The fan laws, also called the affinity laws, are three proportionalities that predict how a fan's performance changes when you alter its speed, its impeller diameter or the density of the air it is moving. They are the reason a small reduction in fan speed produces a startlingly large reduction in energy use, and the reason a fan that worked at sea level behaves differently on a mountain.
This calculator takes a known operating point — flow, pressure and shaft power at a given speed and diameter — and projects it to a new condition. Arb Digital built it around the ratios rather than around any particular fan curve, because the fan laws are relative statements: they tell you how a known point moves, never where that point was to begin with.
What This Fan Calculator Does
You supply a baseline operating point and the new speed, diameter and density ratio. The tool returns the projected airflow, static pressure and shaft power, the percentage change in power, and the fan's static efficiency, which is the ratio of useful air power to the shaft power actually consumed.
The three bars are the fastest way to see what the laws are doing. Set the speed ratio to anything other than one and the bars immediately separate, because the three quantities respond to different powers of that ratio. At eighty per cent speed the flow bar sits at eighty per cent, the pressure bar at sixty-four, and the power bar at fifty-one. That divergence is the entire practical content of the subject.
Efficiency is included because it is the sanity check. The fan laws preserve efficiency: a geometrically similar fan at a different speed sits at the same point on its dimensionless performance curve, so the efficiency figure should not move when you change speed alone. If you expected a speed change to make a fan more efficient, the calculator will show you that it does not — it makes the fan consume less power by doing less work, which is a different thing.
How to Use It
- Enter a real baseline point. The laws scale a known condition. Take the flow, pressure and power from a fan curve, a commissioning record or a manufacturer's rating at a stated speed — not from a guess.
- Use shaft power, not motor nameplate. Nameplate power includes motor efficiency losses and a deliberate margin. Feeding it in as baseline shaft power inflates every projected power figure and produces a nonsensical efficiency.
- Change one thing at a time at first. Speed alone is the common case. Leave both diameters equal unless you are genuinely comparing different impeller sizes in the same fan series.
- Set the density ratio for altitude or temperature. Hot air and thin air are less dense, which reduces pressure and power proportionally while leaving volumetric flow untouched.
- Check the efficiency figure looks plausible. Real fans land somewhere between roughly forty and eighty per cent static efficiency. A figure above one hundred per cent means the baseline power was too small or the baseline pressure too large.
The Formula: One Ratio Raised to Three Different Powers
With N for speed, D for impeller diameter and ρ for air density, the three laws are:
- Flow scales with speed and with the cube of diameter: Q₂ = Q₁ × (N₂/N₁) × (D₂/D₁)³. Density does not appear.
- Pressure scales with the square of speed, the square of diameter, and density: p₂ = p₁ × (N₂/N₁)² × (D₂/D₁)² × (ρ₂/ρ₁).
- Power scales with the cube of speed, the fifth power of diameter, and density: P₂ = P₁ × (N₂/N₁)³ × (D₂/D₁)⁵ × (ρ₂/ρ₁).
The third follows from the first two: power is flow multiplied by pressure, and speed to the first power multiplied by speed squared is speed cubed. Nothing independent is being asserted in the power law, which is why it is safe to trust it exactly as far as you trust the other two.
Work the defaults. A fan moving 5,000 CFM at 2.0 inches water gauge, absorbing 2.5 hp at 1,200 RPM, is sped up to 1,500 RPM. The speed ratio is 1.25. Flow becomes 5,000 × 1.25 = 6,250 CFM. Pressure becomes 2.0 × 1.25² = 3.125 inches. Power becomes 2.5 × 1.25³ = 4.88 hp — a 95 per cent increase in energy for a 25 per cent increase in air. The air power at the baseline is 5,000 × 2.0 ÷ 6,356 = 1.573 hp, giving a static efficiency of 62.9 per cent, and that efficiency is unchanged at the new speed. Efficiency classification for fans of this kind is set out in ANSI/AMCA Standard 205-19, Energy Efficiency Classification for Fans. Unit relationships between the inch of water, the pascal and the horsepower come from the conversion tables in NIST Special Publication 811.
The Laws Assume Geometric Similarity — and That Is a Real Constraint
Every one of these relationships rests on an assumption that is easy to state and easy to violate: the two conditions being compared must describe geometrically similar fans operating at dynamically similar points. Geometric similarity means every dimension scales by the same factor — blade width, blade thickness, inlet cone, tip clearance, hub diameter, everything — so that the two fans are genuine scale models of each other.
Changing the speed of a single fan satisfies this trivially, because the geometry does not change at all. That is why the speed laws are reliable in practice and why variable-speed drive calculations are trustworthy. The diameter laws are the shakier half. They hold across a manufacturer's series where the impellers really are scaled versions of one another, and they fail when you trim an impeller in place, because a trimmed wheel keeps its original blade width and housing while only the outer diameter changes. That is not a scale model, and trimmed-impeller performance departs from the fifth-power law, usually falling short of it.
