At what speed in r/min should the 35 -in-diameter pump of...

At what speed in r/min should the 35 -in-diameter pump of Fig. $11.7 b$ be run to produce a head of 400 ft at a discharge of 20,000 gal/min? What brake horsepower will be required? Hint: Fit $H(Q)$ to a formula.

Key Concepts

Brake Horsepower (BHP)

Brake horsepower is the actual power output required by the pump’s motor to overcome hydraulic losses and deliver the desired head and flow rate. It takes into account mechanical inefficiencies and is calculated based on the product of the flow, head, and constants related to fluid density and gravity, often requiring conversion factors to yield power in horsepower. Assessing BHP is crucial for selecting a motor that can adequately drive the pump under the specified operating conditions.

Dimensional Analysis and Scaling

Dimensional analysis is used to ensure that the equations governing pump performance are physically consistent and to scale the performance of pumps to different sizes or operating conditions. By identifying key dimensionless parameters, engineers can apply similarity principles to predict how changes in geometry or operating conditions (such as speed) will affect performance. This forms the theoretical basis behind the pump affinity laws.

Pump Affinity Laws

These laws relate the performance parameters of a pump (flow rate, head, and power) to its operating speed and impeller diameter. They state that when the impeller diameter is fixed, the flow rate is directly proportional to the speed, the head is proportional to the square of the speed, and the power required is proportional to the cube of the speed. This allows one to predict how changing the speed of a pump will affect its operational performance, which is essential when designing or adjusting a pump system to meet specific operating conditions.

Head-Discharge Relationship and Curve Fitting

In pump design, the head produced by a pump typically decreases as the flow rate increases. This relationship is often represented by an equation such as H = H0 - kQ^2 (or a similar quadratic form), where H0 is the shut-off head, Q is the flow rate, and k is a constant that depends on the pump’s geometry and operating conditions. Curve fitting this relationship allows engineers to derive the necessary constants and accurately model the pump’s performance over a range of flows.

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