# A Study on Mixed Convection Flow in Different Channels

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**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 …………………………………………………………………………………………………….. 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

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

**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.

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).

**REFERENCES**

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Aung, W. and Woyku, G. (1986) Theory of fully developed combined convection including Flow reversal.ASME Journal of Heat Transfer vol.108, pp.485-488

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Beavers, G.S, Joseph, D.D, (1967) Boundary Conditions at a Natural Permeable Wall Fluid Mechanics j. Fluid Mechanics vol. 30 pp. 197-207

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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.

De Lemos, M.J.S., (2006) Turbulence in porous media: Modeling and Applications, Elsevier, Oxford.

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Hamadah, T.T and Wirtz, R.A., (1991) Analysis of laminar fully developed mixed convection in A vertical channel with opposing buoyancy. ASME Journal of Heat Transfer, vol. 113 pp. 507-510.

Havstad, M.A., Burns, P.J., (1982) Convective Heat Transfer in vertical cylindrical annuli filled With a porous medium. Int. J. Heat Mass Transfer vol. 25 pp.1755-1766.

Ingham, D.B., Pop, I.,(Eds) (2005) Transport Phenomena in porous media, Elsevier, Oxford. Jha, B.K., Ajibade, A.O., (2009) Free convective flow of heat generating/absorbing fluid

Between vertical porous plates with periodic heat input. Int. Commun. Heat Mass Trans. Vol. 36 pp. 624-631.

Jha, B.K., Ajibade, A.O., (2010)Free convective flow between vertical porous plates with Periodic heat input. Z. Angew. Math. Mech. Vol. 90 (3) pp. 185-193.

Jha, B.K., Odengle, J.O., Kaurangini, M.L., (2011) Effect of transpiration on free-convective Couette flow in a composite channel. J. Porous Media vol. 14(7) pp. 627-635.

Joshi, H.M., (1987) Fully developed natural convection in an isothermal vertical annular duct. International communication of Heat Transfer, vol.14 pp.657-667.

Kaurangini, M.L., Jha, B.K., (2010) Steady and Transient investigation of generalized couette Flow in a compositechannel with suction/injection. J. Porous Media vol. 13(10) pp.931- 943.

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