Question

A stainless slab with initial temperature T$0$ is heated from both sides at the constant temperature

T$1.$ The temperature profile with time and distance is given by the equation,

The thermal conductivity of the slab is $15\mathrm{.}0$ Wm$-1$K$-1,$ density $\backslash$rho $=7850$ kg m$-3,$ specific heat

capacity Cp $=500$ J kg$-1$ K$-1,$ half slab thickness b $=100$ mm$,$ T$0=30$ oC$,$ and T$1=900$ oC$.$ n

$=2$ is used in the equation for the following calculations.

$(1)$ Estimate how long it will take to increase the temperature at b$/2$ to $600$ oC and the

time required to increase the temperature of the slab centre to $600$ oC$.$

$(2)$ If the slab size is height H$=5000$ mm and width W$=3000$ mm$,$ calculate the heat

transfer rates $($W$)$ at the slab surface at t $=30$ minutes, and how much heat has been

transported into the slab within $30$ minutes.

$(3)$ Calculate the distance of heat penetration depth from the surface at t $=30$s$.$

$(4)$ calculate the average temperature of the slab at $30$ minutes.