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 gap between the plates can be considered to be a blackbody square cylindrical surface as shown in the sketch.
Also, since neither A1 nor A2 can see itself, F11 = F22 = 0.
From Equation (11.46) 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 using Equations (11.69) and (11.66)
From Equation (11.69) [1] A1 G1 = J1 A1 F11 + J2 A2 F21 + J3 A3 F31 = J2 A2 F21
[2] A2 G2 = J1 A1 F12 + J2 A2 F22 + J3 A3 F32 = J1 A1 F12
[3] A3 G3 = J1 A1 F13 + J2 A2 F23 + J3 A3 F33 = J1 A1 F13 + J2 A2 F23
From Equation (11.66) [4] J1 = ??1 G1 + ??1 Eb1?
[5] J2 = ??2 G2 + ??2 Eb2
[6] J3 = 0
Substituting Equations [4] and [5] into [1] and [2]
Substituting G1 from the first equation into the second and using F21 = F12 yields
The shape factor F12 can be determined from Figure 11.32: for x/D = y/D= 5 ? F12 = 0.71
Also
From Equation [5]
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