Two concentric spheres of diameters D1 = 0.5 m and D2 = 1 m are separated by an air space as shown in given figure and have surface temperatures of 400 K and 300 K respectively.
Two concentric spheres of diameters D1 = 0.5 m and D2 = 1 m are separated by an air space as shown in given figure and have surface temperatures of 400 K and 300 K respectively.
(a) If the surfaces are black, what is the net rate of radiation exchange between the spheres?
(b) (i) What is the net rate of radiation exchange between the surfaces if they are diffuse and gray with ε 1= 0.5 and ε2 = 0.5?
(ii) What error would be introduced by assuming blackbody behaviour for the outer surface(ε2 = 1) with all other conditions remaining the same?
(a) For two concentric spheres radiation heat exchange is \[Q = \frac{A_{1}\sigma \left ( T_{1}^{4} – T_{2}^{4}\right )}{\frac{1}{\epsilon _{1}}+\left ( \frac{A_{1}}{A_{2}} \right )\left ( \frac{1}{\epsilon _{2}}-1 \right )}\]
Since sphere is black, therefore \[\epsilon _{1} = \epsilon _{2} = 1\]
So, \[Q = A_{1}\sigma \left ( T_{1}^{4}-T_{2}^{4} \right ) = \pi\left ( 0.5 \right )^{2}\left ( 5.67\times10^{-8} \right )\left [ 400^{4} – 300^{4} \right ]=779\,W\]
(b)(i) Net rate of radiation exchange between the surfaces if they are diffuse and gray with \(\epsilon _{1} = 0.5\) and\(\epsilon _{2} = 0.5\), \[Q = \frac{779}{\frac{1}{0.5}+\left ( \frac{0.5}{1} \right )^{2}\left ( \frac{1}{0.5}-1 \right )}=346\,W\]
(b)(ii) Error introduced by assuming blackbody behaviour for the outer surface , having all other conditions same, since \(\epsilon _{2} = 1\)
\[Q = A_{1}\sigma \epsilon _{1}\left ( T_{1}^{4} – T_{2}^{4}\right )=\frac{\pi}{4}\left ( 0.5 \right )^{2}\left ( 5.67\times10^{-8} \right )\left [ 400^{4} – 300^{4} \right ]= 389.5\,W\]
Error induced is,
\[\frac{389.5-346}{346}\times 100 = 12.57\, \%\]
