ABSTRACT
his dissertation investigatesthe effect of steady-periodic heating at the surfaces of the cylinders on natural convection flow in a tube and an annulus.Two problemsare considered in this work. In both problems, the fluid was assumed to be fully developed and the mathematical equations governing the flow were derived. The first problem presents the impact of time-periodic heating at the surface of a tube on fluid velocity, temperature, rate of heat transfer and skin friction.While the second problem examines the effect of periodic heating at the surfaces ofthe cylinders on natural convection flow in an annulus.Using suitable transformation, the partial differential governing equations were transformed to their corresponding ODE, representing the steady and periodic regimes.Closed-form expressions for velocity, temperature, skin-friction, mass flow rate and rate of heat transfer which was expressed as Nusselt number were obtained in terms of modified Bessel’s function of first and second kinds for both cases.The solutions obtained were graphically represented and the effects of periodic heating (Strouhal number 𝑆𝑡), Prandtl number 𝑃𝑟 and aspect ratio 𝜆 on fluid velocity, temperature, rate of heat transfer and skin-friction were investigated. Result indicates that the role of Strouhal number, Prandtl number and aspect ratio is to decrease fluid velocity, temperature, and skin-friction at the surfaces of the cylinders.Also, the rate of heat transfer between the surfaces of the cylinders and the fluid increases with increase in Strouhal number 𝑆𝑡 , Prandtl number 𝑃𝑟 and aspect ratio 𝜆 .Furthermore, for small Strouhal number 𝑆𝑡 ,velocity, temperature, rate of heat transfer, mass flow rate and skin-friction of the time-periodic regime (periodic part) reduces to steady regime. At this point, the steady and periodic regimes make equal contributions on overall velocity. In addition, for large Strouhal number 𝑆𝑡 , the periodic regime contributes an adverse effect on the overall fluid temperature and velocity.
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TABLE OF CONTENTS
Fly page ………………………………………………………………………………. Error! Bookmark not defined.
Title page…………………………………………………………………………….. Error! Bookmark not defined.
Declaration …………………………………………………………………………………………………………………….ii
Certification …………………………………………………………………………………………………………………. iii
Dedication ……………………………………………………………………………………………………………………. iv
Acknowledgements ………………………………………………………………………………………………………… v
Abstract ………………………………………………………………………………………………………………………. vii
Table of Contents ………………………………………………………………………………………………………… viii
List of Figures ………………………………………………………………………………………………………………. xi
List of Table ………………………………………………………………………………………………………………… xii
List of Appendices ………………………………………………………………………………………………………. xiii
Nomenclatures and Greek Letters ………………………………………………………………………………….. xiv
Dimensionless Quantities ……………………………………………………………………………………………… xvi
Basic Equations ………………………………………………………………………………………………………….. xvii
CHAPTER ONE …………………………………………………………………………………………………………… 1
GENERAL INTRODUCTION ……………………………………………………………………………………….. 1
1.1 Background of the Study ……………………………………………………………………………………….. 1
1.2 Statement of the Problem ………………………………………………………………………………………. 1
1.3 Aim and Objectives of the Study ……………………………………………………………………………. 2
1.4 Research Methodology………………………………………………………………………………………….. 2
1.5 Organization of the Dissertation …………………………………………………………………………….. 3
1.6 Significance of the Study ………………………………………………………………………………………. 3
1.7 Scope of the Study ……………………………………………………………………………………………….. 4
1.8 Definition of Terms ………………………………………………………………………………………………. 4
CHAPTER TWO ………………………………………………………………………………………………………….. 6
LITERATURE REVIEW ………………………………………………………………………………………………. 6
2.1Introduction ………………………………………………………………………………………………………….. 6
2.2 Natural Convection Flow ………………………………………………………………………………………. 6
2.3 Periodic Heat Input in Natural Convection Flow ………………………………………………………. 7
CHAPTER THREE ……………………………………………………………………………………………………… 10
PROBLEM FORMULATIONS AND ANALYSIS …………………………………………………………. 10
3.1 Introduction ……………………………………………………………………………………………………….. 10
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3.2 Natural Convection Flow in a Vertical Tube with Periodic Heat Input ………………………. 10
3.2.1 Mathematical formulation ……………………………………………………………………………… 10
3.3 Natural Convection Flow in a Vertical Annulus with Time-Periodic Boundary Conditions ………………………………………………………………………………………………………………………………. 11
