STUDY ON NATURAL CONVECTION FLOW IN CYLINDRICAL GEOMETRIES WITH TIME-PERIODIC HEAT INPUT
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STUDY ON NATURAL CONVECTION FLOW IN CYLINDRICAL GEOMETRIES WITH TIME-PERIODIC HEAT INPUT
ABSTRACT
His dissertation looks into how steady-periodic heating at the cylinder surfaces affects natural convection flow in a tube and an annulus.This work addresses two problems. In both instances, the fluid was considered to be completely formed, and the mathematical equations governing the flow were calculated.
The first problem examines how time-periodic heating at a tube’s surface affects fluid velocity, temperature, rate of heat transfer, and skin friction.The second problem investigates how periodic heating at the cylinder surfaces affects natural convection flow in an annulus.
The partial differential governing equations were converted to their corresponding ODEs, which represented the steady and periodic regimes, using appropriate transformation.Closed-form equations for velocity,
temperature, skin friction, mass flow rate, and rate of heat transfer expressed as a Nusselt number were found for both cases using modified Bessel’s functions of the first and second kinds.
The solutions were graphically depicted 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 studied. The results show that the Strouhal number, Prandtl number, and aspect ratio all help to reduce fluid velocity, temperature, and skin friction at the cylinder surfaces.
The rate of heat transfer between cylinder surfaces and fluid rises with increasing 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 portion) fall to stable regime. At this time, the steady and periodic regimes have equal contributions to total velocity. Furthermore, for big Strouhal number 𝑆𝑡, the periodic regime has a negative impact on the overall fluid temperature and velocity.
Chapter One: General Introduction.
1.1 Background of the Study
Fluid dynamics has been of great importance over the decades due to its technological, industrial, and engineering applications, which include calculating the forces and moment of aircraft, determining the mass flow-rate of petroleum through pipelines,
predicting weather patterns, modelling fission weapon detonation, traffic engineering, rocket engines, wind turbines, oil pipelines, air condition, turbine system in power generation anFerzigerand Peric (1996); Nishigaki et al. (2013).
The study of laminar natural convection in vertical tubes or annuli has gotten a lot of attention due to its usefulness in many engineering applications. The primary technical applications of these studies are cooling systems for electronic gadgets,
electrochemical processes, nuclear reactors, and solar energy collectors. This interest stems from researchers’ need for a better understanding of heat and mass transfer phenomena in engineering, geophysics, and biology.
1.2 Statement of the Problem
Wang (1998) studied free convection between vertical plates under periodic heat input. In his work, he divided the solutions into steady and unsteady regimes and specified the conditions under which the periodic heat input is relevant.
However, in real life, heat transfer is widespread via cylinders and annuli. It is therefore important to investigate free convection flow in a vertical tube and annulus motivated by periodic heating at the cylinder sides.
This dissertation examines how periodic heating of cylinder surfaces affects natural convection flow generation in a tube and annulus.
1.3 Goals and Objectives of the Study
The purpose of this research is to investigate the flow behaviour of natural convection flow in a tube and an annulus subjected to periodic heat input. The aims are to derive the equations of motion that govern the problems.
ii. examine how periodic heating at the tube’s surface affects velocity, temperature, skin friction, and heat transfer rate.
iii. Investigate how periodic heating and aspect ratio affect velocity, temperature, skin friction, and heat transfer rate in the annulus.
iv. Examine fluid flow behaviour based on Prandtl number (𝑃𝑟). v. Determine the conditions for considerable periodic heating on cylinder surfaces in both tube and annulus.
1.4 Research Methodology
The methodology used to realise the set of objectives is a five-phase strategy. In step one, we analyse current literature and expand it to incorporate various physical geometries. Phase two focuses on solving the mathematical model that detailed the study of natural convectionflow in various geometries.
We employed the indeterminate coefficient approach and the direct integration method to arrive at our analytical results. In phase three, numerical values for analytical solutions from phase two are acquired.
We used the computer package MATLAB 12b to obtain the numerical values of the analytical solutions by building a computer programme for the second phase.
The numerical values collected in phase three are shown graphically in phase four using MATLAB 12b. The final phase involves physical interpretation of the graphs in order to discuss the impact of each controlling parameter on the flow and develop conclusions.
1.5 Organisation of Dissertation
This dissertation is divided into six chapters, including references and appendices. The first chapter essentially serves as the dissertation’s broad introduction. The literature review of the research is primarily featured in chapter two.
The third chapter offers a mathematical explanation of the concerns with non-dimensionalisation. Chapters four and five contain answers to the issues and a discussion of the outcomes, respectively. Chapter six contains the study’s summary and concluding remarks, followed by references and appendices.
1.6 Significance of the Study
This study investigates the effect of periodic heating of cylinder surfaces on natural convection flow in a vertical tube and annulus. The findings of this study will be very useful to design engineers in upgrading electrical devices with thermostats and other appliances that require censors.
Furthermore, it is believed that the findings would not only provide useful information for industrial applications, but also improve on earlier research.
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1.7 Scope of Study
The formulation, analysis, and conclusions reached in this work are theoretical. There has been no experimental work on this work.The governing differential equations were used, and inferences were formed from the illustrated lines and contour graphs.
1.8 Definition of Terms.
i. Annulus: The area bordered by two concentric cylinders.
ii. Boussinessq Approximation: The Boussinessq approximation assumes that fluid flow is considered with little fluctuation in temperature and density.
iii. Compressible and Incompressible Fluids: A compressible flow is one in which the fluid density varies significantly throughout the flow field. In an incompressible flow, the density remains constant.
iv. Cylindrical Geometry: A cylindrical geometry refers to any shape that resembles a cylinder.
v. Dimensionless Quantity: A quantity with no physical dimensions.
vi. Free or Natural Convection: Free or natural convection is a mechanism that generates fluid motion solely through density variations in fluid caused by temperature gradients.It is a method of heat transport in which fluid motion is not caused by an external source (pump, fan, or suction devices).
vii. Laminar Flow: Laminar flow occurs when fluid flows at low velocity in parallel strata with no interruptions between them.
viii. Nusselt Number (𝑵𝒖): This is the rate of heat transfer between the fluid and the surfaces of the cylinders.
ix. Periodic Function: A function 𝑓 is considered periodic with period 𝑃 (where 𝑃 is a nonzero constant) if 𝑓 𝑥+𝑃 = 𝑓(𝑥) for all 𝑥 in the domain.
5. Periodic Heating: This refers to heating on cylinder surfaces that follows a sinusoidal or periodic function.
xi. Skin-friction: Skin-friction is described as the friction (drag) between a fluid and the surface of a solid passing through it, or between a moving fluid and its surrounding surface.
xii. Steady Flow: A flow is said to be steady when its velocity and other flow parameters, such as pressure and acceleration, are independent of time but may vary with position.
xiii. Unsteady Flow: A flow is considered unsteady if its velocity and other flow characteristics, such as pressure and acceleration, vary with time.
xiv. Volume Flow Rate: The volume of fluid passing through the geometry per unit time, indicated by the symbol 𝑄. The SI unit is cubic metres per second (m3/s).
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