Heat transfer by laminar Hartmann's flow in thermal entrance region with uniform wall heat flux: the Graetz problem extended

Abstract

Thermally developing laminar Hartmann flow through a parallel plate channel, with prescribed transversal uniform magnetic field, including both viscous dissipation, Joule heating and axial heat conduction with uniform heat flux, is studied analytically by using a functional analysis method. The analytical expressions for the developing temperature and local Nusselt number in the entrance region are obtained in the general case. However, the solution obtained could be applied to any developed flow with heat generation. Applications to Hagen-Poiseuille and Hartmann flows are made. The associated eigenvalues problem is solved analytically to obtain explicit forms of the eigenfunctions, which correspond to Mathieu's functions. Results show that the effects of viscous and Joule dissipation can be neglected for liquid metal, even for large values of Hartmann number M, for which the corresponding Brinkman number Br is very small. Therefore, the heat transfer characteristics in the entrance region are only influenced by Peclet and Hartmann numbers. Results show that as long as M increases, the heat transfer increases until a certain saturation is reached, for which the effect of magnetic field on Nusselt number vanishes completely. Finally, a practical empirical expression for local Nusselt number is proposed in terms of axial distance ξ and Hartmann number M.