Question

A shaft of diameter 30 mm is used to connect a dynamometer. The rate of heat generation at drum has to be released by shaft which acts as an tip insulated fin of 700 mm length. The root temperature of shaft at drum side is ${600}^{\circ }C$ and environment temperature is ${25}^{\circ }C$. Convective heat transfer coefficient is $20{\text{W/m}}^2\text{K}$ and thermal conductivity of shaft is 132.3 ${\text{W/m}}^2\text{K}$. Heat lost at steady state in one minute is equal to

• A. 14.43 kJ
• B. 11.93 kJ
• C. 12.43 kJ
• D. 15.93 kJ

Answer 1

The correct option is A 14.43 kJ

Given $d=30\text{mm}$,
$L=700\text{mm}$
$T_{\infty }={25}^{\circ }C$,
$T_0={600}^{\circ }C$
$h=20{\text{W/m}}^2\text{K}$,
$k=162.3\text{W/mK}$

$m=\sqrt{\frac{hP}{kA}}=\sqrt{\frac{4h}{kd}}$ (for cylinder)

$m=\sqrt{\frac{4\times 20}{132.3\times 0.03}}=4.49$

Steady state rate of heat lost,

$q=\sqrt{hPkA}\times {\theta }_0\times \tan h\left(mL\right)$

$\sqrt{hPkA}=\sqrt{20\times (\pi \times 0.03)\times 132.3\times (\pi \times 0.25\times {0.03}^2)}=0.4199$

$\tan h\left(\text{mL}\right)=\tan h(4.49\times 0.7)=0.9963$

$q=0.4199\times 0.9963\times (600-25)$

$q=240.57\text{W}$

Heat lost in one minute
$q.t=240.57\times 60\times {10}^{-3}\text{kJ}=14.43\text{kJ}$