Question

Three cylindrical rods $A, \, B$ and $C$ of equal lengths and equal diameters are joined in series as shown in the diagram. Their thermal conductivities are $2K, \, K$ and $0.5 \, K$ respectively. In steady-state, if the free ends of rods $A$ and $C$ are at $100 \,{}^\circ C$ and $0 \,{}^\circ C$ respectively. Assume negligible loss through the curved surface. What will be the equivalent thermal conductivity?

• A. $K_{\text {eq }}=\left(\frac{6}{7}\right) K$
• B. $K_{\text {eq }}=\left(\frac{4}{5}\right) K$
• C. $K_{\mathrm{eq}}=\left(\frac{8}{7}\right) K$
• D. $K_{\mathrm{eq}}=\left(\frac{7}{6}\right) K$

Answer 1

Answer: (A)

As the rods are in series, $R_{e q}=R_{A}+R_{B}+R_{C}$ with $R=(L / K A)$
i.e., $ R_{\mathrm{eq}}=\frac{L}{2 \mathrm{KA}}+\frac{L}{\mathrm{KA}}+\frac{L}{0.5 \mathrm{KA}}=\frac{7 L}{2 \mathrm{KA}} $
Furthermore if $K _{\text {eq }}$ is equivalent thermal conductivity,
$R_{\mathrm{eq}}=\frac{L+L+L}{K_{\mathrm{eq}} A}=\frac{7 L}{2 \mathrm{KA}} \quad$ [from Equation (i)]
$[$ from Equation (i)
i.e., $\Rightarrow \frac{3 L}{K_{\text {cq }}}=\frac{7 L}{2 K A}$
$\Rightarrow K_{ eq }=\frac{6}{7} K$