A turbo-pump which runs at 2000 rpm takes in water at...

A turbo-pump which runs at 2000 rpm takes in water at atmospheric pressure at the rate of $0.010 \mathrm{~m}^3 / \mathrm{s}$ and discharges it at $200,000 \mathrm{~Pa}$ above atmospheric pressure. a. If the pump rpm is increased to 3,000 , find the flow rate and the discharge pressure at which the new operating conditions are completely similar to the first ones. Neglect viscous dissipation b. Find the ratio of the powers necessary to run the pump in the two cases.

A turbo-pump which runs at 2000 rpm takes in water at atmospheric pressure at the rate of $0.010 \mathrm{~m}^3 / \mathrm{s}$ and discharges it at $200,000 \mathrm{~Pa}$ above atmospheric pressure.
a. If the pump rpm is increased to 3,000 , find the flow rate and the discharge pressure at which the new operating conditions are completely similar to the first ones. Neglect viscous dissipation
b. Find the ratio of the powers necessary to run the pump in the two cases.

Key Concepts

Dynamic Similarity

Dynamic similarity ensures that the flow patterns, inertial forces, and other dynamic effects in a pump remain consistent between different operating conditions. This concept is essential in pump design and analysis because it justifies using scaling laws (derived under ideal, often inviscid, conditions) to predict changes in performance when a pump operates at different speeds or sizes while neglecting dissipative effects like viscous losses.

Scaling of Performance Parameters

Scaling relationships in pump engineering provide the mathematical framework to predict changes in flow rate, pressure, and power with variations in speed or size, assuming all other parameters remain similar. These relationships typically specify that the flow rate scales linearly with speed, the pressure head scales with the square of the speed, and the power required scales with the cube of the speed. This allows performance prediction when the pump’s operating conditions are modified.

Pump Affinity Laws

These laws describe how changes in the operating speed of a pump affect its performance parameters. Specifically, the flow rate (Q) is proportional to the rotational speed (N), the pressure head (or discharge pressure) is proportional to the square of the speed (N²), and the power required is proportional to the cube of the speed (N³). These relationships allow engineers to predict how alterations in pump speed will influence the pump’s behavior under dynamically similar conditions.

Similitude and Geometric Similarity

Similitude refers to the condition where two systems have the same dimensionless parameters and therefore behave in a similar manner despite differences in size or operating conditions. Geometric similarity is a subset of similitude where the pump’s geometry is the same, ensuring that scaling laws (like those detailed in the pump affinity laws) can be applied directly. This concept is crucial when changing operating conditions, such as rotational speed, while maintaining dynamically similar performance.

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