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 conductivites 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) gt T gt T_(2) gt T_(3)`, we have
`(DeltaQ)/(Deltat)]_(AB)=(DeltaQ)/(Deltat)]_(BC)+(DeltaQ)/(Deltat)]_(BD)`
i.e. `(KA(T_(1)-T))/(2L)=(2KA(T-T_(2)))/(L)+(3KA(T-T_(3)))/(L)`
or `(T_(1)-T)/(2)=2(T-T_(2))+3(T-T_(3))`
or `T=(1)/(11)(T_(1)+4T_(2)+6T_(3))`