thermal response on nanofluid flow in the presence...
TRANSCRIPT
THERMAL RESPONSE ON NANOFLUID FLOW IN
THE PRESENCE OF MAGNETIC FIELD WITH
VARIABLE STREAM CONDITIONS
VIBHU VIGNESH BALACHANDAR
UNIVERSITI TUN HUSSEIN ONN MALAYSIA
THERMAL RESPONSE ON NANOFLUID FLOW IN THE
PRESENCE OF MAGNETIC FIELD WITH VARIABLE STREAM
CONDITIONS
VIBHU VIGNESH BALACHANDAR
A thesis submitted in
fulfilment of the requirement for the award of the
Degree of Master in Mechanical Engineering
Faculty of Mechanical and Manufacturing Engineering
Universiti Tun Hussein Onn Malaysia
DECEMBER 2015
iii
I would like to dedicate this thesis to
ALMIGHTY “GOD”
(Who gave me strength, knowledge, patience and wisdom)
MY “PARENT”
(Their pure love, devotion, cares and prayers had helped me to achieve this target)
MY “SIBLINGS and FRIENDS”
(Their care, encouragement and motivation made me complete this valuable work)
iv
ACKNOWLEDGEMENT
I am grateful to Almighty GOD who is the most congenital, most sympathetic and
sustainer for the worlds for giving me the potency and the ability to do this research
work.
I would like to express my sincere thanks and cordial appreciation to my
academic supervisor Professor Dr. Sulaiman Bin Haji Hasan, his efforts and sincerity
enable my abilities to achieve this target. His support at every stage of study with
patience and unlimited guidance results the completion of this study within time. I
express my gratitude to my co-supervisor Professor Dr. R. Kandasamy who
supported for the research work, without his support it was impossible to complete
this study.
Sincere thanks to Universiti Tun Hussein Onn Malaysia who provide me
platform where I did this research work for my higher studies. It was impossible
without the financial support to complete this research. The university supported the
research by allocating Fundamental research grant scheme (FRGS) under Vot No.
1208 and Geran Insentif Penyelidik Siswazah (GIPS).
I am also thankful to my parents, siblings, friends and especially Engr. Qadir
Bakhsh Jamali, Engr. Shalini Sanmargaraja and Engr. Ashwin Kumar Erode Natrajan
for their moral support and motivation at every step of this research.
v
ABSTRACT
Nanofluids are electrically conducting fluids which are the suspension of metallic,
non-metallic or polymeric nano-sized material in base liquid such as water, oil or air
are employed to perform tasks such as heat transfer and thermal conductivity. To
overcome limitations in heat transfer, an innovative new class of heat transfer fluid is
engineered as nanofluids by suspending metallic nanoparticles in conventional heat
transfer fluids, which expected to exhibit high thermal conductivities compared to
those of currently used heat transfer fluids, and they have the potential to enhance
heat transfer process. Enhancing the heat capacity of nanofluids in various
applications in industries and in most of the real life application as heat transfer is a
great challenge. This study proposes the analysis for thermal response of nanofluid
flow over the porous surface in the presence of magnetic field in various stream
conditions. The mechanical system of nanoparticle elements is suitable for
stagnation-point flow with convective boundary layer over a vertical porous
permeable surface is framed into a mathematical model. The governing nonlinear
partial differential equations are transformed into a system of coupled nonlinear
ordinary differential equations using similarity transformations and then the framed
mathematical equations are applied numerically using the fourth-fifth order Runge–
Kutta–Fehlberg method by using coded MAPLE 18 software. The work theoretically
investigate and analyse via simulation on the effects of various governing parameters
on flow field and heat transfer and nanoparticle volume concentration characteristics
of the convective boundary layer stagnation point flow of nanofluid towards a porous
stretching and shrinking permeable surface subjected to suction/ injection effect. The
result indicated that flow, heat transfer and nanoparticle concentration can be
controlled by changing the quantity of governing parameters.
vi
ABSTRAK
Nanofluids adalah cecair elektrik yang mengandungi bahan logam dan bukan logam
dan serbok polimerik yang bersaiz nano yang melaksanakan proses seperti
pemindahan haba dan kekonduksian therma. Untuk mengatasi kelemahan dalam
pemindahan haba, cecair inovatif kelas baru bagi pemindahan haba kejuruteraan,
nanofluids dengan campuran nanopartikel logam dalam cecair pemindahan haba
konvensional, yang dijangka mempamerkan keberaliran haba yang tinggi berbanding
dengan cecair yang ada dan cecair ini berpotensi untuk meningkatkan proses
pemindahan haba. Meningkatkan kapasiti haba nanofluids dalam pelbagai aplikasi
dalam industri dan kebanyakan aplikasi kehidupan sebagai pemindah haba masih ada
kelemahan yang perlu diselesaikan. Keperluan industri berorientasikan nanofluid
memerlukan kajian yang banyak. Kajian ini mencadangkan analisis kepada aliran
nanofluid atas permukaan berliang di dalam medan magnet mengikut pelbagai
keadaan aliran. Sistem mekanikal elemen nanopartikel sesuai untuk aliran stagnation
point dan aliran lapisan sempadan atas yang mengecut atau menegak dan permukaan
regangan dibingkaikan ke dalam model matematik. Persamaan separa pembezaan
tidak linear diubah menjadi satu sistem ditambah pula persamaan pembezaan tidak
linear biasa menggunakan transformasi persamaan dan kemudian persamaan
matematik dirangka digunakan secara berangka menggunakan order keempat dan
kelima kaedah Runge-Kutta-Fehlberg oleh dan dikodkan ke dalam perisian MAPLE
18. Kerja-kerja ini secara teori menyiasat melalui simulasi kesan pelbagai parameter
yang mengawal di padang aliran dan pemindahan haba dan ciri-ciri kepekatan jumlah
nanopartikel aliran titik stagnation point nanofluid ke arah regangan dan mengecut
aliran yang berliang tertakluk kepada kesan sedutan dan suntikan. Hasil kajian
menunjukkan aliran, pemindahan haba dan penumpuan nanopartikel boleh dikawal
dengan mengubah parameter kawalan.
vii
TABLE OF CONTENTS
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRACT v
ABSTRAK vi
LIST OF CONTENTS vii
LIST OF TABLES x
LIST OF FIGURES xi
LIST OF APPENDICES xiv
LIST OF PUBLICATIONS xv
LIST OF ABBREVIATIONS xvi
CHAPTER 1 INTRODUCTION 1
1.1 Background 1
1.2 Problem statement 3
1.3 Aim and objectives of the research 3
1.4 Scope of the research 4
1.5 Significance of study 4
1.6 Organization of thesis 5
CHAPTER 2 LITERATURE REVIEW 7
2.1 Introduction 7
2.2 Nanofluids and Nanoparticles 7
2.3 Experimental works on thermal conductivity of
nanofluids 8
2.4 Experimental works on viscosity of nanofluids 9
2.5 Nanofluid flow over stretching and shrinking surface 10
2.6 Classification of flow 11
2.6.1 Stagnation point flow over porous surface 11
viii
2.6.2 Stagnation point 12
2.7 Magnetohydrodynamic (MHD) flow over porous
surface 12
2.8 Conditions 14
2.8.1 Boundary layer 14
2.8.2 Porous medium, Porosity, Porous parameter 18
2.8.3 Suction pressure 19
2.8.4 Injection pressure 20
2.8.5 Prandtl number 21
2.8.6 Chemical reaction 22
2.8.7 Thermal conductivity 23
2.9 Stream Conditions 23
2.9.1 Magnetic effect 23
2.9.2 Thermal radiation 24
2.9.3 Brownian motion 25
2.9.4 Thermophoresis 26
2.9.5 Grashof number 27
2.9.6 Lewis number 28
2.9.7 Reynolds number 28
2.9.8 Thermal radiation 29
2.10 Computational fluid dynamic 30
2.10.1 Continuity equation 31
2.10.2 Momentum equation 32
2.10.3 Energy equation 32
2.11 MAPLE Software 33
2.12 Summary 35
CHAPTER 3 RESEARCH METHODOLOGY 36
3.1 The Algorithm 36
3.2 Research design 37
3.2.1 Stage 1: Identifying parameters 38
3.2.2 Stage 2: Framing code to MAPLE software 38
3.2.3 Stage 3: Optimization of program 39
3.2.4 Stage 4: Analysing the characteristics of nanofluid 39
3.3 Simulation conditions 40
ix
3.3.1 Validity and reliability of research tool 40
3.4 Flow analysis 41
3.5 Summary 46
CHAPTER 4 RESULT AND DISCUSSION 47
4.1 Results and discussions 47
4.1.1 Nanofluid subjected to suction effect 49
4.1.2 Nanofluid subjected to injection effect 71
4.2 Findings for suction 93
4.3 Findings for injection 95
4.4 Discussion 97
4.4.1 Similarities between suction and injection process 97
4.4.2 Differences between suction and injection process 98
4.5 Summary 99
CHAPTER 5 CONCLUSION 100
5.1 Conclusion 100
5.2 Contributions to the research 101
5.3 Future recommendation 102
REFERENCES 103
APPENDIX A 111
APPENDIX B 120
VITA 128
x
LIST OF TABLES
3. 2 Simulation conditions 41
4. 1 Comparison of the present results for �'' with published
works 48
4. 2 Parametric values of stream conditions 50
4. 3 Parametric values of stream conditions 72
4. 4 Differences between suction and injection process 99
xi
LIST OF FIGURES
2. 1 Boundary layer flow 15
2. 2 MAPLE 18 software screen 33
2. 3 MAPLE 18 software simulation screen. 34
3. 1 Research design flow chart 37
4. 1 Co-ordinate system and the physical model of porous
surface 42
4. 2 Brownian motion on temperature 48
