A 15 cm black disk is placed halfway between two black 3-m-diameter disks that are 7 m apart with all disk surfaces parallel to each other. If the surroundings are 0 K, determine the temperature of the two larger disks required to maintain the smaller disk at 540°C.
GIVEN
GIVEN
• A black disk (A1) halfway between two other black disks (A2 & A3)
• Diameter of A1 (D1) = 15 cm = 0.15 m
• Diameter of A2 and A3: (D2 = D3) = 3 m
• Distance between A2 and A3 (2L) = 7 m
• Surrounding temperature (T?) = 0 K
• Temperature of A1 (T1) = 540°C = 813 K
FIND
• The temperature A2 and A3 required ASSUMPTIONS
• A2 and A3 are at the same temperature (T2 = T3)
• Steady state conditions
SKETCH
PROPERTIES AND CONSTANTS
the Stephan-Boltzmann constant (?) = 5.67 × 10–8 W/(m2 K4)
The shape factor for the geometry is
where a = D1/2 and 0.075 m b = D2/2 = 1.5 m L = 3.5 m
By symmetry F13 = F12 = 0.155.
The sum of the shape factors, including the shape factor with the surroundings must be unity
The net rate of heat transfer from A1 to A2 is
Similarly
For steady state, these rates of heat transfer must sum to zero
Solving for T2
The net rate of heat transfer from A1 to A2 is
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