ABSTRACT
This research work titledA STUDY ON MIXED CONVECTION FLOW IN DIFFERENT
CHANNELS, is divided into three parts.The first part analytically simulates the mixed
convection flow in the steady, laminar fully developed region of a parallel porous plate channel
filled with porous material .The Darcy-Brinkman model and Boussinesq are employed. While
the second part investigates analytical solutions for fully developed mixed convection flow in a
vertical channel formed by two infinite vertical parallel plates partially filled with porous
material.The third part studies the steady mixed convection flow in an annulus partially filled
with porous material. In the second and third problems, the Brinkman-extended Darcy model is
used to simulate momentum transfer in the porous region and the Navier-Stokes equation is used
to simulate the momentum transfer in the clear fluid region. The clear fluid and porous regions
are coupled by equating the velocity, temperature, heat flux and by considering shear stress jump
condition at the interface.In the first part, the effect of suction/injection on mixed convection
flow between vertical porous plates filled with porous material is investigated. The second part
investigates the relative significance of the governing parameters on the velocity,interface
velocity and temperature in the channel. While the third part studies the impact of the Darcy
number and the adjustable coefficient in the stress-jump condition at the interface of fluid/porous
layer on flow formation inside the annulus. Moreover,a dimensionless temperature 0 is
introduced which is related to the usual
Re
Gr
parameter. The exact solutions for momentum and
energy equations obtained are presented graphically and discussed.
TABLE OF CONTENTS
Cover Page ……………………………………………………………………………………………………….. i Fly Leaf …………………………………………………………………………………………………………… ii Title page …………………………………………………………………………………………………………. iii Declaration ……………………………………………………………………………………………………….. iv Certification ……………………………………………………………………………………………………… v Dedication ………………………………………………………………………………………………………… vi Acknowledgement …………………………………………………………………………………………….. vii Abstract …………………………………………………………………………………………………………… viii Table of Content ……………………………………………………………………………………………….. ix List of Figures …………………………………………………………………………………………………… xi List of Appendices …………………………………………………………………………………………….. xv Nomenclature and Greek Letters ………………………………………………………………………….. xvi Chapter One …………………………………………………………………………………………………….. 1 General Introduction ………………………………………………………………………………………….. 1
1.1 Introduction …………………………………………………………………………………………………. 1
1.2 Basic Definitions………………………………………………………………………..3
1.3 Dimensionless Quantities……………………………………………………………….4 1.4 Objectives of the Study ………………………………………………………………………………….. 4 1.5 Research Methodology ………………………………………………………………………………….. 5 1.6 Statement of the problem………………………………………………………………..5 1.7 Limitations of the study…………………………………………………………………6 1.8 Significance of the study………………………………………………………………..6 1.9 Organization of the Thesis ……………………………………………………………………………… 6 Chapter Two…………………………………………………………………………………………………….. 7 Literature Review ………………………………………………………………………………………………. 7 2.1 Introduction …………………………………………………………………………………………………. 7 2.2 Mixed Convection ………………………………………………………………………………………… 7 2.3 Porous Media ………………………………………………………………………………………………. 8 Chapter Three ………………………………………………………………………………………………….. 13 Mathematical Analysis ……………………………………………………………………………………….. 13
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3.1 Introduction …………………………………………………………………………………………………. 13 3.2 Role of suction/injection on mixed convection flow in a vertical porous channel filled with porous material …………………………………………………………………………………… 13 3.3 Fully developed mixed convection flow between vertical parallel plates partially filled with porous material……………………………………………………………………………………. 15 3.4 Steady fully developed non-Darcian mixed convection in an annulus partially filled with porous materials and partially filled with clear fluid. ………………………………… 16 3.5 Non-Dimensionlisation of the equations ……………………………………………………………. 18 3.6 Solution of Problems …………………………………………………………………………………….. 21 3.7 Solution of Problem 3.1 ………………………………………………………………………………….. 21 3.8 Solution of Problem 3.2 ………………………………………………………………………………….. 24 3.9 Solution of Problem 3.3 …………………………………………………………………………………. 