Problem 2. Figure 2 shows a centrifugal pump that supplies input volumetric-flow rate into the hydraulic tank. The pump takes water from the sump (reservoir) at atmospheric pressure and increases the pressure according to the equation AP = aw² (in 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 () as the input variable. (b) Repeat part (a) assuming that the flow through valve 1 is turbulent where R₂ = RT. (c) Derive the mathematical model in terms of the variable dynamic / for a laminar and a turbulent flow through valve 1. (d) If the dynamic variable is the gage pressure P = P - Parm rewrite the mathematical models obtained in (a) and (b). P₂(s) (e) Calculate the transfer function - where 2(s) is the Laplace transform of w(t). Ω(5) Valve, R, Pump Input flow Base pressure, P h Sump Area, A (reservoir) Valve, R, Fig. 2. Hydraulic system'.