Fan Affinity Laws Explained: How Speed Changes Flow, Pressure and Power
Slow a fan down by 20%, and it moves 20% less air. That much is intuitive. What surprises most people is the power: the same 20% speed cut drops shaft power by almost half. This is the reason variable frequency drives pay for themselves on fan systems, and the reason an engineer who understands the fan laws can size, troubleshoot and retrofit fans far faster than one who relies on vendor software alone.
This guide explains the three fan affinity laws, shows a worked example with real numbers, covers the diameter version, and lists the situations where the laws stop being accurate. When you want to apply them to your own machine, including the effect of altitude, temperature and humidity on air density, use the density-aware affinity law calculator.
[Add diagram: a fan curve at 100% and 80% speed with a system curve through the origin, showing the two equivalent operating points.]
The Three Fan Laws
The fan laws (also called affinity laws) relate a fan’s flow rate, pressure rise and power at two different speeds, provided the fan geometry and the air density stay the same. With N as rotational speed, the relationships are:
| Quantity | At 1,450 rpm | At 1,160 rpm | Change |
|---|---|---|---|
| Flow | 10,000 m³/h | 8,000 m³/h | −20% |
| Pressure rise | 1,000 Pa | 640 Pa | −36% |
| Shaft power | 5.5 kW | 2.8 kW | −48.8% |
In words: flow changes in direct proportion to speed, pressure changes with the square of speed, and power changes with the cube of speed.
These laws come from dimensional analysis. They assume the fan operates at the same point on its non-dimensional performance curve before and after the change, which engineers call dynamic similarity. In practice, this means the new operating point must lie on the system curve passing through the origin, which is the usual case when the system’s losses are dominated by friction and fittings that scale with the square of flow.
Worked Example: Slowing a Fan by 20%
A fan runs at 1,450 rpm. At its duty point, it delivers 10,000 m³/h at 1,000 Pa and absorbs 5.5 kW at the shaft. A VFD reduces the speed to 1,160 rpm, which is 80% of the original.
| Quantity | Law | If speed is 80% of original |
|---|---|---|
| Flow rate, Q | Q₂/Q₁ = N₂/N₁ | 80% |
| Pressure rise, ΔP | ΔP₂/ΔP₁ = (N₂/N₁)² | 64% |
| Shaft power, P | P₂/P₁ = (N₂/N₁)³ | 51.2% |
The calculation for power is 5.5 × 0.8³ = 5.5 × 0.512 = 2.82 kW. A 20% reduction in airflow saves nearly half of the shaft power.
Why Small Speed Cuts Save So Much Energy
Because power follows the cube of speed, the savings grow quickly:
| Speed reduction | Flow | Shaft power | Power saving |
|---|---|---|---|
| 10% | 90% | 72.9% | 27.1% |
| 20% | 80% | 51.2% | 48.8% |
| 30% | 70% | 34.3% | 65.7% |
| 50% | 50% | 12.5% | 87.5% |
This is why throttling a fan with a damper wastes energy. A damper adds resistance and reduces flow but leaves the fan running at full speed, so most of the power is still consumed. Reducing speed with a VFD cuts the power far more for the same flow reduction.
A caution: these are shaft power figures. Motor and drive efficiency fall at low load, so the savings measured at the electrical input will be a little smaller than the cube law suggests.
Changing the Impeller Diameter
The laws can also be extended to a change in diameter D, for geometrically similar fans running at the same density:
- Flow: Q₂/Q₁ = (N₂/N₁)(D₂/D₁)³
- Pressure rise: ΔP₂/ΔP₁ = (N₂/N₁)²(D₂/D₁)²
- Shaft power: P₂/P₁ = (N₂/N₁)³(D₂/D₁)⁵
These hold for fans of the same design scaled up or down. If you only trim an impeller on an existing fan, the geometry is no longer exactly similar, so treat the results as estimates for small trims and confirm with the manufacturer.
Where the Fan Laws Stop Being Accurate
The laws are a fast screening tool, not a replacement for performance testing. Be careful in these situations:
- Static pressure in the system. If part of the system resistance does not change with flow (for example, a fixed pressure difference across a process), the new operating point is not a similar point, and the laws do not apply directly.
- Large speed changes. Efficiency shifts as the fan moves away from its best efficiency point. Speed changes beyond roughly 20 to 30% deserve a check against the fan curve.
- Mach and Reynolds number effects. At high tip speeds or very low speeds, the similarity assumption weakens.
- High pressure rise. When the pressure rise is more than roughly 10% of absolute inlet pressure, gas compressibility matters. Use manufacturer curves with corrected speed and flow instead.
- Mechanical limits. Raising speed increases centrifugal stress with the square of speed. Never exceed the fan manufacturer’s maximum continuous speed or the motor and bearing limits.
- Changing gas density. The laws assume constant density. See the next section.
When Air Density Changes
When the inlet density changes between the reference and the new condition, because of altitude, temperature or humidity, the pressure and power laws must carry a density term:
- ΔP₂/ΔP₁ = (ρ₂/ρ₁)(N₂/N₁)²(D₂/D₁)²
- P₂/P₁ = (ρ₂/ρ₁)(N₂/N₁)³(D₂/D₁)⁵
For instance, a fan rated at sea level, 15°C and 50% humidity, then moved to a site at 1,500 m and 35°C, sees roughly 22% lower inlet density. At the same speed, it produces about 23% less pressure, and to hold the required pressure, it must run roughly 13.7% faster. You can reproduce this case and test your own conditions in the density-aware affinity law calculator.
Quick Checklist Before You Apply the Fan Laws
- Are both operating points on a system curve that passes through the origin?
- Is the speed change small enough that efficiency stays close to the original?
- Is inlet density the same, or have you corrected for it?
- Is the pressure rise small compared with the inlet absolute pressure?
- Is the new speed within the fan’s maximum continuous speed and the motor rating?
If you can answer yes to all five, the fan laws will give a reliable estimate.
Frequently Asked Questions
- Do the fan laws apply to pumps?
The same affinity relationships apply to centrifugal pumps, using head instead of pressure. The pump version is covered in our guide to pump affinity laws.
2. Why does power fall with the cube of speed?
Air power is flow multiplied by pressure. Flow scales with speed and pressure with speed squared, so their product scales with speed cubed.
3. Can I use the fan laws to size a VFD?
They help you estimate the power at a reduced speed, but VFD selection also depends on the motor current, the starting duty and the speed range. Confirm with the drive manufacturer.
4. What is the difference between throttling and speed control?
Throttling adds resistance and leaves the fan at full speed, so power falls only a little. Speed control moves the fan along its curve and follows the cube law, so the power saving is much larger.
5. Are the fan laws exact?
No. They are an approximation based on dynamic similarity. They work well for modest speed changes on low-pressure fans, and less well outside that range.
References
- ISO 5801, Fans: performance testing using standardised airways.
- AMCA Publication 201, Fans and Systems.
- Your fan manufacturer’s performance curves and technical data.