A laboratory apparatus is used to maintain a horizontal slab of ice at -2.2°C so that specimens can be prepared on the surface of the ice and kept close to 0°C. If the ice is 10 cm by 3.8 cm and the laboratory is kept at 16°C, find the cooling rate in watts that the apparatus must provide to the ice.
GIVEN
• A slab of ice in a laboratory
• Ice temperature (Ti) = -2.2°C
• Ice dimensions: 10 cm by 3.8 cm
• Ambient temperature (T?) = 16°C
FIND
• The cooling rate (q) in watts
ASSUMPTIONS
• Air in the laboratory is still
• Effects of sublimation are negligible
• Effects of moisture in the air are negligible
SKETCH
PROPERTIES AND CONSTANTS
for dry air at the mean temperature of 6.9°C
Thermal expansion coefficient (?) = 0.0036 1/K
Thermal conductivity (k) = 0.0242 W/(m K)
Kinematic viscosity (?) = 14.51*10-6 m2/s
Prandtl number (Pr) = 0.71
The characteristic length for the ice is
The Grashof and Rayleigh numbers based on this length are
This is a case with very low Rayleigh number, which is below 105 by two orders of magnitude (RaL = 5.6×103), and hence poses a problem for estimating the heat transfer coefficient. For RaL ? 105, the natural convection heat transfer coefficient can be determined from the NuL value predicted. However, that is not the case here and such situations are indeed encountered in engineering practice. Thus an engineering solution can be obtained by extrapolating and extending the applicability so as to use it to predict the Nusselt number even for the low Rayleigh number case, and hence an approximation can be obtained as follows:
The cooling load is
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