The pressure developed by a centrifugal pump depends on the fluid density, the diameter of the pump impeller, the rotational speed of the impeller, and the volumetric flow rate through the pump (centrifugal pumps are not recommended for highly viscous fluids, so viscosity is not commonly an important variable). Furthermore, the pressure developed by the pump is commonly expressed as the "pump head," which is the height of a column of the fluid in the pump that exerts the same pressure as the pump pressure.

(a) Perform a dimensional analysis to determine the minimum number of variables required to represent the pump performance characteristic in the most general (dimensionless) form.

(b) The power delivered to the fluid by the pump is also important. Should this be included in the list of important variables, or can it be determined from the original set of variables? Explain.

You have a pump in the field that has a $1.5 \mathrm{ft}$ diameter impeller that is driven by a motor operating at $750 \mathrm{rpm}$. You want to determine what head the pump will develop when pumping a liquid with density of $50 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}$ at a rate of $1000 \mathrm{gpm}$. You do this by running a test in the lab on a scale model of the pump that has a $0.5 \mathrm{ft}$ diameter impeller using water (at $70^{\circ} \mathrm{F}$ ) and a motor that runs at $1200 \mathrm{rpm}$.

(c) At what flow rate of water should the lab pump be operated (in gpm)?

(d) If the lab pump develops a head of $85 \mathrm{ft}$ at this flow rate, what head would the pump in the field develop with the operating fluid at the specified flow rate?

(e) How much power (in horsepower) is transferred to the fluid in both the lab and the field cases?

(f) The pump efficiency is defined as the ratio of the power delivered to the fluid to the power of the motor that drives the pump. If the lab pump is driven by a $2 \mathrm{hp}$ motor, what is the efficiency of the lab pump? If the efficiency of the field pump is the same as that of the lab pump, what power motor (horsepower) would be required to drive it?