Question $5$

$50$ points

A long steel component, the cross section of which is schematically shown in Fig. HTQ$1$ below, is initially maintained at a uniform temperature $T_i=427C.$ Suddenly all surfaces are exposed to cooling by an airstream at $T_{}=29C$ with a heat transfer coefficient $h=96W{m}^{-2}{K}^{-1}.$ Thermal diffusivity $$ and thermal conductivity $k$ of the solid material are given as $=1E-5m^2{s}^{-1}$ and $k=31W{m}^{-1}{K}^{-1}.$ The component is shown schematically in Fig. HTQ$1$ having been spatially discretised on to a mesh where $x=y=0\mathrm{.}01m.$ Because of symmetry only one$-$quarter of the cross$-$section is used for the present analysis and is shown in Fig. HTQ$1.$

$x=y=0\mathrm{.}01m$

Fig. HTQ$1$

$($a$)$ Using the energy balance method, and a mesh size $x=y==0\mathrm{.}01m,$ find out the discretised equation for nodes $1$ to $8$ and determine the maximum allowable time step $t$ for all the nodes. Please provide answers to $3$d$.$p$.$

For Node $1$: $t_{n1}$

Correct response: $2\mathrm{.}424906+-0\mathrm{.}01s$

For Node $2$: $t_{n2}s$

$q,$

Section Attempt $1$ of $2$