One end of a copper rod of uniform cross-section and of length $3.1$ m is kept in contact with ice and the other end with water at $100°C $ . At what point along it's length should a temperature of $200°C$ be maintained so that in steady state, the mass of 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 vaporisation of water are $80 cal/gm$ and $540$ cal/gm respectively
Diffcult
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(a) Rate of flow of heat is given by $\frac{{dQ}}{{dt}} = \frac{{\Delta \theta }}{{l/KA}}$also $\frac{{dQ}}{{dt}} = L\frac{{dm}}{{dt}}$ (where $L$ = Latent heat)
==> $\frac{{dm}}{{dt}} = \frac{{KA}}{l}\,\left( {\frac{{\Delta \theta }}{L}} \right)$. Let the desire point is at a distance $x$ from water at $100°C$ .
( Rate of ice melting = Rate at which steam is being produced
==> ${\left( {\frac{{dm}}{{dt}}} \right)_{Steam}} = {\left( {\frac{{dm}}{{dt}}} \right)_{Ice}}$
==> ${\left( {\frac{{\Delta \theta }}{{Ll}}} \right)_{Steam}} = {\left( {\frac{{\Delta \theta }}{{Ll}}} \right)_{Ice}}$
==> $\frac{{(200 - 100)}}{{540 \times x}} = \frac{{(200 - 0)}}{{80\,(3.1 - x)}}$
==> $x = 0.4 m = 40 cm$
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