Problem: A plate of thickness t is subjected to the imposed surface flux q_s on its top surface with a pitch of 2L as shown in Figure. The plate is infinitely long in the horizontal direction. The in-depth width of the plate is w. The thermal conductivity of the plate is k. The unheated area on the top surface is thermally insulated. The bottom surface faces an air at T_∞. The convective motion of the air gives rise to the heat transfer coefficient h.

If the thermal conductivity k is very large and the thickness t is small, you can assume that it is a 1D steady-state heat conduction problem in the horizontal direction.

(a) Sketch the temperature distribution along the plate.

(b) Describe all the boundary conditions.

(c) Derive the governing equations and obtain the mathematical expressions for the local temperature.

Now, consider poor thermal conductivity and a large thickness for the plate. You need to treat it as a 2D steady-state heat conduction problem this time. Considering the temperature distribution in the both horizontal and vertical direction,

(d) Describe all the boundary conditions.

(e) Obtain the mathematical expression for the local temperature T(x, y).

To work on these problems, you should carefully set up coordinate axes as discussed in class.

Figure. Plate subjected to heat flux with spatial period on its top and to air convection on its bottom.