Chapter 7, Problem 7.16P

In fuel cell stacks, it is desirable to operate under conditions that promote uniform surface temperatures for the electrolytic membranes. This is especially true in high-temperature fuel cells where the membrane is constructed of a brittle ceramic material. Electrochemical reactions in the electrolytic membranes generate thermal energy, while gases flowing above and below the membranes cool it. The stack designer may specify top and bottom flows that are in the same, opposite, or orthogonal directions. A preliminary study of the effect of the relative flow directions is conducted whereby a $150\text{mm}\text{×150}\text{mm}$ thin sheet of material, producing a uniform heat flux of $100{\text{W/m}}^{\text{2}},$ is cooled (top and bottom) by air with a free stream temperature and velocity of 25°C and 2 m/s, respectively.

1. Determine the minimum and maximum local membrane temperatures for top and bottom flows that are in the same, opposite, and orthogonal directions. Which flow configuration minimizes the membrane temperature? Hint: For the opposite and orthogonal flow cases, the boundary layers are subject to boundary conditions that are neither uniform temperature nor uniform heat flux. It is, however, reasonable to expect that the resulting temperatures would be bracketed by your answers based on the constant heat flux and constant temperature boundary conditions.
2. Plot the surface temperature distribution $T\left(x\right)$ for the cases involving flow in the opposite and same directions. Thermal stresses are undesirable and are related to the spatial temperature gradient along the membrane. Which configuration minimizes spatial temperature gradients?