An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface ¡s well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density ρ and specific heat c of aluminum are known to be 2700 Kg / m 3 and 900 J/Kg ⋅ K , respectively. (a) Consider conditions for which the plate ¡s at a temperature of 25°C and its top surface is suddenly exposed to ambient air at T ∞ = 20 ° C and to solar radiation that provides an incident flux of 900 W / m 2 . The convection heat transfer coefficient between the surface and the air is h = 20 W / m 2 ⋅ K . What ¡s the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? (c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for 0.05 ≤ ∈ ≤ 1 , with all other conditions remaining as prescribed. Repeat your calculations for values of α s = 0.5 and 1.0, and plot the results with those obtained for α s = 0.8 . If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation?
Solution Summary: The author calculates the energy balance to be applied at an instant of time to a control about the plate.
An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface ¡s well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density
ρ
and specific heat c of aluminum are known to be
2700
Kg
/
m
3
and
900
J/Kg
⋅
K
, respectively. (a) Consider conditions for which the plate ¡s at a temperature of 25°C and its top surface is suddenly exposed to ambient air at
T
∞
=
20
°
C
and to solar radiation that provides an incident flux of
900
W
/
m
2
. The convection heat transfer coefficient between the surface and the air is
h
=
20
W
/
m
2
⋅
K
. What ¡s the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? (c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for
0.05
≤
∈
≤
1
, with all other conditions remaining as prescribed. Repeat your calculations for values of
α
s
=
0.5
and 1.0, and plot the results with those obtained for
α
s
=
0.8
. If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation?
The emissivity of galvanized steel sheet, a common roofing material, is ε = 0.13 at temperatures around 300 K, while its absorptivity for solar irradiation is αS = 0.65. Would the neighborhood cat be comfortable walking on a roof constructed of the material on a day when GS = 750 W/m2, T∞ = 16°C, and h = 7 W/m2 · K? Assume the bottom surface of the steel is insulated.
A spherical interplanetary probe of 0.5-m diameter contains electronics that dissipate 150 W. If the probe surface has an emissivity of 0.8 and the probe does not receive radiation from other surfaces, as, for example, from the sun, what is its surface temperature?
A group of 25 power transistors, dissipating 1.5 W each, are to be cooled by attaching them to a black-anodized square aluminum plate and mounting the plate on the wall of a room at 30°C. The emissivity of the transistor and the plate surfaces is 0.9. Assuming the heat transfer from the back side of the plate to be negligible and the temperature of the surrounding surfaces to be the same as the air temperature of the room, determine the size of the plate if the average surface temperature of the plate is not to exceed 50°C.
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