$4\mathrm{.}2$ A centrifugal pump impeller is equipped with pump$-$out vanes for reducing the axial thrust. The impeller is mounted on a through shaft that is supported at both sides by ball bearings. The following data provides the pump operating conditions as well as the main dimensions of the impeller:

a$.$ Determine the pressure in the clearance space between the back shroud and the casing at a radius $\backslash ($ r$=80\backslash$mathrm$\{$~mm$\}\backslash )$

b$.$ Calculate the reduction in axial thrust due to the use of pump$-$out vanes.

c$.$ Determine the magnitude and direction of the remaining unbalanced axial thrust.