HVAC
Fan laws and affinity laws field guide for HVAC air systems
Predict what a speed change does to airflow, pressure, and power, read the fan against the system curve, and size the motor so the cube law never bites you.
Direct answer
The fan laws (affinity laws) predict how a fan's airflow, pressure, and power change with speed. Airflow varies directly with RPM, static pressure with the square of RPM, and shaft power with the cube of RPM. Slow a fan 20 percent and it draws roughly half the power, which is why variable speed saves so much.
Key takeaways
- Fan laws: airflow varies directly with RPM, static pressure with the square of RPM, and shaft power with the cube of RPM.
- Cube law dominates: dropping a fan to 80 percent speed cuts power to 0.512 (about half) while still moving 80 percent of the air.
- Speeding a fan up 20 percent raises brake horsepower about 73 percent (1.2 cubed), so recalculate BHP against the motor nameplate before any speed increase.
- Rising static with falling CFM at constant speed is a restriction (dirty filter or closing damper moving the operating point up the curve), not a fan fault.
- Fan laws assume constant density; correct fan pressure and power to actual density at altitude or non-standard temperature before sizing the motor. CFM stays the same.
The fan laws, and why a small speed change is a big deal
The fan laws, also called the affinity laws, are the relationships that tell you what happens to airflow, pressure, and power when a fan's speed changes. Change the RPM and all three move, but they do not move by the same amount, and that difference is the whole point. Airflow tracks speed one for one. Pressure climbs with the square of speed. Power climbs with the cube.
On the job they answer a question you face constantly: I need more air, or less air, what will it cost me and what will it do to the rest of the system. Speed a fan up to get 10 percent more air and you have not added 10 percent to the power bill. You have added about a third. That asymmetry is why a small speed change matters in both directions.
The laws assume the fan and the duct stay the same and the air density stays the same. Change the ductwork, foul the filter, or move to altitude and you are no longer comparing the fan to itself, so the clean ratios bend. Most field mistakes with the fan laws come from applying them across a change they were never meant to cover.
What are the fan laws?
The fan laws are three relationships between fan speed and what the fan produces, written as ratios between an old condition and a new one. Speed is RPM at the fan, or the drive frequency in hertz on a VFD, which tracks RPM directly.
Airflow varies directly with speed: CFM2 = CFM1 x (RPM2 / RPM1). Double the speed, double the air.
Static pressure varies with the square of speed: SP2 = SP1 x (RPM2 / RPM1) squared. Double the speed and pressure goes up four times.
Power varies with the cube of speed: HP2 = HP1 x (RPM2 / RPM1) cubed. Double the speed and shaft power goes up eight times.
The pressure and power laws fall straight out of the first one. Pressure rises with the square of flow through a fixed system, and power is flow times pressure, so power ends up as speed times speed squared, which is speed cubed. You do not memorize three separate facts. You memorize that airflow tracks speed and power is flow times pressure, and the other two follow.
The cube law is the whole story
The cube law is the headline, because it is where the money is. Shaft power follows the cube of speed, so a small cut in speed is a large cut in power. Drop a fan to 80 percent speed and it draws 0.8 cubed, which is 0.512, about half the power, while still moving 80 percent of the air. Drop it to half speed and it draws 0.125, one eighth the power, for half the air.
That is the entire economic case for variable speed. A constant-speed fan throttled down with a damper still burns most of its full-speed power, because the motor is spinning at full RPM and the damper just wastes the pressure as heat. Slow the fan instead and the cube law goes to work. The energy you are not spending is the air you are not moving, cubed.
The cube cuts the other way too, and that is the part that bites. Speed a fan up 25 percent for more air and you have not asked the motor for 25 percent more work. You have asked for 1.25 cubed, about 95 percent more, nearly double. The motor that was fine at the old speed can be in overload at the new one.
Field example: a 10 percent speed trim
Take a fan moving 10,000 CFM at 800 RPM, pulling 1.5 in. wg of static and 7.5 brake horsepower. Slow it 10 percent to 720 RPM and run the three laws.
Airflow: 10,000 x (720 / 800) = 9,000 CFM. Ten percent less air, exactly tracking the speed.
Static pressure: 1.5 x (720 / 800) squared = 1.5 x 0.81 = 1.22 in. wg. About 19 percent less pressure.
Power: 7.5 x (720 / 800) cubed = 7.5 x 0.729 = 5.47 brake horsepower. About 27 percent less power.
