Pulley Ratios and the Fan Laws by Hand

Why this matters

A belt-drive blower's speed is set by two diameters and a motor speed, and its power draw by the cube of that speed. That pairing is why a sheave adjustment that looks like a small tweak on a pad of paper can overload a motor. Speed up a blower 10 percent and it moves about 10 percent more air, pushes about 21 percent more pressure, and asks the motor for about 33 percent more power. This card is the arithmetic for both halves, done by hand with units on every line: the drive ratio that sets the speed, the fan laws that say what the speed costs, and the checks that catch an inverted ratio or a power figure nobody multiplied out. Why the drive trades speed for torque, and the full conditions on the cube law, are owned by How Sheave Diameter Sets Speed and Torque.

Lock it out before you measure or touch a sheave

Measuring pitch diameters, reading a sheave's setting or changing it means hands near the belt. Open and lock the motor's disconnect, tag it, and prove the motor circuit dead with a tester checked on a known live source before and after, with electrical work practices per 29 CFR 1910.333(b)(2) and live-dead-live per NFPA 70E Article 120 (120.5 in the 2021 edition, renumbered in 2024). The wheel itself is stored mechanical energy, controlled under 29 CFR 1910.147: wait for it to stop, confirm by eye that it has, and block it or close and secure a damper so airflow through the unit cannot windmill it while your fingers are at the belt. Motor and drive service on equipment you are not qualified for goes to the person who is.

The drive ratio

On a belt drive, the belt moves at one speed along its whole length, so the rim speed at the pitch line of both sheaves is the same. A small sheave has to turn faster to keep up with a big one. That gives the one formula:

driven RPM = driver RPM x driver pitch diameter / driven pitch diameter

The driver is the motor sheave. The driven is the blower sheave. The pitch diameter is where the belt's load-carrying cords actually ride, which is a little smaller than the sheave's outside diameter; the sheave's marking or the drive maker's tables give it. Using the outside diameter on both sheaves gets you close and, on small sheaves, measurably wrong.

BELT DRIVE, MOTOR TO BLOWER

    MOTOR                         BLOWER
  1,725 RPM                      ? RPM
   driver                        driven
   .----.                       .------.
  ( 3 in )====== belt =========( 6 in   )
   '----'                       '------'
  pitch dia 3 in               pitch dia 6 in

driven RPM = 1,725 x 3 in / 6 in = 862.5 RPM

belt speed, driver side
  3.1416 x 3 in / 12 x 1,725 = 1,355 ft/min
belt speed, driven side
  3.1416 x 6 in / 12 x 862.5 = 1,355 ft/min

The inches cancel in the ratio, so the answer is in RPM. The two belt-speed lines are the check: if the driven RPM is right, the rim speed computed from each side matches. Rough estimate: the blower sheave is twice the motor's, so the blower turns at half the motor's speed, a bit over 860 RPM. Agrees.

The slip that flips the ratio

Inverting the ratio is the common error: 1,725 x 6 / 3 = 3,450 RPM, four times the true 862.5 on this 2:1 drive. The sanity check is physical. A small sheave driving a big one slows the load down, always. If the answer comes out faster than the motor with a bigger sheave on the blower, the fraction is upside down.

The second slip is trusting the motor's nameplate speed as exact. 1,725 RPM is the speed at full rated load on a four-pole motor whose magnetic field turns at 1,800 RPM at 60 Hz. At lighter load the motor slips less and runs faster, closer to 1,800. So a lightly loaded motor turns a blower a little faster than the nameplate calculation says. For a setting that matters, measure the blower shaft with a tachometer, with guards in place and a non-contact instrument, rather than computing from the plate.

The fan laws

For a given fan running in a given system, with the same air density, three quantities follow the fan's speed in fixed proportion:

• Airflow goes with speed. 10 percent faster, 10 percent more CFM.
• Pressure goes with speed squared. 1.10 x 1.10 = 1.21, so 21 percent more.
• Power goes with speed cubed. 1.10 x 1.10 x 1.10 = 1.331, so about 33 percent more.

The conditions in that first sentence are what make it true. The same fan, the same system (nothing changed in the ductwork, the dampers, the filter or the coil), the same air density. Change the filter between the before and after readings and you have changed the system, and the laws no longer connect the two. The fuller conditions, and why a system with a large fixed-pressure component departs from them, are in How a Fan Law Changes What the Drive Has to Deliver. Centrifugal pumps follow the same three relationships, usually called the affinity laws.

