$1.$ AISI $\backslash (316\backslash$left$(\backslash$alpha$=3\mathrm{.}48\backslash$times $10^\{-6\}\backslash$mathrm$\{$~m$\}2/\backslash$mathrm$\{$s$\}\backslash$right$)\backslash )$ stainless steel plate of $4$ cm thickness is maintained at room temperature $\backslash (\backslash$left$(25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash$right$)\backslash )$ initially. This plate is then placed in a furnace, maintained at $\backslash (850^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Considering the convection heat transfer coefficient to be $\backslash ($ h$=120\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$mathrm$\{$~K$\}\backslash ).$ Answer the following using one$-$term approximations. $[$Refer to Appendix or literature for determining properties of AISI $316]$

a$.$ Using one$-$term approximation, determine the temperature at the center of the plate after $3$ min $.$ Show details of the calculations. $[$Ans:$415$ C$]$

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b$.$ Using one$-$term approximation, determine the surface temperature of the plate after $10$ min $.$ Show details of the calculations. $[$Ans: $777$ C$\backslash (]\backslash )$

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Using Heisler and Grober charts, determine the temperature at the center of the plate after $3$ min $.$ Show details of the calculations. Attach a copy of chart with work shown in it to receive full credit.

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Note: Show details of calculations for solving the problems. Show appropriate diagrams wherever necessary.

Using Heisler and Grober charts, determine the surface temperature of the plate when it comes out of the furnace after $10$ min $.$ Show details of the calculations. Attach a copy of chart with work shown in it to receive full credit.