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Question 4.1: A steel shaft having a diameter of 120 mm is initially at a ......

A steel shaft having a diameter of 120 mm is initially at a temperature of 26°C and is placed in a furnace with a temperature of 900°C.

Determine the time required to heat the shaft when the axis acquires a temperature of 820°C.

Also determine the surface temperature at the above determined time.

λ = 21 w/m°C,     α = 6.2 × 10^{-6} m²/sec,     h = 180 w/m²°C

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The Biot number (Bi) is

\mathrm{Bi}=\frac{h \gamma_o}{\lambda}=\frac{180 \times 0.06}{21} \\ =0.51 \\ \begin{aligned}t_f & =900^{\circ} \mathrm{C}, \quad t_o=26^{\circ} \mathrm{C}, \quad t_{\gamma_o}=820^{\circ} \mathrm{C} \\\Theta & =\frac{t_f-t_{\gamma_o}}{t_f-t_o}=\frac{900-820}{900-26}=\frac{80}{874} \\& =0.092\end{aligned}

Referring to Θ, Fourier number Fo, from the graph for cylinders, at Bi = 0.51

Fo = 2.9

\begin{aligned}\tau & =\frac{\gamma_o^2-\mathrm{Fo}}{a}=\frac{0.06^2 \times 2.9}{6.2 \times 10^{-6}}=\frac{3.6 \times 2.9 \times 10^3}{6.2} \\& =1683  \mathrm{sec}=28 \mathrm{~min}=\text { say } 1 / 2 \mathrm{~h}\end{aligned}

For Bi = 0.051 and Fo = 2.9, the surface value of the dimensionless temperature Θ is obtained from Bi, Θ, and Fo graph for surface. Hence

\begin{aligned}\Theta & =\frac{t_f-t_{\gamma=\gamma_o}}{t_f-t_o}=\frac{900-t_{\gamma=\gamma_o}}{900-26}=0.07 \\ \therefore t_{\gamma=\gamma_o} & =838^{\circ} \mathrm{C}\end{aligned}

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