Consider the slab shown for a heat transfer analysis. The boundary conditions are characterized by the ambient temperatures $T_1,$ $T_2$ and the surface coefficients $\alpha_1,$ $\alpha_2.$ Initially, $T_1=T_2,$ then $T_1$ suddenly drops. Calculate the transient response of the slab for the surface coefficients being functions of the wall temperatures $T_{w1},$ $T_{w2}$ and compare these results with the steady-state solution.

$\lambda=20\text{ W/mK},$ $\rho\!\cdot\!c=3\times10^6\text{ J/m$^3$K}.$

$T_1=-20\text{ $^\circ$C},$ $T_2=25\text{ $^\circ$C}$ at time $t>0.$ $\alpha_1=\alpha(T_{w1}),$ $\alpha_2=\alpha(T_{w2}),$ $\alpha(T)=\left\{\begin{align} 30\text{ W/m}^2\text{K}\quad\text{for}\quad T\le0\ ^\circ\text{C} \\ 10\text{ W/m}^2\text{K}\quad\text{for}\quad T>0\ ^\circ\text{C} \end{align}\right..$

$T(x)\equiv25\text{ $^\circ$C}$ at time $t=0.$

The surface coefficients $\alpha_1,$ $\alpha_2$ and the initial temperature field are processed in the same way as in Steady-state analysis of a slab. Numerical analysis is carried out in the time interval $(0,200~\text{h})$ with the initial time step set to $\mathtt{STEP}=10~\text{h}.$ The step length is controlled by the parameter $\mathtt{TOL}$ which is matched with the convergence tolerance $\mathtt{TOL}=\mathtt{EDIF}=0.1~\text{$^\circ$C}.$

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• Hexahedra, BEAM56T2
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