The 11 -in Bell and Gossett pump of Prob. P11.8 is to be...

The 11 -in Bell and Gossett pump of Prob. P11.8 is to be scaled up to provide, at best efficiency, a head of $250 \mathrm{ft}$ and a flow rate of 3000 gal/min. Find the appropriate $(a)$ impeller diameter; $(b)$ speed in $\mathrm{r} / \mathrm{min} ;$ and $(c)$ horsepower required.

Key Concepts

Best Efficiency Point (BEP)

Operating at the best efficiency point (BEP) is a crucial concept in pump design and scaling, as it represents the conditions under which the pump delivers peak performance with minimal energy loss. Maintaining BEP during scaling ensures that the pump operates under optimal hydraulic conditions, which helps in achieving the desired performance in terms of head, flow rate, and power consumption after scaling.

Pump Affinity Laws

The pump affinity laws provide mathematical relationships that describe how key performance parameters—flow rate, head, and power—change with variations in impeller diameter and rotational speed, provided that the pump’s geometry remains similar. Specifically, they state that the flow rate scales as the product of the speed and the cube of the impeller diameter, the head scales as the square of the speed and the square of the impeller diameter, and the power scales as the cube of the speed times the fifth power of the impeller diameter. These relationships are essential for extrapolating pump performance when scaling up or down.

Geometric Similarity

Geometric similarity is the concept that two pumps are considered similar if all their corresponding dimensions are proportional. This assumption allows engineers to use scaled models to predict full-scale pump performance. When a pump is scaled geometrically, the dynamic behaviors such as flow patterns, pressure distribution, and efficiency remain consistent, which is critical for applying the affinity laws accurately.

Dimensional Analysis

Dimensional analysis involves reducing physical quantities to fundamental dimensions to form dimensionless groups that characterize the system’s behavior. In pump scaling, this technique is used to ensure that all important factors, such as head, flow rate, speed, and power, remain proportionally consistent when the size or operating conditions of a pump are changed. It provides a systematic approach for deriving the scaling laws and verifying that similarity conditions hold.

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