4. 3 Brownian motion on nanoparticle volume concentration 49
4. 4 Porosity effects on the velocity distribution 53
4. 5 Porosity effects on the temperature distribution 53
4. 6 Porosity effects on the nanoparticle volume fraction 54
4. 7 Magnetic effects on the velocity distribution 55
4. 8 Magnetic effects on the temperature distribution 56
4. 9 Magnetic effects on the nanoparticle volume concentration 56
4. 10 Grashof number on the velocity distribution 58
4. 11 Grashof number on the temperature distribution 58
4. 12 Grashof number on the nanoparticle volume fraction 59
4. 13 Lewis number on the velocity distribution 60
4. 14 Lewis number on the temperature distribution 60
4. 15 Lewis number on the nanoparticle volume concentration 61
4. 16 Brownian motion effect on the velocity distribution 62
4. 17 Brownian motion effect on the temperature distribution 63
4. 18 Brownian motion effect on the nanoparticle volume
fraction 63
4. 19 Thermophoresis effect on the velocity distribution 65
4. 20 Thermophoresis effect on the temperature distribution 65
xii
4. 21 Thermophoresis effect on the nanoparticle volume
concentration 66
4. 22 Thermal radiation on the velocity distribution 67
4. 23 Thermal radiation on the temperature distribution 68
4. 24 Thermal radiation on the nanoparticle volume
concentration 68
4. 25 Suction effect on the velocity distribution 70
4. 26 Suction effect on the temperature distribution 70
4. 27 Suction effect on the nanoparticle volume concentration 71
4. 28 Porosity effects on the velocity distribution 75
4. 29 Porosity effects on the temperature distribution 75
4. 30 Porosity effects on the nanoparticle volume fraction 76
4. 31 Magnetic effects on the velocity distribution 77
4. 32 Magnetic effects on the temperature distribution 78
4. 33 Magnetic effects on the nanoparticle volume concentration 78
4. 34 Grashof number on the velocity distribution 80
4. 35 Grashof number on the temperature distribution 80
4. 36 Grashof number on the nanoparticle volume fraction 81
4. 37 Lewis number on the velocity distribution 82
4. 38 Lewis number on the temperature distribution 82
4. 39 Lewis number on the nanoparticle volume concentration 83
4. 40 Brownian motion effect on the velocity distribution 84
4. 41 Brownian motion effect on the temperature distribution 85
4. 42 Brownian motion effect on the nanoparticle
volumefraction 85
4. 43 Thermophoresis effect on the velocity distribution 87
4. 44 Thermophoresis effect on the temperature distribution 87
4. 45 Thermophoresis effect on the nanoparticle volume
concentration 88
4. 46 Thermal radiation on the velocity distribution 89
4. 47 Thermal radiation on the temperature distribution 90
4. 48 Thermal radiation on the nanoparticle volume
concentration 90
4. 49 Injection effect on the velocity distribution 92
xiii
4. 50 Injection effect on the temperature distribution 92
4. 51 Injection effect on the nanoparticle volume concentration 93
xv
LIST OF PUBLICATIONS
Journal Articles
1. Magnetohydrodynamic and heat transfer effects on stagnation flow of an
electrically conducting nanofluid flow past a porous vertical
shrinking/stretching sheet in the presence of variable stream conditions.
Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, R. Kandasamy.
Journal of Applied Mechanics and Technical Physics. (In review)
2. Influence of injection and heat transfer effects on stagnation flow of an
electricaly conducting nanofluid flow over a porous shrinking/stretching
sheet in the presence of variable stream conditions
Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, R. Kandasamy.
Internaational Journal of Applied Engineering Research. (In Press)
3. Thermal response of convective boundary layer stagnation flow of nanofluid
over shrinking surface influencing suction and variable stream conditions.
Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, Ashwin kumar,
R. Kandasamy.
ARPN Journal of Engineering and Applied Sciences. (In Press)
4. Nanofluid thermal response through stagnation point flow over shrinking
surface influencing injection and variable stream conditions.
Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, Ashwin kumar,
R. Kandasamy.
ARPN Journal of Engineering and Applied Sciences. (In Press)
xvi
LIST OF ABBREVIATIONS
a Velocity component on porous surface B Uniform transverse magnetic field strength C Concentration of nanofluid Cw Concentration of nanofluid at wall C∞ Concentration of nanofluid at infinity
CFD Computational fluid dynamics
c Constants (+ve) cp Specific heat at constant pressure, DB Brownian diffusion coefficient DT Thermophoresis diffusion coefficient � Force � ' Dimensionless velocity � '' Dimensionless skin friction Gr Grashof number Gm Modified Grashof number � Gravitational force k Permeability of the porous medium k Mean absorption coefficient �� Lewis number � Length M Magnetic parameter
MHD Magnetohydrodynamics m Mass NB Brownian motion parameter Nt Thermophoresis parameter
xvii
ODE Ordinary differential equations qr Radiative heat flux and S Suction / Injection parameter T Temperature of nanofluid Tw Temperature of nanofluid at wall T∞ Temperature of nanofluid at infinity U Velocity of nanofluid Uw Velocity of nanofluid at wall U∞ Velocity of nanofluid at infinity U x Free stream velocity u Velocity components along the x direction uT Tangential velocity v Velocity components along the y direction
2-D Two-dimensional
3-D Three-dimensional
Greek symbols αf Fluid thermal diffusivity αm Thermal diffusivity
η Similarity variable
Dimensionless temperature of the fluid
w Wall temperature excess ratio parameter ' Dimensionless heat transfer rate
Dynamic viscosity v Kinematic viscosity ρ Density ρf Density of base fluid σ Electrical conductivity of the fluid δ Stefan–Boltzmann constant τ Ratio of the effective heat capacity of the nanoparticle material to the
heat capacity of the ordinary fluid φ Nanoparticles volume fraction
ψ Stream function
xviii
Porous medium v Mass flux
Superscripts ' Differentiate with respect to y, x, η correspondingly
Subscripts
� Fluid
s Solid
T Tangential
w Wall ∞ Free stream
CHAPTER 1
INTRODUCTION
1.1 Background
Heat transfer is indispensable to maintain the reliability of desired performance of a
wide variety of equipment’s in industries and consumer products. High-tech
industries face technical challenges day by day. Heat transfer is one of them
especially in industries such as manufacturing, metrology, microelectronics, nuclear
reactors, hybrid-powered engines, vehicle thermal management, domestic
refrigerator, chillers, heat exchanger, nuclear reactor coolant, grinding, machining,
space technology, ships, and boiler flue gas temperature reduction. These industries
faces heat transfer problems because of unprecedented heat loads and heat fluxes. Air
cooling, water cooling systems, coolant, fins, liquid nitrogen are commonly used as
coolants in industries. In optoelectronic devices in power electronics devices are also
found to have heat transfer problems. Increasing in heat flux necessitates the use of
liquid cooling technologies such as heat pipes, spray cooling for chips, direct
immersion cooling. In manufacturing industries conventional heat transfer fluids oil,
water, ethylene glycol, toluene are used for heat transfer and cooling purpose. But
these cooling systems are not able to satisfy the required heat transfer requirements
while dispersed solid particles with heat transfer fluid shows better cooling rate
compared with traditional cooling systems.
Heat transfer fluid having millimetre or micrometre sized particles have some
weakness. Rapid settling of millimetre or micrometre sized particle is the problem
and if the fluid is kept circulating to prevent particle settling, these particles would
2
wear out bearings, pumps and pipes. Furthermore such sized particles are not
applicable in microsystems because it can clog micro channels. Significantly this
type of cooling system results in pressure drop. To overcome disadvantages and
improve heat transfer characteristics an innovative idea of implementing nanometer
sized particle (Choi & Eastman, 1995) became a conventional method for cooling
purpose. Many research are still on going for nanofluids and its heat transfer
characteristics.
Nanofluid is a new kind of fluid containing small quantity of nano-sized
particles (usually less than 100nm) that are uniformly and stably suspended in a
liquid. The dispersion of a small amount of solid nanoparticles in conventional fluids
changes their thermal conductivity remarkably. Nanofluids are characterized by base
fluid like water, toluene, ethylene glycol or oil with nanoparticles in variety of types
like metals, oxides, carbides, carbon, nitrides and others. Some benefits of nanofluids
that make them useful are:
1. The nanoparticles surface to volume ratio is × times larger than
that of microparticles. Surface area of the particles conducts heat. Since
nanoparticles have larger surface area, it enhances the heat conduction of
nanofluids.
