Abstract We perform a systematic analysis of heat transfer in a counter-current three dimensional convective exchanger, when the inlet/outlet influence is fully taken into account. The analysis, carried out for constant fluid properties, considers the various influences of the fluid/solid conductivity, the imposed convection, inlet/outlet far-field conditions, and lateral boundary conditions. Using a generalized Graetz mode decomposition which permits to consider, both transverse and longitudinal diffusion influence in the exchanger as well as in the inlets/outlets, we put forward several salient generic features of convection/conduction heat transfer. In all cases we found an optimal P\'clet number for the cold or hot effectiveness. Even if, as expected, the larger the P\'clet the larger the Nusselt number, high transfer performances are found to be poorly efficient and/or to necessitate non-compact elongated exchangers. Performance degradation arising at high P\'clet number are found to be related to â "convective leaksâ" taking place within outlets. A fully developed regime occurs at large P\'clet and/or for long exchangers, which is fully determined by the first eigenvalue of the generalized Graetz mode decomposition, which is an extension of classical Graetz analysis. Numerical results are found consistent with a generalized linear relation between effectiveness and the number of heat transfer units asymptotically established in the convection dominated regime. This study opens new perspectives for micro-heat exchangers where moderate convection provides the best effectiveness and compactness. This contribution is also useful for giving reference solutions to counter-flow exchangers with realistic inlet/outlet boundary conditions.