Question

Five rods with identical geometries are arranged as shown. Their thermal conductivities are shown. Only A and C are maintained at ${100}^{\circ }C$ and $0^{\circ }C$ respectively. In other words, heat flows in from A and flows out of C. If temperature difference between ends D and B can be written as $10x^{\circ }C$, where $x$ is an integer. Then find $x$.

Answer 1

$R_{TH}=\frac{2L}{kA}+\frac{L}{kA}+\frac{2L}{kA}=\frac{5L}{kA}$

$\frac{dH}{dt}=\frac{100}{5L}kA=20\frac{kA}{L}$
$\frac{dH}{dt}=\frac{100-T_1}{\frac{2L}{kA}}=20\frac{kA}{L}$

$100-T_1=40\Rightarrow T_1={60}^{\circ }C$
$\frac{dH}{dt}=\frac{T_2-0}{\frac{2L}{kA}}=20\frac{kA}{L}\Rightarrow T_2={40}^{\circ }C$
$T_B=T_2,T_D=T_1\Rightarrow \Delta T={20}^{\circ }C$