A furnace wall $30$ cm thick has k$=1\mathrm{.}4$ W$/$mK$.$ The heat transfer coefficient of outer surface is given as h$=8\mathrm{.}1+0\mathrm{.}09$ AT where AT is the temperature difference between outside wall surface and surrounding. If the inner surface temperature is $1450\backslash$deg C$,$ calculate the rate of heat loss per unit area. The fumace wall is insulated such that the heat losses do not exceed $500$ W$/$m$?.$ This is done by putting two layers of insulation on the outer surface: the first one is of heat resisting brick of k$=0\mathrm{.}6$ W$/$mK and the second one is of silica brick of k$=0\mathrm{.}15$ W$/$mK$.$ If the thickness of the silica brick is $30$ cm$.$ find the thickness of the heat resisting brick. Assume the surrounding temperature to be $40\backslash$deg C$.$