ABSTRACT
This dissertation presents an analytical study on thermo-solutal natural convection flow in an
inclined channel filled with porous material. Two problems are considered in this work. In both
problems, the flow is assumed to be hydrodynamically and thermally fully developed and the
mathematical equations governing the flow were formulated. The first problem focused on Soret
effect on steady natural convection of double-diffusive flow of heat generating fluid. While the
second problem examines the transpiration and Soret effects on thermo-solutal convection flow
of heat generating fluid in an inclined porous channel filled with porous material. Using
appropriate transformations, the differential equations governing the flow were transformed and
solved analytically to obtain an expression for velocity, concentration, temperature, skin friction,
Nusselt number, Sherwood number as well as volumetric flow rate. The effects of various
governing parameters on fluid velocity, concentration, temperature, skin friction, Nusselt
number, local Sherwood number and volumetric flow rate are shown in graphs and tables and
analyzed in detail. During the course of this investigation, it is found that Soret effect, angle of
inclination and viscosity of the porous material are very effective in controlling the
hydrodynamics within the channel.Furthermore, there is a rise in the skin friction on both plates
due to increase in heat generation parameter , permeability parameter 𝐷𝑎 and angle of
inclination 𝑎 while a fall is observed in skin friction on both plates due to increase in Prandtl
number 𝑃𝑟, Soret number 𝑆𝑜 and Schmidt number 𝑆𝑐.
TABLE OF CONTENTS
Cover page
Fly leaf ……………………………………………………………………………………………… Error! Bookmark not defined.
Title page ………………………………………………………………………………………….. Error! Bookmark not defined.
Declaration ………………………………………………………………………………………………………………………………… iii
Certification……………………………………………………………………………………………………………………………….. iv
Dedication ………………………………………………………………………………………………………………………………….. v
Acknowledgement ……………………………………………………………………………………………………………………… vi
Abstract ……………………………………………………………………………………………………………………………………. vii
Tables of Contents ……………………………………………………………………………………………………………………. viii
List of Figures …………………………………………………………………………………………………………………………….. xi
Lis of Tables ……………………………………………………………………………………………………………………………….xiii
Nomenclatures and Greek Letters ……………………………………………………………………………………………….. xiv
Greek Letters …………………………………………………………………………………………………………………………….. xv
Dimensionless Quantities. ………………………………………………………………………………………………………….. xvi
CHAPTER ONE …………………………………………………………………………………………………………………………….. 1
GENERAL INTRODUCTION …………………………………………………………………………………………………………….. 1
1.1 Background of the Study ………………………………………………………………………………………………… 1
1.2 Statement of the Problem ……………………………………………………………………………………………… 2
1.3 Aim and Objectives of the Study ……………………………………………………………………………………… 2
1.4 Research Methodology ………………………………………………………………………………………………….. 3
1.5 Organization of the Dissertation ……………………………………………………………………………………… 3
1.6 Significance of the Study ………………………………………………………………………………………………… 3
1.7 Scope of the Study ………………………………………………………………………………………………………… 4
1.8 Basic Definitions ……………………………………………………………………………………………………………. 4
1.9 Basic Equations ……………………………………………………………………………………………………………… 5
CHAPTER TWO ……………………………………………………………………………………………………………………………. 8
LITERATURE REVIEW ……………………………………………………………………………………………………………………. 8
2.1 Introduction…………………………………………………………………………………………………………………….. 8
2.2 Natural Convection Flow in Channel ………………………………………………………………………………….. 8
ix
2.3 Heat Generation …………………………………………………………………………………………………………….. 9
2.4 Soret Effect on Double-diffusive Free Convection Flow ……………………………………………………… 10
2.5 Suction/Injection …………………………………………………………………………………………………………… 12
CHAPTER THREE ………………………………………………………………………………………………………………………… 14
PROBLEM FORMULATIONS AND SOLUTION ………………………………………………………………………………….. 14
3.1 Introduction …………………………………………………………………………………………………………………. 14
3.2 Natural convection double-diffusive flow of heat generating fluid in an inclined channel filled with porous material ………………………………………………………………………………………………………………. 14
3.2.1 Mathematical formulation ………………………………………………………………………………………… 14
3.3 Transpiration and Soret effects on thermo-solutal convection flow of heat generating fluid in an inclined channel filled with porous material ……………………………………………………………………………… 16
3.3.1 Mathematical formulation …………………………………………………………………………………………. 16
3.4 Dimensionless Analysis …………………………………………………………………………………………………… 18
3.5 Governing Equations in Dimensionless Form …………………………………………………………………….. 19
3.5.1 Governing equations in dimensionless form of problem 3.2 ………………………………………….. 19
3.5.2 Skin-friction of problem 3.2………………………………………………………………………………………. 20
3.5.3 Volumetric flow rate of problem 3.2 ………………………………………………………………………….. 21
3.5.4 Nusselt number of problem 3.2 ………………………………………………………………………………… 22
3.5.5 Sherwood number of problem 3.2 …………………………………………………………………………….. 22
3.5.6 Governing equations in dimensionless form of problem 3.3 ………………………………………….. 22
3.5.7 Skin-friction of problem 3.3………………………………………………………………………………………. 23
3.5.8 Volumetric flow rate of problem 3.3 ………………………………………………………………………….. 24
3.5.9 Nusselt number of problem 3.3 ………………………………………………………………………………… 24
3.5.10 Sherwood number of problem 3.3 …………………………………………………………………………… 24
CHAPTER FOUR …………………………………………………………………………………………………………………………. 25
