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GATE 2024 CH – Question 39

Heat Transfer · Conduction: steady and unsteady, equation of energy · 2 marks · Multiple choice

A solid slab of thickness $H_1$ is initially at a uniform temperature $T_0$. At time $t = 0$, the temperature of the top surface at $y = H_1$ is increased to $T_1$, while the bottom surface at $y = 0$ is maintained at $T_0$ for $t \ge 0$. Assume heat transfer occurs only in the $y$-direction, and all thermal properties of the slab are constant. The time required for the temperature at $y = H_1/2$ to reach 99% of its final steady value is $\tau_1$. If the thickness of the slab is doubled to $H_2 = 2H_1$, and the time required for the temperature at $y = H_2/2$ to reach 99% of its final steady value is $\tau_2$, then $\tau_2/\tau_1$ is

  1. 2
  2. $\frac{1}{4}$
  3. 4
  4. $\frac{1}{2}$

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Correct answer: (C) 4

Explanation

In unsteady conduction the dimensionless time is the Fourier number $\frac{\alpha t}{H^2}$. The same relative temperature at the same relative position needs the same Fourier number, so $t \propto H^2$. Doubling the thickness gives $\frac{\tau_2}{\tau_1} = 4$.