Soil
under an Opaque Cylindrical Enclosure. Agricultural and Forest Meteorology,
1998. 90: p. 27-38.
27. Akamatsu, M., Higano, Mitsuo, Takahashi, Yoshio and Ozoe, Hiroyuki,
Numerical Computation of Magnetothermal Convection of Water in a Vertical
Cylindrical Enclosure International Journal of Heat and Fluid Flow, 2005. 26: p.
622-634.
28. Walid, A.a.A., Omri, Buoyancy Induced Heat Transfer and Fluid Flow Inside a
Prismatic Cavity International Centre for Heat and Mass Transfer (ICHMT),
2009. DOI: 10.1615/ICHMT, 2009, conv. 1200, : p. 15 pages.
29. Covaro, F.a.P., Massimo The Natural Convective Heat Transfer in a Partially
Divided Enclosure: A study on the Influence of the Source Position. Journal of
Thermodynamics, 2009. 2009(DOI: 10.1155/2009/792370): p. 10 pages.
30. Paroncini, M., Corvaro, Francesco, Maddalena de Padova, Maria, Study and
Analysis of the Influence of a Small Heating Source Position on the Natural
Convective Heat Transfer in a Square Cavity Proceedings of the 4th International
Conference on Heat Transfer, Thermal Engineering and Environment, Greece,
August 21-23, 2006: p. 305-310.
31. Aydin, O.a.Y., Wen-Jei, Natural Convection in the Enclosures with Localized
Heating from Below and Symmetrical Cooling from Sides International Journal of
Numerical Methods for Heat and Fluid Flow 2000. 10(5): p. 518-529.
100
32. Brito, R.F., Menon, Genesio Jose and Pirani, Marcelo Jose, Turbulent Natural
Convection in Enclosures Using Large-Eddy Simulation with Localized Heating
from Horizontal Bottom Surface and Cooling from Vertical Surfaces, . Journal of
Brazilian Society of Mechanics, Science and Engineering, July-September 2009.
XXXI(3): p. 199-209.
33. Saha, G., Saha, Sumon, Islam, M. Quamrul and Razzaq Akhanda, M. A.,
Natural Convection in Enclosure with Discrete Isothermal Heating from Below
Journal of Naval Architecture and Marine Engineering, June 2007. 4: p. 1-13.
34. Al-Bahi, A., Al-Hazmy, M. and Zaki, G. M., Natural Convection in a Tilted
Rectangular Enclosure With a Single Discrete Heater Journal of King Abdul Aziz
University, 2005. 16(2): p. 117-136.
35. Aswatha, G., C. J. Gangadhara, Sridhara, S. N. and Seetharamu, K. N., Effect of
Different Thermal Boundary Conditions at Bottom Wall on Natural Convection in
Cavities Journal of Engineering Science and Technology, 2011. 6(1): p. 109-130.
36. Amara, T., Slimi, Khalifa, and Ben Nasrullah, Sassi, Free Convection in a
Vertical Cylindrical Enclosure. International Journal of Thermal Sciences, 2000.
39: p. 616-634.
37. Lai, F.C., Mathew, J., and Zhang, J. M., Effects of Buoyancy on EHD-Enhanced
Forced Convection in a Horizontal Channel American Institute of Aeronautics
and Astronautics, Inc., 1997: p. 11 pages.
38. Lemembre, A.a.P., J. P., Laminar Natural Convection in a Laterally Heated and
Upper Cooled Vertical Cylindrical Enclosure. International Journal of Heat and
Mass Transfer, 1998. 41(16): p. 2437-2454.
39. Sharma, A.K., Velusamy, K. and Balaji, C., Conjugate Transient Natural
Convection in a Cylindrical Enclosure with Internal Volumetric Heat Generation
Annals of Nuclear Energy, 2008. 35: p. 1502-1514.
40. Bairi, A., Transient Natural 2D Convection in a Cylindrical Cavity with the Upper
Face Cooled by Thermoelectric Peltier Effect Following an Exponential Law
Applied Thermal Engineering, March 2003. 23(4): p. 431-447.
