Question

A steam cylindrical pipe of inner radius $5\text{cm}$ and outer radius $7\text{cm}$, having conductivity $0.07\text{W/mK}$, carries steam at ${100}^{\circ }\text{C}$. If the temperature at the outer wall of the pipe jacket is ${20}^{\circ }\text{C}$, then find the heat lost through the jacket per metre length in an hour.

• A. $1.28\times {10}^5\text{J}$
• B. $9.72\times {10}^5\text{J}$
• C. $3.76\times {10}^5\text{J}$
• D. $7.84\times {10}^5\text{J}$

Answer 1

The correct option is C $3.76\times {10}^5\text{J}$


Consider an elemental cylindrical shell of thickness $dr$ and radius $r$.
Thermal resistance per metre length of the element is
$dR=\frac{dr}{K\left(2\pi r\right)}$ $[\because R=\frac{l}{KA}]$
$\therefore$ Total resistance $R=\int dR$
$=\frac{1}{2\pi K}{\int }_{5.0\times {10}^{-2}\text{m}}^{7.0\times {10}^{-2}\text{m}}\frac{dr}{r}$
From the data given in the question, we get
$R=\frac{1}{2\pi K}\ln \left(\frac{7}{5}\right)$
$=\frac{1}{\left(2\pi \right)\left(0.07\right)}\ln \left(1.4\right)=0.765\text{K/W}$
Heat current, $H=\frac{\text{Temperature difference}}{\text{Thermal resistance}}$
$H=\frac{(100-20)}{0.765}=104.6\text{W}$
Heat lost in one hour = Heat current $\times$ time $=\left(104.6\right)\left(3600\right)=3.76\times {10}^5\text{J}$
Thus, option (c) is the correct answer.