Question

The ends of a cylindrical steel rod are maintained at two different temperatures. The rod conducts heat from one end to the other at a rate of 10 cal/s. At what rate would a steel rod twice as long and twice the diameter conduct heat between the same two temperatures?

Answer 1

Answer:20 cal/s

Explanation:

Given

Heat transfer rate is $\dot{Q}=10 cal/s$

Also heat rate is given by

$\dot{Q}=kA\frac{dT}{dx}$

where $k=thermal\ conductivity$

$A=area\ of\ cross-section$

$dT=change\ in\ temperature$

$dx=change\ in\ length$

$10=k\frac{\pi }{4}d^2\frac{dT}{L}----1$

For $d'=2d, Length L'=2L$

$\dot{Q}=k\frac{\pi }{4}(2d)^2\frac{dT}{2L}---2$

dividing 1 and 2 we get

$\frac{10}{\dot{Q}}=\frac{2d^2}{4d^2}$

$\dot{Q}=20 cal/s$