$10\mathrm{.}6.$ Estimate the following quantities for a typical direct$-$injection diesel fuel spray. The injection pressure is $500$ bar ; the cylinder pressure during injection is $50$ bar $.$

$($a$)$ Assuming that the flow through the nozzle orifice is incompressible and quasi$-$steady, estimate the liquid fuel velocity at the orifice exit. At this velocity, how long would the fuel take to reach the cylinder wall? The bore is $125$ mm $.$

$($b$)$ Each nozzle orifice diameter $\backslash ($ d$\_\{$n$\}\backslash )$ is $0\mathrm{.}34$ mm and $\backslash ($ L$\_\{$n$\}/$ d$\_\{$n$\}=4\backslash ).$ Determine the spray spreading angle and plot spray tip penetration versus time.

$($c$)$ Use Eq$.(10\mathrm{.}28)$ to estimate the initial average drop size assuming that the injection process in $($a$)$ above continues for $1$ millisecond and the injector nozzle has four orifices.

Note: Because $\backslash (\backslash$mathrm$\{$A$\}2/\backslash$mathrm$\{$Amathrm$\{$Cd$\}=0\mathrm{.}8\backslash )$ in Eq$.10\mathrm{.}14$