Question

Three rods AB, BC and BD having thermal conductivities in the ratio 1:2:3 and lengths in the ratio 2:1:1 are joined as shown in Fig. The ends A, C and D are at temperature `T_1`,`T_2` and `T_3` respectively Find the temperature of the junction B. Assume steady state.

Answer 1

Let the thermal conductivity of the rod AB, BC and BD ber K, 2K, and 3K respectively. Also let the length be 2L, L and L.
If T be the required temperature of the junction B and assuming `T_1gtTgtT_2` and `TgtT_3`, we have
`(DeltaQ)/(Deltat)]_(AB)=(DeltaQ)/(Deltat)]_(AB)+(DeltaQ)/(Deltat)]_(BD)` (Point rule)
`(KA(T_1-T))/(2L)=(2KA(T-T_2))/(L)+(3KA(T-T_3))/(L)`
`(T_1-T)/(2)=2(T-T_2)+3(T-T_3)`
`T=(1)/(11)(T_1+4T_2 + 6T_3)`