Question

One end of a copper rod of uniform cross-section and of length $1.5m$ is kept in contact with ice and the other end with water at ${100}^{o}C$. At what point along its length should a temperature of ${200}^{o}C$ be maintained so that in steady state, the mass of the ice melting be equal to that of the steam produced in the same interval of time? Assume that the whole system is insulated from the surroundings. Latent heat of fusion of ice and vaporization of water are $80cal/gmand540cal/gm$ respectively.

Answer 1

Let the source of heat at ${200}^{o}C$ be at a distance $x$ from ice. Let $m$ gm of ice melts and $m$ gm of water evaporates in same time.

Rate of heat transfer to ice $=Q_{ice}=m{L}_{ice}=\frac{kA(200-0)t}{x}$

Rate of heat transfer to water $=Q_w=m{L}_{vap}=\frac{kA(200-100)t}{1.5-x}$

$\frac{Q_w}{Q_{ice}}=\frac{kA(200-100)t}{1.5-x}\times \frac{x}{kA(200-0)t}=\frac{m\cdot 540}{m\cdot 80}\Rightarrow \frac{x}{1.5-x}\times \frac{100}{200}=\frac{540}{80}\Rightarrow \frac{x}{1.5-x}=\frac{27}{2}\Rightarrow 2x=40.5-27x\Rightarrow 29x=40.5\Rightarrow x=1.396m=139.666cm$