Transmission Of Heat Question 341
Question: A rod of length i and cross section area A has a variable thermal conductivity given by $ k=\alpha T, $ where a is a positive constant and T is temperature in kelvin. Two ends of the rod are maintained at temperatures $ T _{1} $ and $ T _{2}(T _{1}>T _{2}) $ . Heat current flowing through the rod will be
Options:
A) $ \frac{A\alpha (T _{1}^{2}-T _{2}^{2})}{\ell } $
B) $ \frac{A\alpha (T _{1}^{2}+T _{2}^{2})}{\ell } $
C) $ \frac{A\alpha (T _{1}^{2}+T _{2}^{2})}{3\ell } $
D) $ \frac{A\alpha (T _{1}^{2}-T _{2}^{2})}{2\ell } $
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Answer:
Correct Answer: D
Solution:
[d] Heat current $ i=-kAdT $ $ id=-kA,dT $ $ i\int\limits _{0}^{\ell }{dx}=-A\alpha \int\limits _{T _{1}}^{T _{2}}{T}dT $
$ \Rightarrow $ $ i,\ell =-A,\alpha \frac{( T _{2}^{2}-T _{1}^{2} )}{2} $
$ \Rightarrow $ $ i=\frac{\Alpha \alpha ( T _{1}^{2}-T _{2}^{2} )}{2\ell } $
BETA