RESULTS AND DISCUSSION………………………………………………………………………………………………………….. 25
4.1 Introduction ………………………………………………………………………………………………………………….. 25
4.2 Natural convection double-diffusive flow of heat generating fluid in an inclined channel filled with porous material ………………………………………………………………………………………………………………. 26
4.3 Transpiration and Soret effects on thermo-solutal convection flow of heat generating fluid in an inclined channel filled with porous material ……………………………………………………………………………… 38
CHAPTER FIVE …………………………………………………………………………………………………………………………… 59
SUMMARY, CONCLUSION AND RECOMMENDATIONS ……………………………………………………………………. 59
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5.1 Summary …………………………………………………………………………………………………………………………. 59
5.2 Conclusion ……………………………………………………………………………………………………………………….. 60
5.3 Recommendations ……………………………………………………………………………………………………………. 61
References ……………………………………………………………………………………………………………………………….. 62
APPENDICES ……………………………………………………………………………………………………………………………… 66
APPENDIX I ……………………………………………………………………………………………………………………………. 66
APPENDIX II …………………………………………………………………………………………………………………………… 69
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background of the Study
Natural convection flow is caused by density differences induced by the temperature gradients, chemical composition gradients and material compositions. It is a type of heat transport in which the fluid motion is not generated by an external source (pump, fan, suction devices). The driving force is buoyancy. Fluid flows generated by combined temperature and concentration gradients are referred to as thermo-solutal convection. Considerable attention has been paid to natural convection for over ten decades as it plays important roles in different applications, such as the early stages of melting and in the transient heating of insulating air gap by heat input at the start-up of furnaces. Also, time dependent laminar natural convection is likely to find wider use as it could provide the flow mechanism in some types of solar heating and ventilating passive systems. In modern electronic equipment, the vertical circuit boards include heat generating elements, and this situation can be modeled by parallel heated plates with upward flow in the intervening space. Heat generation/absorption plays significant role in engineering problems such as plasma studies, nuclear reactors, geothermal energy extraction and the boundary layer control in the field of aerodynamics.
The diffusion of mass due to temperature gradient is called Soret or thermal diffusion effect. Soret effect arises when the mass flux contains a term that depends on the temperature gradient. The major focus of our study is the effect on natural steady convection flow with injection of some diffusing species. Convection in binary fluids is considerably more complicated than that
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in pure fluids. Both temperature and concentration gradients contribute to the initiation of convection and each may have stabilizing or destabilizing effects. Even when a concentration gradient is not externally imposed (thermosolutal problem) it can be created by applied thermal gradient via the Soret effect.
1.2 Statement of the Problem
Several studies have been conducted on fluid flow involving heat and mass transfer and many of the works have neglected thermal diffusion effects. However, when the temperature of fluid increases its concentration is not affected.Therefore, this study investigates the thermal diffusion (Soret effect) on steady natural convection double diffusive flow of heat generating/absorbing fluid in an inclined channel filled with porous material. This dissertation will further consider the opposing flow between heat and mass transfer with suction/injection.
1.3 Aim and Objectives of the Study
This work is aimed at studying the hydrodynamics, thermodynamics and solutal behaviour of fluid in a channel formed by two inclined parallel plates. This will be achieved through the following objectives which are to:
i. investigate the influence of suction/injection in the control of fluid flow and the thermodynamics in an inclined channel.
ii. investigate the effects of Soret parameter on the hydrodynamics in inclined channel.
iii. determine the influence of porous material on skin friction, mass flux, Nusselt number and Sherwood number within the inclined channel.
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1.4 Research Methodology
The methodology adopted in the realization of the set of objectives is characterized as follows. Firstly, we review existing literatures and extend them to include different physical geometries. To obtain the analytical solutions, we use the method of undetermined coefficients and direct integration. We then obtain the numerical values of the analytical solutions and the computer package, MATLABR2015a is used to plot the graphs. The last part has to do with physical interpretation of the graphs so as to discuss the influence of the governing parameters on the flow and draw conclusions
1.5 Organization of the Dissertation
This dissertation is grouped into five chapters with references and appendices. Chapter one contains the general introduction of the dissertation. Chapter two is the literature review while chapter three contains the problem formulations and solution. The results and discussion are contained in chapter four. Chapter five is the summary, conclusion and recommendation, while references and appendices follow.
