An aluminium sphere weighning 5.5 kg and initially at a temperature of 290°C is suddenly immersed in a fluid at 15°C. The convective heat transfer coefficient is 58 W/m²K.
An aluminium sphere weighning 5.5 kg and initially at a temperature of 290°C is suddenly immersed in a fluid at 15°C. The convective heat transfer coefficient is 58 W/m²K. Estimate the time required to cool the aluminium to 95°C, using the lumped capacity method of analysis. [rho=2700 kg/m³; c = 900 J/kg K; k = 205 W/mK]
Time required to cool aluminium to 95°C.
\[T = 95^{\circ}C, T_{0} = 290^{\circ}C, T_{\infty} = 15^{\circ}C\]
\[\frac{T-T_{\infty }}{T_{0}-T_{\infty}}=\textrm{exp}\left [ -\left ( \frac{hA}{\rho cV} \right )t \right ]\]
\[V = \frac{m}{\rho}=\frac{5.5}{2700}=2.037\times10^{-3}m^{3}\]
\[V = \frac{4}{3}\pi R^{3}\Rightarrow R = 0.0786\,m\]
\[\Rightarrow L_{c} = \frac{R}{3} = 0.0262\,m\]
\[\frac{hA}{\rho cV} = \frac{3h}{\rho cR} = \frac{3\times58}{2700\times900\times0.0786}=9.1\times10^{-4}/s\]
\[\frac{95-15}{290-15} = \frac{80}{275}=\textrm{exp}\left ( -9.1\times10^{-4} t\right )\]
\[3.4375 = \textrm{exp}\left ( 9.1\times10t \right )\]
\[\Rightarrow t = 1357\,s\]