41. Hamady, F.J., Lloyd, J. R., Yang, K. T., Yang, H. Q., A Study of Natural
Convection in a Rotating Enclosure Journal of Heat Transfer, Transactions of
the ASME, May 1986. 108: p. 136-143.
101
42. Ho, C.J., Chang, W. S. and Wang, C. C., Natural Convection between Two
Horizontal Cylinders in an Adiabatic Circular Enclosure Journal of Heat
Transfer, Transactions of the ASME, February 1993. 115: p. 158-165.
43. Sparrow, E.M., Effect of Rotation and Coolant throughout on the Heat Transfer
and Temperature Field in an Enclosure Journal of Heat Transfer, Transactions of
the ASME, August 1976: p. 387-394.
44. Smith, T.F., Shen, Z. F., and Alturki, A. M., Radiative and Convective Transfer in
a Cylindrical Enclosure for a Real Gas Journal of Heat Transfer, Transactions
of the ASME, May 1985. 107: p. 482-485.
45. Ben Salah, M., Askri, F., Slimi, K. and Ben Nasrullah, S., Numerical Resolution
of the Radiative Transfer Equation in a Cylindrical Enclosure with the Finite-
Volume Method, . International Journal of Heat and Mass Transfer, 2004. 47: p.
2501-2509.
46. Oliveski, R.C.a.H., F., Fully Non-Dimensional Nusselt Number Correlation for
Transient Natural Convection in Tanks Physics of Fluid Dynamics, 18 October
2010: p. 16 pages.
47. Guellal, M., and Abdesselam, Hamlaoui, Alternating Direction Implicit Method
for Free Convection Simulation in a Cylindrical Enclosure Contemporary
Engineering Sciences, 2008. 1(2): p. 51-62.
48. Keyhani, M., Kulacki, F. A. and Christensen, R. N., Experimental Investigation of
Free Convection in a Vertical Rod Bundle ------- A General Correlation for Nusselt
Numbers Journal of Heat Transfer, Transactions of the ASME, August 1985.
107: p. 611-623.
49. Niu, F., Zhao, Haihua, Peterson, Per F., Woodcock, Joel, and Henry, Robert E. ,
Investigation of Mixed Convection in a Large Rectangular Enclosure Nuclear
Engineering and Design, 2007. 237: p. 1025-1032.
50. Prud`homme M., a.B.H., Linear Stability of Free Convection in a Vertical Cavity
Heated by Uniform Heat Fluxes International Communication of Heat and Mass
Transfer, 2001. 28(6): p. 743-750.
51. Trevisan, O.V.a.B., A., Combined Heat and Mass Transfer by Natural Convection
in a Vertical Enclosure Journal of Heat Transfer, Transactions of the ASME,
February 1987. 109: p. 104-112.
102
52. Das, S.P., Chakraborty, S., and Dutta, P., Natural Convection in a Two-
Dimensional Enclosure Heated Symmetrically from Both Sides, . International
Communication of Heat and Mass Transfer, 2002. 29(3): p. 345-354.
53. Joly, F., Vasseur, P., and Labrosse, G., Soret-Driven Thermosolutal Convection in
a Vertical Enclosure International Communication of Heat and Mass Transfer,
2000. 27(6): p. 755-764.
54. Sigey, J.K., Gatheri, F. K. and Kinyanjui, M., Numerical Study of Free
Convection Turbulent Heat Transfer in an Enclosure, , . Energy Conversion and
Management, 2004. 45(15-16): p. 2571-2582.
55. Chen, S.A.H., J. R., Humphrey, J. A. C. , Steady, Two-Dimensional, Natural
Convection in Rectangular Enclosures With Differently Heated Walls Journal of
Heat Transfer, Transactions of the ASME, May 1987. 109: p. 400-406.
56. Yu, E.a.J., Y. K., Heat Transfer in Discretely Heated Side-Vented Compact
Enclosures by Combined Conduction, Natural Convection and Radiation. Journal
of Heat Transfer, Transactions of the ASME, November 1999. 121: p. 1002-1010.
57. Adams, V.H., Joshi, Y. and Blackburn, D. L., Three-Dimensional Study of
Combined Conduction, Radiation and Natural Convection from Discrete Heat
Sources in a Horizontal Narrow-Aspect-Ratio Enclosure Journal of Heat Transfer,
Transactions of the ASME, November 1999. 121: p. 992-1000.
