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The purpose of this paper is to numerically investigate the heat transfer enhancement in a tube filled partially with porous media under non-uniform porosity distribution and thermal dispersion effects. The optimum porous thickness ratio [R_(r,Nu)] for the heat transfer enhancement under these conditions with and without considering required pumping power is evaluated.

The local thermal non-equilibrium and Darcy–Brinkman–Forchheimer models are used to simulated thermal and flow fields in porous region. The tube wall and flow regime are assumed to be isothermal and laminar, respectively. The impacts of Darcy number (Da = 10^-6 - 10^-1) and inertia parameter (F = 0 − 2) on the Nusselt number and friction factor are studied for non-uniform porosity distribution.

First, the effect of Nusselt number indicates that there are two different behaviors with respect to uniform and non-uniform porosity for partially and fully filled porous pipe. Second, variable porosity in porous region has significant influence on the optimum thickness ratio with considering required pumping power. For negligible inertia term, it depends on the Darcy number, whereas it is 0.9 at F > 1. Third, the plug flow assumption cannot be valid even at lower Darcy number under non-uniform porosity, while this assumption is applicable at Da < 10^-3 for constant porosity distribution in porous region.

According to the best knowledge of authors, the optimum porous thickness ratio for the heat transfer enhancement considering the pressure loss effects under variable porosity has not reported up to now. Also the plug flow assumption in such physics is not discussed.

The purpose of this paper is to investigate the hydromagnetic third-grade non-Newtonian fluid flow and heat transfer between two coaxial pipes with a variable radius ratio.

To solve the approximate nonlinear and linear problems with variable coefficients, a trial function was applied. Methods include collocation, least square and Galerkin that can be applied for obtaining these coefficients.

It is revealed that an increase of the non-Newtonian parameter, Hartmann number, and radius ratio leads to an augmentation of the absolute value of the dimensionless velocity, temperature, velocity gradient, and temperature gradient of about 10-60%. Further, the augmentation of Bi₁ reduces the absolute value of the dimensionless temperature profile and dimensionless temperature gradient about three to four times; hence, the dimensionless heat transfer rate reduces. However, the growth of Bi₂ has a contrary impact. Besides, the increase of Pr and Ec leads to an increase in the dimensionless temperature profile and dimensionless temperature gradient; therefore, the dimensionless heat transfer rate increases.

The convection heat transfer on the walls of the pipes is considered, and the nonlinear coupled momentum and energy equations are solved using the least squared method and collocation methods, respectively.