2. Nanoparticles stay suspended longer time compared to microparticles.
3. High stability and high thermal conductivity.
4. Nanoparticles reduce erosion, clogging and abrasion dramatically.
Thermal conductivity of nanofluid has been measured with several
nanoparticles volume fraction, material and dimension in several base fluids and all
findings showed that thermal conductivity of nanofluid is higher than the base fluids
and have great potential and more efficient for a cooling system.
The novel concept of nanofluids has been proposed as a route to analyse the
performance of heat transfer fluids currently available(Bachok et al., 2010). But the
characteristic of nanoparticle volume concentration, nanofluid temperature and flow
field velocity of nanofluid differs on the basis change in stream conditions. It is in a
great interest of this research to analyse the characteristics of fluid flow over vertical
porous medium in the presence of magnetic field with variable stream conditions.
3
1.2 Problem statement
Heat transfer is one of the challenges in industries such as manufacturing, metrology,
microelectronics, nuclear reactors, heat exchanger, space technology, ships, and
boiler flue gas temperature reduction. Heat transfer failure caused by unprecedented
increase in heat loads and heat flux caused challenges in high tech industries.
Conventional heat transfer methods like air cooling, water cooling systems, coolant,
fins and also conventional heat transfer liquids like water, toluene, ethylene glycol or
oil unable to fulfil the requirements of industries because of its low thermal
conductivity (Choi 1995) and (Eastman et al., 2001). To overcome the heat transfer
issues nanoparticles disperse with base fluids as coolant can achieve the desirable
performance due to high heat transfer capabilities (Gharagozloo & Goodson 2011).
Nanofluids ability has already provide but just that enough to enable nanofluid in real
life application. Analysing its thermal behaviour on various situation is important
(Bachok et al., 2010). Owing to this application, this research will try to find
solutions following questions arise.
1. What are nanofluid characteristics that provide optimum heat transfer with
the conditions involved?
2. How to formulate numerical model presenting the nanofluid characteristics?
3. How to optimize nanofluid model using simulation software?
4. What are the effects of various stream conditions towards the temperature,
velocity and nanoparticle volume concentration profiles?
5. How does the fluid flow characteristic differs between stretching and
shrinking surface?
6. How does the fluid flow characteristic differs between injection and
suction?
1.3 Aim and objectives of the research
The overall aim and objectives of present study is to analyse the characteristics of
heat transfer and mass transfer by constructing mathematical model and generating
the numerical algorithms on convective boundary layer stagnation point flow over
porous surface. The specific objectives are as follows:
4
1. To frame numerical algorithm of nanofluid model with various streaming
conditions involved.
2. To analyse the nanofluid model simulation using software MAPLE 18
software.
3. To analyse thermal response, velocity, concentration of nanofluid flow on
stretching/shrinking surface subjected to various physical effects in suction
process through simulation.
4. To analyse thermal response, velocity, concentration of nanofluid flow on
stretching/shrinking surface subjected to various physical effects in injection
process through simulation and comparing the results with the results of
objective 3.
1.4 Scope of the research
This work will be limited to the algorithm generated and its simulation using
MAPLE 18. The simulation will be focus on various thermal response, velocity,
concentration in suction and injection process. The simulation model will focus on
various streaming conditions and will study nanofluid response on velocity flow,
temperature distribution and concentration on stretching and shrinking surface
subjected to injection and suction process on convective boundary layer stagnation
point flow over porous surface.
1.5 Significance of study
Nanofluids in the presence of magnetic effect have great potential for heat transfer
enhancement and it is highly suited in application in heat transfer processes. Due to
magnetic field Lorentz force acts opposite to the flow, velocity of nanofluid
decreases, which tends to increase the thermal conductivity of nanofluid. Surface to
volume ratio of nanoparticles, stability, and are added advantages of using
nanofluids. This provides promising ways for engineers to develop highly compact
and effective heat transfer fluids. When addressing the nanofluids, it is foremost
important to establish its flow over surface. Nanofluids flow in porous surface play a
very important role on engineering applications. In diverse applications nanofluid
5
over porous surface has been an efficient active field. Nanofluids flow over porous
surface varies with respect to stream conditions. Brownian motion and
thermophoresis holds important handy in thermal conductivity and nanoparticle
volume fraction. Providing nanofluids with proper stream conditions can achieve
efficient heat transfer.
Heat transfer by convection in numerous examples of naturally occurring
fluid flow, such as wind, oceanic currents, and movements within the Earth's mantle.
Porous surface splits into stretching and shrinking surface, each one implies its
necessity on their particular applications. Flow over stretching/shrinking differs from
each other due to the characteristics of nanofluids and surface.
1.6 Organization of thesis
This thesis consists of an introductory chapter and four main chapters dealing with
the following problems.
Chapter 1 provides brief introduction of the subject, problems, physical
features involved in the study of stagnation flow of magnetohydrodynamic boundary
layer flow of nanofluids with Brownian motion, thermophoresis, thermal radiation,
magnetic effect, suction/ injection and various dimensionless numbers.
Nanofluids have high heat transfer ability to fulfil the necessity of heat
transfer fluid in industries. It is also a need to increase the efficiency of nanofluid, so
that performance will increase. The study of thermal response of nanofluid in the
presence of magnetic field is of great practical importance to engineers and scientists
because of its almost universal occurrence in many branches of science and
engineering. Nanofluids with magnetic effect change its response to heat with
variable stream conditions. Therefore, Chapter 2 presents the literature review that
leads to overall study in this thesis.
Chapter 3 presents an overview of the research methodology involved in
choosing appropriate steps and procedures to give the solution of problem to achieve
the objectives of study. In order to ensure the effectiveness of nanofluid in heat
transfer applications.
Chapter 4 presents the effects of thermal response of nanofluid flow in the
presence of magnetic field and suction/injection effect with variable stream
6
conditions on a vertical porous surface. The main objective is to investigate the
thermal response of nanofluid flow on various physical effects, nanofluids energy
transmission regarding its volume concentration and to analyse stability of nanofluid.
Here the stagnation flow over porous surface (stretching/shrinking surface) on
vertical plate is to investigate numerically and analyse using MAPLE software
(version 18). The physical aspects of fluid flow are governed with governing
equations of Computational fluid dynamics. Governing nonlinear boundary-layer
equations are converted by similarity transformation to coupled higher order
nonlinear ordinary differential equation. The output nanoparticle volume
concentration, temperature and velocity profiles show the clear objectives of thesis
with flow and characteristics when subjected to various physical effects.
Finally, concluded the thesis with Chapter 5 which consists of two main
parts: contributions and future work.
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
A number of researchs relating to effects of nanoparticles usage in heat transfer
fluids have been carried out by a number of researchers recently. This chapter will
discuss their findings. The chapter will also include parameters and factors that
relates to heat transfer characteristics and properties that have been studied. The
literatures presented here have influenced the work and methodology used.
2.2 Nanofluids and Nanoparticles
Choi (1995) introduced the term nanofluid. It refers to the fluids with suspended
nanoparticles. He engineered and innovated suspending metallic nanoparticles in
conventional heat transfer fluids. The results showed that the addition of a very small
amount of nanoparticles to convectional heat transfer fluids, increased the thermal
conductivity of the fluid up to approximately two times. The reason was the high
surface area of nanoparticles enhances the heat conduction of nanofluids since heat
conduction occurs on the surface of the particle. The surface to volume ratio of
nanoparticles is 1000 times larger than that of microparticles so heat transfer rate is
also high.
Eastman et al., (2001) indicated that a small amount (<1% volume fraction)
of Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil is
reported to increase the thermal conductivity of a liquid by 40% and 150%
8
respectively. However, a convectional solid particle-liquid suspensions with larger
size and higher density requires more concentrations (>10%) of particle to achieve
such enhancement. This amount of concentration induces a problem of settling, flow
resistance and possible erosion. Masuda et al., (1993) observed a phenomenon that
nanofluids has a characteristic feature of thermal conductivity enhancement, which
indicates the possibility of using nanofluids in advanced nuclear systems due to high
stability and good conductivity. The research therefore will study this phenomenon
through simulation and study the characteristics of heat conduction when certain
parameters are changed.
2.3 Experimental works on thermal conductivity of nanofluids
From Eastman et al., (1997) report, nanoparticles induced in water made a promising
turning point of using nanofluids in heat transfer process. The article showed that 5%
volume of nanocrystalline copper oxide CuO particles suspended in water resulted in
an improvement in thermal conductivity of almost 60% compared to water without
nanoparticles. Murshed et al., (2005) found that nanofluids prepared by dispersing
5% volume fraction Titanium Dioxide TiO2 nanoparticles in deionized water, the
thermal conductivity enhancement is observed through hot wire technique to be
nearly 33% respectively over the base fluid. Putnam et al., (2006) used an optical
beam deflection technique to measure the thermal conductivity of ethanol-water
mixtures. The enhancement was in a factor of 2 larger than predicted by effective
medium theory.