3.3.1 Mathematical formulation ……………………………………………………………………………… 12
3.4 Non- dimensionalization ……………………………………………………………………………………… 13
3.4.1 Non-dimensionalization of problem 3.2…………………………………………………………… 13
3.4.2 Non-dimensionalization of problem 3.3…………………………………………………………… 15
CHAPTER FOUR ……………………………………………………………………………………………………….. 16
SOLUTION OF THE PROBLEMS ……………………………………………………………………………….. 16
4.1 Introduction ……………………………………………………………………………………………………….. 16
4.2 Natural Convection Flow in a Vertical Tube with Periodic Heat Input ………………………. 16
4.2.1 Phase angle and amplitude of velocity and temperature for problem 3.2 ……………… 16
4.2.2 Skin-friction of problem 3.2 …………………………………………………………………………… 17
4.2.3 Rate of heat transfer for problem 3.2 ………………………………………………………………. 17
4.2.4 Volume flow rate for problem 3.2 …………………………………………………………………… 18
4.2.5 Limiting case of Prandtl number (𝐏𝐫) for velocity, skin-friction and volume flow rate of problem 3.2 ……………………………………………………………………………………………………… 18
4.3 Natural Convection Flow in a Vertical Annulus with Time-Periodic Boundary Conditions ………………………………………………………………………………………………………………………………. 18
4.3.1 Skin-friction of problem 3.3 …………………………………………………………………………… 19
4.3.2 Rate of heat transfer for problem 3.3 ………………………………………………………………. 19
4.3.3 Volume flow rate for problem 3.3 …………………………………………………………………… 20
4.3.4 Limiting case of Prandtl number (𝑷𝒓) for velocity and skin-friction of problem 3.3 20
CHAPTER FIVE …………………………………………………………………………………………………………. 21
RESULTS AND DISCUSSION ……………………………………………………………………………………. 21
5.1Introduction ………………………………………………………………………………………………………… 21
5.2 Natural Convection Flow in a Vertical Tube with Periodic Heat Input ………………………. 21
5.3 Natural Convection Flow in a Vertical Annulus with Time-Periodic Boundary Conditions ………………………………………………………………………………………………………………………………. 29
CHAPTER SIX …………………………………………………………………………………………………………… 43
SUMMARY, CONCLUSION AND RECOMMENDATIONS …………………………………………. 43
6.1Summary ……………………………………………………………………………………………………………. 43
6.2Conclusion …………………………………………………………………………………………………………. 43
6.3 Recommendations ………………………………………………………………………………………………. 44
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6.4 Contribution to Knowledge ………………………………………………………………………………….. 44
6.5 Suggestions for Further Research …………………………………………………………………………. 45
REFERENCES……………………………………………………………………………………………………………. 46
APPENDICES ……………………………………………………………………………………………………………. 50
APPENDIX I…………………………………………………………………………………………………………… 50
APPENDIX II …………………………………………………………………………………………………………. 52
APPENDIX III ………………………………………………………………………………………………………… 68
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background of the Study
With advancement in science and technology, fluid dynamics has been of great importance over the decades due to its technological, industrial and engineering applications, which include calculating forces and moment of aircraft, determining the mass flow-rate of petroleum through pipelines, predicting weather pattern, modeling fission weapon detonation, traffic engineering, rocket engines, wind turbines, oil pipelines, air condition, turbine system in power generation and air condition.Ferzigerand Peric (1996), Nishigaki et al. (2013)
The investigation of laminar natural convection in vertical tubes or annuli has received a lot of attention because of its importance in many engineering applications. The main technical applications of these researches concern cooling systems for electronic devices, electrochemical processes, nuclear reactors, and in solar energy collectors. This interest is due to the fact that researchers need a better understanding of the phenomena of heat and mass transfer in Engineering, Geophysical and Biological sciences.
1.2 Statement of the Problem
Wang (1998) investigated free convection between vertical plates with periodic heat input. In his work, he separated the solutions into steady and unsteady regime and stated the condition on which the periodic heat input is significant. But in real life situations, heat transfer is common through cylinders and annuli. It is therefore significant to study free convection flow in a vertical tube and annulus inspired by periodic heating at the surfaces of the cylinders.Hence, this dissertation endeavors to investigate the impact of periodic
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heating at the surfaces of the cylinders on natural convection flow formation in a tube and an annulus.
