Question

Four identical rods AB, CD, CF and DE are joined as shown in figure (28-E8). The length, cross-sectional area and thermal conductivity of each rod are l, A and K respectively. The ends A, E and F are maintained at temperature T1, T2 and T3 respectively. Assuming no loss of heat to the atmosphere, find the temperature at B.
Figure

Answer 1

The temperature at B is $\mathrm{T}=\left(3 T_{1}+2 T_{2}+2 T_{3}\right) / 7$

Explanation:

Given,

Temperature at end A = T1

Temperature at end E = T2

Temperature at end F = T3

Let the temperature at end B = T

Length, area and thermal conductivity of all four rods are equal

As there is no heat loss in the atmosphere, the heat must be distributed as  

$\mathrm{q}_{1}=\mathrm{q}_{2}+\mathrm{q}_{3}$

$\frac{K A\left(T_{1}-T\right)}{l}=\frac{K A\left(T-T_{2}\right)}{l+l / 2}+\frac{K A\left(T-T_{3}\right)}{l+l / 2}$        (the bent portion have half length of original length of rod)

Step 1:

$T_{1}-T=\frac{2\left[T-T_{2}+T-T_{3}\right]}{3}$

Step 2:

$3\left(T_{1}-T\right)=2\left(2 T-T_{2}-T_{3}\right)$

Step 3:

$\mathrm{T}=\left(3 T_{1}+2 T_{2}+2 T_{3}\right) / 7$

This must be the temperature at end B.