The following design data apply to a double$-$sided centrifugal compressor: Outer diameter of impeller Speed Mass flow Inlet temperature Inlet pressure Isentropic efficiency of impeller only Radial gap of vaneless space Axial depth of vaneless space Slip factor Power input factor $50$ cm $270$ rev$/$s $16\mathrm{.}0$ kg$/$s $288$ K $1\mathrm{.}01$ bar $0\mathrm{.}904\mathrm{.}0$ cm $5\mathrm{.}0$ cm $0\mathrm{.}91\mathrm{.}04($a$)$ Calculate the stagnation pressure and temperature at the outlet of the impel ler, assuming no prewhirl. $($b$)$ Show that the radial outlet velocity at the impeller tip is about $96$ m$/$s and hence find the Mach number and air leaving angle at the impeller tip. $($In calculating the circumferential area at the tip, the thickness of the impeller disc may be neglected.$)($c$)$ Assuming isentropic diffusion in the vaneless space, find the correct angle of the leading edges of the diffuser vanes, and also find the Mach number at this radius.