A 1-m-square copper plate is placed horizontally on 2-m-high legs. The plate has been coated with a material that provides a solar absorptivity of 0.9 and an infrared emissivity of 0.25. If the air temperature is 30°C, determine the equilibrium temperature on an average clear day in which the solar radiation incident on a horizontal surface is 850 W/m2

Question

GIVEN

• A horizontal copper plate in air

• Plate dimensions (s × s) = 1 m × 1 m

• Solar absorptance (?s) = 0.9

• Infrared emittance (?) = 0.25

• Air temperature (T?) = 30°C = 303 K

• Incident solar radiation (qs/A) = 850 W/m2

FIND

• Equilibrium Temperature (Ts)

ASSUMPTIONS

• The sky behaves as a black body at 0 K

• The effect of the legs is negligible

• Air is quiescent

• Radiative heat transfer from the bottom of the plate is negligible

SKETCH



PROPERTIES AND CONSTANTS

The Stephan-Boltzmann constant (?) = 5.67 ? 10–8 W/(m2 K4

Answer 1

Since the air properties must be evaluated at the mean of the surface and ambient temperatures, an iterative process must be used

1. Guess at the surface temperature.

2. Evaluate the air properties and calculate the Grashof number.

3. Use the appropriate correlation to find the average convective heat transfer coefficients on the top and bottom of the plate.

4. Calculate a new surface temperature.

This process must be repeated until the temperature converges within an acceptable tolerance. Iteration #1

1. Let Ts = 90°C = 363 K

2. for dry air at he mean temperature of (60°C)

Thermal expansion coefficient (?) = 0.00300 1/K

Thermal conductivity (k) = 0.0279 W/(mK)

Kinematic viscosity (?) = 19.4 × 10–6 m2/s

Prandtl number (Pr) = 0.71

The characteristic length of the plate (L) = A/P = (1 m2)/(4 m) = 0.25 m

The Grashof and Rayleigh numbers based on the characteristic length are



3. For the top of the plate, gives the average Nusselt number



For the bottom of the plate, gives the average Nusselt number



4. For equilibrium, the rate of solar gain must equal the total rate of convective heat transfer from top and bottom and radiative heat transfer from the top surface.



Checking the units then eliminating them for clarity



By trial and error: Ts = 362 K = 89°C.

Since this is very close to the initial guess, another iteration is not necessary. The equilibrium temperature is about 89°C.