Two 1.5 m-square and parallel flat plates are 30 cm apart. Plate A1 is maintained at a temperature of 1100 K and A2 at 500 K. The emissivities of the plates are 0.5 and 0.8, respectively. Considering the surroundings black at 0 K and including multiple inter-reflections, determine (a) the net radiant exchange between the plates and (b) the heat input required by surface A1 to maintain its temperature. The outer-facing surfaces of the plates are adiabatic.
GIVEN
• Two square parallel flat plates, 30 cm apart
• Temperature of A1 (T1) = 1100 K
• Temperature of A2 (T2) = 500 K
• Emittances: ?1 = 0.5 ?2 = 0.8
• Surroundings are black at (T3) = 0 K
FIND
Including multiple inter-reflections, determine:
(a) The net radiant exchange (q1–2)
(b) The heat input at surface A1 (q1) required to maintain its temperature
SKETCH
PROPERTIES AND CONSTANTS
the Stephan-Boltzmann constant (?) = 5.67 × 10–8 W/(m2 K4)
The gap between the plates can be considered to be a blackbody square cylindrical surface as shown in the sketch.
?3 = 0 and ?3 = 1 ? J3 = Eb3 = ? T34 = 0
Also, since neither A1 nor A2 can see itself, F11 = F22 = 0.
A1 F12 = A2 F21 ? F12 = F21 (This is apparent from the symmetry of the problem). The radiosities J1 and J2 are needed to calculate the rate of radiant heat transfer and can be determined
Substituting Equations [4] and [5] into [1] and [2]
Substituting G1 from the first equation into the second and using F21 = F12 yields
since A1 = A2 and Ebi = ? Ti4
The shape factor F12 can be determined for x/D = y/D= 5 ? F12 = 0.71
Also
G1 = F21 J2 = F12 J2 = 0.71( 9192 /W m 2 ) = 6435 W/m2
(a) The net radiant exchange is
(b) The required input to surface A1 is equal to the rate of radiative loss from surface A1 which is
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