Three identical heat conducting rods are connected in series as shown in the figure. The rods on the

Question

Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity $2 K$ while that in the middle has thermal conductivity $K$. The left end of the combination is maintained at temperature $3 T$ and the right end at $T$. The rods are thermally insulated from outside. In steady state, temperature at the left junction is $T_1$ and that at the right junction is $T_2$. The ratio $T_1 / T_2$ is

Professor PC Thomas Classes · Answer

(a) : According to heat transfer, at junction \ $T_1$ $ \frac{2 K A\left(3 T-T_1ight)}{L}-\frac{K A\left(T_1-T_2ight)}{L} $ \ Since, all identical rods are connected in series, So, $A$ and $L$ are same. $ \begin{aligned} & 	herefore \quad 6 T-2 T_1=T_1-T_2 \ & 6 T=3 T_1-T_2 \end{aligned} $ \ At junction $T_2$, $ \frac{K A\left(T_1-T_2ight)}{L}=\frac{2 K A\left(T_2-Tight)}{L} $ \ or $\quad T_1-T_2=2 T_2-2 T$ \ or $\quad 2 T=3 T_2-T_1$ From eqn. (i) and (ii) $ \frac{T_1}{T_2}=\frac{5}{3} $