$1.$ Derive an expression for the rate of heat flow in the case of a plane wall.

$2.$ A furnace is constructed with $225$ mm thick fire brick, $120$ mm of insulating brick, and $225$ mm of the building brick. The inside temperature is $1200$K$,$ and the outside temperature is $330$K$.$ Find the heat loss per unit area and the temperature at the junction of the fire brick and insulating brick.

$[$Data: K for fire brick: $1\mathrm{.}4$ W$/$m$.$K; K for insulating brick: $0\mathrm{.}2$ W$/$m$.$K; K for building brick: $0\mathrm{.}7$ W$/$m$.$K$.]$