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GATE 2024 ME – Question 17

Heat Transfer · Modes of heat transfer, one-dimensional conduction, resistance concept · 1 mark · Multiple choice

A plane, solid slab of thickness $L$, shown in the figure, has thermal conductivity $k$ that varies with the spatial coordinate $x$ as $k = A + Bx$, where $A$ and $B$ are positive constants ($A > 0$, $B > 0$). The slab walls are maintained at fixed temperatures of $T(x = 0) = 0$ and $T(x = L) = T_0 > 0$. The slab has no internal heat sources. Considering one-dimensional heat transfer, which one of the following plots qualitatively depicts the steady-state temperature distribution within the slab?

A slab between $x = 0$ at temperature 0 and $x = L$ at temperature $T_0$, followed by four plots (A) to (D) of $T$ against $x$ from 0 to $T_0$. (A) is a straight line. (B) rises steeply at first and then flattens, so it is concave. (C) rises slowly at first and then steeply, so it is convex. (D) has an S shape with a flat middle.
  1. Option A in the figure
  2. Option B in the figure
  3. Option C in the figure
  4. Option D in the figure

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Show answer and explanation

Correct answer: (B) Option B in the figure

Explanation

With no heat source the heat flux $q = -k\frac{dT}{dx}$ is the same at every section. So $\frac{dT}{dx} = -\frac{q}{A + Bx}$, and its magnitude falls as $x$ increases because the conductivity grows. The temperature therefore rises steeply near $x = 0$, where the conductivity is lowest, and flattens towards $x = L$. That is the concave curve in plot B.