Penas et al., (2008) conducted an experiment using the multi current hot wire
technique by dispersing nanoparticles of silica (SiO2) and Copper Oxide (CuO) in
water and ethylene glycol at various concentrations up to 5% in mass fraction. Good
agreement was found in recommended and published articles about the enhancement
in the thermal conductivity of the fluids due to presence of dispersed nanoparticles
using this technique. Han (2008) reported that 52% enhancement in the thermal
conductivity was found in water-Fluorocarbons (FC72) nanofluids. At room
temperature, the thermal conductivity of FC72 is 0.057 W/m.K, only one tenth of the
thermal conductivity of water (0.58 W/m.K). This water FC72 nanoemulsion fluid
9
was prepared by using high- intensity ultrasonic homogenizer to emulsify water
droplets (<10nm in radius) into FC72.
A benchmark study on the thermal conductivity of nanofluids was made by
(Buongiorno et al., (2009). The nanofluids tested were comprised of aqueous and
nanoaqueous basefluids, metal and metal oxide particles, near-spherical and
elongated particles, at low and high particle concentrations. The thermal conductivity
of the nanofluids was found to increase with particle concentration and aspect ratio
through a variety of experimental approaches, including the transient hot wire
method, steady state methods, and optical methods. The resulted thermal
conductivities were then validated through the effective medium theory developed
for dispersed particles by Maxwell in 1881 and generalized by (Nan et al., 1997).
Evans et al., (2006) have demonstrated that based on kinetic theory analysis of heat
flow in fluid suspensions of solid nanoparticles in nanofluids, the hydrodynamics
effects associated with Brownian motion have only a minor effect on the thermal
conductivity of the nanofluid which was also discussed in (Gharagozloo & Goodson
2011). All the above research have shown that thermal conductivity of nanofluids are
much higher than normal fluids.
2.4 Experimental works on viscosity of nanofluids
Einstein was the first to calculate the effective viscosity of a suspension of spherical
solids using phenomenological hydrodynamic equations. A benchmark study on the
viscosity of nanofluids was made by (Venerus et al., 2010). The viscosity
measurements were on the colloidal dispersions (nanofluids) for heat transfer
applications. They examined the influence of particle shape and concentration on the
viscosity of the same nanofluids tested by (Buongiorno et al., 2009). They compared
data to prediction from classical theories on suspension rheology. The result showed
that the lower concentration ( = . ) fluid is Newtonian, while the higher
concentration ( = . ) fluid is non-Newtonian. The addition of solid particles to a
liquid can significantly alter its rheological behaviour.
A study on viscosity was conducted by Duan et al., (2011) two weeks after
aluminium oxide Al2O3-water nanofluids were prepared at the volume concentration
of 1-5%. A higher nanoparticle aggregation had been observed in the nanofluids
10
resembled non-Newtonian fluids. After ultrasonic agitation treatment, the nanofluids
resumed as a Newtonian fluids. The relative viscosity increases about 60% as the
volume concentrations increases to 5% in comparison with the base fluid. These
experiments have shown that relative viscosity of nanofluids increases as the results
of the addition of nanoparticles.
2.5 Nanofluid flow over stretching and shrinking surface
Nazar et al., (2004) studied unsteady two-dimensional stagnation point flow of an
incompressible viscous fluid over a deformable sheet. He discussed the analysis
when the flow is started impulsively from rest and the sheet is suddenly stretched in
its own plane with a velocity proportional to the distance from the stagnation point.
Wang (2008) studied two dimensional stagnation point flow on a two dimensional
shrinking sheet and axisymmetric stagnation point flow on an axisymmetric
shrinking sheet. It was noted that the line of stagnation flow is perpendicular to the
stretching surface.
Pop et al., (2004) presented radiation effects on the flow near the stagnation
point of a stretching sheet. They show that a boundary layer is formed and its
thickness increases with the radiation, velocity and temperature parameters and
decreases when the Prandtl number is increased. Ishak et al., (2006) analysed steady
two dimensional stagnation point flow of an incompressible viscous and electrically
conduction fluid subjected to a uniform magnetic field, over a vertical stretching
surface, he stated velocity of nanofluid decreases as Lorentz force acting opposite to
the flow, which tends to increase the thermal conductivity of nanofluid
proportionally.
Dulal (2011) numerically analysed the flow and heat transfer in laminar flow
of an incompressible Newtonian fluid past an unsteady stretching sheet in the
presence of a non-uniform heat source/sink and thermal radiation. The time-
dependent stretching velocity and surface temperature resulted to the unsteadiness in
the flow and temperature fields. Hamad et al., (2012) studied heat and mass transfer
for boundary layer stagnation-point flow over a stretching sheet in a porous medium
saturated by nanofluid with internal heat generation/absorption and suction/injection.
He stated that suction tends to stabilize the boundary layer flow and injection can
11
reduce the friction drag. Therefore, one can conclude that there is some effect on
flow of nanofluids over stretching and shrinking surfaces. This work will also take
the nanofluid flow over stretching and shrinking surface into considerations.
2.6 Classification of flow
Flow are generally classified into uniform and non-uniform flow, compressible and
incompressible flow, steady and unsteady flow, and laminar and turbulent flow. A
steady flow is one in which the conditions (velocity, pressure and cross-section) may
differ from point to point but do not change with time (Shaughnessy et al., 2005). If
at any point in the fluid, the conditions change with time, the flow is described as
unsteady. Density of all fluids will change if pressure changes. Liquids are quite
difficult to compress, so under most steady conditions they are treated as
incompressible. In some unsteady conditions very high pressure differences can
occur and it is necessary to take these into account.
2.6.1 Stagnation point flow over porous surface
The proposed algorithm will also investigate the thermal characteristics of nanofluid
at stagnation point over porous surface. Mahapatra & Gupta (2001) numerically
studied boundary-layer and magnetohydrodynamics stagnation point flow towards a
stretching sheet. Their analysis showed that velocity at a point increase with an
increase in the magnetic field when the free stream velocity is greater than the
stretching velocity. Mahapatra et al. (2002) extend their investigation to a power-law
fluid and studied the magnetohydrodynamic stagnation point flow of a power-law
fluid towards a stretching surface. The result signifies that for a given magnetic
parameter, the dimensionless shear stress coefficient increases in magnitude with an
increase in power-law index when the values of the ratio of free stream velocity and
stretching velocity are close to 1.
Makinde et al., (2013) analysed the effects of buoyancy force, magnetic field
and convective heating on stagnation-point flow and heat transfer due to nanofluid
flow towards a stretching sheet. The combined effects of Brownian motion and
thermophoresis on nanofluid over a stretching sheet were investigated. He concluded
12
response of nanofluid depends mainly on Brownian motion and thermophoresis.
These findings will also be trailed through simulation. Porous surface and stagnation
point will be taken into consideration.
2.6.2 Stagnation point
In general, there are steady and unsteady flow, laminar and turbulent flow. In steady
flow, the flow properties at any given point in space are constant in time such as
velocity, pressure and cross-section. In unsteady flow, the flow properties at any
given point in space change with time. As for laminar flow, fluid particles move in
smooth, layered fashion (no substantial mixing of fluid occurs) whereas for turbulent
flow, the fluid particles move in a chaotic, tangled fashion (significant mixing of
fluid occurs). There is another classification of flow available namely compressible
and incompressible flow. In in compressible flow, the volume of a given fluid
particle does not change which implies that the density is constant everywhere. In
compressible flow, the volume of a given fluid particle can change with position
which implies that the density will vary throughout the field.
In fluid dynamics, a stagnation point is a point in a flow field where the local
velocity of the fluid is zero. In the flow field at the surface of objects stagnation
points exist, where the object brought fluid to rest. When the velocity is zero the
static pressure will be highest says Bernoulli equation and hence at stagnation points
static pressure is at its maximum value. This static pressure is called as stagnation
pressure. The Bernoulli equation applicable to incompressible flow shows that
addition of dynamic pressure and static pressure is equal to the stagnation pressure.
So in incompressible flows, total pressure is equal to stagnation pressure. Since
addition of dynamic pressure and static pressure is equal to the total pressure
providing the fluid entering the stagnation point is brought to rest. This phenomenon
is also considered in this work.
2.7 Magnetohydrodynamic (MHD) flow over porous surface
The word magnetohydridynamics (MHD) is derived from magneto which means
magnetic field, hydro means liquid and dynamics which mean movement. In
13
addition, MHD nanofluid flow is the study of electrically conducting nanofluids
movement. The field of MHD was initiated by Hannes Alfven, an astrophysicist,
who received the Noble Prize in physics in 1970. He found that a magnetic field line
can transmit transverse initial wave (Davidson, 2001).
Prior to this, an engineer called J.Hartmann invented an electromagnetic
pump in 1918, with this pump he investigated the flow of mercury, a conducting
liquid in a magnetic field (Davidson, 2001). It was during the 1960s that the
development of MHD came into the field of engineering. The result was three
technological innovations: (i) a liquid sodium pump cooler for fast-breeder reactor;
(ii) a controlled thermonuclear fusion that require the hot plasma separated from
material surface by magnetic force; and (iii) a MHD power generator that use the
magnetic field to propel the ionized gas to improve power station efficiencies
(Davidson, 2001).
The set of equations which describe MHD are a combination of Navier-
Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism.
