Magnetic Effects on Fluid Flow through a Rotating Straight Rectangular Duct

Abstract

Numerical study has been performed to investigate the fluid flow through a rotating rectangular straight duct in the presence of magnetic field under various flow conditions. Spectral method is applied as a main tool for the numerical calculation technique, where the Chebyshev polynomial, the Collocation method and the Newton-Raphson method are also used as secondary tools. The characteristics of the flow are described in chapter 4.

The Magnetohydrodynamics incompressible viscous steady fluid flow through a straight duct of rectangular cross-section rotating at a constant angular velocity about the center of the duct cross-section is investigated numerically to examine the combined effects of Magnetic parameter (M_g ), Taylor number (T_Y ), Dean number (D_n ) and aspect ratio

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where a is the half width of the duct cross-section, b is the half height of the duct,

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… is the angular velocity, … is the viscosity, .. is the kinematic viscosity. One of the interesting phenomena of the flow is the solution curve and the flow structure. We examine the flow structures in case of rotation of the duct axis and the Dean numbers with large Magnetic force number as well as large Taylor number while other parameters are remain constant.

The calculation are carried out for

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D_n = 500, 1000, 1500 and 2000 where the aspect ratio Y = 1.0, 2.0 and 3.0. For high magnetic parameter and large Taylor number, almost all the fluid particles strength are weak. The maximum axial flow will be turn into the ring shape.

Table of Content

Abstract .. ii

Acknowledgement .. iii

Nomenclature .. iv

List of Figure .. vi

Introduction .. xiii

Chapter 1 Literature Review .. 1-12

1.1 Magnetohydrodynamics .. 1

1.2 Definition of Useful Parameters .. 3

1.3 Fully Developed flows in Rectangular Curved Duct .. 5

1.4 Developing Flow in Pipes and Rectangular Duct .. 6

1.5 Rotating Duct .. 9

1.6 MHD Flow in a Duct and Pipe .. 11

Chapter 2 .. 13-15

2.1 Governing Equation .. 13

Chapter 3 Numerical Technique .. 16-18

Chapter 4 Flow through a Rotating Rectangular Straight Duct

with Magnetic Field along Center Line .. 19-106

4.1 Introduction .. 19

4.2 Governing Equation .. 20

4.3 Numerical Solution .. 24

4.4 Results and Discussion .. 25

Chapter 5 Conclusion .. 107

References .. 109

Introduction

Fluid flow in a straight duct is of great importance. It has large applications in chemical and mechanical engineering. A lot of research works regarding fully developed flow have been carried out at different times. The aim of this thesis is to make some numerical calculations on the fluid flow in a rotating rectangular straight duct in presence of magnetic field which has been interested to the engineering communication and to the investigators dealing with the problem in many industrial applications. The results of this investigation may not have direct practical applications but are relevant to the problems mentioned above. The fluid flowing through a rectangular straight duct to rotate at a constant angular velocity about an axis normal to a plane including the duct is subjected to both Coriolis and centrifugal forces. Such rotating passages are used in cooling systems for conductors of electric generators. Flow in a rotating straight pipe is of interest because the secondary flows in this case are qualitatively similar to those in stationary curved system in view of the similar centrifugal mechanism including the secondary curved systems (Ishigaki (1955)). The earliest work on the flow in rotating straight pipe was carried out for the asymptotic limits of weak and strong rotations by Barua (1955). Benton Baltimore (1956) used a perturbation expansion to the Hagen-Poiseuille flow. The study of Mori and Nakayama (1983), Ito and Nanbu (1971), Wanger and Velkoff (1994) for small rotational speed and high axial pressure gradient resulted good agreement with experiments, showing an increases in friction factor with rotational speed. Alam, Begum and Yamamoto (2007) have used spectral method to describe the flow through a rotating straight pipe with large aspect ratio. MHD flow in an insulating rectangular duct under a non-uniform magnetic field is studied by Moreau et.al (2010). Numerical simulations of MHD flows past obstacles in a duct under externally applied magnetic field is studied by Dousset.V (2009). Zengyu et.al (2005) investigates the study of surface and bulk instabilities in MHD duct flow with imitation of insulator coating imperfection.

In chapter 1, literature review has been regarding flow in a straight duct, curved duct, MHD duct flow, rotating duct with various effects have summarized and discussed from both analytical and numerical point of view. In chapter 2, the basic governing equation has taken as the mathematical model related to the problems considered there after shown in standard forms. In chapter 3, the calculation technique for this problem is discussed, in chapter 4, a particular problem of the fluid flow through a rotating rectangular straight duct with magnetic field along center line has considered.

