Discussion
Consider a rod of uniform thermal conductivity whose one end (\(x = 0\)) is insulated and the other end (\(x = L\)) is exposed to flow of air at temperature \(T_{\infty}\) with convective heat transfer coefficient \(h\). The cylindrical surface of the rod is insulated so that the heat transfer is strictly along the axis of the rod. The rate of internal heat generation per unit volume inside the rod is given as
\[ \dot{q} = \cos\frac{2\pi x}{L} . \]
The steady state temperature at the mid-location of the rod is given as \(T_{A}\). What will be the temperature at the same location, if the convective heat transfer coefficient increases to \(2h\)?
Answer: C
The temperature stays \(T_{A}\).
The total heat generated in the rod, per unit cross-section, is
\[ \int_{0}^{L}\cos\frac{2\pi x}{L}\,dx = 0 \]
The end \(x = 0\) and the sides are insulated, so in steady state the heat leaving at \(x = L\) must equal this net generation, zero. The convective flux \(h(T(L) - T_{\infty})\) is therefore zero, so \(T(L) = T_{\infty}\) whatever the value of \(h\).
The temperature inside follows from \(kT'' = -\dot{q}\) with \(T'(0) = 0\) and \(T(L) = T_{\infty}\), none of which involves \(h\); at mid-length it is \(T_{\infty} - L^{2}/(2\pi^{2}k)\). Doubling \(h\) changes nothing, so the temperature stays \(T_{A}\).
The other options assume a net heat flow through the convective end, which is zero here.
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