Question$-2(35$ points$)$: Two large flat porous horizontal plates are separated by a relatively small distance $\backslash ($ H $\backslash ).$ The upper plate at $\backslash ($ z$=$H $\backslash )$ is at temperature $\backslash ($ T$\_\{$H$\}\backslash ),$ and the lower one at $\backslash ($ z$=0\backslash )$ is to be maintained at a higher temperature $\backslash ($ T$\_\{0\}\backslash$left$($T$\_\{0\}>$T$\_\{$H$\}\backslash$right$)\backslash ).$ To reduce the amount of heat that must be removed from the lower plate, an ideal gas at $\backslash ($ T$\_\{0\}\backslash )$ is blown upward with velocity $\backslash ($ U$\_\{0\}\backslash )$ through both plates at a steady rate. Because of the low gas velocity, conduction and convection heat transfer may be comparable.

a$)$ Obtain a differential equation for the temperature distribution in the space between two plates and specify the boundary conditions.

b$)$ Recast the equation and boundary conditions in terms of dimensionless quantities using proper dimensionless variables $(\backslash (\backslash$theta$=\backslash )$ dimensionless temperature, $\backslash (\backslash$eta$=\backslash )$ dimensionless distance$)$ and obtain the temperature profile.

c$)$ Obtain an equation for the amount of heat that must be removed from the cold plate per unit area.