4.5 Figure P4.5 shows a centrifugal pump that supplies input volumetric-flow rate Qin to the hydraulic tank. The pump takes water from the sump (reservoir) at atmospheric pressure and increases the pressure according to the equation AP = aw² Pa where is the angular velocity of the centrifugal pump in revolutions per minute (rpm) and a is a constant derived from empirical data. Flow through output valve 2 is assumed to be laminar. a. Assuming that the flow through valve 1 is laminar (i.e., R₁ = R₁) derive the mathematical model with pressure P as the dynamic variable and pump speed oo as the input variable. b. Repeat part (a) assuming that the flow through valve I is turbulent where R₁ = RT. Valve, R₁ Pump Input flow Lin Base pressure, P h Sump (reservoir) Area, A Valve, R₂ Figure P4.5 lout