A STUDY ON MHD NATURAL CONVECTION FLOW FORMATION IN A TUBE WITH PERIODIC HEAT INPUT AND TRANSVERSELY APPLIED MAGNETIC FIELD.
This project presents an analytical solution for magnetohydrodynamics natural convection flow in a vertical tube due to time-periodic heating of tube surfaces. The fluid is assumed to be fully developed and the mathematical equations governing flow formation and heat transfer are derived, the governing partial differential equations were transformed to their corresponding ordinary differential equation (ODE), representing the steady and periodic regimes. Closed-form expressions for velocity, temperature, skin-friction, mass flow rate and rate of heat transfer which is expressed as Nusselt number are obtained in terms of modified Bessel’s function by solving the present mathematical model after separating into steady component and periodic regime. The effects of pertinent parameters such as Strouhal number (St), Prandtl number (Pr) and radius ratio (λ) are shown with the aid of line graphs. Results indicate that the role of magnetic field is to decrease fluid velocity and skin-friction. Furthermore, increase in Strouhal number increases the temperature phase-lag.
TABLE OF CONTENTS
GREEK SYMBOLS vi
1.1 Background of the study 1
1.2 Statement of Problem 2
1.3 Aim and objectives 2
1.4 Significance of the Study. 3
1.5 Definition of Terms 5
1.6 Justification 7
2.1 A Comprehensive review on the study of Magnetohydrodynamics (MHD) 9
2.2 Periodic Heat Input in Natural Convection Flow 12
3.1 Introduction 15
3.2 Tools used 16
3.3 Mathematical analysis 17
3.3.1 skin friction 24
3.3.2 Mass flow rate 24
3.3.4 Amplitude of velocity and temperature for the problem 25
3.3.5 Rate of heat transfer 25
ANALYSIS AND DISCUSSION OF RESULTS
4.1 Result and discussion 26
4.2 Discussion of the problem 26
SUMMARY, RECOMMENDATIONS AND CONCLUSIONS
5.1 Summary 35
5.2 Recommendations 35
5.3 Conclusion 36
a radius of inner cylinder
b radius of outer cylinder
B_o constant magnetic flux density
u^' velocity of the fluid
g acceleration due to gravity
M magnetic field
r dimensionless radial distance
v kinematic viscosity of liquid
σ electrical conductivity
β coefficient of volume expansion
μ coefficient of viscosity
λ radius ratio
α thermal diffusivity
β coefficient of thermal expansion
κ thermal conductivity
ψ phase of temperature (dimensionless)
ρ fluid density
ρ0 initial fluid density
χ phase of velocity (dimensionless)
ν kinematic viscosity
ω frequency of time-periodic heating
τ skin-friction (dimensionless)
LIST OF FIGURES
Figure 4. 1 Temperature distribution Real(G(R)) varying St at Pr = 0.71, λ = 2.0 27
Figure 4. 2 Velocity profiles Real(B(R)) varying St at Pr=0.71 28
Figure 4. 3 Skin-friction on the tube varying St and Pr at λ=2.0 29
Figure 4. 4 Phase of temperature for different St at Pr=0.71, λ=2.0 30
Figure 4. 5 Phase of velocity for different St at Pr=0.71, λ=2.0 31
Figure 4. 6 Amplitude of temperature for different St at Pr=0.71 32
Figure 4. 7 Mass flowrate for different St at Pr=0.71 33
Figure 4. 8 Rate of heat transfer of the tube varying St and λ 34
1.1 Background of the study
Magnetohydrodynamics, commonly referred to as MHD, represents a fascinating and interdisciplinary field that amalgamates the principles of electromagnetism and fluid dynamics. It provides a profound insight into the behavior of conducting fluids when exposed to magnetic fields. This field finds its roots in both astrophysical phenomena like solar flares and terrestrial applications such as nuclear fusion research and industrial processes involving molten metals and plasmas. MHD's ubiquity is a testament to its vital role in shaping our understanding of natural processes and its practical applications across a myriad of domains. Natural convection, a captivating fluid dynamics phenomenon, arises from the innate tendency of fluids to move due to density variations driven by temperature gradients. This phenomenon is ubiquitous in nature and engineering, influencing diverse systems ranging from the circulation of Earth's atmosphere to the cooling of electronic components in advanced computing devices. The intricate interplay of thermal gradients, buoyancy forces, and fluid motion characterizes the essence of natural convection, making it a subject of profound scientific and engineering significance.
