Transmission Of Heat Question 351
Question: Direction: Consider a spherical body A of radius R which placed concentrically in a hollow enclosure H, of radius 4R as shown in the figure. The temperature of the body A and H are $ T _{A} $ and $ T _{H}, $ respectively.
Emissivity, transitivity and reflectivity of two bodies A and H are $ \text{(}e _{A},e _{H})\text{ (}t _{A},t _{H}\text{)} $ and $ (r _{A},r _{H}) $ respectively. For answering following questions assume no absorption of the thermal energy by the space in-between the body and enclosure as well as outside the enclosure and all radiations to be emitted and absorbed normal to the surface. [Take $ \sigma \times ,4\pi R^{2},\times ,300^{4}=,\beta J{{s}^{-1}} $ ] Consider two cases, first one in which A is a perfect black body and the second in which A is a non-black body. In both the cases, temperature of body A is same equal to 300K and H is at temperature 600K. For H, $ t=0 $ and $ a\ne 1 $ . For this situation, mark out the correct statement.
Options:
A) The bodies lose their distinctiveness inside the enclosure and both of them emit the same radiation as that of the black body.
B) The rate of heat loss by A in both cases is the same and is equal to$ \beta ,J{{s}^{-1}} $ .
C) The rates of heat loss by A in both the cases are different.
D) From this information we can calculate exact rate of heat loss by A in different cases.
Show Answer
Answer:
Correct Answer: C
Solution:
[c]
If $ \beta =\sigma \times 4\pi R^{2}\times ,300^{4}, $
Then $ \sigma \times ,4\pi ,{{(4R)}^{2}}\times ,600^{4},=256\beta ,=\gamma $
Let $ a _{H},=e _{H}=0.5 $
and for A in 2nd case, $ e _{A}=,a _{A}=,0.5 $
For 1st case, $ P _{emitted},=,\beta ,J{{s}^{-1}} $
$ P _{absorbed},=,\frac{\gamma }{2},+\frac{\beta }{2} $
Rate at which energy is lost, $ P=( \beta -\frac{\gamma }{2}-\frac{\beta }{2} ),J{{s}^{-1}} $
For 2nd case,
$ P _{emitted}=( \frac{\beta }{2}+,\frac{\beta }{8}+\frac{\beta }{32}+…. ),+( \frac{\gamma }{4}+\frac{\gamma }{16}+… ) $
$ =,\frac{2\beta }{3}+,\frac{\gamma }{3} $
$ P _{absorbed}=,( \frac{\beta }{8}+,\frac{\beta }{32}+…. ),+,( \frac{\gamma }{4}+,\frac{\gamma }{16}+… ),,=\frac{\beta }{6}+\frac{\gamma }{3} $
Rate at which heat is lost, $ P=,\frac{\beta }{2} $ .
BETA
