Extended Graetz problem in laminar duct flows with Robin boundary conditions: Sherwood/Nusselt correlations and averaged transport closures

We investigate the extended Graetz problem for heat and mass transfer in laminar duct flows with Robin wall boundary conditions. By retaining axial diffusion, the transfer closure depends on both the Péclet number and the wall-exchange parameter, represented by the Damköhler number for mass transfer or the Biot number for heat transfer. Four canonical configurations are considered: parallel plates and a circular tube, each with plug and Poiseuille flow. For each case, we compute the fully developed Sherwood/Nusselt number as a function of the Péclet and Damköhler/Biot numbers. We also introduce a cross-sectional averaging factor, χ, relating area-averaged andmixing-cup concentrations/temperatures, which enables a consistent one-dimensional closure of the advection--diffusion--reaction problem. The results show that, for plug flow, the Sherwood/Nusselt number is independent of the Péclet number and depends only on wall exchange, recovering the expected weak- and strong-exchange limits. For Poiseuille flow, both the Sherwood/Nusselt number and the averaging factor χ vary nontrivially with the Péclet and Damköhler/Biot numbers. Across all cases, we provide compact, asymptotically constrained correlations that accurately reproduce the numerical solutions and are suitable for reduced one-dimensional modeling. These results provide a consistent closure framework for averaged transport equations with axial diffusion and finite wall exchange.