Three rods of same cross-sectional area AB, BC and BD having thermal conductivities in the ration $1 : 2 : 3$ and lengths in the ration 2:1:1 are joined as shown is Figure. The ends A, C and D are at temperatures $T_{1}, T_{2}$ and $T_{3}$ respectively.
The temperature of the junction B is $T = \frac{1}{d} (a T_{1} + b T_{2} + c T_{3}) .$ Assume steady state. Find the value of $(a + b + c) / d$ value.

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Detailed Solution

Let the thermal conductivities of the rods AB, BC and BD be K, 2K and 3K respectively. Also, let their lengths be 2L, L and L.
If T be the required temperature of the junction B and assuming $T_{1} > T > T_{2}, T_{3}$ ,
we have
${\frac{Δ Q}{Δ t}]}_{AB} = {\frac{Δ Q}{Δ t}]}_{BC} + {\frac{Δ Q}{Δ t}]}_{BD}$ (Point rule)
i.e., $\frac{K A (T_{1} - T)}{2 L} = \frac{2 K A (T - T_{2})}{L} + \frac{3 K A (T - T_{3})}{L}$