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/m^{2}. The front of the
plate loses heat to thesurroundings which are at 20^{o}C but radiation
heat loss from the plate is negligible. 
The convection heat transfer
coefficient is 25 W/m^{2}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 steadystate. 


Read : 
The key to this problem is to recognize
that at steadystate,
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 
^{} 



T_{air} 
20 
^{o}C 

q_{max} 
650 
W/m^{2} 



h 
25 
W/m^{2}K 










Find: 
T_{s} 
??? 
^{o}C 
















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 steadystate, 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 steadystate,
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 T_{s}: 

















Eqn 4 











Plug numbers into Eqn 3 to answer the question: 

T_{s} 
38.72 
^{o}C 










Verify: 
The assumptions cannot
be verified from the information in the problem statement alone. 










Answers : 
T_{s} 
38.7 
^{o}C 















