Affinity Laws in Pump and Fan Systems and the Motor

When you change the speed of a motor in a pump or fan system, knowing in advance how much the flow, pressure and power will change is one of the fundamental questions of engineering. The answer to this question is given by the affinity laws. These laws describe, with simple ratios, how centrifugal pumps and fans respond to a change in speed, and they explain in particular why energy savings are so large in inverter-driven (variable-speed) systems. In this article we cover what the affinity laws are, why a drop in speed is reflected in power as a cubic effect, and how this affects motor selection. To think about the topic together with the energy side, our article on energy saving with a frequency inverter is a good complement.

What Are the Affinity Laws?

The affinity laws are three basic proportionality rules that define how centrifugal machines (pumps, fans, blowers) respond to a change in speed. These rules state that flow varies directly with speed, pressure varies with the square of speed, and power varies with the cube of speed. This simple but powerful relationship is the mathematical basis for why variable-speed drive provides such large savings.

The Three Basic Proportions

We can write the essence of the affinity laws as follows: flow varies with speed (Q ∝ N), pressure with the square of speed (H ∝ N²), and power with the cube of speed (P ∝ N³). So if you halve the speed, flow halves, pressure drops to a quarter, and power drops to an eighth. The cubic behavior of power is the most critical point in the energy saving of speed control.

Affinity Laws Table

The table below summarizes how flow, pressure and power change in response to a percentage change in speed. The values are for ideal centrifugal behavior; in real systems, static pressure and efficiency curves alter the result slightly.

Why Does Power Drop ~50% When Speed Drops 20%?

As the table shows, when you lower the speed to 80 percent (that is, reduce it by 20 percent), power drops to 51 percent; in other words, almost halves. The reason is that power is proportional to the cube of speed: 0.8³ ≈ 0.51. This cubic relationship means that even a small speed reduction turns into a very large energy gain. This is exactly where the appeal of variable-speed pumps and fans comes from.

The Linear Relationship of Flow With Speed

Flow is directly proportional to rotational speed. If you reduce the speed of a pump or fan by 10 percent, the amount of fluid it carries also decreases by about 10 percent. This linear relationship shows that the most natural way to adjust the required flow is speed control. Lowering the speed instead of throttling with a valve or damper both adjusts the flow and saves energy.

The Square Relationship of Pressure

Pressure (or head in a pump) varies with the square of speed. If you lower the speed to 70 percent, pressure drops to 49 percent. This means that at low speed the system can produce less pressure; for this reason, in systems with high static head (for example, the height to which water must be lifted), there is a limit to lowering the speed. The system curve determines this limit.

The Cubic Relationship of Power and Energy

The most valuable result of the affinity laws concerns power. Because power varies with the cube of speed, the most economical way to reduce flow is to lower the speed. In traditional throttling methods, the motor keeps spinning at full speed and the excess energy is wasted as heat at the valve or damper. In speed control, the motor actually draws less power.

The Difference Between Throttling (Valve/Damper) and Speed Control

There are two ways to reduce flow: throttle the flow or lower the speed. In the throttling method a valve or damper adds resistance to the flow; because the motor keeps spinning at full speed, the power draw decreases very little. In speed control the motor itself slows down and the power it draws falls cubically. For the same flow reduction, speed control consumes far less energy.

The Energy Waste of Throttling

Suppose you reduce flow by 20 percent with a valve. The motor still spins at full speed and its power barely drops; the difference turns into heat as a pressure drop across the valve. When you make the same 20 percent reduction with speed, power drops by half. This difference between the two methods makes up the bulk of the annual energy bill in most pump and fan systems.

The System Curve and Operating Point

A pump or fan operates at the point where its own characteristic curve intersects the system curve. When speed changes, the pump curve shifts according to the affinity laws and a new operating point forms. The lower the static component in the system curve, the greater the saving obtained from speed control. Systems consisting entirely of friction losses are ideal for speed control.

The Effect of Static Head

If there is significant static head in the system (for example, water being pumped to a high tank), the affinity laws cannot be applied directly. This is because once the pump falls below a certain minimum pressure, it can deliver no flow at all. In such systems speed control still provides benefit, but the saving is more limited than in fully frictional systems. Correct analysis is done by deriving the system curve.

Affinity in Pump Applications

In water pumps, flow demand changes throughout the day. A fixed-speed pump either continually starts and stops or is throttled with a valve to meet this change. A variable-speed pump, by adjusting its speed to demand, both saves energy and reduces pressure fluctuations. For choosing the right pump motor, you can look at our article on water pump electric motor selection.

Affinity in Fan and Blower Applications

Fans and blowers are also true centrifugal machines and obey the affinity laws. In ventilation systems the required air flow frequently changes; lowering the fan speed instead of throttling with a damper provides a large energy gain. This is because reducing the air flow by 20 percent cuts power almost in half. Our article on fan and blower motor selection details this application.

Speed Control With an Inverter

The way to truly benefit from the affinity laws is the frequency inverter. The inverter continuously adjusts the speed by changing the frequency given to the motor. When demand drops the frequency falls, the motor slows, and by the affinity laws the power recedes cubically. Thus the system spends exactly as much energy as needed at every moment; the excess is not wasted.

The Additional Benefits an Inverter Provides

An inverter does not only save energy. Thanks to a soft start it reduces mechanical shocks and water hammer, and limits the starting current on the motor and the grid. It also makes closed-loop control scenarios such as keeping pressure constant possible. All these benefits make the inverter almost mandatory in variable-flow pump and fan systems.

