Question

A rod of length l and cross section area A has a variable thermal conductivity given by K = a T, where a is a positive constant and T is temperature in Kelvin. Two ends of the rod are maintained at temperatures T₁ and T₂ (T₁> T₂). Heat current flowing through the rod will be –

• A. $\frac{A\alpha(T_{1}^{2} - T_{2}^{2})}{3\mathcal{l}}$
• B. $\frac{A\alpha(T_{1}^{2} + T_{2}^{2})}{\mathcal{l}}$
• C. $\frac{A\alpha(T_{1}^{2} + T_{2}^{2})}{3\mathcal{l}}$
• D. $\frac{A\alpha(T_{1}^{2} - T_{2}^{2})}{2\mathcal{l}}$

Answer 1

Answer: (D)

$\frac{A\alpha(T_{1}^{2} - T_{2}^{2})}{2\mathcal{l}}$

Answer 2

Heat current i = – KA $\frac{dT}{dX}$

idX = – KA dT

$i\int_{0}^{\mathcal{l}}{dX = - A\alpha\int_{T_{1}}^{T_{2}}{TdT}}$

̃ il = – A a$\frac{(T_{2}^{2} - T_{1}^{2})}{2}$

̃ i = A a $\frac{(T_{1}^{2} - T_{2}^{2})}{2\mathcal{l}}$