This paper presents an analytical investigation of the thermal transport in a parallel-plate channel comprised of superhydrophobic walls. An analytical solution is obtained for the thermally developing state, however, it is the condition far downstream from the entrance where the temperature field exhibits repeating periodic streamwise variation that is of primary interest here. The superhydrophobic walls considered in this paper exhibit alternating microribs and cavities positioned perpendicular to the flow direction and the transport scenario analyzed is that of constant wall heat flux through the rib surfaces with negligible thermal transport through the vapor cavity interface. Axial conduction is neglected in the analysis and the problem is one of Graetz flow with apparent slip-flow and periodicity of constant heating. Closed form solutions for the local Nusselt number and wall temperature are presented and are in the form of infinite series expansions. Previously, it has been shown that significant reductions in the overall frictional pressure drop can be expected relative to the classical smooth channel laminar flow. The present results reveal that the overall thermal transport is markedly influenced by the relative cavity region (cavity fraction), the relative rib/cavity module width, and the flow Peclet number. The following conclusions can be made regarding thermal transport for a constant heat flux channel exhibiting the superhydrophobic surfaces considered: (1) Increases in the cavity fraction lead to decreases in the average Nusselt number; (2) Increasing the relative rib/cavity module length yields a decrease in the average Nusselt number; and (3) as the Peclet number increases the average Nusselt number increases. For all parameters explored, the limiting upper bound on the fully developed average Nusselt number corresponds to the limiting case scenario of classical laminar flow through a smooth-walled channel with constant heat flux.
