Example Problem with Complete Solution

4B-1 : Radiation Heating and Convective Cooling of a Flat Plate 3 pts
A thin, metal plate is insulated on the back surface and exposed to solar radiation on the front surface, as shown here. The exposed surface of the plate has an absorptivity of 0.6 for solar radiation. If solar radiation is incident on the plate at a rate of 700 W/m2 and the surrounding air temperature is 25°C, determine the temperature of the exposed surface of the plate at steady-state. Assume the convection heat transfer coefficient is 50 W/m2-K and disregard heat loss by radiation.
  Figure 1    
Given: a 0.6 Find:
qmax 700 W/m2 Ts ??? oC
Tair 25 oC
h 50 W/m2-K
- Radiation heat losses from the plate are negligible.
- Heat losses through the edges of the plate are negligible.
- The back of the plate is perfectly insulated.  Thus, at steady-state, the temperature of the plate is uniform.
- The incident radiation, the convection heat transfer coefficient and the absorptivity of the surface are all uniform over the surface of the plate.
Solution: 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.
Equation 1
Eqn 1
a is the fraction of the incident radiation that is absorbed by a surface.  Therefore:
Equation 2 Eqn 2
Newton's Law of Cooling gives us the convection heat transfer rate at the surface of the plate.
Equation 3 Eqn 3
Set Eqn 1 equal to Eqn 2 and solve for Ts:
Equation 4 Eqn 4
Plug numbers into Eqn 3 to answer the question:
Ts 33.4 oC