A long $20-$cm$-$diameter cylindrical shaft $($illustrated in Figure $2)$ is made of stainless steel

$304$ comes out of an oven at a uniform temperature of $600\backslash$deg C $($as given in Figure $2).$ The

shaft is then allowed to cool slowly in an environment chamber at $200\backslash$deg C with an average

heat transfer coefficient of h $=80$ W$/$m$2\backslash$deg C$.$

Determine:

$($a$)$ the temperature at the centre of the shaft $45$ min after the start of the cooling

process.

And

$($b$)$the heat transfer per unit length of the shaft during this time period.

Please consider that the Fourier number is $\backslash$tau $>0\mathrm{.}2,$ so that the one$-$term approximate

solutions are applicable. Please take the properties of stainless steel $304$ at room

temperature, i$.$e$.$ k $=14\mathrm{.}9$ W$/$m $\backslash$deg C$,\backslash$rho $=7900$ kg$/$m$3,$ Cp $=477$ J$/$kg $\backslash$deg C$,$ and $\backslash$alpha $=3\mathrm{.}95$ x$10-$

$6$ m$2/$sec$.$