An analytical solution to electromagnetically coupled duct flow in MHD

Abstract

The flow of an electrically conducting fluid in an array of square ducts, separated by arbitrary thickness conducting walls, subject to an applied magnetic field is studied. The analytical solution presented here is valid for thick walls and is based on the homogeneous solution obtained by Shercliff (Math. Proc. Camb. Phil. Soc., vol. 49 (01), 1953, pp. 136-144). Arrangements of ducts arise in a number of applications, most notably in fusion blankets, where liquid metal is used both as coolant and for tritium generation purposes. Analytical solutions, such as those presented here, provide insight into the physics and important benchmarking and validation data for computational magnetohydrodynamics (MHD), as well as providing approximate flow parameters for 1D systems codes. It is well known that arrays of such ducts with conducting walls exhibit varying degrees of coupling, significantly affecting the flow. An important practical example is the so-called Madarame problem (Madarame et al., Fusion Technol., vol. 8, 1985, pp. 264-269). In this work analytical results are derived for the relevant hydrodynamic and magnetic parameters for a single duct with thick walls analogous to the Hunt II case. These results are then extended to an array of such ducts stacked in the direction of the applied magnetic field. It is seen that there is a significant coupling affect, resulting in modifications to pressure drop and velocity profile. In certain circumstances, counter-current flow can occur as a result of the MHD effects, even to the point where the mean flow is reversed. Such phenomena are likely to have significant detrimental effects on both heat and mass transfer in fusion applications. The dependence of this coupling on parameters such as conductivities, wall thickness and Hartmann number is studied.

Numerical Study of the Buoyancy Effect on Magnetohydrodynamic Three-Dimensional LiPb Flow in a Rectangular Duct

“…At this time, there are a few studies about analytical solution of the MHD flow. Recent work was presented by Bluck and Wolfendale [25] about an analytical solution to the laminar flow of an electrically conducting fluid in an array of partially conducting ducts subjected to an applied magnetic field. These solutions provide an understanding of the physics of such flows and an important benchmarking and validation data for computational MHD.…”

Analytical Solutions to Nonuniform Surface Heat Transfer With Volumetric Sources in Magnetohydrodynamic Duct Flow

“…The magnetic field within the wall satisfies a Laplace equation. The solution to this Hunt-II problem is given in [8]. Note that this solution reduces to the Shercliff solution for wholly insulating walls by setting P n = 0.…”

“…Multiplying (28) by U, integrating over the cross-section and using (19) and the results in [8] and [10], it can be shown that…”

“…Further analytical solutions were obtained for arbitrary, thin conducting walls by Hunt [6]. More recently, analytical solutions for certain relevant Hunt-type flows have been developed for arbitrarily thick walls [8]. The so-called Hunt-II case, considered here consists of conducting Hartmann walls and insulating side-walls (those walls parallel to the applied magnetic field).…”

Numerical solution through mathematical modelling of unsteady MHD flow past a semi-infinite vertical moving plate with chemical reaction and radiation

“…An electrically conducting fluid flow in an array of square ducts, separated by conducting walls of arbitrary thickness, subject to an applied magnetic field has been investigated [14]. An investigation of unsteady hydromagnetic natural convection heat and mass transfer flow past an impulsively moving infinite vertical plate embedded in a uniform porous medium when the temperature of the plate has a temporarily ramped profile has been carried out [15].…”