The Effect on Motor Selection

The affinity laws directly affect motor power selection. The motor must be selected with enough power to meet the system's highest flow-pressure requirement (the peak point), because power is at its maximum at this point. However, since the system will most of the time operate below the peak point, an inverter drive reveals the saving potential. For correct power selection, our article on high and low kW motors will be useful.

The Oversizing Trap

A common mistake in pump and fan systems is choosing the motor too large "just in case." An oversized motor usually runs at low efficiency and low power factor at partial load. In a system that, by the affinity laws, already operates below the peak point most of the time, having the motor be larger than necessary is a double waste. Our article on the oversized motor and partial load trap covers this pitfall.

Pole Count and Base Speed

The pole count of the motor determines the base speed without an inverter. A two-pole motor spins around 3000 rpm and a four-pole motor around 1500 rpm. Choosing the pole count that gives the base speed closest to the desired nominal speed of the pump or fan keeps the inverter operating in a reasonable frequency range most of the time. Our article on the relationship between pole count and speed explains this choice.

Efficiency and Affinity Must Be Considered Together

The affinity laws assume ideal conditions; in real life pump and motor efficiency change slightly as speed changes. At very low speeds motor and pump efficiency may fall; for this reason efficiency curves must also be taken into account when calculating savings. Even so, the cubic power relationship is so dominant that speed control is almost always more efficient than throttling. Our article on electric motor efficiency losses deepens the topic.

Combining With a High-Efficiency Motor

When you combine the saving provided by the affinity laws with a high-efficiency motor, the gain multiplies. IE3, IE4 and IE5 class motors run with fewer losses at every speed; when used together with an inverter, the total energy consumption of the system drops markedly. Our article on high-efficiency electric motors explains this synergy.

The Power Factor Dimension

Speed control also affects power factor. While the inverter applies a variable frequency on the motor side, it generally shows a high power factor on the grid side. Even so, reactive power management across the facility is important. Our article on power factor (cosφ) covers this topic.

A Practical Saving Scenario

Imagine a fan that runs at partial flow for most of the day. When throttled with a damper at fixed speed, the motor always draws full power. When the speed is lowered to 80 percent with an inverter, power drops to 51 percent; that is, almost a half saving. Over a year this difference can reach a magnitude that pays back the inverter and motor investment in a short time on its own.

Water Hammer and Mechanical Load

In fixed-speed pumps, sudden stops and starts cause water hammer and mechanical stress. The smooth speed change based on the affinity laws largely eliminates these shocks. This in turn extends the life of the pipeline, valves and pump and reduces maintenance costs.

From a Noise and Vibration Standpoint

Lowering the speed also reduces fan- and pump-related noise, because aerodynamic and hydraulic noise rise strongly with speed. A system running at low speed is usually quieter. To reduce motor-related noise and vibration you can look at our article on reducing electric motor noise and vibration.

Affinity at Industrial Scale

In large facilities many pumps and fans operate together. Applying the affinity laws and variable-speed drive across the facility provides very large gains in total energy consumption. Correct motor selection, correct pole count and correct inverter matching are the foundation of industrial efficiency. Our article on industrial electric motors offers a broad view.

Parallel-Operating Pumps

In many facilities several pumps are connected in parallel to meet flow demand. In this arrangement, pumps are switched in one by one as demand rises. The affinity laws apply separately to each pump; however, because the system curve changes in parallel operation, the flow increase brought by each additional pump is not linear. Using a variable-speed lead pump together with fixed-speed assist pumps both covers a wide flow range and preserves energy efficiency. This strategy is common in water supply systems where demand varies greatly throughout the day.

Stepped Speed Control or Continuous Control?

In some systems the pump or fan is run at only a few fixed speed steps; for example, with two- or three-speed motors. This approach is simpler and cheaper than an inverter but cannot fully use the continuous optimization the affinity laws offer. In applications where demand varies continuously and over a wide range, the stepless speed control provided by a frequency inverter delivers much greater savings. If demand varies at a few points within a narrow band, a stepped solution may be sufficient.

Low Speed From a Bearing and Cooling Standpoint

Standard induction motors cool with their own shaft-mounted fan. As speed drops, the cooling this fan provides also decreases. If the motor continues to produce high torque at low speed, inadequate cooling can cause a temperature rise. With pump and fan loads that obey the affinity laws, torque also drops with speed, so this problem usually does not occur; but with constant-torque loads, an external (forced) cooling fan may be needed. For this reason it is important to evaluate the load type and speed range together.

The Minimum Speed Limit

The affinity laws imply in theory that speed can be lowered as much as desired; in practice, however, there is a lower limit. At very low speeds a centrifugal pump cannot produce sufficient pressure, a cooling fan cannot cool its own motor enough, and bearing lubrication can deteriorate. For this reason a safe minimum speed is defined for each system. In systems where static head is dominant, this lower limit is higher; in fully frictional systems it can be taken lower.

The Limits of the Affinity Laws

The affinity laws apply to centrifugal machines; positive-displacement pumps (for example, gear or piston types) do not obey these laws. In addition, over very wide speed ranges, efficiency and system curve effects alter the result. For this reason the affinity laws are a powerful first approach; the final design should be verified with the actual pump/fan curves and a system analysis.

Use the Power of Affinity With the Right Motor

The affinity laws show mathematically why energy saving in pump and fan systems comes through speed control. But to fully reveal this potential, a motor of the right power, the right pole count and a high efficiency class is needed. DRG Motor supplies induction motors in the IE3, IE4 and IE5 efficiency classes, compatible with frequency inverters and designed for variable-speed pump and fan applications. For the motor and drive solution most suitable for your system, you can contact the DRG Motor team and review our motor portfolio on our DRG electric motor product page. To reinforce the fundamentals, you can also look at our article on what an electric motor is.