1.6 Significance of the Study
This study reveals the role of Soret on steady natural convection of double-diffusive flow of heat generating fluid. It would provide further details on the behavior of fluid in an inclined channel with heat generation as well as suction and injection. This research work has many applications in geophysics, oil reservoir, ground water system and boundary layer control in the field of aerodynamics. Further, it is hoped that the results obtained will not only provide useful information for engineering applications but also serve as an improvement on the previous studies.
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1.7 Scope of the Study
The formulation, analysis and results obtained in this dissertation are theoretical. No experimental work has been carried out on this dissertation. The governing ordinary differential equations were employed and conclusions were drawn based on the behavior of the graphs and tables.
1.8 Basic Definitions
i. Buoyancy:is the force that causes objects to float.
ii. Boussinesq approximation: is the assumption that the fluid is considered under variations of temperature and density.
iii. Double diffusive convection: fluid flow generated by combined temperature and concentration gradients is referred to as double-diffusive convection or thermol-solutal convection.
iv. Dimensionless quantity: is a quantity without an associated physical dimension.
v. Free or Natural convection: Free convection arises in the fluid when temperature changes cause density variation leading to buoyancy forces acting on the fluid elements.
vi. Heat generation/Absorption: refers to the energy transferred as a result of a difference in temperature.
vii. Heat flux: is defined as the amount of heat transferred per unit area per unit time from or to a surface.
viii. Injection: is an infusion method of putting fluid in to the body.
ix. Inclined channel: is a channel inclined at an angle to the horizontal.
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x. Mass flux: is the rate of mass flow per unit area.
xi. Nusselt number: is the rate of heat transfer.
xii. Porous medium or porous material: is a material containing pores (voids).
xiii. Suction: is the flow of a fluid in to partial vacuum or region of low pressure.
xiv. Skin friction: is the shear stress that occurs between the fluid and the solid surface of
boundaries.
xv. Sherwood number: is a dimensionless number used in mass transfer operation.
xvi. Soret or thermal diffusion effect: the diffusion of mass due to temperature gradient
is called Soret or thermal diffusioneffect.
xvii. Isothermal/Isoflux thermal boundary condition: is one that takes place at constant
temperature.
1.9 Basic Equations
These four equations all together fully described the fundamental characteristic of fluid flow in
an inclined channel.
i. Continuity equation
the continuity equation in Cartesian coordinates is given as:
0
v
t
(1.1)
For viscous incompressible fluid flow, the continuity equation (1.1) reduces to:
v 0
(1.2)
ii. Momentum equation
The general expression for conservation of momentum (The Navier-Stokes equation)
is:
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u u p u F
t
u
1 1 2
(1.3)
Where is the kinematic viscosity, 𝜌 is fluid density and 𝐹 is the sum of any body
forces. In the present problem, a steady flow is considered so that time derivative of
the velocity, 0
t
u
. In addition, the flow is hydrodynamically fully developed so that
the advection terms are ignored and the momentum equation (1.3) reduces to:
0
1 2 u F
(1.4)
Where the pressure gradient is ignored in a natural convection problem. Here, the
density is assumed to be partially fixed and partially dependent on the fluid
temperature as well as concentration so that the equation of state is given as:
0 0 0 0 0 0 T T C C
(1.5)
Where 0 is the fluid density at temperature 𝑇0 and concentration 𝐶0.
From (1.5), it is observed that
0 0 0 0 0 T T C C
(1.6)
In natural convection problem, the body force term F g 0 is the gravitational
body force, acting in the opposite direction of the buoyancy induced flow so that
0 2 0
g
u (1.7)
So that:
0 2 0 0 0 0
T T C C g
u (1.8)
Invoking Boussinesq approximation in which the change in density is negligible so
that 0 , then the momentum equation becomes:
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0 0 0
2 u g T T g C C (1.9)
iii. Energy equation
The energy equation that describes the conservation of energy in the system is given
as:
k T
Dt
DT
Cp 2 𝜻(1.10)
Where Cp is the specific heat constant pressure, k is the thermal conductivity and𝜻is
the dissipation function. Here, the dissipation function is ignored since the effect of
pressure on the density is neglected. Also, the flow is steady and hydrodynamically
fully developed, the energy equation (1.10) reduces to:
0 2 T
Cp
k
(1.11)
Equation (1.11) can be written as:
0 2 T where
Cp
k
(1.12)
iv. Concentration equation
The general equation of concentration is:
D c vc R
t
c
(1.13)
Where c is the variable of interest (species concentration for mass transfer,
temperature for heat transfer), 𝐷 is the diffusivity (also called diffusion coefficient),
such as mass diffusivity for particle motion or thermal diffusivity for heat transport.
v
is the velocity field that the quantity is moving with. It is also a function of time and
space. 𝑅 describes “ sources” or “ sinks” of the quantity 𝑐. Here, the diffusion
coefficient is constant, there are no sources or sinks, and the velocity field describes
an incompressible flow (i.e., it has zero divergence). Then the formula simplifies to:
D c v c
t
c
2 (1.14)
Since the flow is steady and hydrodynamically fully developed, 0
t
c
, so equation
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(1.14) reduces to:
0 2 D c (1.15)
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