Question

The three rods shown in figure have identical dimensions. Heat flows from the hot end at a rate of $40\, W$ in the arrangement (a). Find the rates of heat flow when the rods are joined as in arrangement (b). (Assume $K_{ Al }=200 \,W / m ^{\circ} C$ and $\left.K_{ cu }=400\, W / m ^{\circ} C \right)$

• A. 75 W
• B. 200 W
• C. 400 W
• D. 4 W

Answer 1

Answer: (C)

$\because$ The rods have identical dimensions.
Let their area of crossection be $=A$ and length be $=L$
So each rod would have heat resistance of
$R=\frac{L}{K A}$
$R_{\text{eff}}=R_{1}+R_{2}+R_{3}$
$=\frac{L}{K_{A l} A}+\frac{L}{K_{C u} A}+\frac{L}{K_{A l} A}$
$R_{\text {eff }}=\frac{L}{A} \times \frac{5}{400}$
$\left[\because K _{ Al }=200, K _{ Cu }=400\right]$
$H_{1}=\frac{\Delta T}{R_{\text {eff }}}=\frac{100}{\frac{L}{A} \times \frac{5}{400}}\,\,\,...(i)$
(b) When rods are connected in parallel
$\frac{1}{R_{ eff }}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}$
$R_{ eff }=\frac{L}{A} \times 800$
$H_{2}=\frac{100}{\frac{L}{A} \times 800}\,\,\,...(2)$
Dividng (2) by (1)
$\frac{40}{H_{2}}=\frac{400}{800 \times 5} \Rightarrow H_{2}=400\, W$