# Find the Thermal Conductivity of an Infinitely Long Horizontal Plate

Question

A plate of thickness t is subjected to the imposed surface flux q," on its top surface with a pitch of 2L as shown in the 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 an ID 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).

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

Summary

The question belongs to Mechanical Engineering and it is about thermal conductivity of a plate which has infinitely long in the horizontal direction. The questions are about finding the temperature distribution along the plate, describe the boundaries of the plate and to find the governing equations for the plate.

Note: The solution is in handwritten format.

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