A circular ice skating rink, 20 m in diameter, is enclosed by a large hemispherical dome of diameter 30 m. Assuming that the ice is 2 cm thick, estimate the time it takes for the ice to melt if the refrigeration system of the rink fails. Make this calculation first with both the dome and the ice behaving as black bodies at respective temperatures of 20°C and 0°C. Then repeat with the water-ice having an emissivity of 0.3 and losing heat by natural convection to the air in the dome with a convection heat transfer coefficient of 15 W/(m2 K).

Question

GIVEN

• Circular skating rink enclosed by hemispherical dome

• Diameter of the rink (D1) = 20 m

• Diameter of hemispherical dome (D2)= 30 m

• Temperature of dome and surrounding air (T2)= 200C

• Temperature of ice (T1) = 00C

FIND

Time it takes for ice to melt if the refrigeration system of rink fails considering (a) both dome and ice behaving as black bodies. (b) ice having emissivity of 0.3 and loosing heat by natural convection to air in dome with 2 15 W/(m K)

SKETCH



PROPERTIES AND CONSTANTS

Latent Stefan-Boltzman’s heat of fusion constant for ice (Lf)= (? 334 KJ/kg )=5.67*10-8 W/(m2 K4)

Answer 1

(a) F12= 1 for circular ice rink.



Net heat transfer from circular rink to dome is given by



Thus, heat loss if refrigeration system fails is 32.343 kW.

Mass of ice in circular rink is

m = A1*t*? = 314.2*0.02*917 kg/m3 = 6257 kg

Thus total energy required to melt the ice is



Thus total time required to melt the ice is



Thus the time required to completely melt all ice if refrigeration system fails is 18 hrs.

(b) Now, with emissivity of 0.3

Net heat transfer form circular rink to dome is given by.



Thus total heat loss rate to when the refrigeration system fails q= q12+qc = -103.96 kW

Thus total time required to melt the ice is



Thus the time required to completely melt all ice if refrigeration system fails is 5.6 hrs.