How a Fan Law Changes What the Drive Has to Deliver

Why this matters

"Put a smaller sheave on it and get a bit more air" is one of the cheapest-sounding jobs in the trade and one of the most expensive things you can do to a drive. The reason is that the three fan relationships are not linear together: flow follows speed, pressure follows the square, and shaft power follows the cube, so a change that reads as a modest airflow gain arrives at the motor, the belts and the fan bearings as something much larger. The second reason, which costs more callbacks than the first, is that these relationships hold under a specific condition, and the most common field situation - a system that got dirtier or leakier - violates that condition completely. Applied there, the laws give an answer that is not just imprecise, it points the wrong way.

Before the sheaves come off

A sheave change is stopped, isolated work. Lock and tag the energy-isolating device under 29 CFR 1910.147, then watch the wheel until it has fully stopped, because a fan wheel stores enough rotational energy to keep turning long after the contactor drops out and that stored energy is precisely what the standard addresses. Set and check belt tension only on the stopped, locked-out drive against the manufacturer's force-deflection figure, never against a moving belt. Reinstall the guard before the first run and make the direction check from outside the plane of rotation. Any current reading taken afterwards at the starter is energized electrical work under 29 CFR 1910.333(b)(2), in arc-rated PPE and voltage-rated gloves after an arc-flash risk assessment.

The relationships, each with the condition it was derived under

For the same fan, at the same air density, operating against the same system curve, changing speed moves the operating point in this locked pattern:

The torque row is the one people leave out, and it is the row the belt drive cares about. A speed change that raises power by half raises torque by less than that, because part of the power increase is bought by turning faster rather than by pushing harder.

A separate relationship covers density, and it is not part of the same family: at the same fan, same speed and same system, volumetric flow is unchanged while pressure and shaft power scale directly with density. Density itself varies inversely with absolute temperature at constant pressure, and falls with elevation.

The condition that gets violated

Every row above assumes the system curve did not move. The laws describe what happens when you change the fan's speed and nothing else changed.

When the system resistance changes - a loading filter, a closing damper, a blocked coil, a duct leak, a blank-off left out - the fan speed has not changed at all. The fan simply slides along its own performance curve to a new intersection with the new system curve. The fan laws say nothing about that move, and the only correct source is the fan's own published curve. Cubing a flow ratio in that situation applies a coefficient outside the geometry it was derived under, and on most centrifugal fans it gets the sign of the power change backwards.

Direction, stated by fan family because it genuinely differs by family: on centrifugal fans, both backward-inclined and forward-curved, restricting flow moves the operating point toward shutoff and generally reduces shaft power, which is why a forward-curved wheel overloads its motor on an under-resisted system rather than a dirty one. On axial and vaneaxial fans, restricting flow generally increases shaft power as the blades move toward stall. So the reflex sentence "the filter is dirty so the motor is working harder" is wrong on the centrifugal unit in most air handlers and right on an axial fan, and the fan type decides it.

Worked example: a 1.5-inch sheave change

A belt-driven supply fan needs more air. The driver sheave on the motor stays as it is; the driven sheave on the fan shaft goes from a 12.0 inch pitch diameter to a 10.5 inch pitch diameter. The system is untouched, so the fan laws genuinely apply here.

Speed ratio: 12.0 divided by 10.5 is 1.143, so the fan turns about 14.3 percent faster.

Carry that through:

• Flow: 1.143, about 14 percent more air. That is what was asked for.
• Pressure: 1.143 squared, 1.31, about 31 percent more.
• Shaft power: 1.143 cubed, 1.49, about 49 percent more.
• Fan-shaft torque: 1.143 squared, 1.31.

At the motor. The fan was drawing about 0.75 of motor rating. At 1.49 times that, it now draws 0.75 times 1.49, about 1.12 of rating. On a 1.15-service-factor nameplate that looks survivable, and it is only survivable if the motor is at rated voltage, rated frequency and below its nameplate ambient; the service-factor article in this library covers why those conditions are not optional, and this motor sits in a warm plenum.

At the belts. The effective tension the belts must carry is torque divided by sheave radius, and it is the same tension throughout the span. Check it from the motor side, where nothing geometric changed: motor speed is unchanged and motor power rose 49 percent, so motor torque rose 49 percent, and with the driver sheave diameter unchanged the effective tension rose about 49 percent. Checking from the fan side gives the same answer: fan torque up 1.31 divided by a sheave radius that shrank to 0.875 of its former size gives 1.31 divided by 0.875, which is 1.49. The two routes agree, which is the check worth doing.