58. Khalilollahi, A.a.S., B., Unsteady Natural Convection Generated by a Heated
Surface Within an Enclosure Numerical Heat Transfer, 1986. 9: p. 715-730.
59. Bouali, H., Mezrhab, A., Amouli, H. and Bouzidi, M., Radiation – Natural
Convection Heat Transfer in an Inclined Rectangular Enclosure International
Journal of Thermal Sciences, 2006. 45: p. 553-566.
60. Ganguli, A.A., Pandit, A. B., and Joshi, J. B., Simulation of Heat Transfer in a
Two-dimensional Vertical Enclosure. Chemical Engineering Research and
Design, 2009. 87: p. 711-727.
61. Guimaraes, P.M., and Da Silva, C. E. S., , A parametric Study of Forced
Convection in an Enclosure with Stationary Heated Cylinders, . International
Communication of Heat and Mass Transfer, 2010. 37: p. 469-475.
62. Khanafer, K., and Vafai, K., Buoyancy-driven Flow and Heat Transfer in Open-
ended Enclosure: Elimination of the Extended Boundaries. International Journal
of Heat and Mass Transfer, 2000. 43: p. 4087-4100.
103
63. Wilkes, J.O.a.C., S. W. , The Finite-Difference Computation of Natural
Convection in a Rectangular Enclosure. American Institute of Chemical
Engineers Journal, January 1966. 12(1): p. 161-166.
64. Oosthuizen, P.H., Kalendar, A. and Simko, T. M., , Three-Dimensional Natural
Convective Flow in a Rectangular Enclosure with a Rectangular Heated Section on
One Vertical Wall and a Cooled Horizontal Upper Wall, . 5th European Thermal
Sciences Conference, The Netherlands, 2008, 2008: p. 8 pages.
65. Dagtekin, I.a.O., H. F., Natural Convection Heat Transfer by Heated Partitions
within Enclosure International Communication Heat and Mass Transfer, 2001.
28( 6): p. 823-834.
66. Xia, Q., Yang, K. T., Mukutmoni, D., Effect of Imposed Wall Temperature
Oscillations ori the Stability of Natural Convection in a Square Enclosure Journal
of Heat Transfer, Transactions of the ASME, February 1995. 117: p. 113-120.
67. Kim, D.M.a.V., R., Effect of Wall Heat Conduction on Natural Convection Heat
Transfer in a Square Enclosure. Journal of Heat Transfer, Transactions of the
ASME, February 1985. 107: p. 139-146.
68. Kwak, H.S., Kuwahara, Kunio and Hyun, Jae Min, Resonant Enhancement of
Natural Convection Heat Transfer in a Square Enclosure International Journal of
Heat and Mass Transfer, 1998. 41: p. 2837-2846.
69. Aktas, M.K., Farouk, Bakhtier, Numerical Simulation of Developing Natural
Convection in an Enclosure due to Rapid Heating International Journal of Heat
and Mass Transfer, 2003. 46: p. 2253-2261.
70. Refai Ahmed, G., and Yovanovich, M. M., Numerical Study of Natural
Convection from Discrete Heat Sources in a Vertical Square Enclosure Journal of
Thermodynamics, January-March 1992. 6(1): p. 121-127.
71. Aminossadati, S.M.a.G., B., The Effects of Orientation of an Inclined Enclosure
on Laminar Natural Convection Heat and Technology, 2005. 23(2): p. 43-49.
72. Habibzadeh, A., Habibollah, Sayehvand and Mekanik, Abolghasem, Numerical
Study of Natural Convection in a Partitioned Square Cavity Filled with Nanofluid
International Journal of Chemical Engineering and Applications, August 2011.
2(4): p. 261-267.
73. Warrington, R.O.J.a.C.G., Jr., Natural Convection Heat Transfer Between
Cylindrical Tube Bundles and a Cubical Enclosure. Journal of Heat Transfer,
Transactions of the ASME, February 1981. 103: p. 103-107.
