The fan laws (also called affinity laws) are proportional relationships that screen how airflow, static pressure, and shaft power may change when fan speed or impeller diameter changes. They are useful everyday planning relationships in HVAC and industrial ventilation, but they do not replace certified fan curves, field measurements, density corrections, system-effect review, or motor/VFD checks.
Understanding the system curve, and where it intersects the fan curve to establish the operating point, is equally important. Without that curve intersection, you only have a preliminary estimate of what may happen when speed or resistance changes. This guide covers the three fan laws, the system curve concept, operating-point shifts, energy context, and where the simplified model breaks down.
The Three Fan Laws
The fan laws relate a known operating condition (speed N1, airflow Q1, pressure P1, power W1) to a new condition at a different speed (N2). They assume the system curve shape does not change, the air density remains constant, and the fan is operating in a stable region of its curve.
Law 1 (Airflow): Q2 = Q1 × (N2/N1). Airflow is directly proportional to speed. Cut the speed in half, and airflow drops to 50%.
Law 2 (Pressure): P2 = P1 × (N2/N1)². Static pressure (or total pressure) varies with the square of the speed ratio. Cut speed in half, and pressure drops to 25% of the original.
Law 3 (Power): W2 = W1 × (N2/N1)³. Shaft power varies with the cube of the speed ratio. Cut speed in half, and power drops to 12.5% of the original. This cubic relationship is the reason VFDs save so much energy on variable-volume fan systems.
These laws also apply to impeller diameter changes (substitute D2/D1 for N2/N1), but diameter changes are less common in practice and limited to small trims (no more than 10-15% reduction is typical for centrifugal fans before the volute becomes mismatched).
Fan Laws Calculator
Apply AMCA 201 fan affinity laws to predict flow, pressure, and power changes from speed or system resistance changes. Includes system curve estimate and VFD energy savings analysis.
The System Curve Concept
A system curve plots the pressure loss through a duct system as a function of airflow. For a fixed duct layout, the pressure loss at any given airflow is the sum of friction losses (proportional to flow squared) and fixed losses (stack effect, hood entry losses that are essentially flow-squared dependent too). The result is a parabola passing through the origin: ΔP = k × Q², where k is the system resistance constant determined by duct geometry, fittings, filters, coils, and damper positions.
The system resistance constant k changes when you modify the ductwork: adding a filter, closing a damper, extending a run, or adding a branch. Each change creates a new system curve. A dirtier filter increases k and steepens the curve. Opening a damper decreases k and flattens it. Understanding this is critical because the fan does not operate at a fixed point on its performance curve; it operates wherever the fan curve and the system curve intersect.
In practice, you rarely know k precisely. Instead, you measure airflow and pressure at one operating condition and use that point to define the system curve: k = ΔP_measured / Q_measured². Then you can predict the pressure at any other airflow using that k value, assuming the system configuration has not changed.
The Operating Point
The operating point is where the fan performance curve intersects the system resistance curve. At this point, the pressure developed by the fan exactly equals the pressure lost by the system, and the airflow is the steady-state delivery. If you overlay the fan curve (from the manufacturer's catalog, plotted at a given RPM) on the system curve, the intersection defines the airflow, pressure, and (from the power curve) the shaft horsepower at that operating condition.
When fan speed changes (via VFD, pulley change, or variable-pitch blades), the entire fan curve shifts. At a lower speed, the fan curve drops and to the left, and the new operating point is usually at lower airflow and lower pressure, following the system curve downward. The fan laws estimate that movement as Q2 = Q1 × (N2/N1) and P2 = P1 × (N2/N1)² when the same fan, density, efficiency region, and system curve assumptions hold.
Problems arise when the system curve changes simultaneously. For example, a VAV system with terminal box dampers modulates as fan speed changes, altering the system resistance. In this case, the simple fan law prediction is only approximate because the system curve shape has shifted. This is one reason modern VAV systems use static pressure sensors and PID control to find the actual operating point rather than relying purely on fan law calculations.
VFD Energy Savings
The cubic power relationship explains why VFDs can be attractive on variable-air-volume systems. If average airflow demand is 75% of a known point and the assumptions hold, the shaft-power screen is roughly 0.75³ = 42% of that known point. In a throttled system, some of the unused pressure is often lost across dampers, filters, or controls instead of being removed by reducing fan speed.
Real-world savings depend on the load profile, fan curve, control strategy, minimum speed, static pressure reset, motor efficiency, VFD efficiency, utility tariff, maintenance condition, and installed cost. Systems that run at full capacity most of the time may see little benefit, while variable-load systems need measured kW or manufacturer data before a business case is trusted.
One caution: the cubic power law assumes the system curve does not change. If the VFD reduces fan speed on a system with significant fixed-pressure losses, or if dampers and terminal boxes change position at the same time, actual savings may differ from the cube-law screen. Set a minimum VFD speed limit based on the fan manufacturer's recommendation and field commissioning data.
Common Pitfalls
Assuming constant density: The fan laws assume constant air density. If temperature or altitude changes significantly between the known and predicted conditions, you must correct for density. Fan airflow (CFM) is volumetric and does not change with density at a given speed, but the mass flow rate (lbm/min) and the pressure developed (in. w.g.) do change. A fan running at 200°F moves the same CFM as at 70°F, but the pressure capability and power consumption both decrease proportionally with the density ratio.
Applying fan laws outside the stable range: Every fan curve has a stall or surge region (to the left of the peak pressure point on centrifugal fans, or beyond the stall angle on axial fans). The fan laws are not valid in this region because the flow is unstable. If your speed reduction pushes the operating point into the stall zone, the actual airflow and pressure will be unpredictable and the fan may vibrate or hunt. Always check that the predicted operating point is in the stable portion of the fan curve.
Confusing fan static pressure with total pressure: Fan manufacturers may rate fans on either static pressure (SP) or total pressure (TP = SP + velocity pressure). Make sure you are using the correct curve and matching units when overlaying the system curve. The system curve is typically plotted in total pressure, so using a static pressure fan curve without adding the velocity pressure correction will give incorrect operating point predictions.
Neglecting motor and drive efficiency changes: The fan laws predict shaft power. Actual electrical power also depends on motor efficiency (which varies with load) and VFD efficiency (typically 96-98%). At reduced speeds, the motor operates at a lower percentage of rated load, where efficiency may drop, partially offsetting the cubic savings. For motors below 30% load, efficiency can fall significantly.