These differential equations have to be solved simultaneously, either analytically or
numerically because MHD is a fluid theory, it cannot treat kinetic phenomena. There
are many studies on MHD flow such as Kafoussias & Nanoousis (1997) suction or
injection pressure on MHD laminar boundary layer flow and ( Chamkha & Khaled.,
2000) on hydromagnetic combined heat and mass transfer by natural convection
from a permeable surface embedded in a fluid saturation porous medium while
(Chamkha et al. 2003) on thermal radiation effect on MHD forced convection flow
in the presence of heat source or sink. Ishak et al., (2008) have investigated MHD
flow and heat transfer on stretching surface and stated the flow of nanofluids. It can
be argued that similar conditions are applicable for nanofluids on shrinking surface.
Consequently, the studies had entered a new phase where nanofluid was
included. Chamkha & Aly (2010) have analysed MHD free convection flow of a
nanofluid past a vertical plate in the presence of heat generation or absorption
effects. Kandasamy et al., (2011) investigated MHD boundary layer flow of a
nanofluid past a vertical stretching surface in the presence of suction and injection.
Nourazar et al., (2011) have studied MHD nanofluid flow over a horizontal
stretching surface. Hamad & Pop (2011) have examined the MHD free convection
flow past a vertical permeable flat plate in rotating frame of reference with constant
14
heat source in a nanofluid. All the above research has shown that MHD flow over
porous surface is must in considering the study of thermal response of nanofluid.
2.8 Conditions
2.8.1 Boundary layer
Ludwig Prandtl in 1904 pointed out that the flow of liquid with a small viscosity in
the neighbourhood of fixed body can be divided into two regions. Acheson (1990)
stated the region of very thin layer near the boundary layer, where viscosity forces
can be neglected. The influences of viscosity forces in the boundary layer can be
explained by the liquid adhering to the surface of the body, and the adhesion of the
liquid cause on account of friction a dragging of thin layer adjoining the walls. In this
thin layer, the velocity of the flow past the body at rest changes abruptly increasing
from zero to the value of the velocity in the external current (Acheson, 1990;
Anderson, 2005).
Prandtl set up an equation satisfied in the first approximation by the velocity
of flow of fluid in the boundary layer. These equations are called prandlt’s system.
They form the basis of boundary layer theory. He theorized that an effect of friction
was to cause the fluid immediately adjacent to the surface to stick to the surface. In
other words, he assumed the no slip condition at the surface. Outside the boundary
layer, the flow was essentially the inviscid flow (Achenson, 1990; Anderson, 2005).
The concept of boundary layer is sketched in Figure 2.1 are velocity, thermal,
velocity, concentration boundary layers respectively.
A thermal boundary layer will develop if the surface temperature and free
stream temperature are different. In Figure 2.1, at the leading edge, the temperature
profile is uniform with T = T∞. The wall is maintained the thermal boundary layer
thickness in x-direction. The fluid particles coming into contact with the surface
exchange thermal energy with those in the neighbouring layers and a thermal
gradient is setup. With increasing distance, y, from the surface, the fluid temperature
approaches the free stream temperature. The effects of heat transfer penetrate further
into free stream resulting in the growth of thermal boundary layer thickness. Like the
15
velocity boundary, the thermal boundary will also be defined as laminar or turbulent
depending upon the critical value of Reynolds number (Kahar, 2014).
In addition to the lack of consensus on physical properties there is not yet an
accepted model for nanofluid flow. A number of models and physical mechanisms
are framed is described (Das et al., 2003). This research focus on a particular form of
model shown in Figure 2.1. In the real world situation, the heat exchanger for an
example, the fluid is in contact with the walls and tubes of the exchanger and the
bulk of the fluid moves relative to these surfaces. As the fluid moves, the layer of the
fluid in contact with the metal surface is in fact stationary. Away from the metal
surface the fluid begins to move faster and faster. This flow is known as laminar flow
and is characterised by the fluid moving parallel to the surface of exchanger. Further
into the fluid there comes a point where the flow is in longer laminar and is
composed of an ever increasing amount of turbulence. This flow is known as
turbulent flow.
Figure 2. 1 Boundary layer flow (Kandasamy et al., 2011)
where,
Velocity of nanofluid
Velocity of nanofluid at wall
16
∞ Velocity of nanofluid at infinity
Temperature of nanofluid
Temperature of nanofluid at wall
∞ Temperature of nanofluid at infinity
Concentration of nanofluid
Concentration of nanofluid at wall
∞ Concentration of nanofluid at infinity
These regions of laminar and turbulent flow are extremely important in heat
transfer. The layer of fluid that exhibits laminar flow is termed the boundary layer.
There is a critical velocity of the fluid at which the laminar boundary layer becomes
turbulent. As the fluid velocity increases, the change from laminar to turbulent flow
occurs closer and closer to the metal surface. In other words, the boundary layer
becomes thinner as the fluid velocity increases (Mahapatra et al., 2011). The equation
developed takes steady laminar flow as an important part of the work. The boundary
layer shows the general flow and response of temperature and velocity in the Figure
2.1. The above condition is taken into consideration in the theoretical flow analysis
of this work.
2.8.1.1 Convective boundary layer flow over porous surface
The study of convective flow, heat transfer in porous surface has been an active field
of research as it plays a crucial role in diverse applications, such as thermal
insulation, extraction of crude oil and chemical catalytic reactors. More over our
research is with convective boundary layer flow. Numerous models and group theory
methods have been proposed by different authors to study convective boundary layer
flows of fluid, which are (Birkfoff 1998, 1960; Yurusoy & Pakdemirli1997, 1999a,
1999b; Yurusoy et al., 2001).
17
Kuznetsov & Nield (2010) have examined the influence of nanoparticles on natural
convection boundary layer flow past a vertical plate, using a model in which
Brownian motion and thermophoresis are accounted. In this study it is assumed the
simplest possible boundary conditions, namely those in which both the temperature
and the nanoparticle fraction are constant along the wall. Nield & Kuznetsov (2009b)
have analysed the effect of nanoparticles on natural convective boundary layer flow
in a porous medium past a vertical plate and employed the Darcy model for the
momentum equation. Bachok, et al., (2010) have studied theoretically the problem of
steady boundary layer flow of a nanofluid past a moving semi-infinite flat plate in a
uniform free stream flow move in the opposite directions. The problem of laminar
fluid flow resulting from the stretching of a flat surface in a nanofluid has been
investigated numerically (Khan & Pop, 2010). Mankinde & Aziz (2011) studied the
boundary layer flow of a nanofluid past a stretching sheet with a convective heating
boundary condition. Kuznetsov & Neild (2011) studied the double diffusive natural
convective boundary layer flow of a binary nanofluid past a vertical surface
incorporated it with the effects of Brownian motion and thermophoresis.
Maiga et al., (2005) investigated the problem of laminar forced convection
flow of nanofluids for two geometrical configurations, namely a uniformly heated
tube and a system of parallel, coaxial and heated disks. For the case of tube flow, the
heat transfer enhancement increases considerably with an augmentation of the flow
Reynolds number. For the case of radial flow, both the Reynolds number and the
distance separating the disks do not seem to considerably affect in one way or
another the heat transfer enhancement of nanofluids (i.e., when compared to the base
fluid at the same Reynolds number and distance). Hamad & Ferdows (2012) have
investigated a two dimensional laminar forced convection flow over a permeable
stretching surface in a porous medium saturated by a nanofluid. Evans et al., (2006)
have demonstrated that the Brownian motion have only minor effect on the thermal
conductivity of the nanofluid which was also shown in (Gharagozloo & Goodson,
2011).
Fully developed laminar mixed convection of a nanofluid consisting of water
and Al2O3 in a horizontal curved tube has been studied numerically (Akbarinia &
Behzadmehr, 2007). For a given Reynolds number, Buoyancy force has a negative
effect on the Nusselt number while the nanoparticles concentration has a positive
effect on the heat transfer enhancement and also on a skin friction reduction. Ahmad
18
& Pop (2011) studied a steady mixed convection boundary layer flow past a vertical
plate embedded in a porous medium filled with nanofluids of Cu, Al2O3 and TiO2. It
is shown that the solution has two branches in a certain range of the parameters.
Gorla et al., (2011) presented a boundary layer analysis of mixed convection past a
vertical plate in a porous medium saturated with a nanofluid. The equation in this
study will take into consideration of all these findings.
2.8.2 Porous medium, Porosity, Porous parameter
Porous materials are encountered literally everywhere in everyday life, in technology
and in nature. With the exception of metals, some dense rocks, and some plastics,
virtually all solids and semi-solid materials are porous to varying degrees. Porous
medium is characterized by a very large surface area to a volume ratio. This peculiar
feature of the porous media can be utilized to either distribute heat energy uniformly
or to enhance the heat transfer in heat exchange systems (Seigen, 1972). Porous
material contain relatively small spaces, so called pores or voids, free of solids,
imbedded in the solid or semi-solid matrix. The pores usually contain some fluid,
such as air, water, oil. etc., or a mixture of different fluids. Pores material are
permeable to a variety of fluids, that is fluids should be able to penetrate through one
face of septum made of the material and emerge on the other side (Bear & Bachmat,
1990). In this research the study of transport phenomena in porous media have
primarily been initiated by the research activity in geophysical and chemical
engineering. The study of transport phenomena in porous medium has attracted
considerable attentions and has been motivated by broad range of engineering
applications.
