Finite element method solution for steady magnetohydrodynamic flow in a straight horizontal pipe of elliptical cross section

Abstract

Velocity profile and temperature distribution for Magnetohydrodynamic (MHD) flow in a straight horizontal pipe of elliptical cross section has been investigated. Many researchers have carried out research on pipes of circular, square, rectangular, annular and elliptical cross sections in magnetohydrodynamics because there are many applications. Their studies concentrated on a given cross section as a different entity with fluid being driven by pumps. In this study, investigation is done on a circular pipe as it changes into an elliptical pipe when fluid is propelled by gravitational force. The main purpose of the study is to find out which pipe between one whichhasacircularcrosssectionandanotherofellipticalcrosssectionismorebeneficial. Effects of velocity profile and temperature distribution on the pipe as it changes cross section from circular to elliptical are investigated. Governing equations, partial differential equations (pdes), are formulated, non dimensionalised, expressed in terms of stream function and transformed into ordinary differential equations (odes) using similarity transformation. The odes are solved by Finite Element Method in conjunction with Mathematica version 12.0. The objectives of the study are: To model Finite Element Method solution for steady Magnetohydrodynamic flow in a straight horizontal pipe of elliptical cross section. To formulate governing equations (pdes) in cylindrical coordinates (r,θ,z) comprising Navier-Stokes equations, Ohm’s law of electromagnetism, equation of continuity, cross section of elliptical pipe and heat energy equation. To solve by Finite Element Method the ordinary differential equations (odes) formed when non dimensionalisation and similarity transformation are carried out on the governing equations. To determine the effects of dimensionless numbers of Hartmann number, Reynolds number, Eckert number and Prandtl number as well as other physical quantities of gravitational force and aspect ratio on fluid velocity and temperature. To find out the repercussions of velocity and temperature on a pipe as it transits from circular to elliptical cross section. Finite Element Method (FEM) is embraced instead of other methods like Finite Difference Method (FDM) because FEM is able to handle complicated geometries and boundaries with relative ease while other methods are restricted to handle rectangular shapes. Also many of the real life medical, engineering, astrophysics, etc problems can be solved in weak form, which FEM encompasses compared to strong form, which other methods employ. Results are displayed as tables and graphs and reveal that: Increase in Hartmann number, 1.0≤Ha≤40.0, increases temperature but retards velocity. Rise in Reynolds number, 0.5≤Re≤8.0 and aspect ratio, 1≤α ≤1.6, leads to rise in both velocity and temperature. An upsurge in gravitational force, 0.00002≤λθ ≤0.00008, results in an upsurge in velocity. Temperature increases when Eckert number, 1≤Ec≤40 , increases but decreases when Prandtl number, 0.5≤Pr≤2.0, is raised. In all scenarios, velocity and temperature are maximum at the centre of pipe but diminish to zero at the periphery. Spike in aspect ratio leads to rise in velocity which results in increase in temperature. A pipe of elliptical cross section will be more convenient where there is limited space in the vertical direction due to existing structures yet there is demand in increase in productivity. This is in comparison to circular shape. A pipe of elliptical cross section has greater capacity for the same depth of flow. It is envisaged that the conducting fluid is flowing as a coolant at a nuclear power plant or as molten metal at a metallurgical process. A pipe of elliptical cross section would therefore be moreproductive inindustrialprocessesthanone whichiscircularaccordingto thefindingsofthis dissertation.