Eleven identical rods are arranged as shown in figure. Each rod has length $L$ , cross-sectional area $A$ and thermal conductivity of material $K$ . Ends A and F are maintained at temperatures $T_{1}$ and $T_{2} (T_{2} < T_{1})$ respectively. If lateral surface of each rod is thermally insulated, the rate of heat transfer $(\frac{d Q}{d t})$ in each rod is

{(\frac{d Q}{d t})}_{A B} = {(\frac{d Q}{d t})}_{C D}

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{(\frac{d Q}{d t})}_{B E} = \frac{2}{7} \frac{(T_{1} - T_{2}) K A}{t}

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{(\frac{d Q}{d t})}_{C H} \neq {(\frac{d Q}{d t})}_{D G}

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{(\frac{d Q}{d t})}_{B C} = {(\frac{d Q}{d t})}_{D C}

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Solution

The correct option is D ${(\frac{d Q}{d t})}_{B C} = {(\frac{d Q}{d t})}_{D C}$
Resistance of each identical rod $R = \frac{l}{K A}$ .

Due to the symmetry of the network, heat currents in each of the branches will be as shown below.

In steady state: $T_{B} = T_{D}$
$T_{E} = T_{G}$

Thermal current $(\frac{d Q}{d t}) = i$

For path $A B E F$ ;
$i_{1} R + i_{3} R + i_{1} R = (T_{1} - T_{2})$
$2 i_{1} + i_{3} = \frac{(T_{1} - T_{2})}{R}$ (i)

For the path $A B C H G F$ ,
$i_{1} R + (i_{1} - i_{3}) R + 2 (i_{1} - i_{3}) R + (i_{1} - i_{3}) R + i_{1} R = (T_{1} - T_{2})$
$\Rightarrow 6 i_{1} - 4 i_{3} = \frac{(T_{1} - T_{2})}{R}$ ... (ii)

From (ii) and (iii),
$6 i_{1} - 4 i_{3} = 2 i_{1} + i_{3}$ (iv)
$\Rightarrow 4 i_{1} = 5 i_{3} \Rightarrow i_{3} = \frac{4}{5} i_{i}$ ...(v)

Putting back in eq. (i)
$2 i_{i} + \frac{4}{5} i_{i} = \frac{(T_{1} - T_{2})}{R}$
$\Rightarrow \frac{14 i_{i}}{5} = \frac{(T_{1} - T_{2})}{R}$

Since total current entering and leaving the network is $2 i_{1}$ ,
Equivalent thermal resistance $= \frac{(T_{1} - T_{2})}{2 i_{1}} = \frac{7}{5} R$

Rates of heat transfer across each branch is given by

${(\frac{d Q}{d t})}_{A B} = \frac{5 (T_{1} - T_{2}) K A}{14 l}$

${(\frac{d Q}{d t})}_{B E / D G} = \frac{2}{7} \frac{(T_{1} - T_{2}) K A}{l}$

${(\frac{d Q}{d t})}_{B C / D C / H G} = \frac{(T_{1} - T_{2}) K A}{14 l}$

${(\frac{d Q}{d t})}_{C H} = \frac{(T_{1} - T_{2}) K A}{7 l}$

Hence, options (b), (c) and (d) are correct.

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Similar questions

Q. Eleven identical rods are arranged as shown in figure. each rod has length

l

, cross-sectional area A and thermal conductivity of material K. ends A and F are maintained at temperatures

T_{1}

and

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, respectively. if lateral surface of each rod is thermally insulated, the rate of heat transfer

(\frac{d Q}{d t})

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Four identical rods AB, CD, CF and DE are joined as shown in figure. The length, cross -sectional area and thermal conductivity of each rod are l, A and K respectively. The ends A, E and F are maintained at temperatures $T_{1}, T_{2}$ and $T_{3}$ respectively. Assuming no loss of heat to the atmosphere, find the temperature at B.