A multi$-$stage centrifugal pump is discharging $90,000$ liters of water per minute against a total manometric head of $60$ m$.$ There are four identical impellers keyed to the same shaft, which is running at $350$ RPM$.$ The vanes are curved backward at an angle of $60$ to the tangent at the outer periphery. The velocity of flow at the outlet is $0\mathrm{.}27$ times the corresponding peripheral velocity and is constant along the radial direction to avoid radial thrust. The hydraulic loss in the impeller is $1/3$ of the velocity head at the exit of the impeller. Find:

$($a$)$ Diameter of the impeller

$($b$)$ Manometric efficiency