# Example Problem with Complete Solution

4B-1 : Radiation Heating and Convective Cooling of a Flat Plate 3 pts
The sun shines on the front of a thin plate that is insulated on the back surface, as shown below. The surface of the plate exposed to the sun has an absorptivity of 0.72 for solar radiation. The total solar radiation incident on the front of the plate is 650 W/m2. The front of the plate loses heat to thesurroundings which are at 20oC but radiation heat loss from the plate is negligible.
The convection heat transfer coefficient is 25 W/m2-K. The plate warms up until the solar heat gained is balanced by the convection heat lost. Determine the temperature of the front surface of the plate at steady-state.

Read : The key to this problem is to recognize that at steady-state, the rate at which heat is transferred into the plate from the sun by radiation must be equal to the rate at which heat is lost from the plate to the surrounding air by convection.
Diagram: The diagram in the problem statement is adequate.
Given: a 0.72 Tair 20 oC
qmax 650 W/m2 h 25 W/m2-K
Find: Ts ??? oC
Assumptions:
1 - Radiation heat losses from the plate are negligible.
2 - Heat losses through the edges of the plate are negligible.
3 - The back of the plate is perfectly insulated.  Thus, at steady-state, the temperature of the plate is uniform.
4 - The incident radiation, the convection heat transfer coefficient and the absorptivity of the surface are all uniform over the surface of the plate.
Equations / Data / Solve:
The key to this problem is to recognize that at steady-state, the rate at which heat is transferred into the plate from the sun by radiation must be equal to the rate at which heat is lost from the plate to the surrounding air by convection. Eqn 1
Absorptivity, a, is the fraction of the incident radiation that is absorbed by a surface.  Therefore: Eqn 2
Newton's Law of Cooling gives us the convection heat transfer rate at the surface of the plate. Eqn 3
Set Eqn 1 equal to Eqn 2 and solve for Ts: Eqn 4
Plug numbers into Eqn 3 to answer the question: Ts 38.72 oC
Verify: The assumptions cannot be verified from the information in the problem statement alone. 