Porous media are proven to operate in most of the corresponding free fluid
ranges. They can be used as an insulator for all temperature ranges and can be used
as a heat transfer promoter for either sensible or latent heat transfer. Different
transport models are used to model energy and momentum transport in porous media.
These models are phonologically based upon governing equations which are
inherited from the corresponding free-fluid flow. Porosity is the factor often
characterizes the porous medium. Porosity is a fraction between 0 and 1. pores
structure and the media porosity with respective properties can sometimes be derive
19
some of the other properties of porous medium like tensile strength, permeability
and electrical conductivity, but such a derivation is usually complex. Porous medium
the porosity is defined as the fraction of the total volume of the medium that is
occupied by void space. It is shown that under conditions convective flow may occur
in fluid which permeates a porous medium and is subject to a vertical temperature
gradient, on the assumption that flow obeys Darcy’s law. This phenomenon is also
considered in this work and the equation (2.2) is an integral part of the algorithm. � = (2.2)
where, � Permeability of the porous medium � Porous medium (dimensionless)
Kinematic viscosity
� Velocity component on porous surface
2.8.3 Suction pressure
Flow of a fluid into a low pressure region, or a partial vacuum region is called as
suction pressure. The regions differ of pressure gradient between them and the
ambient pressure will pull liquid toward area of low pressure. Mostly suction is
considered as an attractive effect or vacuum effect. The production of a vacuum or
partial vacuum in a cavity or over a surface so that the external atmospheric pressure
forces the surrounding fluid, particulate solid, etc. into the cavity or causes
something to adhere to the surface. Vacuum cannot attract matter but the atmospheric
high pressure of the surrounding fluid can propel liquid into a vacuum area (Calvert
& James, 2000).
Typically pumps have both an inlet as well as an outlet. The inlet location is
said to be at the suction side where the fluid enters the pump and the outlet location
is said to be at the discharge side of the pump where the fluid comes out. Operation
differs on the basis of pump. Pump creates a low pressure region (suction pressure) at
the inlet side so that fluid can enter the pump, at the discharge side pump operation
20
causes higher pressure by forcing the fluid out at the outlet (Calvert & James, 2000).
Under normal conditions of atmospheric pressure suction can draw pure water up to
a maximum height of approximately 10.3 m. This is the same as the maximum height
of a siphon, which operates by the same principle. = √ (2.3)
where,
is the suction parameter >
Mass flux ∗
� Velocity component on porous surface
Kinematic viscosity
Suction effect is given in one part of the work and following with injection
effect on another part of the work. Both played major role in this work. The suction
parameter equation (2.3) is included in this work.
2.8.4 Injection pressure
Flow of a fluid into a high pressure region is called as injection pressure. The regions
differ of pressure gradient between them and the ambient pressure will force liquid
toward area of high pressure. Mostly injection is considered as a blowing effect. In
injection, with the help of external force the fluid is forced into high pressure region
(Calvert & James, 2000).
A similar pump characteristic as discussed in section 2.9.3 is applicable.
Forcing a fluid to a region is termed as injection or blowing (Calvert & James, 2000).
Injection force is given in the later part of the work by replacing suction effect. As in
equation 2.3, similar equation can be used here. The difference is in value which is
now an injection parameter < .
21
= − √ (2.4)
where,
Injection parameter <
Mass flux ∗
� Velocity component on porous surface
Kinematic viscosity
2.8.5 Prandtl number
The Prandtl number is a dimensionless number, defined as the ratio of momentum
diffusivity (kinematic viscosity) to thermal diffusivity. That is, the Prandtl number is
given as:
� = � = � � � �ℎ � � (2.5)
where,
Kinematic viscosity
Thermal diffusivity
� means thermal diffusivity dominates � means momentum
diffusivity dominates. In heat transfer problems, the Prandtl number controls the
relative thickness of the momentum and thermal boundary layers. When Pr is small,
it means that the heat diffuses very quickly compared to the velocity (momentum).
This means that for liquid metals the thickness of the thermal boundary layer is much
bigger than the velocity boundary layer (Sherman, 1990). The dimensionless Prandtl
22
number is used as the integral part of the algorithm. The equation (2.5) is used in the
algorithm
Gases typically have Prandtl numbers in the range 0.7 -1 , while the Prandtl
number for most liquids is much larger than unity. The Prandtl number for water
ranges from 5-10, while that for an oil might be of the order of 50-100. It is not
uncommon to encounter Prandtl numbers for viscous liquids that are of the order of
several thousand or even larger. One exception to the general rule for liquids is a
liquid metal. Liquid metals conduct heat very efficiently, and therefore have a
relatively large value of the thermal diffusivity when compared with ordinary liquids.
On the other hand, viscosities are about the same as that of ordinary liquids. As a
consequence, the Prandtl number of a liquid metal is typically of the order of 10-2
(White, 2006; Hewitt et al., 1994). Since many researchers like Gururaj & Devi
(2014), Pal et al. (2014) and Rohni et al. (2012) have chosen Prandtl number around
7 for water nanofluids at 20℃. In this research Prandtl number is chosen as 6.2 for
nanofluids as most researchers use 6.2, the author has decided to use Prandtl number
as 6.2.
2.8.6 Chemical reaction
A chemical reaction is a process involving one or more substances called reactants,
characterised by a chemical reaction change and yielding one or more products
which or different from the reactants. A chemical change is defined as molecules
attaching to each other form large molecules, molecules breaking apart to form two
or more, smaller molecules or rearrangement of atoms within molecules (The
Encyclopaedia of Earth, 2011). The study of heat and mass transfer with chemical
reaction in the presence of nanofluids is of considerable importance in chemical and
hydrometallurgical industries. Chemical reaction can be codified as either
heterogeneous or homogeneous process. This depends on whether they occur at an
interface or as single phase volume reaction.
Homogenous reaction or chemical reaction in which the reactants are in the
same phase, while heterogeneous reactions have reactants in two or more phases.
Reactions that take place on the surface of a catalyst of a different phase are also
heterogeneous. A reaction between two gases, two liquids or two solids is
23
homogenous. A reaction between a gas and a liquid, a gas and a solid or a liquid and
a solid is heterogeneous. Practical application of heterogeneous reaction are in
catalytic converter, fuel cells and chemical vapour deposition among others, recently,
manufacturing engineers have surface reactions for synthesis of micro and nanoscale
in biomedical devices. In this research water is considered as a base fluid so chemical
reaction is not considered. If other base fluids like oil, glycol, and petrol are used
then chemical reaction should be considered.
2.8.7 Thermal conductivity
Choi & Eastman (1995) states thermal conductivity is the property of a material to
conduct heat. It is the quantity of heat transmitted through a unit thickness in a
direction normal to a surface of unit area, due to a unit temperature gradient under
steady state conditions. The ratio of nanofluid thermal conductivity � to the base
fluid thermal conductivity � (Buongiorno, 2009).
2.9 Stream Conditions
There are various other factors including stream conditions that influenced the
characteristics of the boundary layer flow over porous surfaces. Stream conditions
have some dimensionless parameters. Several physical parameters, which
characterised the flow often called as dimensionless numbers arise in the study of
nanofluid flows (Vandiver & Marcollo, 2003). In this section, few of these
dimensionless parameters are presented which is more importantly used in this
research. In this thesis, nanofluid flow velocity, temperature and nanoparticle volume
concentration on porous vertical stretching and shrinking surface are analysed. Some
of the factors to be considered while analysing nanofluid flow. Some of the factors
and stream conditions are as follows:
2.9.1 Magnetic effect
Magnetic field has two important mathematical properties that relate it to its sources.
The sources are currents and changing electric fields. These two properties, along
24
with the two corresponding properties of the electric field, make up Maxwell's
Equations. Maxwell's Equations together with the Lorentz force law form a complete
description of classical electrodynamics including both electricity and magnetism
(Huray & Paul, 2009). Magnetic field tends to produce Lorentz force to the flow
field. Lorentz force induced by the magnetic field plays a major role in flow field. In
this work, magnetic effect is given to observe the electrically conducting nanofluids
flow in the presence of magnetic field. The equation (2.6) is an integral part of
algorithm. = �� (2.6)
where,
Magnetic parameter Ω
� Electrical conductivity of the fluid Ω ∗
� Density of base fluid
Uniform transverse magnetic field strength �
� Velocity component on porous surface
2.9.2 Thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of
charged particles in matter. All matter with a temperature greater than absolute zero
emits thermal radiation. When the temperature of the body is greater than absolute
zero, interatomic collisions cause the kinetic energy of the atoms or molecules to
change. This results in charge-acceleration and/or dipole oscillation which produces
electromagnetic radiation, and the wide spectrum of radiation reflects the wide
spectrum of energies and accelerations that occur even at a single temperature
(Sciuto, 2012). Heat and power can be harvested from the solar radiation or radiation
or from sun. this term is denoted by radiation. Unlike heat transfer methods
REFERENCES
Akbarinia, A. & Behzadmehr, A. (2007). Numerical study of laminar mixed
convection of a nanofluid in horizontal curved tubes. Applied Thermal
Engineering, 27(8), 1327-1337.
