Derive an empirical expression for the for the actual shaft power, (Pp)in input to a pump, assuming incompressible, turbulent flow. The inertia force (FI = Ma = ρv2 L2 ) and the actual shaft power,...

Derive an empirical expression for the for the actual shaft power, (Pp)in input to a pump, assuming incompressible, turbulent flow. The inertia force (FI = Ma = ρv2 L2 ) and the actual shaft power, (Pp)in will always be important. Note that v = ω D, which is the characteristic velocity (where ω is the pump impeller rotational speed), and D, which is the characteristic length, L, and is the diameter of the pump impeller blades. Using Tables 11.8, 11.9, and 7.8 and Section 11.7.8 as a guide, the additional dominant/significant forces and physical quantities in the flow situation include the definition of the actual shaft power input to a pump, (Pp)in = ωTshaft,in and the absolute pump impeller blade surface roughness height, ɛ. Thus, the important physical quantities include: ρ, ω, D, (Pp)in, T, and ɛ. What are the important π terms used to study and describe this flow process for incompressible, turbulent flow through a pump?