Note $2$: You will need to use Appendix A in your textbook to find the thermophysical properties of material.

$1)$ Molten metal spheres with a diameter of $30$ mm are dropped in a tank of water $\backslash (\backslash$left$(4^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash$right$)\backslash ).$

$($a$)$ Find the approximate height of water tank needed to cool the spheres from $\backslash (80^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ to $\backslash (40^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )?$

$($b$)$ What is the height of the tank if engine oil is used instead of water? Comment on the validity of the correlations for this case.

Hint $1$: Assume the spheres reach terminal velocity upon impact with water; this means the velocity of the spheres in water is constant. Find this terminal velocity by equating the drag and gravitational forces acting on the sphere. For the purposes of calculating transient conduction, treat the molten metal as a solid object $($you haven't learned equations for molten metal$),$ with the thermophysical properties of Lead.

The drag force acting on the sphere can be described by:

$\backslash [$

F$\_\{$D$\}=$C$\_\{$D$\}$ A$\_\{$f$\}\backslash$rho $\backslash$frac$\{$V$^\{2\}\}\{2\}$

$\backslash ]$

where $\backslash ($ C$\_\{$D$\}\backslash )($drag coefficient$)$ can be found from Table $7\mathrm{.}9$

Boundary layer separation angles are for a cylinder. Adapted with permission.

Keep in mind that you will need the velocity of the sphere $($unknown$)$ to calculate $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$D$\}\}\backslash ),$ which means you will have to use an iterative process.

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Hint $2$: If Fo you can use Table B$.3$ to calculate four terms of the series in equation $5\mathrm{.}51$ and solve for Fo $($use excel or MATLAB$),$ or move forward the same way as you would if Fo$>0\mathrm{.}2,$ basically assuming the other $3$ terms do not make a significant contribution to the final result.