The aim of the present work is to numerically simulate laminar forced convection in the fully developed flow through the finned annulus of a double-pipe subject to constant heat flux boundary condition (H1) and to investigate the effects of various geometric designs and arrangements of fins on the thermal performance of the finned duct. The designs of the finned annulus considered in the present work are longitudinal triangular fins with equal and unequal heights, and fins with variable fin-tip thickness. The governing partial differential equations of the convection problems are numerically solved by employing the finite element method (FEM) and the discontinuous Galerkin finite element method (DG-FEM). The hydraulic and thermal characteristics like the friction factor, the Nusselt number and the surface flow area goodness factor are studied against various geometric design parameters. Moreover, the validity of various assumptions in simpler models is investigated in more realistic models. The behaviour of the flow and thermal characteristics has been found to match with the physics of the problems and therefore, substantially validates the present simulations. For highly conductive triangular fins of equal heights, the results show upto more than 4 i times gain in the heat transfer rate relative to increase in the pressure loss because of the attachment of fins when the equivalent diameter is used as the characteristic length. Using fins in two groups of different heights, the velocity and temperature distributions can be significantly altered to have more favourable flow and thermal characteristics. For specified values of the number of fins, the thermal conductivity ratio and the size of the inner pipe, a combination of unequal heights of the two fin groups rendering maximum heat transfer coefficient exists in many cases. Such combinations of fin heights are very useful in reducing pressure loss and promoting heat transfer rate. The present study recommends the use of unequal fin heights depending on the cost, weight and heat duty requirements. For longitudinal fins with specified base angle and variable tip thickness, no single tip-angle renders maximum values of the heat transfer coefficient and surface flow area goodness factor for all values of the height and number of fins indicating that the choice of fin-tip angle strongly depends on the number of fins and their height. It is found that highly significant gain in the Nusselt number can be achieved by considering variation in the thickness of fin tip for various values of the conductivity parameter. This gain for trapezoidal fin is upto 38% over the triangular fin and 163% over the rectangular fin. Therefore, the consideration of the fin-tip thickness as a design parameter is significantly important for the optimal design of the present heat exchange system. All the computed results are in excellent comparison with the available literature results.