26 Chapter Four ……………………………………………………………………………………………………. 30 Discussion of the Results …………………………………………………………………………………….. 30 4.1Introduction …………………………………………………………………………………………………… 30 4.2 Discussing the Results of Problem 3.1 ………………………………………………………………. 30 4.3 Discussing the Results of Problem 3.2 ……………………………………………………………… 44 4.4 Discussing the Results of Problems 3.3 …………………………………………………………….. 49 Chapter Five …………………………………………………………………………………………………….. 55 Summary and Conclusions …………………………………………………………………………………… 55 5.1 Summary …………………………………………………………………………………………………….. 55 5.2 Conclusions ………………………………………………………………………………………………….. 56 5.3 Recommendations……………………………………………………………………….57 References ……………………………………………………………………………………………………….. 58 Appendices ………………………………………………………………………………………………………. 64 Appendix I ………………………………………………………………………………………………………… 64 Appendix II ……………………………………………………………………………………………………….. 66 Appendix III ……………………………………………………………………………………………………… 70
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CHAPTER ONE
GENERAL INTRODUCTION
INTRODUCTION
Fluid flow occurs in a wide range of practical applications which include crude oil extraction, thermal insulation, chemical catalytic convertors, storage of grains, pollutant dispersion in aquifers, buried electrical cables, food industry, chemical transport simulations and many other biological, geophysical, engineering and environmental applications.
A porous medium is a material consisting of a solid matrix with an interconnected void. The interconnection of the void allows the flow of fluids through the material. The flow and the spread of fluid through random porous media such as soils, bed packing’s, ceramic and concrete is an important topic of research because of its wide range of application in environmental and technological processes (Sahimi,1994). Darcy was the first to observe that under certain conditions the volume rate of water through a pipe packedwith sand was proportional to the negative of the pressure gradient. This relationship is known as Darcy’s law. Darcyflow is an expression of the dominance of viscous forces applied by the solid porous matrix on the interstitial fluid and is oflimited applicability. At higher fluid velocities by increasing inertial forces, the ratio of pressure drop to velocity gradually deviates from Darcy’s law. Many other results were presented on Darcy (1856) as compiled and presented by (Narasimhan ,2006). The flow through porous media capability enables engineers to simulate fluid flow through media such as ground rock, filters and catalyst beds. The simulation of underground flow through porous rock can enable engineers to predict the movement of contaminated fluid from a solid
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waste landfill into a drinking water supply. Moreover, porous media can provide sites for chemical catalyst or absorption of components of the fluid. Additionally, the standard Darcy’s law material model (which relates volumetric flow and pressure drop with properties of the fluid and media), the fractional power Darcy’s law is also supported. This latter material model incorporates inertial effects for high Reynolds number applications. Fluid flow in a composite channel partially filled with a porous medium and partially filled with a clear fluid, occurs in practical applications such as geophysical, biomedical, engineering and environmental applications(Kakac et al.;1991;, Kuznetsov; 1998;1996). If a solidifying alloy does not have a eutectic composition, the frozen part of the casting is separated from the liquid part by a mushy zone, which can be viewed as a porous medium with variable permeability (Kakac et al. ,1991). In addition, the use of porous substrates to improve forced convection heat transfer in chemicals, which is considered as a composite of fluid and porous layers, finds applications in heat exchangers, electronic cooling, heat pipes, filtration and chemical reactions etc. In these applications engineers avoid filling the entire channel with a solid matrix to reduce the pressure drop. The problem of fluid flow in the porous medium/clear fluid interface was first investigated by (Beavers Joseph ,1967). Comprehensive literature survey concerning this subject is amply documented in the monograph by (Nield and Bejam ,1992) and (Kaviany ,1991), as reported by (Kuznetsov ,1996).
