Entrance length effects on Graetz number scaling in laminar duct flows with periodic obstructions: Transport number correlations for spacer-filled membrane channel flows

Self-similarity and scaling laws are powerful tools in engineering and thus useful for the design of apparatus. This self-similarity is well understood for the heat and mass transfer in laminar empty channel flows, including the fully developed region as well as inlet length effects in the developing region (Graetz problem). In this study, we examine the validity of the scaling behavior arising from the Graetz solution for channel flows disturbed by periodic obstructions. Simulation results show that entrance length effects and scaling laws do not change due to the presence of obstructions if the flow field remains steady in time and the dimensionless inlet length is given by XT/Dh≈Cinl.·Re·Pr, where Cinl.≈0.01 for the local and Cinl.≈0.03 for the average Nusselt number. The Nusselt number in the inlet region for an internal flow scales by Nu=(Re·Pr)1/3, similar to the empty channel flow (Shah and London, 1978). If the analogy between heat and mass transfer holds, same conclusions and relations are valid for the Sherwood number, Sh∝(Re·Sc)1/3, where Sc denotes the Schmidt number. In the fully developed region, the Nusselt number depends slightly on the Reynolds and Prandtl numbers owing to the loss in self-similarity of the velocity field (contrary to the empty channel flow). The limit of the classical self-similarity is the onset of temporal oscillations (instability) in the flow field. Beyond this limit, the length of the thermal entrance region is strongly reduced. Furthermore, a strong dependency of the Nusselt number in the fully developed region on the Prandtl number is found.