Tubes, with their cylindrical symmetry, are emblematic geometric configurations found in numerous engineering applications. These configurations are not merely academic abstractions but integral components of heat exchangers, pipelines, and cooling systems. Understanding the nuances of fluid flow and heat transfer within tubes is paramount for optimizing the performance of these systems. The cylindrical confinement of tubes adds an intriguing dimension to the study of natural convection, introducing complexities that require meticulous investigation. Periodic heat input, characterized by cyclic variations in temperature, stands as a hallmark feature in countless real-world scenarios. Systems such as solar collectors, nuclear reactors, and certain chemical processes inherently exhibit periodic heat input. The dynamic nature of these temperature fluctuations can exert a profound influence on the fluid flow and heat transfer characteristics within tubes. This temporal dimension adds a layer of complexity to the study, necessitating a deeper exploration of the interplay between periodic heat input and natural convection.
The structure of this project reflects a systematic approach to unraveling the complexities of MHD natural convection within tubes with periodic heat input. It comprises several meticulously crafted chapters, each contributing to the overarching goal of advancing our knowledge in this field. The chapters encompass a comprehensive literature review, mathematical modeling of the phenomena, in-depth numerical simulations, and rigorous discussions of results. The sequential flow of the thesis ensures a logical progression, facilitating a profound exploration of the subject matter.
1.2 Statement of 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 cylindrical tube. It is therefore significant to study the role of magnetic on free convection flow in a vertical tube inspired by periodic heating at the surfaces of the cylinders. Hence, this current work is aimed to investigate the impact of periodic heating at the surfaces of the cylinders on magnetohydrodynamics natural convection flow formation in a cylindrical tube.
1.3 Aim and objectives
The primary aim of this research is to achieve a comprehensive understanding of the complex interplay between magneto hydro dynamics (MHD), natural convection, and periodic heat input within a tube. This understanding will encompass the dynamic behavior of fluid flow, temperature distribution, and heat transfer rates in this intricate we'll achieve this by pursuing these objectives:
i. Derive the equations of motion governing the problems.
ii. Investigate the role of periodic heating at the tube surface on the flow formation and skin friction.
iii. Examine the impact of magnetic field on flow formation and drag reduction.
1.4 Significance of the Study.
The significance of this study is multifaceted, encompassing both theoretical advancements and practical applications. It addresses a complex and relatively unexplored problem within the realm of fluid dynamics and magneto hydro dynamics, offering profound contributions to several key areas. This study significantly advances our scientific understanding of the intricate interplay between magneto hydro dynamics (MHD), natural convection, and periodic heat input within a confined tube. It adds depth to the existing body of knowledge by elucidating the underlying physical mechanisms and providing insights into how these phenomena synergistically influence each other.
The findings of this project have direct implications for the optimization of engineering systems that involve tubes, including heat exchangers, pipelines, and cooling systems. By enhancing our understanding of how MHD and natural convection interact with periodic heat input, this study empowers engineers and designers to develop more energy-efficient and reliable systems. In era where sustainability is paramount, this research contributes to the development of sustainable energy solutions. By improving the efficiency of systems such as solar collectors and geothermal heat exchangers, it plays a role in harnessing renewable energy sources more effectively, ultimately reducing our reliance on fossil fuels.
The insights gained from this study are pertinent to nuclear reactor cooling systems, where MHD effects and periodic heat input can impact reactor safety and efficiency. Understanding these dynamics is crucial for ensuring the safe and reliable operation of nuclear power plants. In the aerospace industry and space exploration missions, where extreme conditions prevail, understanding the behavior of fluids in the presence of magnetic fields and varying heat inputs is crucial. This research can aid in the design and operation of spacecraft propulsion systems and thermal management solutions.
This study bridges the gap between various scientific disciplines, bringing together concepts from fluid dynamics, electromagnetism, and heat transfer. This interdisciplinary approach fosters knowledge transfer and collaboration among researchers from different fields. In a world striving for sustainability and energy efficiency, the outcomes of this study have the potential to make a global impact. By optimizing energy systems and engineering solutions, it contributes to the broader goal of mitigating climate change and conserving vital resources.
In summary, the significance of this study extends far beyond the confines of academic research. It offers a pathway to advancing knowledge, improving engineering systems, fostering sustainability, and addressing real-world challenges in various industries. It stands as a testament to the power of interdisciplinary research in shaping a more sustainable and technologically advanced future.
1.5 Definition of Terms
Magnetohydrodynamics (MHD): MHD is a multidisciplinary field of study that combines principles from electromagnetism and fluid dynamics. It deals with the behavior of electrically conducting fluids (e.g., plasmas, molten metals, or certain liquids) in the presence of magnetic fields.
Natural Convection: Natural convection is a mode of heat and mass transfer in fluids driven by density differences due to temperature variations. It occurs without any external mechanical forces and is characterized by fluid motion driven by buoyancy forces.
