Fan Laws Explained: Speed, Flow, Pressure and Power With Worked Examples

The fan laws say that for a given fan in a given system, airflow is proportional to speed, pressure to the square of speed, and power to the cube of speed. They are the most useful shortcuts in fan engineering, because they let you predict what happens when you change speed, size, or air density, without new testing. They also have limits, and knowing those limits is what separates a sound estimate from a costly mistake.

The Three Laws for Speed

For the same fan, in the same system, at the same air density:

• Flow: Q₂ = Q₁ × (N₂ ÷ N₁)
• Pressure: P₂ = P₁ × (N₂ ÷ N₁)²
• Power: W₂ = W₁ × (N₂ ÷ N₁)³

where N is rotational speed. The relationship for power follows from air power, which is flow multiplied by pressure. If flow scales with speed and pressure scales with its square, power scales with the cube.

Worked Example: Changing Speed

A fan delivers 40 m³/s at 500 Pa total pressure. Air power is 40 × 500 = 20 kW. At 80% total efficiency, shaft power is 25 kW.

Speed change	Flow	Total pressure	Shaft power
Base	40 m³/s	500 Pa	25.0 kW
Speed up 10%	44 m³/s	605 Pa	33.3 kW
Slow down 20%	32 m³/s	320 Pa	12.8 kW

Reducing flow by 20% through speed control cuts power by about half. This is the energy argument for variable-speed drives on variable duties. It also has a warning in the other direction: speeding a fan up by only 10% raises power by a third, which can overload a motor that was sized with little margin.

Scaling With Fan Size

For geometrically similar fans of different diameter D, at the same speed:

• Flow scales with D³
• Pressure scales with D²
• Power scales with D⁵

Example. A 1,000 mm fan running at 1,450 rpm delivers 15 m³/s at 600 Pa, with 11.25 kW shaft power. A geometrically similar 1,250 mm fan at the same speed would deliver about 29.3 m³/s at 937.5 Pa and absorb about 34.3 kW.

Fan diameter	Flow	Pressure	Shaft power	Tip speed
1,000 mm	15.0 m³/s	600 Pa	11.3 kW	75.9 m/s
1,250 mm	29.3 m³/s	937.5 Pa	34.3 kW	94.9 m/s

The tip speed column explains why real designs rarely keep speed constant as size grows. Tip speed rises with diameter, and with it comes noise and mechanical stress. Larger fans are normally run slower.

The Density Law

At constant speed and constant flow in the same system, pressure and power are proportional to air density. This matters at altitude and high temperature.

Site condition	Density (kg/m³)	Ratio to standard	A 500 Pa fan develops
Sea level, 20 °C	1.204	1.00	500 Pa
Sea level, 45 °C	1.109	0.92	460 Pa
1,800 m, 20 °C	0.968	0.80	402 Pa
2,400 m, 20 °C	0.899	0.75	373 Pa

Two practical consequences follow. First, a fan selected from a sea-level catalogue at 2,400 m develops about three-quarters of its rated pressure at the same speed, so the speed or fan size may need to increase. Second, motor power is highest at the densest, coldest condition, so a motor sized only for hot conditions can overload at start-up or on a cold morning.

Where the Fan Laws Stop Working

The laws assume the operating point moves along a system curve that passes through the origin, meaning resistance is purely proportional to flow squared. Several real conditions break this.

• A static head or fixed resistance. If part of the resistance does not vary with flow, pressure does not scale simply with the square of speed, and the savings from slowing down are smaller than the cube law suggests.
• Control dampers. Changing a damper changes the system curve, so the same fan-law relationships do not apply between the two states.
• Very large speed changes. Efficiency changes when the blades run far from their design conditions.
• Stall and instability. At low speeds against high static head the operating point may move into the stall region, where the laws give no warning.
• Different fan types. The similarity laws apply to geometrically similar fans, not to fans with different blade shapes.
• Mechanical limits. Increasing speed raises stress, bearing loads, and tip speed. Never speed a fan up without the manufacturer’s approval.

Check any significant change against the actual fan curve. See how to read an axial fan curve.

Using the Laws in Practice

Retrofit. If a plant needs 10% more air, the laws show the pressure and power penalty before anyone changes a pulley.

Variable-speed drives. They let you follow load with the cube-law saving. Our guide to adjustable pitch versus VFD control compares this with blade-pitch control.

Commissioning. Measured speed, flow, and power let you check whether the installed fan sits where the curve predicts.

Noise. Speed reductions cut noise as well as power. A common rule of thumb is that sound power changes by roughly 50 times the base-10 logarithm of the speed ratio, so a 20% speed cut lowers noise by about 4.8 dB. See axial fan noise.

A Short Method for Any Speed Change

1. Write down the base flow, pressure, power, and speed.
2. Compute the speed ratio.
3. Apply flow × ratio, pressure × ratio², and power × ratio³.
4. Check whether the resulting operating point is on the real system curve.
5. Check the motor, drive, and mechanical limits.
6. Correct for air density if conditions differ.

Check a Speed or Density Change Before You Make It

Tell us your fan, duty, and site conditions, and we’ll check the fan-law effect and the operating point. Contact us for a free quote.

Frequently Asked Questions