A thick opaque plate is facing from below a radiating wall that is maintained at a high temperature (Tw > 1,500 K), as illustrated below. The space between the plate and the wall is vacuumed. The plate is cooled from the top by blowing ambient air (T = 300 K, and u = U) along its upper surface in the x-direction.

1- Provide the complete governing differential equations (along with the required boundary conditions) that if solved will yield the temperature profile in the plate. Do not solve the governing equations; however, you will need to clearly state all your imposed acceptable assumptions.

2- Explain how you would calculate the local Nu number along the upper surface of the plate. Draw the air flow viscous and thermal boundary layers along x-direction on one common figure.

3- Illustrate the plate upper surface (exposed to air) and lower surface (exposed to radiating wall) temperature profiles along x-direction. Show the two profiles on one common figure.

4- Illustrate the temperature profile in the y-direction within the plate at x = L/2.

5- Describe how the formulation of the problem in hand will change if the gap size, H, significantly increases. Do not provide the formulation.

Assume the following conditions.

a- Consider the problem to be at steady state.

b- The air flow is laminar.

c- All properties (fluid and the plate) are constant except for the thermal conductivity of the plate, k, which is a strong function of temperature.

d- The two sides between the plate and heated wall play a negligible role on the transfer of heat.

e- The two sides of the plate are perfectly insulated.

f- The spectral emissivities of the wall (on the top surface) and the plate (on the bottom surface) are given below.

Hint: The heat flux at the plate/air interface is not uniform.