A 10 percent speed trim gave back 10 percent of the air but 27 percent of the power. That is the lever every variable-air-volume system and every EC-motor retrofit is pulling. The reverse is the warning: to push that same fan from 9,000 back up to 10,000 CFM, you pay the 27 percent in power to get the 10 percent in air.
| Quantity | Law | Result at 720 RPM (from 800) |
|---|---|---|
| Airflow | Direct with speed | 10,000 to 9,000 CFM (10 percent less) |
| Static pressure | Square of speed | 1.5 to 1.22 in. wg (19 percent less) |
| Brake horsepower | Cube of speed | 7.5 to 5.47 BHP (27 percent less) |
What is the operating point?
The operating point is where the fan curve and the system curve cross, and it is the only place the fan can actually run. Everything else is a wish.
The fan curve is the manufacturer's plot of pressure against airflow for a given fan at a given speed. It slopes down: the harder the fan has to push, the less air it moves. The system curve is the ductwork's plot of how much pressure it takes to force a given airflow through that specific arrangement of duct, fittings, coil, and filter. It slopes up, because more flow takes more pressure. The two curves cross at exactly one point, and that crossing is the airflow and pressure the installed fan delivers into the installed duct.
This is the same external static pressure you read at the air handler, covered in the duct static pressure guide. The fan laws move the fan curve up or down with speed. The system curve stays put until you change the duct. Find the new crossing and you have found the new operating point.
The system curve is itself a square law
The system curve is a square law, and it is the same square that shows up in the second fan law. Pressure loss through a fixed duct rises with the square of the airflow through it: double the CFM through the same duct and the pressure to push it goes up about four times. That is why the system curve is a rising parabola, not a straight line.
This is the partner to the fan law. The fan's pressure rises with the square of speed, and the system's required pressure rises with the square of flow, and because flow tracks speed, the fan rides its own curve up and down a path that matches the system's shape. When the duct does not change, slowing the fan walks the operating point straight down the existing system curve, and the clean fan-law ratios hold.
Change the duct and you change the curve. Close a damper, load a filter, or crush a flex run and the system curve gets steeper, which is a different situation entirely and the next section.
A dirty filter moves the point up the curve
A dirty filter or a closing damper does not slow the fan. It moves the operating point up the fan curve, and the result surprises people: less air at more pressure. The added resistance steepens the system curve, the crossing point slides up and to the left along the fan curve, and the fan settles at higher static pressure and lower airflow.
This is the opposite of the fan laws, and confusing the two is a common error. The fan laws describe moving the fan curve by changing speed. A dirty filter leaves the fan curve alone and moves the system curve. Same fan, same RPM, different operating point.
You read it in the field as rising static and falling CFM at constant speed. The static climbs while the air at the registers drops off. On a PSC blower the airflow falls hard as static rises. On an ECM the motor tries to hold airflow by speeding up, so the tell is rising watts and RPM at the same delivered CFM, until the motor runs out of capacity. Either way, rising static on a constant-speed fan is a restriction, not a fan problem.
Why you cannot just turn up the fan
You cannot make a fan do something off its curve. The curve is the complete set of pressure-and-airflow combinations that fan can produce at a given speed, so if the system needs more air than the curve offers at the system's pressure, no amount of wishing gets it. Your only moves are to speed the fan up, which lifts the whole curve, or to lower the system resistance, which lets the existing curve deliver more.
This is where the fan laws and reality meet. People say turn up the fan as if airflow were a volume knob. On a belt drive there is no knob. You change the sheave to change the speed, and then the new curve has to cross the system curve where you need it. On a direct-drive ECM you can command more speed, but you are still bound by the cube law, the motor's horsepower, and the duct's resistance.
If a system is starved for air, the honest answers are a faster fan within its rating, larger or cleaner duct, or a bigger fan. Throttling harder never adds air. It only adds static and noise.
Changing speed: sheaves vs VFD
There are two ways to change a fan's speed, and they suit different jobs.
On a belt-drive fan you change speed by changing the sheaves, the pulleys on the motor and fan shafts. A smaller motor sheave or a larger fan sheave slows the fan, a larger motor sheave speeds it up. Adjustable sheaves let you dial speed in during balancing, then you lock them or swap to a fixed sheave for the final setting. The catch is that a sheave change is a one-time mechanical setting, and the cube law means a small sheave change can swing the motor amps a lot, so you clamp the amps after every change.
A variable frequency drive changes the motor's speed electrically by changing the frequency it feeds the motor, and it can change continuously while the system runs. That is what makes variable-air-volume and demand-based control possible. The VFD is also where the cube-law savings get captured automatically, because the drive slows the fan whenever full air is not needed.
For a fixed airflow set once, a sheave is simple and cheap. For airflow that needs to vary, the VFD pays for itself in the power it does not spend.
Why do VFDs save so much energy?
