 Example B.12

(a) Consider a large steel vessel, wall thickness 16 mm protected by 40 mm of vermiculite cement. Find the metal temperature after 1 and 2 h in a flux of 100 kW m−2 for surface emissivities of 1 (~black), 0.3 (~aluminum paint), and 0.1 (~white). Properties of steel are density 7800 kg m−3, specific heat 0.5 kJ kg−1 K−1 and properties of vermiculite are density 430 kg m−3, specific heat 0.84 kJ kg−1 K−1 and thermal conductivity 9.5 × 10−5 kW m−1 K−1.

 Surface temperature $\begin{array}{ll}100\hfill & =5.6697×{10}^{-11}\left({{T}_{\mathrm{s}}}^{4}-{293}^{4}\right)+\frac{0.01}{\epsilon }\left({T}_{\mathrm{s}}-293\right)\hfill \\ {T}_{\mathrm{s}}\hfill & =856,797,640°\mathrm{C}\mathrm{for}\epsilon =1,0.3,\mathrm{and}0.1\hfill \end{array}$ Insulation time constant $\begin{array}{ll}{t}_{\mathrm{c}}\hfill & =\left(\frac{{W}_{\mathrm{R}}{s}_{\mathrm{R}}}{{A}_{\mathrm{R}}}+\frac{{x}_{\mathrm{I}}{\rho }_{\mathrm{I}}{s}_{\mathrm{I}}}{2}\right)\frac{{x}_{\mathrm{I}}}{{k}_{\mathrm{I}}}\hfill \\ =\left(\frac{7,800×0.016{A}_{\mathrm{R}}×0.5}{}\hfill \end{array}$