A Study on Mixed Convection Flow in Different Channels

A Study on Mixed Convection Flow in Different Channels.

ABSTRACT

This research work titled A 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 introduced which is related to the usual 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 General Introduction

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 Literature Review

2.1 Introduction .. 7
2.2 Mixed Convection ….. 7
2.3 Porous Media …….. 8

Chapter Three Mathematical Analysis

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 Discussion of the Results

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 Summary and Conclusions

5.1 Summary … 55
5.2 Conclusions…. 56
5.3 Recommendations…….57
References …. 58
Appendices …….. 64

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 packed with 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 of limited 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

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.

REFERENCES

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Cheng, C.H, Huag, H.S, Huang, W.H, (1990) Flow reversal and heat transfer of fully developed Mixed convection in vertical channels. Journal of Thermophysics Heat Transfer,vol.4 pp.375-383.

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