Acheson, D.J. (1990). Elementary fluid dynamics. NewYork: Oxford University
Press.
Anderson, J.D. (2005). Ludwig Prandtl’s boundary layer. Physics Today, 58(12), 42-
48.
Anderson, J.D. (2009). Basic Philosophy of Computational Fluid Dynamics. Berlin
Heidelberg: Springer. 3-14. doi: 10.1007/978-3-540-85056-4_1
Bachok, N., Ishak, A. & Pop, I. (2010). Boundary layer flow of nanofluids over a
porous surface in a flowing fluid. International Journal of Thermal Sciences,
49(9), 1663-1668.
Bitwise magazine. (2005). Power of two. Retrieved April 2, 2015, from
http://www.bitwisemag.com/copy/reviews/software/maths/maple10_mathema
tica52.html
Beer, J. & Bachmat, Y. (1990). Introduction to modelling of transport phenomena in
porous media. 1st
ed. Netherlands: Springer. doi: 10.1007/978-94-009-1926-6
Bouery, C. (2012). Contribution to algorithmic strategies for solving coupled
thermo-mechanical problems by an energy-consistent vibrational approach.
Ecole Centrale de Nantes: Ph.D. Thesis.
Brikhoff, G. (1948). Lattice theory. 25th
ed. New York: American Mathematical
Society.
Brikhoff, G. (1960). Hydrodynamics. New Jersey : Princeton University Press.
Buongiorno, J. (2006). Convective transport in nanofluids. Journal of Heat Transfer,
128(3), 240–250.
104
Buongiorno, J., Venerus, D.C., Prabhat, N., McKrell, T., Townsend, J., Christianson,
R. & Leong, K.C. (2009). A benchmark study on the thermal conductivity of
nanofluids. Journal of Applied Physics, 106(9), 094312-094312.
Chamkha, A.J. & Khaled, A.R.A. (2000). Hydromagnetic combined heat and mass
transfer by natural convection from a permeable surface embedded in a fluid-
saturated porous medium. International Journal of Numerical Methods for
Heat and Fluid Flow, 10(5), 455-477.
Chamkha, A.J., Mujtaba, M., Quadri, A. & Iss, C. (2003). Thermal radiation effects
on MHD forced convection flow adjacent to a non-isothermal wedge in the
presence of heat source or sink. Heat and Mass Transfer, 39(4), 425-441.
Chamkha, A.J., & Aly, A.M. (2010). MHD free convection flow of a nanofluid past
a vertical plate in the presence of heat generation or absorption effects.
Chemical Engineering Communications, 198(3), 2040-2044.
Chiam, T.C. (1994). Stagnation-point flow towards a stretching plate. Journal of
Physics Society, 63(6), 2443–2444.
Choi., S.U.S., (1995). Enhancing thermal conductivity of fluids with nanoparticles.
Fluids Engineering Division, 231, 99-103.
Choi, S.U.S., Zhang, Z.G., Yu, W., Lockwoow F.E. & Grulke, E.A. (2001).
Anomalous thermal conductivities enhancement on nanotube suspension.
Applied Physics Letters, 79(10), 2252–2254.
Das, S., Putra, N., Thiesen, P. & Roetzel, W. (2003). Temperature dependence of
thermal conductivity enhancement for nanofluids. Journal of Heat Transfer,
125(4), 567–574
Davidson, P.A. (2001). An Introduction to magnetohydrodynamics.1st ed. New York:
Cambridge University Press.
Duan. F., Kwek, D. & Crivoi, A. (2011). Viscosity affected by nanoparticle
aggregation in Al2O3-water nanofluids. Nanoscale Research Letters, 6(1), 1-
5.
Dulal, P. (2011). Combined effects of non-uniform heat source/sink and thermal
radiation on heat transfer over an unsteady stretching permeable surface,
Communications in Nonlinear Science and Numerical Simulation, 16(4),
1890–1904.
Eastman, J.A., Choi, S.U.S., Li, S. & Thompson, L.J. (1997). Enhanced thermal
conductivity through the development of nanofluids. Proc. of the fifth
105
Symposium on Nanophase and Nanocomposite Materials II, 457. USA.
Materials Research Society. 3–11.
Eastman, J.A., Choi, S.U.S., Yu, W. & Thompson, L.J. (2001). Anomalously
increased effective thermal conductivity of ethylene glycol-based nanofluids
containing copper nanoparticles. Applied Physics Letters, 78(6), 718–720.
Evans, W., Fish, J. & Keblinski, P. (2006). Role of Brownian motion hydrodynamic
on nanofluid thermal conductivity. Applied Physics Letters, 88(9), 5-8. doi:
10.1063/1.2179118.
Fang, T.G., & Zhang, J. (2010). Thermal boundary layers over a shrinking sheet. An
Analytical Solution of Mechanical, 209, 325–343.
Gharagozloo, P.E. & Goodson, K.E. (2011). Temperature-dependent aggregation and
diffusion in nanofluids. International Journal of Heat and Mass Transfer,
54(4), 797-806.
Gururaj, A.D.M. & Devi, S.P.A. (2014). MHD boundary layer flow with forced
convection past a nonlinearly stretching surface with variable temperature and
nonlinear radiation effects. International Journal of Development Research,
4(1), 75-80.
Hamad, M.A.A. (2011). Analytical solution of natural convection flow of a nanofluid
over a linearly stretching sheet in the presence of magnetic field.
International Communication in Heat and Mass Transfer, 38(4), 487-492.
Hamad, M.A.A. & Ferdows, M. (2012). Similarity solution of boundary layer
stagnation-point flow towards a heated porous stretching sheet saturated with
a nanofluid with heat absorption/generation and suction/blowing: A lie group
analysis. Communications in Nonlinear Science and Numerical Simulation.
17(1), 132–140. doi: 10.1016/j.cnsns.2011.02.024
Han, Z. (2008). Nanofluids with enhanced thermal transport properties. University
of Maryland: Ph.D. Thesis.
Hewitt, G. F., Shires, G. L. & Bott, T. R. (1994). Heat Transfer. Florida: CRC Press.
Huray, P.G. (2009). Maxwell’s Equation. 2nd ed. Hoboken: Wiley-IEEE Press.
Ishak, A., Nazar, R. & Pop, I. (2006). Magnaetohydrodynamic stagnation point flow
towards a stretching vertical sheet. Magnaetohydrodynamic, 42(1), 17-30.
Ishak, A., Nazar, R. & Pop, I. (2008). Magnetohydrodynamic flow and heat transfer
due to stretching surface. Energy Conversation and Management, 49(11),
3265-3269.
106
Incropera, F.P., Lavine, A.S. & DeWitt, D.P. (2011). Fundamentals of heat and mass
transfer. Danvers: John Wiley & Sons Incorporated.
Kafoussias, N.G. & Nanousis, N.D. (1997). Magnetohydrodynamic laminar
boundary layer flow over a wedge with suction or injection. Canadian
Journal of Physics, 75(10), 733-741.
Kandasamy, R., Loganathan, P. & Arasu, P.P. (2011). Scaling group transformation
for MHD boundary-layer flow of a nanofluid past a vertical stretching surface
in the presence of suction/injection. Nuclear Engineering and Design, 241,
2053–2059.
Kandasamy, R., Muhaimin, I. & Mohamad, R. (2013). Thermophoresis and
Brownian motion effects on MHD boundary-layer flow of a nanofluid in the
presence of thermal stratification due to solar radiation. International Journal
of Mechanical Sciences, 10(3), 760-771.
Kahar, A.R. (2011). Scaling group transformation for boundary-layer flow of a
nanofluid past a porous vertical stretching surface in the presence of chemical
reaction with heat radiation. Computers and Fluids, 52 (1).15-21.
Kahar, A.R. (2014). Heat transfer of nanofluid flow over porous surfaces with
variable stream conditions. Universiti Tun Hussein Onn Malaysia: Ph.D.
Thesis.
Khan, W.A. & Pop, I. (2010). Boundary layer flow of a nanofluid past a stretching
sheet. International Journal of Heat and Mass Transfer, 53(11-12), 2477-
2483.
Khan, M.S., Karim, I., Ali, LE. & Islam, A. (2012). Unsteady MHD free convection
boundary-layer flow of a nanofluid along a stretching sheet with thermal
radiation and viscous dissipation effects. International Nano Letter, 24(2),
760-771. doi:10.1186/2228-5326-2-24.
Kuznetsov, A.V. & Nield, D.A. (2010). Natural convective boundary-layer flow of a
nanofluid past a vertical plate. International Journal of Thermal Science,
49(2), 243–247.
Kuznetsov, A.V. & Nield, D.A. (2011). Double diffusive natural convective
boundary-layer flow of a nanofluid past a vertical plate. International Journal
of Thermal Science, 50(5), 712-717.
107
Lapwood, E.R. (1948). Convection of fluid in a porous medium. Mathematical
Proceedings of the Cambridge Philosophical Society, 44(4), 508-521. New
York: Cambridge University Press.
Lok, Y.Y., Ishak, A. & Pop, I. (2011). MHD stagnation point flow with suction
towards a shrinking sheet. Sains Malaysia, 40(10), 1179–1186.
