A vertical brick duct (tube) is filled with cast iron. The lower end of the duct is maintained at a temperature `T_1` which is greater than the melting point `T_m` of cast iron and the upper end at a temperature `T_2` which is less than the temperature of the melting point of cast iron. It is given that the conductivity of liquid cast iron is equal to k times the conductivity of solid cast iron. Determine the fraction of the duct filled with molten metal.