Solar radiation at 600 W/m2 is absorbed by a black roof inclined at 30°C as shown. If the underside of the roof is well insulated, estimate the maximum roof temperature in Tinf= 20°C air.
GIVEN
• Inclined roof, well insulated on the underside
• Incline angle (?) = 30 degrees
• Air temperature = 20°C
• Solar radiation absorbed (qs) = 600 W/m2
FIND
• The maximum roof temperature
ASSUMPTIONS
• The roof behaves as a black body (? = 1.0)
• The sky behaves as a black body at 0 K
SKETCH
PROPERTIES AND CONSTANTS
The Stephan-Boltzmann constant (?) = 5.67 × 10–8 W/(m2 K4).
The maximum roof temperature will occur when the air is quiescent. Since the air properties must be evaluated at the mean of the ambient and surface temperatures, as iterative procedure must be used. Iteration #1 Let Ts = 60°C = 333 K
for dry air at the mean temperature of 40°C
Thermal expansion coefficient (?) = 0.00319 1/K
Thermal conductivity (k) = 0.0265 W/(m K)
Kinematic viscosity (?) = 17.6 × 10–6 m2/s
Prandtl number (Pr) = 0.71
The Grashof number is
The average convective heat transfer coefficient for this geometry. Although GrL Pr cos ? is slightly larger than 1011, will be extrapolated by this problem
For steady state, the solar gain must equal the convective and radiative losses
Checking the units then eliminating them for clarity
By trial and error: Ts = 314 K.
Repeating this procedure for another iteration
The maximum roof temperature:
COMMENTS
The procedure converges quickly because of the 1/4 power in the Nusselt number correlation.
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