We focus on the entropy generation minimization analysis associated with the thermo-fluidic transport of a Newtonian fluid through a hyper-porous channel formed between two asymmetrically heated parallel plates. Employing an analytical method, which is consistent with the perturbation analysis, we solve the governing transport equations taking the effects of nonlinear Forchheimer drag and conjugate heat transfer into account. We also invoke to the thermal boundary conditions of third kind at the outer boundaries of channel for the conjugate analysis of heat transfer. We bring out the alterations in the entropy generation rate in the system as attributable to the nonlinear interactions between the heat transfer rate as modulated by the fluid temperature, temperature gradient, the dissipative heat originating from the non-linear Forchheimer drag, Darcy frictional effect and the viscous shearing stress in the flow field. Also, we unveil optimum values of wall thickness, wall to fluid conductivity ratio and other thermophysical parameters, leading to a minimum entropy generation rate in the system for a given set of the other different parameters considered. We believe that the inferences obtained from this analysis may improve the design and optimization of thermodynamic systems/devices typically used in different engineering applications. (C) 2016 Elsevier Ltd. All rights reserved.