Engineering · Mechanical Engineering · Fluid Power
Fan Affinity Laws Calculator
Calculate changes in fan flow rate, pressure, and power when rotational speed or impeller diameter changes using the Fan Affinity Laws.
Calculator
Formula
Q₁ and Q₂ are the initial and new volumetric flow rates (m³/s or CFM); N₁ and N₂ are the initial and new rotational speeds (RPM); D₁ and D₂ are the initial and new impeller diameters (m or in); Ps₁ and Ps₂ are the initial and new static pressures (Pa or in w.g.); W₁ and W₂ are the initial and new shaft power values (W or hp). When only speed changes, set D₂ = D₁. When only diameter changes, set N₂ = N₁.
Source: ASHRAE Handbook — Fundamentals (2021), Chapter 21; AMCA Publication 201 — Fans and Systems.
How it works
The Fan Affinity Laws are grounded in dimensional analysis of fluid machinery. When a centrifugal fan's impeller rotates faster or is replaced with a geometrically similar impeller of a different diameter, every performance parameter scales in a predictable, mathematically exact way — provided the fan operates at the same efficiency point and the fluid density remains constant. The three laws cover the three most critical performance variables: volumetric flow rate, static pressure (or total pressure), and absorbed shaft power.
The First Law states that flow rate (Q) varies linearly with speed ratio and as the cube of diameter ratio: Q₂/Q₁ = (N₂/N₁)(D₂/D₁)³. The Second Law states that static pressure (Ps) varies as the square of both ratios: Ps₂/Ps₁ = (N₂/N₁)²(D₂/D₁)². The Third Law — the most consequential for energy auditing — states that shaft power (W) varies as the cube of speed and the fifth power of diameter: W₂/W₁ = (N₂/N₁)³(D₂/D₁)⁵. The steep cubic relationship between power and speed means that even a modest speed reduction yields dramatic energy savings; reducing fan speed by just 20% cuts power consumption to approximately 51% of its original value.
In practice, the fan laws are applied in variable air volume (VAV) HVAC systems where variable speed drives (VSDs) modulate fan speed to match building load. They are equally important when an engineer substitutes a different-diameter impeller (re-wheeling) into an existing fan housing to shift the duty point. Industrial process engineers use the laws to match fans to changed duct system resistances after a plant retrofit, while energy auditors use the power law to quantify expected electricity savings from VSD installations.
Worked example
An existing centrifugal supply fan operates at N₁ = 1,450 RPM, delivering Q₁ = 2.5 m³/s at a static pressure of Ps₁ = 400 Pa while consuming W₁ = 1,200 W of shaft power. The impeller diameter remains unchanged (D₁ = D₂ = 0.5 m). A variable speed drive is installed and the fan speed is increased to N₂ = 1,750 RPM to meet a higher ventilation demand. We want to find the new duty point.
Step 1 — Speed ratio: N₂/N₁ = 1,750 / 1,450 = 1.2069
Step 2 — New flow rate (First Law): Q₂ = 2.5 × 1.2069 × (1.0)³ = 3.017 m³/s
Step 3 — New static pressure (Second Law): Ps₂ = 400 × (1.2069)² × (1.0)² = 400 × 1.4566 = 582.6 Pa
Step 4 — New shaft power (Third Law): W₂ = 1,200 × (1.2069)³ × (1.0)⁵ = 1,200 × 1.7588 = 2,110.6 W
The speed increase of approximately 20.7% results in a 20.7% increase in flow, a 45.7% increase in static pressure, and a 75.9% increase in power demand. This illustrates how the cubic power law means oversizing a fan and running it at full speed is far more energy-intensive than running a correctly sized fan.
Limitations & notes
The Fan Affinity Laws assume geometric and dynamic similarity between the two operating conditions. They are accurate only when: (1) the fluid density is constant — changes in air temperature or altitude will alter results, and a density correction factor must be applied; (2) the fan operates at the same point on its non-dimensional performance curve (same efficiency point) — significant changes in system resistance that shift the operating point away from the design point introduce errors; (3) the fan is centrifugal or mixed-flow — axial fans can deviate from the laws more significantly at off-design conditions; and (4) no compressibility effects are present, which is generally valid for pressure rises below approximately 5 kPa in air systems. Additionally, mechanical friction losses do not scale perfectly with the affinity laws, so at very low speeds the laws overpredict efficiency. Bearing and seal losses, motor slip, and belt drive losses must be accounted for separately when computing overall system efficiency. The laws also do not account for resonance, surge, or stall behaviour that can occur at extreme speed ratios.
Frequently asked questions
What are the Fan Affinity Laws used for?
The Fan Affinity Laws are used to predict how a centrifugal fan's flow rate, pressure, and power consumption change when its speed or impeller diameter is altered. They are essential tools for HVAC system design, variable speed drive sizing, energy auditing, and fan re-wheeling or re-drive projects without requiring full system retesting.
Do the affinity laws apply to pumps as well as fans?
Yes. The same three relationships — also called the Pump Affinity Laws — apply identically to centrifugal pumps, with flow rate, head, and power playing the same roles as fan flow, pressure, and power. The underlying physics is the same dimensional analysis of rotating turbomachinery.
How much energy can I save by reducing fan speed by 10%?
Using the Third Affinity Law, reducing speed by 10% (N₂/N₁ = 0.9) reduces power to 0.9³ = 0.729, meaning approximately 27% energy savings. This significant reduction for a modest speed change is the main justification for installing variable speed drives on large fans and pumps in commercial and industrial buildings.
Why does air density matter for the fan laws?
The affinity laws assume constant fluid density. Static pressure and power both scale with fluid density, so if a fan is tested at sea level (ρ ≈ 1.2 kg/m³) but installed at high altitude where air is less dense, the pressure and power will be lower than predicted. A correction factor of ρ_site/ρ_standard must be applied to pressure and power outputs.
Can I use the fan laws to change both speed and diameter simultaneously?
Yes. The full form of each law includes both the speed ratio and the diameter ratio, so you can apply combined changes in a single calculation. However, ensure the new impeller is geometrically similar to the original — same blade angles, number of blades, and proportional dimensions — otherwise the laws will not accurately predict performance.
Last updated: 2025-01-15 · Formula verified against primary sources.