A long vertical brick duct is filled with cast iron. The lower end of the duct is maintained at a temperature $T_{1} > T_{m e l t}$ ( $T_{m e l t}$ is the melting point of cast iron), and the upper end at a temperature $T_{2} < T_{m e l t}$ . The thermal conductivity of molten (liquid) cast iron is k times higher than that of solid cast iron.
Determine the fraction of the duct filled with molten metal

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Solution

In steady-state, $\frac{d Q}{d t}$ is across the boundary of molten cast iron and solid cast iron.
Hence,
$K_{m o l t e n} A \frac{(T_{1} - T_{m e l t})}{l_{m o l t e n}} = K_{s o l i d} A \frac{(T_{m e l t} - T_{2})}{l_{s o l i d}}$
$∴ \frac{l_{s o l i d}}{l_{m o l t e n}} = \frac{(T_{m e l t} - T_{2})}{k ((T_{1} - T_{m e l t})}$
$∴ \frac{l_{s o l i d}}{l_{m o l t e n}} + 1 = \frac{((T_{m e l t} - T_{2})}{k (T_{1} - T_{m e l t})} + 1$
$∴ \frac{l_{m o l t e n}}{l} = \frac{k (T_{1} - T_{m e l t})}{k ((T_{1} - T_{m e l t}) + (T_{m e l t} - T_{2})}$

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In steady-state,dQdt is across the boundary of molten cast iron and solid cast iron.Hence,KmoltenA(T1−Tmelt)lmolten=KsolidA(Tmelt−T2)lsolid∴lsolidlmolten=(Tmelt−T2)k((T1−Tmelt)∴lsolidlmolten+1=((Tmelt−T2)k(T1−Tmelt)+1∴lmoltenl=k(T1−Tmelt)k((T1−Tmelt)+(Tmelt−T2)