Dynamic similarity adds a second condition: the flow regime must be comparable. At very low speeds or very small sizes, viscous effects become proportionally larger and the fan does not behave like a scaled version of its bigger self. In practice this matters at extremes rather than across ordinary HVAC operating ranges, but it is the reason the laws are approximations rather than identities.
Why Slowing a Fan Saves So Much More Than It Seems
The cube law on power is the single most valuable fact in ventilation energy management. Reduce a fan to eighty per cent speed and the power falls to 0.8³ = 51 per cent. A twenty per cent reduction in air movement halves the energy. Reduce to fifty per cent speed and power falls to twelve and a half per cent — an eightfold reduction from full speed.
This is why variable-speed drives pay back so quickly on systems that were oversized or that spend most of their hours at part load, and why throttling with a damper is so wasteful by comparison. A damper reduces flow by adding resistance, which moves the operating point up the fan curve to a higher pressure. The fan keeps spinning at full speed and keeps drawing something close to full power while delivering less air. Slowing the fan instead moves the whole fan curve down and collects the cube-law saving.
The reverse is the warning. Speeding a fan up to fix an underperforming system is expensive in a way that catches people out. The default case above needs almost twice the power for a quarter more air, and the motor that was adequate before will now be overloaded. Always check the projected shaft power against the motor rating before changing a pulley or a drive setting.
Density, Altitude and the Flow That Does Not Change
Air density appears in the pressure and power laws but not in the flow law, and that asymmetry is the source of persistent confusion. A fan is a constant-volume machine: it sweeps a fixed volume per revolution regardless of what is in it. Move the same fan to altitude and it still delivers the same cubic feet per minute.
What changes is everything that depends on the mass of air being moved. Thinner air means less pressure developed and less power absorbed, both falling in direct proportion to density. So a fan at 5,000 feet, where density is roughly eighty-six per cent of sea level, produces eighty-six per cent of its rated static pressure and draws eighty-six per cent of its rated power while moving exactly the same volume.
The practical trap is that the job usually needs mass, not volume. Ventilation for heat removal or combustion air is a mass requirement, and delivering the same volume of thinner air delivers less of it. Correcting for that means increasing the volumetric flow, which takes you back up the speed law and into the cube law on power. The air density calculator gives the density ratio to enter here, and the air changes per hour calculator converts a ventilation requirement into a volumetric target.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Applying the diameter law to a trimmed impeller — trimming changes diameter without scaling blade width or housing, so geometric similarity is broken and the fifth-power law overstates the result.
- Using motor nameplate power as the baseline — the laws scale shaft power. Nameplate includes motor losses and margin, which inflates every projection and corrupts the efficiency figure.
- Expecting density to change airflow — a fan moves a fixed volume. Altitude and heat cut pressure and power, not cubic feet per minute.
- Assuming a slower fan is a more efficient fan — efficiency is preserved by the laws. A slower fan uses less power because it does less work, not because it wastes less.
- Forgetting to recheck the motor after a speed increase — the cube law means a modest speed rise can overload a motor that was comfortably sized before.
Related Free Tools From Arb Digital
To find the density ratio for altitude or temperature, use the air density calculator. Ventilation targets are set with the air changes per hour calculator, and volumetric rates converted with the flow rate calculator and the flow rate converter. Static pressure in other units is handled by the pressure converter. On the drive side, the horsepower calculator and the motor torque calculator take the shaft power figure through to the motor, and belt drives are sized with the belt length calculator. The complete free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
Airflow varies with fan speed and with the cube of impeller diameter. Static pressure varies with the square of speed, the square of diameter and with air density. Shaft power varies with the cube of speed, the fifth power of diameter and with density. The power law is simply the product of the other two.
Yes. They apply only between fans that are true scale models of one another operating at dynamically similar points, with every dimension scaled by the same factor. Changing the speed of one fan satisfies this automatically, which is why the speed laws are the most reliable.
Power follows the cube of speed, so running at eighty per cent speed uses about fifty-one per cent of the power and running at fifty per cent uses about twelve and a half per cent. That cube relationship is why variable-speed drives save far more energy than the speed reduction alone suggests.
No. A fan is a constant-volume machine and moves the same cubic feet per minute regardless of air density. What falls with density is the static pressure it develops and the power it absorbs, both in direct proportion.
Not reliably. Trimming reduces the outer diameter while leaving blade width, hub and housing unchanged, so the fan is no longer a scale model of its original. The diameter laws hold across a properly scaled fan series, not across a machined-down wheel.
It is the useful air power divided by the shaft power absorbed. Air power is airflow multiplied by static pressure, converted to the same units as the shaft power. Real fans typically fall between about forty and eighty per cent.
A damper reduces flow by adding resistance, which pushes the operating point to a higher pressure while the fan continues to turn at full speed and draw close to full power. Reducing speed instead collects the cube-law saving on power.
This tool is provided for educational and preliminary engineering use. The fan laws are approximations valid only for geometrically and dynamically similar conditions, and no projection here substitutes for the manufacturer's certified performance data for a specific fan and system.
