Question

A long rod has one end at $0^{\circ }C$ and other end at a high temperature. The coefficient of thermal conductivity varies with distance from the low temperature end as $k=k_0(1+ax),where{k}_0={10}^2$ SI unit and $a=1m^{-1}$. At what distance from the first end the temperature will be ${100}^{\circ }C$? The area of cross-section is 1 $c{m}^2$ and rate of heat conduction is 1W.

• A. $2.7m$
• B. $1.7m$
• C. $3m$
• D. $1.5m$

Answer 1

The correct option is B $1.7m$

using fourier's law of heat conduction,

$\stackrel{\bullet }{Q}=-kA\frac{dT}{dx}=-k_0(1+ax)A\frac{dT}{dx}$

Given $\stackrel{\bullet }{Q}$=$1W$ and $dT$=${100}^{\circ }C$

$a=1$, $k_0=100$

on integrating,

${\int }_0^L\frac{dx}{(1+ax)}=-100\times (1\times {10}^{-4})\times {\int }_0^{100}dT\Rightarrow ln(1+L)=1\Rightarrow L=2.73-1=1.73m$