Periodic Heat Input: Periodic heat input refers to the cyclic variation of heat or thermal energy added to a system over time. This phenomenon is often encountered in systems with time-varying heat sources, such as alternating heating and cooling cycles.
Cylindrical Tube: A cylindrical tube is a three-dimensional geometric shape characterized by its cylindrical symmetry, typically with a circular cross-section. In the context of this study, it represents the confined space within which fluid flow and heat transfer are analyzed.
Modified Bessel function: Is a special form of Bessel Functions, which were defined by Friedrich Bessel and Daniel Bernoulli as solutions.
Periodic Solution: A solution that periodically depends on the independent variable.
Steady-State: Steady-state refers to a condition in which a system's properties (e.g., temperature, fluid velocity) do not change with time. In the context of this study, steady-state conditions may be reached after the system has undergone transient changes due to periodic heat input.
Skin Friction: Skin friction, also known as wall friction, is a type of drag force that occurs when a fluid (such as air or water) flows over a solid surface. It is caused by the viscous or sticky nature of the fluid, and it results from the friction between the fluid particles and the surface. Skin friction is an important factor in the study of fluid dynamics and is a key component in the calculation of drag forces on objects moving through a fluid.
Mass Flow Rate: Mass flow rate is a measurement of the amount of mass that passes through a given point in a fluid system per unit of time. It is typically expressed in units such as kilograms per second (kg/s) or pounds per hour (lb/hr). Mass flow rate is a fundamental concept in fluid dynamics and is used to quantify the rate at which a fluid substance, like a gas or a liquid, moves through a pipe, channel, or any other part of a system.
Strouhal Number: The Strouhal number (St) is a dimensionless number used in fluid mechanics and aerodynamics to describe the behavior of fluid flow around solid objects. It is defined as the ratio of the frequency of vortex shedding (alternating flow patterns) behind an object to the object's characteristic length and the velocity of the fluid flow. The Strouhal number is often used to predict the occurrence of flow-induced vibrations and oscillations, such as in the study of vortex shedding behind cylinders and the aerodynamics of bodies like airfoils and flags.
Prandtl Number: The Prandtl number (Pr) is a dimensionless parameter used in fluid dynamics to characterize the relative importance of momentum diffusivity (kinematic viscosity) to thermal diffusivity (thermal conductivity) in a fluid. It is defined as the ratio of the kinematic viscosity of the fluid to its thermal diffusivity. The Prandtl number is essential in heat transfer and convection problems, as it helps predict the relative thickness of the thermal and velocity boundary layers in fluid flow.
Nusselt Number: The Nusselt number (Nu) is a dimensionless number used to characterize the convective heat transfer between a solid surface and a fluid. It is defined as the ratio of the convective heat transfer rate to the conductive heat transfer rate for a given system. The Nusselt number is particularly useful in understanding and analyzing heat transfer in situations involving forced convection, natural convection, and boiling or condensation. It depends on the fluid properties, flow conditions, and the geometry of the system.
MATLAB: MATLAB, which stands for "MATrix LABoratory," is a high-level programming language and software environment primarily used for numerical computing, data analysis, and visualization.
The justification for studying MHD natural convection flow in a tube with periodic heating input lies in the multifaceted significance and broad-reaching implications of this research. Several compelling reasons support the need to address the problems outlined in the previous statement:
1. Fundamental Understanding: This research provides an opportunity to deepen our understanding of complex fluid dynamics and heat transfer phenomena. By investigating the interplay of magnetic fields, natural convection, and periodic heating, we can expand the fundamental knowledge base in fluid mechanics and heat transfer, shedding light on previously unexplored interactions.
2. Efficiency Enhancement: In various engineering applications, optimizing heat transfer efficiency is of paramount importance. Understanding how MHD effects and periodic heating can enhance or hinder heat transfer allows us to design more efficient heat exchangers, improve energy generation processes, and manage thermal loads in a more sustainable and cost-effective manner.
3. Nuclear Reactor Safety: In the context of nuclear reactors, safety and efficient cooling are of critical concern. An in-depth study of MHD natural convection can help assess and mitigate potential issues, ensuring the safe and stable operation of nuclear reactors, which is vital for both energy production and environmental safety.
4. Sustainable Energy Generation: Research in this area has relevance in sustainable energy generation, particularly in the harnessing of geothermal and renewable energy sources. By understanding and controlling MHD natural convection, we can optimize energy extraction processes and contribute to sustainable energy solutions, reducing our reliance on fossil fuels.
5. Economic Implications: Energy efficiency and sustainability are not only environmentally important but also have significant economic implications. Efficient energy processes and reduced operating costs can lead to economic savings and improved competitiveness for industries and organizations.
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