104
74. Tagawa, T.a.O., H., Volume, Enhancement of Heat Transfer Rate by Application
of a Static Magnetic Field During Natural Convection of Metal in a Cube Journal
of Heat Transfer, Transactions of the ASME, May 1997. 119: p. 265-271.
75. Newport, D.T., Dalton, T. M., Davies, M. R. D., Whelan, M., Forno, C., On the
Thermal Interaction Between an Isothermal Cylinder and Its Isothermal Enclosure
for Cylinder Rayleigh Numbers of Order 104 Journal of Heat Transfer,
Transactions of the ASME, December 2001. 123: p. 1052-1061.
76. Bohn M. S. and Anderson, R., Temperature and Heat Flux Distribution in a
Natural Convection Enclosure Flow Journal of Heat Transfer, Transactions of
the ASME, May 1986. 108: p. 471-476.
77. Kee, R.J., Landram, C. S. and Miles, J. C., Natural Convection of a Heat
Generating Fluid within Closed Vertical Cylinders and Spheres Journal of Heat
Transfer, Transactions of the ASME, February 1976. 98(1): p. 55-61.
78. Natarajan, E., Basak, Tanmay, and Roy, S., Heatline Visualization of Natural
Convection Flows within Trapezoidal Enclosures Proceedings of 5th
IASME/WSEAS International Conference on Fluid Mechanics and
Aerodynamics, Greece, August 25-27, 2007: p. 59-64.
79. Fluent, Fluent 6.3 User's Guide. Fluent Inc., 2006.
80. Rolf H. Sabersky, A.J.A., Edward G. Hauptmann, Fluid Flow - A First Course in
Fluid Mechanics. Macmillan Publishing Co., Inc., New York, Collier Macmillan
Publishers, London, 1971. Second Edition.
81. White, F.M., Fluid Mechanics. 1986. Second Edition.
82. Robert L. Daugherty, J.B.F.a.E.J.F., Fluid Mechanics with Engineering
Applications. McGraw Hill Book Company, Singapore, 1996. S. I. Metric Edition.
83. Papanicolaou, E.a.B., V., Transient Natural Convection in a Cylindrical Enclosure
at High Rayleigh Numbers International Journal of Heat and Mass Transfer,
March 2002. 45(7): p. 1425-1444.
84. Lin, W.a.A., S. W., Natural Convection Cooling of Rectangular and Cylindrical
Containers, . International Journal of Heat and Fluid Flow 2001. 22: p. 72-81.
85. Wrobel, W., Fornalik-Wajs, E., and Szmyd, J. S., Experimental and Numerical
Analysis of Thermo-magnetic Convection in a Vertical Annular Enclosure, .
International Journal of Heat and Fluid Flow, 2010. DOI:
10.1016/j.ijheatfluidflow.2010.05.012, . p. 13 pages.
105
86. Chin, Y., et al., Convective Heat Transfer in Vertical Asymmetrically Heated
Narrow Channels. Journal of Heat Transfer, 2002. 124(6): p. 1019-1025.
87. Yu, E.a.J., Y.K., Heat Transfer in Discretely Heated Side-Vented Compact
Enclosures by Combined Conduction, Natural Convection and Radiation Journal
of Heat Transfer, Transactions of the ASME, November 1999. 121: p. 1002-1010.
88. Cengel, Y.A., Heat Transfer a Practical Approach. McGraw Hill Companies, Inc,
1221 Avenue of the Americas, New York, 2003. second edition.
89. Sukhatme, S.P., A Textbook of Heat Transfer Universities press (India) private
limited, Hyderguda, Hyderabad, India, 2005. 4th edition.
90. Moffat, R.J., Contributions to the Theory of Single-Sample Uncertainty Analysis.
Journal of Fluids Engineering, 1982. 104(2): p. 250-258.
91. G. Lydon , H.S., Momentum and Heat Transport Inside and Around a Cylindrical
Cavity in Cross Flow. 2nd Southeastern Europe Fluent Users Group Meeting,
November 1-2, 2001: p. 7 pages.
92. Wrobel, W., Fornalik-Wajs, E., and Szmyd, J. S., Experimental and Numerical
Analysis of Thermo-magnetic Convection in a Vertical Annular Enclosure
International Journal of Heat and Fluid Flow, 2010: p. 13 pages.
