$8\mathrm{.}7$ A centrifugal compressor has an impeller equipped with $21$ vanes that are radial at exit. The compressor has a vaneless diffuser and no inlet guide vanes. The stagnation pressure and temperature at the suction nozzle inlet are $100$ kPa abs$.$ and $300$ K $.$

a$.$ Given that the mass flow rate is $\backslash (2\mathrm{.}3\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash ),$ the impeller tip speed is $\backslash (500\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ and the mechanical efficiency is $\backslash (96\backslash \%\backslash ),$ determine the shaft power.

b$.$ Determine the static and stagnation pressures at the diffuser exit, knowing that the velocity at that point is $\backslash (100\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Consider the compressor efficiency to be $\backslash (82\backslash \%\backslash ).$