Two more geometric effects arrive with that same smaller sheave, and neither is in any fan law:

• Bending stress per pass rises on the smaller driven sheave. Every belt section carries a minimum recommended datum diameter below which life falls off sharply, and that minimum lives in the manufacturer's table, not in a rule of thumb. Check the new diameter against it before ordering.
• Arc of contact on the smaller sheave falls, because the diameter difference between the two sheaves grew at the same center distance. The drive's transmittable power for a given tension falls with it, and the manufacturer's selection tables apply an arc-of-contact correction factor for exactly this.

Belt speed, incidentally, is unchanged, because it is set by the driver sheave and that one did not move. That is one limit this change did not touch.

At the fan bearings. On a belt-driven fan the shaft bearing load is normally dominated by belt tension, so a 49 percent tension increase is close to a 49 percent load increase at those bearings. Using the ISO 281 basic rating life relationship for a ball bearing, where life in revolutions goes as the load ratio cubed at constant load and speed under catalog conditions, the life multiplier is 1 divided by 1.49 cubed. That is 1 divided by 3.33, about 0.30: roughly 30 percent of the previous rating life, and that is before accounting for the 14 percent higher revolution rate, which shortens the hours further. If those bearings carry an unusually heavy wheel or high aerodynamic thrust, tension is not the dominant term and this estimate is too pessimistic; the load-path method for checking that is covered in the bearing-load article.

What the job actually delivered: 14 percent more air, half again the shaft power, half again the belt tension, a motor pushed past its rating, a sheave that may be under its section's minimum diameter, a reduced arc of contact, and a fan bearing with roughly a third of the rating life it had that morning. Every one of those is a direct consequence of the one number the customer asked for.

What would change the answer

If the speed change came from a variable-frequency drive instead of a sheave swap, the three fan relationships apply identically, so the 49 percent power increase is the same. But the sheave diameters, the arc of contact, the bending radius and the installed belt tension are all untouched, so the belt and bearing penalties that came from the geometry do not appear. When a drive is already close to a limit, that difference alone is a reason to prefer speed control over a sheave change, and it is worth saying to the customer before the sheave is ordered.

If the air is not at the density the selection assumed, both the pressure and the power move with it. A fan selected on warm air and then started on a cold morning is the classic case: density varies inversely with absolute temperature at constant pressure, so air at 70 F against air at 0 F, using the 460 offset to Rankine which is close enough at these temperatures, gives 530 divided by 460, about 1.15. Fifteen percent more shaft power on a cold start, at the same speed, against the same system, for no reason the operator can see. That is the mechanism behind cold-weather trips on fans that ran all summer. At elevation the effect runs the other way: around 5,000 to 6,000 feet the density is roughly four fifths of sea-level standard, so the same fan at the same speed moves the same volume, develops about four fifths the pressure and asks for about four fifths the shaft power.

How to verify you got this right

• State which case you are in before you calculate. Did the speed change with the system unchanged, or did the system change with the speed unchanged? One is a fan law, one is a curve read, and there is no third option that lets you cube a flow ratio.
• Compute the effective tension from both sheaves and confirm the two agree. Disagreement means you mixed up which quantity was held constant.
• Check the new sheave against the belt section's minimum diameter and the arc-of-contact correction before you order it, not after the belts are on.
• Confirm the motor's new load against rating, and confirm the service factor's conditions are actually met before you count on it.
• Verify with paired readings after the change: flow and pressure together, plus running current. Flow alone confirms only that the sheave is smaller.

References

• AMCA published fan performance and rating practice, for fan curves, fan-law conditions and the differing power characteristics of centrifugal and axial fan types
• ISO 281, rolling bearing dynamic load ratings and rating life, for the load-to-life relationship and the conditions it is defined under
• 29 CFR 1910.147, control of hazardous energy, for isolation and stored rotational energy on fan and drive work
• 29 CFR 1910.333(b)(2), electrically safe work practices for energized current readings; NFPA 70E-2021 for arc-flash risk assessment and PPE
• Manufacturer drive-selection data for minimum sheave diameters, arc-of-contact correction factors and force-deflection tension specifications
• See related: What a Service Factor Actually Buys You; How Load and Life Relate on a Rotating Component; What a Bearing Load Actually Is