As above mentioned the problem has been solved by Spectral method as a main tool for numerical technique and Newton-Raphson method, Collocation method and Arc length method are used as a secondary tools. Finally, conclusion has discussed in chapter 5.

Chapter 1

Literature Review

1.1. Magnetohydrodynamics (MHD)

Magnetohydrodynamics (MHD) is a branch of magneto fluid dynamics i.e. continuum mechanics, which deals with the flow of electrically conducting fluids in electric and magnetic fields. The largest advance towards an understanding of such phenomena probably comes from the field of astrophysics. It has been long suspected that most of the matter in the universe is in the plasma or the state of highly ionized gases and much of the basic knowledge in the area of electromagnetic fluid dynamics evolved from these studies.

The field of Magnetohydrodynamics consists of the study of a continuous, electrically conducting fluid under the influence of electromagnetic fields, as a branch of plasma physics. Originally, MHD included only the study of strictly incompressible fluid but today the terminology is applied to studies of partially ionized gases as well as the other names have been suggested, such as magneto-fluid-mechanics or magneto-aero-dynamics, but original nomenclature has persisted. The essential requirement for problems to be analyzed under the laws of MHD is that the continuum approach be applicable.

There are many natural phenomena and engineering problems are susceptible to MHD analysis. It is useful in astrophysics because much of the universe is filled with widely spaced charged particles and permeated by magnetic fields and so the continuum assumption becomes applicable. Geophysicists encounter MHD phenomena in the interaction of conducting fluids and magnetic fields that are present in and around heavenly bodies. Engineers employee have been used MHD principles to design of heat exchangers, pumps and flow meters, space vehicle propulsion, control and re-entry problems, designing communications and radar system, creating novel power generating systems and developing confinement schemes for controlled fusion.

The most important application of MHD is the generation of electrical power with the flow of an electrically conducting fluid through a transverse magnetic field. Recently, experiments with ionized gases have been performed with the hope of producing power on a large scale in stationary plants with large magnetic fields. Cryogenic and superconducting magnets are required to produce these very large magnetic fields. Generation of MHD power on a smaller scale is interested for space applications.

Generally it is known that, several intermediate transformations are necessary to convert the heat energy into electricity. Each of these steps mean a lose of energy. This naturally limits are the overall efficiency, reliability and compactness of the conversion process. Methods for direct conversion to energy are now increasingly receiving attention. Of these, the fuel cell converts the chemical energy of fuel directly into electrical energy, fusion energy utilizes the energy released when two hydrogen nuclei fuse into a heavier one, and thermoelectrical power generation uses a thermocouple. Magnetohydrodynamic power generation is another important new process that is receiving worldwide attention.

In the experiment of Faraday (1832), the principal MHD effects were first demonstrated. He discovered a voltage that was induced by across the tube due to the motion of the mercury across the magnetic fields, perpendicular to the direction of flow and to the magnetic field by the experiment of the flow of mercury in glass tubes placed between poles of a magnet. Faraday (1832) also suggested that electrical power could be generated in a load circuit by the interaction of a flowing conducting fluid and a magnetic field. Alfven (1942) discovered MHD waves in the sun. These waves are produced by disturbances which propagate simultaneously in the conducting fluid and the magnetic field.

The analogy that explains the generation of an Alfven wave is that of a harp string plucked while submerged in a fluid. The string provides elastic force and the fluid provides inertia force and they combine to propagate a perturbing wave through the fluid and the string.

In summary, MHD phenomena result from the mutual effect of a magnetic field and conducting fluid flowing across it. Thus an electromagnetic force is produced in a fluid flowing across a transverse magnetic field and the resulting current and magnetic field combine to produce a force that resists the fluid’s motion. The current also generates its own magnetic field which distorts the original magnetic field. An opposing or pumping force on the fluid can be produced by applying an electric field perpendicularly to the magnetic field. Disturbance in either the magnetic field or the fluid can propagate in both to produce MHD waves as well as upstream and downstream wave phenomena. The science of magnetohydrodynamics is the detailed study of these phenomena, which occur in nature and are produced in engineering devices.

1.2 Some useful Parameters:

Magnetic Parameter (M_g)

This is obtained from the ratio of the magnetic force to the inertia force and is defined as,

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