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Standard XII

Mathematics

Homogenization

Three identic...

Question

Three identical rods $A . B$ and $C$ of equal lengths and equal diameters are joined in series as shown in figure. Their thermal conductivities are $2 k, k$ and $k / 2$ respectively. The temperatures at two junction points are

85.7, 57.1

^{\circ}

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80.85, 50.3

^{\circ}

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77.3, 48.3

^{\circ}

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75.8, 49.3

^{\circ}

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Solution

The correct option is C 85.7, 57.1 $^{\circ}$ C
Governing equation:
$˙ Q = K A \frac{Δ T}{l}$
Heat flowing through all the rods is the same, and all rods have same lengths and cross sectional areas.
I.e. $\frac{˙ Q l}{A} = K Δ T = c o n s t a n t$ ;
I.e. $0.5 K (T_{2} - 0) = K (T_{1} - T_{2}) = 2 K (100 - T_{1})$
Combining any two of those equations and solving we get $T_{1} = 85.7$
and $T_{2} = 57.1$

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