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GATE 2022 ME (ME1) – Question 45

Heat Transfer · Modes of heat transfer, one-dimensional conduction, resistance concept · 2 marks · Multiple choice

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 $2h$?

  1. $T_A + \frac{\dot{q}L}{2h}$
  2. $2T_A$
  3. $T_A$
  4. $T_A\left(1 - \frac{\dot{q}L}{4\pi h}\right) + \frac{\dot{q}L}{4\pi h}T_\infty$

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Correct answer: (C) $T_A$

Explanation

The total heat generated in the rod is $A\int_0^L\cos\frac{2\pi x}{L}dx = 0$. The end $x = 0$ is insulated, so the heat leaving at $x = L$ is zero, which means $h(T_L - T_\infty) = 0$ and $T_L = T_\infty$ whatever the value of $h$. The temperature profile therefore does not depend on $h$, and the temperature at the mid-location remains $T_A$.