A small gray sphere having an emissivity of 0.5 and a surface temperature of 5370C is located in a blackbody enclosure having a temperature of 37°C. Calculate for this system: (a) the net rate of heat transfer by radiation per unit of surface area of the sphere, (b) the radiative thermal conductance in W/K if the surface area of the sphere is 95 cm2 , (c) the thermal resistance for radiation between the sphere and its surroundings, (d) the ratio of thermal resistance for radiation to thermal resistance for convection if the convective heat transfer coefficient between the sphere and its surroundings is 11 W/(m2 K), (e) the total rate of heat transfer from the sphere to the surroundings, and (f) the combined heat transfer coefficient for the sphere.
GIVEN
• Small gray sphere in a blackbody enclosure
• Sphere emissivity (?s) = 0.5
• Sphere surface temperature (T1) = 3370C= 810 K
• Enclosure temperature (T2) = 37°C = 310 K
• The surface area of the sphere (A) = 95 cm2=9.5*10-3 m2
• The convective transfer coefficient ( ch ) = 11 W/(m2 K)
FIND
(a) Rate of heat transfer by radiation per unit surface area (b) Radiative thermal conductance (Kr) in W/K (c) Thermal resistance for radiation (Rr) (d) Ratio of the radiative and conductive resistance (e) Total rate of heat transfer (qT) to the surroundings (f) Combined heat transfer coefficient ( cr h )
ASSUMPTIONS
• Steady state prevails
• The temperature of the fluid in the enclosure is equal to the enclosure temperature
SKETCH
(a) For a gray body radiating to a blackbody enclosure the net heat transfer is
(b) The radiative thermal conductance must be based on some reference temperature. Let the
reference temperature be the enclosure temperature. Then, from Equation (1.20), the radiative
thermal conductance is
(c) The thermal resistance for radiation is given by
(d) The convective thermal resistance is given by
Therefore the ratio of the radiative to the convective resistance is
(e) The radiative and convective resistances are in parallel, therefore the total resistance,
The total heat transfer is given by:
(f) The combined heat transfer coefficient can be calculated from
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