Mahapatra, T.R. & Gupta, A.G. (2002). Heat transfer in stagnation point flow
towards a stretching sheet. Heat Mass Transfer, 38(6), 517–521.
Mahapatra, T.R. & Gupta, A.S. (2001). Magnetohydrodynamics stagnation-point
flow towards a stretching sheet. Acta Mechanica, 152(1-4), 191–196.
Mahapatra, T.R., Nandy, S.K. & Gupta, A.S. (2009). Magnetohydrodynamic
stagnation point flow of a power-law fluid towards a stretching sheet.
International Journal of Non-Linear Mechanic, 44(2), 124–129.
Mahapatra, T.R. & Nandy, S.K. (2011). unsteady stagnation-point flow and heat
transfer over an unsteady shrinking sheet. International Journal of Applied
Mathematics & Mechanics, 16(7), 11-26.
Mahapatra, T.R. & Nandy, S.K. (2013). Stability of dual solutions in stagnation-
point flow and heat transfer over a porous shrinking sheet with thermal
radiation. Meccanica, 48, 23-32.
Maiga, S. E. B., Nguyen, C. T., Galanis, N. & Roy, G. (2004), "Heat transfer
behaviour of nanofluids in uniformly heated tube", Superlattice and
Microstructure 35(3-6), 543-557.
Roy, G., S.J. Palm, S.J. & Nguyen, C.T. (2005). Heat Transfer enhancement by
using nanofluids in forced convection flows. International Journal of Heat
and Fluid Flow, 26(4), 530-546.
Makinde, O., Khan, W. & Khan, Z. (2013). Buoyancy effects on MHD stagnation
point flow and heat transfer of a nanofluid past a convectively heated
stretching/shrinking sheet. International Journal of Heat and Mass Transfer,
62(0), 526-533.
Maplesoft. (2015). Maplesoft Services. Retrieved on June 24th
, 2014, from
http://www.maplesoft.com/brochures/PDFs/maple18/Maplesoft_SolutionsEng
ineers.pdf
Masuda, H., Ebata A., Teramae K. & Hishinuma N. (1993). Alteration of thermal
conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu
Bussei, 7(4), 227–233.
108
Mankinde, O.D. & Aziz, A. (2011). Boundary layer flow of a nanofluid past a
stretching sheet with convective boundary conditions. International Journal
on Thermal Science, 50(5), 1326-1332.
Miklavcic, M. & Wang, C.Y. (2006). Viscous flow due to a shrinking sheet. Applied
Mathematics, 64(2), 283-290.
Mori, H. (1965). Transport, collective motion, and Brownian motion. Progress of
Theoretical Physics, 33(3), 423-455.
Murthy, P.V. & Singh, P. (1997). Effect of viscous dissipation on a non-Darcy
natural convection regime. International Journal of Heat and Mass Transfer,
40(6), 1251-1260.
Murshed, S.M.S., Leong, K.C. & Yang, C. (2005). Enhanced thermal conductivity of
TiO2-water based nanofluids. International Journal of Thermal Sciences,
40(4), 367-373.
Murshed, S.M.S., Leong, K.C. & Yang, C. (2009). A combined model for the
effective thermal conductivity of nanofluids. Applied Thermal Engineering,
29(11), 2477-2483.
Nan, C.W., Birringer, R., Clark, D.R. & Glieter, H. (1997). Effective thermal
conductivity of particulate composite with interfacial thermal resistance.
Journal of Applied Physics, 81(5), 6692 -6699.
Nazar, R., Amin, N., Filip, D. & Pop, I. (2004). Unsteady boundary layer flow in the
region of the stagnation point on a stretching sheet. International Journal of
Engineering and Science, 42, 1241–1253.
Nield, D.A. & Kuznwtsov, A.V. (2009). Thermal instability of porous medium layer
saturated by a nanofluid. International Journal of Heat and Mass Transfer,
52(25), 5796-5801.
Nourazar, S.S., Habibi, M. M. & Simiari, M. (2011). The HPM applied to MHD
nanofluid flow over a horizontal stretching plate. Journal of Applied
Mathematics, 1-17. doi:10.1155/2011/876437.
Nguyen, C.T., Desgranges, F., Roy, G., Galanis, N., Mare, T., Boucher, S. & Angue,
M. H. (2007). Temperature and particle-size dependent viscosity data for
water based nanofluids-hysteresis phenomenon. International Journal of Heat
and Fluid Flow, 28(6), 1492-1506.
Oosthuizen, P.H. & Naylor, D. (1999). An introduction to convective heat transfer
analysis. New York: WCB/McGraw Hill.
109
Pal, D., Mandal, G. & Vajravelu, K. (2014). Flow and heat transfer of nanofluids at a
stagnation point flow over a stretching/shrinking surface in a porous medium
with thermal radiation. Applied Mathematics and Computation, 238, 208–
224.
Penas, J.R.V., Ortiz de Zarare, J.M. & Khayet, M. (2008). Measurement of the
thermal conductivity of nanofluids by the multi current hotwire method.
Journal of Applied Physics, 104(4), 044314.
Pop, S.R., Grosan, T. & Pop, I. (2004). Radiation effects on the flow near the
stagnation point of a stretching sheet. Technische Mechanik, 25(2), 100-106.
Purcell, E.M. (1997). Life at low Reynolds number. American Journal of Physics,
45(1), 3-11.
Putnam, S.A., Cahill, D.G., Braun, P.V., Ge, Z. & Shimmim, R.G. (2006). Thermal
conductivity of nanoparticle suspensions. Journal of Applied Physics, 99(8),
084308-084308.
Rohni, A.M., Ahmad, S. & Pop, I. (2012). Flow and heat transfer over an unsteady
shrinking sheet with suction in nanofluids. International Journal of Heat and
Mass Transfer, 55(7–8), 1888–1895.
Sanders, C.J. & Holman, J.P. (1972). Franz Grashof and Grashof number.
International Journal of Heat and Mass Transfer, 15(3), 562-563.
Sandnes, B. (2003). Energy efficient production, storage and distribution of solar
energy. University of Oslo: Ph.D. Thesis.
Samir, K. N. & Ioan Pop, (2014). Effects of magnetic field and thermal radiation on
stagnation flow and heat transfer of nanofluid over a shrinking surface.
International Communication in Heat and Mass Transfer, 53(4), 50-55.
Sciuto, G. (2012). Innovative latent heat thermal storage elements design based on
nanotechnologies. University of Trieste: Ph.D Thesis.
Siegel, R. & Goldstein, M.E. (1972) Theory of heat transfer in a two-dimensional
porous cooled medium and application to an eccentric annular region. Journal
of Heat Transfer, 94(4), 425-431.
Shaughnessy, E.J., J., Katz, I. M., & Schaffer, J. P. (2005). Introduction to Fluid
Mechanics. New York: Oxford University Press.
Sherman, F.S. (1990). Viscous Flow. New York: McGraw-Hill, Inc.
Sparrow, E.M. & Cess, R.D. (1978). Radiation Heat Transfer. Washington:
Hemisphere.
110
The Encyclopaedia of Earth. (2011). Chemical reaction. Retrieved on January 10th
,
2015, from http://www.eoearth.org/view/article/171508/
Vandiver, J.K., & Marcollo, A. (2003). High mode number VIV experiments. In
IUTAM symposium on integrated modelling of fully coupled fluid structure
interactions using analysis, computations and experiments, 75, 211-231.
Venerus, D., Buongiorno, J., Christianson, R., Townsend, J., Bang, I.C., Chen, G. &
Zhou, S. (2010). Viscosity measurements on colloidal dispersions
(nanofluids) for heat transfer applications. Journal of Applied Rheology,
20(4), 44582. doi: 10.3933/ApplRheol-20-44582.
Wang. C.Y. (2008). Stagnation point flow towards a shrinking sheet. International
Journal of Non-Linear Mechanical systems, 43, 377–382.
Welty, J.R., Wicks, C.E., Rorrer, G. & Wilson, R.E. (2009). Fundamentals of
Momentum, Heat and Mass Transfer. Hoboken: John Wiley &Sons.
White, F. M. (2006). Viscous Fluid Flow. 3rd
ed. New York: McGraw-Hill.
Xuan, Y. & Li, Q. (2000). Heat transfer enhancement of nanofluids. International
Journal of Heat and Fluid Flow, 21(1), 58-64.
Yurusoy, M. & Pakdemirili, M. (1997). Symmetry reductions of steady two
dimensional boundary layers of some non-Newtonian Fluids. International
Journal of Engineering Science, 35(2), 731-740.
Yurusoy, M. & Pakdemirili, M. (1999a). Group classification of a non-Newtonian
fluid model using classical approach and equivalence transformations.
International Journal of Non-linear Mechanics, 34(2), 341-346.
Yurusoy, M. & Pakdemirili, M. (1999b). Exact solutions of boundary layer equations
of a special non-Newtonian fluid over a stretching surface. Mechanics
Research communications, 26(1), 171-175.
Yurusoy, M., Pakdemirili, M. & Noyan. (2001). Lie Group analysis of creeping flow
of a second grade nanofluid. International Journal of Non-Linear Mechanics,
36(6), 955-960.