Worked example: 10 percent more air

An air handler is moving too little air. The drive is the one in the diagram: a 1,725 RPM motor, 3 in driver, 6 in driven, blower at 862.5 RPM. The fan table for this blower (illustrative values) puts its absorbed power at 0.70 brake horsepower at this operating point, on a motor rated 1 hp. External static pressure reads 0.50 in WC. The goal is 10 percent more airflow.

Step 1, the new speed. 10 percent more airflow needs 10 percent more speed: 862.5 x 1.10 = 948.75 RPM.

Step 2, the sheave that gets it. Solve the drive formula for the driver: driver diameter = driven RPM x driven diameter / motor RPM = 948.75 x 6 / 1,725 = 3.30 in. Check: 1,725 x 3.30 / 6 = 948.75 RPM. A 3.3 in pitch-diameter driver, or an adjustable driver opened down to that pitch diameter.

Step 3, what the pressure does. 0.50 in WC x 1.21 = 0.605, about 0.61 in WC. The ductwork now sees about a fifth more static pressure than before.

Step 4, what the motor is asked for. 0.70 bhp x 1.331 = 0.93 bhp. Under the 1 hp rating, with less than a tenth of the motor's capacity left: 1.00 - 0.93 = 0.07 hp.

Step 5, check it against the motor, not the paper. The 0.70 bhp came from a table and the cube law is a model. After the change, restore power with the guard and access panels back on, stand clear of the belt line as the motor starts, and read running current on each leg against the nameplate full-load amps at the measured voltage. Motor current is not proportional to power at partial load, so the clamp reading checks the motor's heating limit directly instead of trying to recover horsepower from amps. Near or over nameplate current, the change is too much for this motor whatever the arithmetic said.

Now change one input. Ask for 20 percent more air instead: speed x 1.20, power x 1.20 x 1.20 x 1.20 = 1.728, so 0.70 x 1.728 = 1.21 bhp on a 1 hp motor. That is over the rating, and the motor heads for its overload or an early failure. The same drive arithmetic that gave a reasonable answer at 10 percent gives an unacceptable one at 20, and the difference is entirely in the cube. A shortfall that large usually points back at the system (a loaded filter, a closed damper, undersized return) rather than at the blower speed.

Going the other way

Slowing a blower saves power by the same cube. 10 percent slower: 0.90 x 0.90 x 0.90 = 0.729, about 27 percent less power, for 10 percent less air. That is the arithmetic behind slowing an oversized blower for quieter operation or better dehumidification, and the airflow it gives up still has to satisfy the equipment's minimum per its installation manual; a coil or heat exchanger below its minimum airflow is a different failure.

Changing the driven sheave instead

The same 948.75 RPM can come from a smaller blower sheave rather than a bigger motor sheave: driven diameter = 1,725 x 3 / 948.75 = 5.45 in. Both give the same blower speed and the same fan-law consequences, because the fan only knows its speed. They differ in the belt: the center distance and belt length change, and a smaller sheave bends the belt harder and wraps less of it. Which to change is a question of what is available and what the drive maker's tables allow for that belt section, not a question this arithmetic can settle.

Adjustable sheaves

Many motors carry an adjustable-pitch driver: the two halves of the groove screw closer or farther apart, and the belt rides higher or lower. Opening the sheave lowers the belt and shrinks the pitch diameter; closing it raises the belt and grows it. Each turn changes the pitch diameter by an amount the sheave's maker publishes, and it is rarely the same for every turn. Count the turns from fully closed, write the count on the ticket, and read the resulting pitch diameter off the maker's table rather than measuring a groove with a ruler. Every groove of a multi-groove adjustable sheave is set to the same count, or the belts share the load unevenly.

The habit

Write the drive as a line: motor RPM, driver diameter, driven diameter, result, belt speed both sides. Write the fan-law change as the three factors (speed, speed squared, speed cubed) multiplied out before anything is ordered. Then compare the power line with the motor's nameplate, and compare the result after the change with a clamp meter and a tachometer. The paper says what should happen; the meter says what did.

References

• 29 CFR 1910.147 and 29 CFR 1910.333 for isolating the drive and the motor circuit
• NFPA 70E, Article 120, current edition, for the live-dead-live proving sequence
• Belt drive manufacturers' engineering tables for pitch diameters and adjustable-sheave settings
• Fan manufacturers' performance tables for brake horsepower at an operating point
• See related: How Sheave Diameter Sets Speed and Torque; How a Fan Law Changes What the Drive Has to Deliver; Common Belts + Pulleys Reference