ABSTRACT
This paper presents natural convection flow in a vertical channel with suction/injection and isothermal boundaries. The governing equations are solved using undetermined coefficient. The temperature equation and velocity equation for this article are presented and transformed to their corresponding dimensionless form using suitable parameters. The equations are solved exactly and graphical representations are conferred, the skin friction is thereby obtained from the exact solutions. Throughout the cause of the graphical representations, it is observed that from temperature profile, suction increases the temperature gradient in the channel, while injection decreases the temperature gradient, suction decreases the velocity in the channel, while injection increases the velocity, then suction and injection parameters have the same influence on the skin friction
TABLE CONTENTS
DECLARATION i
CERTIFICATION ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
ABSTRACT v
CHAPTER ONE
INTRODUCTION
1.1 Background of the study 1
1.2 Aim and Objectives of the Study 2
1.3 Research Methodology 2
1.4 Definitions of basic terms 3
CHAPTER TWO
LITERATURE REVIEW
2.1 Introduction 5
2.2 Natural Convection in Vertical Channels 7
2.3 Suction and Injection 8
2.4 Natural Convection with Suction/Injection 8
CHAPTER THREE
MATHEMATICAL ANALYSIS
3.2 Suction/injection (S) 10
3.3 Prandtl number (pr) 11
3.4 Mathematical description 13
CHAPTER FOUR
ANALYSIS AND DISSCUSION OF RESULT
4.1 Introduction 19
4.2 Result and discussion 19
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMENDATION
5.1 Summary 27
5.2 Conclusion 27
5.3 Recommendation 27
REFERENCES 28
APPENDICES 30
LIST OF FIGURES
Fig 3. 1: Schematic diagram of the unified solution 14
Fig 4.1.Temperature Distribution for Different Values of S at Pr = 0.71 20
Fig 4. 2 Temperature Distribution for Different Values of S at Pr = 7.0 21
Fig 4. 3 Temperature Distribution for Different Values of S at Pr = 1 22
Fig 4. 4 Velocity Profile for Different Values of S at Pr= 0.71 23
Fig 4. 5 Velocity Profile for Different Values of S at Pr = 7.0 24
Fig 4. 6 skin friction on y= -1 25
Fig 4. 7 skin friction on y= 1 26
CHAPTER ONE
INTRODUCTION
1.1 Background of the study
Fluid dynamics and heat transfer are fundamental aspects of engineering and natural phenomena. The study of fluid flow and heat transfer has significant implications in a wide range of engineering and environmental applications, from the design of efficient heat exchangers to the prediction of weather patterns and climate modeling. Among the many modes of heat transfer, natural convection stands out as a phenomenon of great importance.
Importance of Natural Convection
Natural convection is a heat transfer process driven solely by temperature differences within a fluid. This phenomenon plays a pivotal role in various industrial, environmental, and engineering contexts. One particularly intriguing scenario within the realm of natural convection is the flow in vertical channels with suction/injection and isothermal boundaries. Understanding this complex flow is essential for several reasons:
Energy Efficiency: Natural convection can have a significant impact on the energy efficiency of various systems, including cooling systems in electronic devices, solar collectors, and building HVAC (Heating, Ventilation, and Air Conditioning) systems. Optimizing these systems requires a deep understanding of natural convection patterns.
Environmental Considerations: As we strive to reduce energy consumption and minimize environmental impact, gaining insights into natural convection processes becomes crucial. For example, understanding the natural convection in environmental reservoirs can aid in predicting the dispersion of pollutants or the thermal behavior of aquatic ecosystems.
Industrial Applications: In industrial processes, vertical channels are commonly encountered in heat exchangers, chemical reactors, and cooling systems. Controlling and enhancing heat transfer in such systems is essential for efficient and cost-effective operations.
Scientific Exploration: Beyond practical applications, studying natural convection in vertical channels with suction/injection and isothermal boundaries provides valuable insights into the fundamental principles of fluid dynamics and heat transfer. It challenges researchers to develop mathematical models and computational tools to simulate and predict these complex flows.
1.2 Aim and Objectives of the Study
The overall aim of this research is to study natural convection flow in a vertical channel with section/injection and isothermal boundaries
The objectives to attain the set aim are to:
i. Investigate Natural Convection Flow
ii. Understand the Influence of Suction/Injection
iii. Examine the Impact of Isothermal Boundaries
iv. Develop Mathematical Models
v. Conduct Numerical Simulations
1.3 Research Methodology
To attain the above set objectives, a survey of the existing works on natural convection flow in a vertical channel under different phenomena is conducted and stretched to capture newly identified physical situations where the previous works is limited. The mathematical problems of “natural convection flow in a vertical channel with suction/injection and isothermal boundaries” were solved analytically using method of undetermined coefficients then applying the boundary conditions. The numerical values and graphs of these solutions were obtained using a computer package MATLAB 2016. The numerical values and graphs obtained are analyzed to ascertain the impact of each of the controlling parameters on the flow.
1.4 Definitions of basic terms
i. Natural Convection: Natural convection is a mode of heat transfer in fluids (liquids or gases) where heat is transferred due to density differences caused by temperature variations. It occurs without the use of mechanical devices, driven solely by buoyancy forces.
ii. Vertical Channel: A vertical channel refers to a confined space with a predominantly vertical orientation, often used in engineering and fluid dynamics contexts. In this study, it represents the physical domain where natural convection is analyzed.
iii. Suction/Injection: Suction and injection refer to the processes of removing or introducing fluid from or into a confined space, respectively. In the context of your research, these processes are applied at the boundaries of the vertical channel and can significantly influence the flow patterns.
iv. Isothermal Boundaries: Isothermal boundaries are boundaries that maintain a constant temperature. In your study, these boundaries are kept at a fixed temperature, and they play a critical role in driving the natural convection flow by creating temperature gradients.
v. Buoyancy Forces: Buoyancy forces arise from the density difference between a fluid and its surroundings due to temperature variations. In natural convection, warmer fluid rises and cooler fluid descends, creating buoyant forces that drive the flow.
vi. Temperature Gradient: A temperature gradient refers to the rate of change of temperature with respect to position. In the context of your study, temperature gradients are crucial as they drive fluid movement through buoyancy-driven natural convection.
vii. Fluid Dynamics: Fluid dynamics is the study of the behavior of fluids (liquids and gases) in motion. It includes the analysis of fluid flow, pressure, velocity, and related phenomena.
viii. Heat Transfer: Heat transfer is the process by which thermal energy is transferred from one body or system to another due to a temperature difference. It can occur through conduction, convection, or radiation.
ix. Skin friction: is the shear stress that occurs between the fluid and the solid surface of boundaries.
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