Viscous heating of a laminar flow in the thermal entrance region of a rectangular channel with rounded corners and uniform wall temperature

The numerical solution of Graetz-Brinkman problem is obtained for channels having rectangular cross section with rounded corners, under T boundary condition applied on the heated duct wall and adiabatic condition elsewhere, assuming an adiabatic preparation of the fluid at the inlet section. Several simulations are conducted and both Poiseuille and Nusselt numbers calculated, based on the computed velocity and temperature profiles. The numerical method is first verified with the resulting Nusselt and Poiseuille numbers with literature data, available for simplified configurations and fully developed flow, showing an excellent agreement. Comparison with numerical data is also conducted in case of fully developed flow and non-negligible viscous dissipation. A further validation and verification is carried out comparing current computations with both experimental and numerical data in case of thermally developing flow inside a rectangular channel with negligible viscous heating (i.e. the well-known Graetz problem). The effects of duct cross section geometry and Brinkman number are investigated and new correlations, useful for the design of microchannel heat sinks, are presented in order to predict the Poiseuille number and average Nusselt numbers.