What are the pump affinity laws?

The affinity laws state that for a geometrically similar centrifugal pump, flow rate Q scales linearly with speed n (Q∝n), head H scales with n squared (H∝n²), and shaft power P scales with n cubed (P∝n³). These relationships allow prediction of pump performance at any speed from a single known operating point.

How much energy is saved by reducing pump speed to 80%?

Using the affinity laws, reducing speed to 80% (ratio 0.8) reduces power by the cube: 0.8³ ≈ 0.512, meaning about 49% energy savings. This dramatic reduction is why variable-frequency drives (VFDs) are so widely adopted in pump systems.

What is a pump H-Q curve?

The H-Q curve (head-flow curve) plots head H (m) on the y-axis against flow rate Q (m³/h) on the x-axis. At zero flow (shutoff), head is maximum. Head decreases as flow increases, giving a drooping curve. The operating point is where the pump curve intersects the system resistance curve.

When do the affinity laws break down?

The affinity laws assume geometric similarity and no cavitation. At very low speeds or partial loads, pump efficiency drops significantly and the cubic power law may underestimate actual power consumption. Real-world VFD savings calculations should include efficiency variation with speed.

› Pump Affinity Laws Back

Fluid Analysis

Pump Affinity Laws Simulator

Interactively explore centrifugal pump scaling laws Q∝n, H∝n², P∝n³. Overlay H-Q and power curves at rated and reduced speeds to quantify variable-frequency drive (VFD) energy savings.

Rated Flow Q₁ (m³/h)

100 m³/h

Rated Head H₁ (m)

30 m

Rated Power P₁ (kW)

15 kW

Speed Ratio n₂/n₁

0.80 (80%)

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New Flow Q₂ (m³/h)

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New Head H₂ (m)

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New Power P₂ (kW)

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Power Reduction (%)

Affinity Laws (Similarity Laws)

$$\frac{Q_2}{Q_1}= \frac{n_2}{n_1}$$ $$\frac{H_2}{H_1}= \left(\frac{n_2}{n_1}\right)^2$$ $$\frac{P_2}{P_1}= \left(\frac{n_2}{n_1}\right)^3$$

Power scales as the cube of speed ratio. Reducing to 80% speed saves ~49% power (0.8³ ≈ 0.512). This is the basis for VFD energy savings in pumping systems.

H-Q Curve (Head vs. Flow)

P-Q Curve (Shaft Power vs. Flow)

What are Pump Affinity Laws?

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What exactly are the "affinity laws" for a pump? I've heard they're important for saving energy.

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Basically, they're a set of three simple rules that predict how a pump's performance changes when you change its rotational speed. In practice, if you slow a pump down, you don't just get less flow—the pressure it can generate and the power it needs drop by even more. Try moving the "Speed Ratio" slider in the simulator above from 1.0 down to 0.8 and watch what happens to Power.

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Wait, really? The power drops that much? So if I need less flow, it's way more efficient to slow the pump than to just throttle a valve?

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Exactly! Throttling a valve wastes energy by creating a pressure drop. Slowing the pump with a Variable Frequency Drive (VFD) reduces the energy going into the fluid from the start. For instance, in the simulator, set your rated power to 100 kW and the speed ratio to 0.7. You'll see the new power is only about 34 kW—that's a 66% savings!

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That's huge. But are these laws always accurate? What if I change the impeller diameter instead of the speed?

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Great question. The laws are very accurate for a given pump at different speeds, assuming the fluid and system don't change. They are the foundation for VFD control. For impeller trimming, similar but slightly different laws apply—the exponents change. The simulator here focuses on speed change, which is the most common way to achieve these dramatic savings. Play with different Rated Flow and Head values to see how the savings scale universally.

Physical Model & Key Equations

The affinity laws are derived from the principles of fluid dynamics and fan/pump similarity. The first law states that the volumetric flow rate (Q) is directly proportional to the rotational speed (n).

$$\frac{Q_2}{Q_1}= \frac{n_2}{n_1}$$

Where $Q_1$, $n_1$ are the initial flow and speed, and $Q_2$, $n_2$ are the new values.

The pressure or head (H) developed by the pump is proportional to the square of the speed. This comes from the centrifugal force ($\omega^2 r$). The required shaft power (P) is the product of flow and head, leading to a cubic relationship with speed.

$$\frac{H_2}{H_1}= \left(\frac{n_2}{n_1}\right)^2 \quad ; \quad \frac{P_2}{P_1}= \left(\frac{n_2}{n_1}\right)^3$$

$H$ is pump head in meters, $P$ is power in kW. The cubic law for power is why energy savings are so significant with even small speed reductions.