93. Mazumder, S., On the Use of the Fully Compressible Navier Stokes Equations for
the Steady-State Solution of Natural Convection Problems in Closed Cavities
Journal of Heat Transfer, Transactions of the ASME, March 2007. 129: p. 387-
390.
94. Nazrul Islam, G., U.N., and Sharma, G.K., Mixed Convection Heat Transfer in the
Entrance Region of Horizontal Annuli International Journal of Heat and Mass
Transfer, June 2001. 44(11): p. 2107-2120.
95. Chang, T.S.a.T., Y.L., Natural Convection Heat Transfer in an Enclosure with a
Heated Background Step, . International Journal of Heat and Mass Transfer,
2001. 44: p. 3963-3971.
96. Massimo Paroncini, F.C., Maria Maddalena de Padova, Study and Analysis of the
Influence of a Small Heating Source Position on the Natural Convective Heat
Transfer in a Square Cavity. Proceedings of the 4th International Conference on
Heat Transfer, Thermal Engineering and Environment, August 21-23, 2006: p.
305-310.
106
97. Yang, O.A.a.W.-J., Natural Convection in the Enclosures with Localized Heating
from Below and Symmetrical Cooling from Sides. International Journal of
Numerical Methods for Heat and Fluid Flow, 2000. 10(5): p. 518-529.
98. M. A. I. El-Shaarawi, a.A.S., Free Convection Effects on the Developing Laminar
Flow in Vertical Concentric Annuli. Journal of Heat Transfer, November 1980.
102: p. 617-622.
99. Adachi, T. and S. Imai, Three-dimensional linear stability of natural convection in
horizontal concentric annuli. International Journal of Heat and Mass Transfer,
2007. 50(7-8): p. 1388-1396.
100. D. M. Kim, a.R.V., Effect of Wall Heat Conduction on Natural Convection Heat
Transfer in a Square Enclosure. Journal of Heat Transfer, February 1985. 107: p.
139-146.
101. M. Amoura, N.Z., A. Smati and , M. Gareche, Finite Element Study of Mixed
Convection for Non-Newtonian Fluid Between Two Coaxial Rotating Cylinders.
International Communications in Heat and Mass Transfer, 2006. 33: p. 780-789.
102. M. Sankar, a.M.V., Numerical Investigation of Combined Buoyancy and Surface
Tension Driven Convection in an Axi-symmetric Cylindrical Annulus. Nonlinear
Analysis: Modeling and Control, 2007. 12(4): p. 541-552.
107
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
9 433 Stainless steel O1 33.3826 18.557
10 353 Aluminum O2 17.6935 24.645
11 353 Mild steel O2 21.011 27.892
12 353 Stainless steel O2 6.6351 9.207
13 393 Aluminum O2 23.2227 18.033
14 393 Mild steel O2 29.8577 22.384
15 393 Stainless steel O2 35.3869 28.505
16 433 Aluminum O2 37.5984 19.793
17 433 Mild steel O2 46.4454 24.346
18 433 Stainless steel O2 35.387 19.553
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
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 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
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-7: Experimental temperature data on outer surface of inner cylinder with outer cylinder O2
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 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
Axial distance, mm 5 120 236 352 468 584 700 816 932 1048 1164 1280 1396 1480
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
Axial distance, mm 5 120 236 352 468 584 700 816 932 1048 1164 1280 1396 1480
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
Axial distance, mm 5 120 236 352 468 584 700 816 932 1048 1164 1280 1396 1480
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
Axial distance, mm 5 120 236 352 468 584 700 816 932 1048 1164 1280 1396 1480
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
Axial distance, mm 5 120 236 352 468 584 700 816 932 1048 1164 1280 1396 1480
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
Axial distance, mm 5 120 236 352 468 584 700 816 932 1048 1164 1280 1396 1480
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
steel inner cylinder, for configuration O2 b) aluminum inner cylinder, d). mild steel inner cylinder, f), stainless steel inner cylinder at
433 K.
117
Figure B-4: Velocity vectors 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.
118
Figure B-5: Velocity vectors 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.
119
Figure B-6: Velocity vectors 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
433 K
120
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