$4)$ It is proposed to construct a furnace wall of three layers of brickwork: firebrick $($FB$)$ on the inside, then a layer of insulating brick $($IB$)$ and an outer casing of common building brick $($CB$).$ The hot face temperature of the firebrick will be $1370\backslash$deg C and the heat loss is not to exceed $600$ W m$-2$; it is estimated that this will result in the outer surface temperature being $65\backslash$deg C$.$ Maximum Operating Temperature Thermal ConductivityFirebrick $($FB$)1500$oC $0\mathrm{.}78$ W m$-1$ K$-1$Insulating brick $($IB$)1200$oC $0\mathrm{.}31$ W m$-1$ K$-1$Common brick $($CB$)400$oC $0\mathrm{.}66$ W m$-1$ K$-1($i$)$ Determine the thickness of each layer of brick. $[$HINT fix interface temperatures at the maximum operating temperatures then apply Fouriers Law to each layer separately, heat flux is constant $(600$ W m$-2)]$ Ans: FB $=0\mathrm{.}221$ m minimum, IB $=0\mathrm{.}413$ m minimum, CB $=0\mathrm{.}369$ m maximum $($ii$)$ Due to cost if it were decided to make the thickness of the firebrick and insulating brick integral multiples of $100$ mm$,$ recommend suitable thickness, determine the thickness of common brick required and calculate the interface temperatures. $[$HINT what happens to the interface temperatures if you recalculate them using brick thicknesses in multiples of $0\mathrm{.}1$m$?]$ Ans: either i$)$ FB $=0\mathrm{.}30$ m$,$ IB $=0\mathrm{.}50$ m$,$ CB $=0\mathrm{.}12$ mor ii$)$ FB $=0\mathrm{.}30$ m$,$ IB $=0\mathrm{.}40$ m$,$ CB $=0\mathrm{.}33$ m