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1.1 BASIC DEFINITIONS
1. Free or Natural Convection: Is a type of heat transfer, in which the fluid motion is not generated by any external source but only by density differences in the fluid occurring due to temperature gradients.
2. Forced Convection: Is a heat transfer between a moving fluid and a solid surface, as a result of external source.
3. Mixed Convection: Is a type of heat transfer between a moving fluid and a solid surface, as a result of both natural and external source
4. Suction/injection: Is the increase/decrease of fluid flowing through the walls of the channel.
5. Boussinessq approximation: Is the assumption that the fluid flow is considered under little variations of temperature and density.
6. Porous Media: A porous medium is a material consisting of a solid matrix with an interconnected void. The interconnected of the void allows the flow of fluids through the material.
7. Permeability: Is the ability of a material to transmit fluid through it, which is the property of the porous media only and not the fluid.
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1.2 DIMENSIONLESS QUANTITIES
A dimensionless quantity is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all the units cancel out.A quite number of these quantities were used in this thesis and brief notes of the quantities used are:
1. Grashof Number (Gr): Grashof Number is the ratio of the buoyancy force to the viscous force acting on the fluid:
Gr =𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑒𝑐𝑒𝑠𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒𝑠
2. Reynolds number (Re): is the ratio of inertial forces to viscous forces.
1.3Aims And Objectives of The Study
The main aims and objectives of the work are:
1. Investigation of the effect of suction/injection on mixed convection flow between vertical porous plates filled with porous material
2. Investigation of the relative significance of the governing parameters on the interface velocity, the velocity and the temperature on mixed convection flow in a channel partially filled with porous fluid and partially filled with clear fluid.
3. Study of the effect of Darcy number and the stress jump condition at the interface of fluid/porous layer for the steady fully developed mixed convection flow in a vertical annulus partially filled with porous material.
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1.4Research Methodology
The methodology employed in order to achieve the set objectives is that, firstly, existing literature was reviewed. Secondly, the parameters involved in the governing equations with their boundary conditions in the entire models will be non-dimensionalized. The non-dimensional models are solved analytically. Thirdly, the various boundary conditions were tested on the solutions obtained using MATLAB computer package to present the graphs of the solutions obtained. Finally, the graphs were interpreted so that the impact of the governing parameters can be discussed and draw conclusions. 1.5 Statement of The Problem The study would seek to simulate analytically mixed convection flow in the steady, laminar, fully developed region of
a) parallel porous plate channel filled with porous material:
b) vertical parallel plates partially filled with porous material:
c) an annulus partially filled with porous material.
The Oberbeck-Boussinesq approximation would be employed to formulate the model and the reference temperature would be such that the pressure gradient vanishes (Rossi di Schio,2010); it involves the coupling of fluid flow and temperature flux. The one-dimensional Navier-Stokes equation with appropriate boundary conditions would describe the problem. Matlab codes would also be developed to obtain the behavior of the results.
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1.6 Limitation of the Study This study considered the analytical investigation of steady, laminar, fully developed flow of non-Darcian fluid in a porous parallel vertical plates and a vertical annulus with a dimensionless reference temperature such that the pressure gradient vanishes. The results obtained in this research may have been different if the walls of the channel are kept at a uniform temperature. A numerical approach can also be considered of the mixed convection in a porous parallel vertical plates and in an annulus partially filled with porous material. 1.7 Significance of the Study The significance of this study are; The solutions of natural convection obtained by Wiedman and Medina (Wiedman and Medina, 2008) can be generated using mixed convection with suction/injection in porous medium. Also, the choice of the reference temperature such that the pressure gradient vanishes can be applied in channel partially filled with porous medium.
1.8 Organization of The Thesis
This thesis is arranged and organized in five chapters, reference and appendix. Chapter one presents the general introduction, while chapter two gives the review of related literature. Chapter three presents the mathematical analysis and the non-dimensionalization of the three problems and their solutions. Chapter four presents the discussions of the results obtained in chapter three. Then chapter five presents the summary and conclusions of the research.
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