Question

What is the value of ${\theta }_1$, the temperature of junction $B$?

• A. ${70.9}^o$C
• B. ${50.4}^o$C
• C. ${40.4}^o$C
• D. ${30.4}^o$C

Answer 1

Answer: (A)

${70.9}^o$C

The junction law states that through a junction, The heat flowing in is equal to heat going out.

In the steady state there is no accumulation of heat at any point. Hence at B the incoming heat current must be equal to the outgoing heat current.

Let $2K$ be the thermal conductivity of material $x$ and $K$ the thermal conductivity of material $y$

$\frac{2KA({90}^{\circ }-{\theta }_1)}{L}=\frac{2KA({\theta }_1-{\theta }_2)}{L}+\frac{KA({\theta }_1-{\theta }_2)}{L}$

$2({90}^{\circ }-{\theta }_1)=2({\theta }_1-{\theta }_2)+({\theta }_1-{\theta }_2)$

${180}^{\circ }=5{\theta }_1-3{\theta }_2$ .....(1)

Again applying the same rule to junction $C$

$\frac{2KA({\theta }_1-{\theta }_2)}{L}+({\theta }_1-{\theta }_2)\frac{KA}{L}=\frac{KA({\theta }_2-{20}^{\circ })}{L}$

$2({\theta }_1-{\theta }_2)+({\theta }_1-{\theta }_2)={\theta }_2-{20}^{\circ }$

$3{\theta }_1-4{\theta }_2=-{20}^{\circ }$ .....(2)

Solving equation (1) and (2)

${\theta }_1={70.9}^{\circ }$ C

${\theta }_2={58.2}^{\circ }$