First slide

Conduction

Question

Three rods each of same length and cross section are joined in series. The thermal conductivity of the materials are k, 2k and 3k respectively. If one end is kept at 300 ⁰C and the other at 100 ⁰C. What would be the temperature of the junction which comes before the end at 100 ⁰C in the steady state. Thermal conductivity of copper is 400 WmK^-1.

Moderate

Solution

Thermal resistance $R = \frac{l}{k A}$
$\frac{1}{k_{e q}} = \frac{1}{k} + \frac{1}{2 k} + \frac{1}{3 k} = \frac{6 + 3 + 2}{6 k} = \frac{11}{6 k}$
$\frac{T_{j 1} - T_{i}}{T_{f} - T_{i}} = \frac{k_{eq}}{3 k} \frac{T_{j 1} - 100}{300 - 100} = \frac{6 k}{11} \frac{1}{3 k} = \frac{2}{11} T_{j 1} = \frac{400}{11} + 100 = 136.4 {}^{0}C$

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