Question

A wire whose resistance per cm length is $0.1\Omega$ is embedded along the axis of a cylindrical cement tube of radii $0.06cm$ and $12.cm$. A current of $6Amps$ flow in the wire to keep a steady state temp difference of ${130}^{\circ }C$ between the inner and outer surfaces. The thermal conductivity (W/m K) of the cement is

• A. 0.32
• B. 1.32
• C. 132
• D. 13.2

Answer 1

The correct option is B 1.32

$\begin{array}{cc}\text{Electrical resistance}\left(R\right)& =0.1\Omega /cm\\ & =\frac{0.1\Omega }{0.01m}=10\Omega /mI=6amph.\\ r_1=0.06cm,r_2& =1.2cm\\ \text{Temperature difference}(T_1-T_2)& ={130}^{\circ }C\\ \text{Heat generated in the wire due to flow of}& \\ \text{electricity}& =I^2.R=6^2\times 10\\ & =360W/m\end{array}$
$\text{Heat generated in the wire}=\text{Heat conducted}\text{form the cemented material.}360=\frac{(T_1-T_2)}{\frac{\ell \left(\frac{r_2}{r_1}\right)}{2\pi kL}}=\frac{130}{\frac{\ell \left(\frac{1.2}{0.06}\right)}{2\pi \times k\times 1}}360=\frac{2\pi \times k\times 130}{\ell \left(20\right)}k=1.32W/m-K$