1.3 Aim and Objectives of the Study
The aim of this work is to study flow behavior of natural convection flow in a tube andan annulus subject to periodic heat input. The objectives to attain the set aim are to:
i. derive the equations of motion governing the problems.
ii. investigate the effect of periodic heating at the surface of the tube on velocity, temperature, skin-friction and rate of heat transferin the tube.
iii. investigate the effect of periodic heating and aspect ratio on velocity, temperature, skin-friction and rate of heat transfer in the annulus.
iv. investigate fluid flow behaviour for different types of fluid dictated by the Prandtl number (𝑃𝑟)
v. give the condition on which periodic heating on the surfaces of the cylinders is significant in both tube and annulus.
1.4 Research Methodology
The methodology adopted in the realization of the set of objectives is a five-phase approach. In phase one, we review existing literatures and extend them to include different physical geometries. Phase two is concerned with solving the mathematical model that described the study on some natural convectionflow in different geometries. To obtain the analytical solutions, we used the method of undetermined coefficient and direct integration method. In phase three, numerical values of the analytical solutions in phase two are
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obtained. Here, we used a computer package, MATLAB 12b to obtain the numerical values of the analytical solutions by writing a computer program for phase two. The numerical values obtained in phase three are presented graphically in phase four by the use of MATLAB 12b. The last phase has to do with physical interpretation of the graphs so as to discuss the influence of each of the governing parameters on the flow and draw conclusions.
1.5 Organization of the Dissertation
This dissertation is grouped into six chapters with references and appendices. Chapter one comprises basically the general introduction of the dissertation. Literature review in the research is basically contained in chapters two. Chapter three contains the mathematical analysis of the problems with the non-dimensionalisation. Chapters four and five contain the solutions of the problems and discussion of the results respectively. Chapter six gives the summary and concluding remarks of the study followed by references and appendices.
1.6 Significance of the Study
This study reveals the role of periodic heating at the surfaces of cylinders on natural convection flow in a vertical tube and annulus. The results in this research will be of great important to design engineers in improving electrical devices with thermostat and other appliances that require censors. Moreover, it is hoped that the results will not only provide useful information for industrial 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 work are theoretical. No experimental work has been carried on this work.The governing differential equations were employed and conclusions were drawn based on depicted lines and contour graphs.
1.8 Definition of Terms
i. Annulus: Annulus is the area bounded by two concentric cylinders
ii. Boussinessq Approximation: Boussinessq approximation is the assumption that the fluid flow is considered under little variations of temperature and density.
iii. Compressible and Incompressible Fluids: A compressible flow is a flow whose fluid density varies significantly within the flow field. While in an incompressible flow the density does not change.
iv. Cylindrical Geometry: A cylindrical geometry is any shape like a cylinder.
v. Dimensionless Quantity: Is a quantity without an associated physical dimension.
vi. Free orNatural Convection: Free or Natural convection is a mechanism in which fluid motion is generated only by density differences in fluid cause by temperature gradients.It is a type of heat transport in which the fluid motion is not generated by an external source (pump, fan, suction devices).
vii. Laminar Flow: Laminar flow occurs when fluid flows at low velocity, in parallel layers with no disruption between the layers.
viii. Nusselt Number (𝑵𝒖) : Nusselt numberis the rate of heat transfer between the fluid and the surfaces of the cylinders.
ix. Periodic Function:A function 𝑓 is said to be periodic with period 𝑃 (𝑃 being a nonzero constant) if we have𝑓 𝑥+𝑃 =𝑓(𝑥) for all 𝑥 in the domain.
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x. Periodic Heating: This is when the heating on the surfaces of the cylinders is sinusoidal or follows any periodic function.
xi. Skin-friction: Skin-friction is defined as the friction (drag) between a fluid and the surface of a solid moving through it or between a moving fluid and its enclosing surface.
xii. Steady Flow: A flow is said to be steady if the velocity and other flow parameters such as pressure and acceleration are independent of time but may depend on position.
xiii. Unsteady Flow: A flow is said to be unsteady if the velocity and other flow parameters such as pressure and acceleration are dependent on time.
xiv. Volume Flow Rate: Is the volume of fluid which passes through the geometry per unit time; usually represented by the symbol 𝑄. The SI unit is m3/s (cubic metres per second).
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