Real-World Applications

HVAC Building Systems: In large office towers, cooling water pumps run continuously. Using VFDs to slow pumps at night or during mild weather, following the affinity laws, can cut their energy consumption by over 50%, drastically reducing operating costs.

Water & Wastewater Treatment: Plant inflow varies daily. Instead of constantly turning large pumps on and off, treatment plants use VFDs to modulate pump speed, matching flow demand precisely and avoiding water hammer while saving energy.

Industrial Process Control: In a chemical plant, a process may require different flow rates for different batches. Adjusting pump speed via the affinity laws provides accurate, energy-efficient flow control without needing complex valve networks or bypass lines.

Irrigation Systems: Large agricultural irrigation pumps can be slowed during periods of lower water demand or at the ends of irrigation lines where pressure requirements are lower. This reduces power demand and protects pipelines from excessive pressure.

Common Misconceptions and Points to Note

The affinity laws are powerful, but there are several pitfalls in their application. First, the fundamental assumption is "similarity". Significantly changing the rotational speed alters the flow conditions inside the pump (Reynolds number), changing its efficiency. For example, reducing the speed of a pump from its rated 1750 min⁻¹ down to 500 min⁻¹ tends to result in actual shaft power being slightly higher than predicted by the cube law. The simulator assumes ideal similarity, so detailed efficiency changes in actual machines require separate consideration.

Next, do not overlook the relationship with the system resistance curve. Even if the pump performance curve shifts according to the affinity laws, the actual operating flow rate and head are determined by the intersection with the piping system's resistance curve. For instance, if you lower the speed too close to shut-off (zero flow), a situation can occur where the head becomes insufficient and no water is delivered at all. When using the tool to manipulate the performance curve, always imagine, "Where will this new curve intersect with my actual piping system?"

Finally, note that "reducing speed does not always save energy". While the pump's power consumption alone decreases dramatically, the story is different when looking at the entire process. For example, if the cooling water flow rate drops too much, the heat exchanger's thermal performance may decline, causing the chiller to work harder and potentially increasing the overall power consumption. Always use the results from this tool as part of a system optimization effort.

Related Engineering Fields

The concept of pump affinity laws is actually an application of "similarity laws" and "dimensionless numbers" that connect to various engineering fields. First, the same concept applies to fluid machinery in general. For example, fans and blowers follow exactly the same laws (fan affinity laws). The logic of this simulator can be directly used to evaluate the energy-saving effects of inverter-controlled air conditioning fans in data centers.

Furthermore, in the field of naval architecture, the "propeller similarity laws" used to predict thrust from rotational speed and diameter are similar. Expanding on this, it connects to the important dimensionless number called "specific speed" in turbomachinery design. Specific speed is an index used to classify the shape of pumps or turbines (whether they are centrifugal or axial flow) and is derived by combining the affinity law equations.

From a control engineering perspective, it is deeply related to V/f control of motors using inverters. Since a pump's load torque is proportional to the square of the speed, understanding the affinity laws provides foundational knowledge for comprehending V/f control patterns that adjust voltage and frequency accordingly. Thus, deeply understanding one law can significantly broaden your perspective into adjacent technical domains.

For Further Learning

As a next step, we recommend thoroughly learning about the "system resistance curve". Understanding that frictional losses in pipes and pressure losses from valves are proportional to the square of the flow rate ($H = K Q^2$) will allow you to predict how the operating point (the intersection with the pump curve) moves. If this simulator could be enhanced to also plot a "system curve," it would become an even more practical tool.

Regarding the mathematical background, studying dimensional analysis (Buckingham π theorem) will help you grasp "why these three equations are derived." It's a method to organize the relationships between physical quantities determining pump performance (flow rate, head, power, speed, diameter, density, etc.) using dimensionless numbers, with the affinity laws being one conclusion. Understanding this will clarify what to consider when conditions change, such as when pumping high-viscosity liquids.

If you aim for practical application, look into pump efficiency curves and part-load operation. When you reduce the speed using the affinity laws, the best efficiency point (BEP) also moves. The true energy-saving effect is determined by how far the actual operating point is from this new BEP. Practicing by looking at a pump's catalog performance curve and imagining how the efficiency moves when you change the speed in this simulator should elevate your skills in design and retrofit projects.