Conjugate heat transfer is an important area of research which has been in demand due to its applications related to various scientific and engineering fields. The current research is focused to study the heat transfer in a porous medium sandwiched between two solid walls of an annular vertical cylinder. The prime focus of the current study was to evaluate the effect of solid wall thickness, the influence of variable wall thickness at inner and the outer radius, the conductivity ratio and the solid wall conductivity ratio on the heat transfer characteristics of the porous medium. The surface at inner and outer radii of the annulus is maintained isothermally at and such that . The governing partial differential equations for the conjugate heat transfer in porous medium and that of solid walls are converted into a set of algebraic equations with the help of finite element method and then solved simultaneously to predict the temperature variation in the solid wall as well as the porous region of the annular domain.