Two rectangular blocks, having identical dimensions, can be arranged either in configuration-I or in configuration-II as shown in the figure. One of the blocks has thermal conductivity and the other [2013]

Question

Two rectangular blocks, having identical dimensions, can be arranged either in configuration-I or in configuration-II as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2 k$ . The temperature difference between the ends along the $x$ -axis is the same in both the configurations. It takes 9 s to transport a certain amount of heat from the hot end to the cold end in the configuration-I. The time to transport the same amount of heat in the configuration-II is [2013]

Smart Achievers · Accepted Answer

(1)

Equivalent thermal resistance in configuration-I

$R_{s} = R_{1} + R_{2} = \frac{L}{K A} + \frac{L}{2 K A} = \frac{3}{2} \frac{L}{K A}$

Equivalent thermal resistance in configuration-II

$\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} = \frac{K A}{L} + \frac{2 K A}{L}$ or, $R_{p} = \frac{L}{3 K A} = \frac{R_{s}}{4.5}$

i.e., thermal resistance in configuration-II, $R_{p}$ is 4.5 times less than thermal resistance in configuration-I, $R_{s}$ .

$∴$ $4.5 t_{p} = t_{s}$ $\Rightarrow t_{p} = \frac{t_{s}}{4.5} = \frac{9}{4.5} s = 2 s$

Solution

1 2.0 s

2 4.5 s

3 3.0 s

4 6.0 s

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