VFDs save energy through the cube law, plain and simple. Most air systems run at full design airflow only a few hours a year. The rest of the time the building needs less air, and a variable speed drive gives less air by slowing the fan, where the power falls with the cube of speed. A fan loafing at 70 percent speed for most of the season draws about 0.34 of full power, a two-thirds cut, for air the building does not miss.
The old way was to run the fan flat out and throttle the excess with dampers or inlet vanes. That wastes the pressure across the damper and leaves the motor near full draw. Variable-air-volume systems and EC (electronically commutated) motors take the other path: move only the air that is needed and spend only the cubed power that air costs.
This is the energy half of test and balance, covered in the air balancing report guide. Setting the fan to design and then letting controls back it off when load drops is where the modeled savings actually show up. A system balanced to design but stuck at full speed never collects the cube-law dividend.
The pump affinity laws are the same idea
The water side follows the same laws, because the affinity laws are not specific to air. A centrifugal pump obeys the identical relationships: flow tracks speed, head rises with the square of speed, and power rises with the cube. GPM2 = GPM1 x (RPM2 / RPM1), head with the square of the ratio, brake horsepower with the cube.
That is why variable speed pumping saves the same way variable speed fans do, and why a chilled-water or hydronic system designed for variable flow leans on the cube law to cut pump energy at part load. A primary-secondary loop or a variable-primary system slows the pumps as the building's flow demand drops, and the pump power falls off a cliff in the good direction.
The system curve idea carries over too. A piping system has its own square-law curve, pump head rises with the square of flow through fixed pipe, and the pump rides its curve to the operating point where it crosses the system curve. If you balance air, the pump side feels familiar, because it is the same physics with water in the pipe instead of air in the duct.
How do you tell where a fan is running?
To tell where a fan is on its curve, read its speed and its current and bring those to the manufacturer's data. A tachometer or strobe gives RPM at the fan shaft, which on a belt drive is not the motor RPM, so you account for the sheave ratio. A clamp meter gives motor amps, which stand in for power once you know the voltage and a reasonable motor power factor and efficiency.
Speed plus amps puts you on the fan's published curves. The fan tables list, for a given RPM and static pressure, the airflow and brake horsepower. Read your RPM, read your static at the unit, and the table tells you roughly the CFM and the power the fan should be making. If your measured amps translate to far more power than the table predicts for that point, something does not match: the static is higher than you think, the speed is off, or the reading is wrong.
This is triage, not a flow measurement. To prove airflow you traverse the duct or read it at the outlets, which is the balancing guide's job. But RPM and amps against the fan curve get you close enough to know whether the fan is where the design put it or somewhere it should not be.
One field shortcut: if you know the fan was at design once, the fan laws let you scale from that baseline instead of reading the curve cold. Today's RPM divided by the baseline RPM, cubed, times the baseline brake horsepower, tells you the power now. The catch is that it only holds if the duct has not changed, so check the static first.
Selecting a fan near its best efficiency, not far out on the curve
A fan should be selected to run near the best efficiency point of its curve, not far out on either end. The curve has a sweet spot, a region where the fan turns the most shaft power into air and pressure and the least into noise and heat. Pick a fan whose design operating point lands in that region and it runs quiet, cool, and cheap. Pick one running far to the right, near free delivery, or far to the left, near shutoff, and you pay for it.
Two efficiencies show up on the data. Total efficiency uses the fan's total pressure, static plus velocity. Static efficiency uses static pressure alone and is the lower, more conservative number. Know which one the manufacturer is quoting before you compare fans, because comparing a total-efficiency number to a static one flatters the wrong fan.
The far-left end of the curve is the dangerous one. Many fans, especially backward-inclined and airfoil centrifugals, have a surge or stall region near the peak of the pressure curve where flow becomes unstable and the fan pulses, vibrates, and can damage itself. The manufacturer marks the do-not-select zone. Stay out of it. A fan oversized and then choked down with a damper can be driven into that region, which is one more reason oversizing and throttling is a bad habit.
Density correction: the laws assume the air stays the same
The fan laws assume the air density does not change, and that assumption breaks at altitude and at temperature. A fan is a constant-volume machine: it moves the same CFM regardless of density, but the pressure it develops and the power it draws both scale with density. Thinner air means less pressure and less power for the same RPM and CFM.
At altitude the air is less dense, so a fan at 5,000 ft develops less static and draws less power than the same fan at sea level, even at the same speed and airflow. The same goes for hot air: a fan handling 600 degree F process exhaust pulls far less power than the catalog point at standard conditions, because hot air is light. The fan tables are published at standard air, commonly taken as 0.075 lb per cubic foot at sea level and 70 degree F.
This ties to psychrometrics, because density depends on temperature, altitude, and moisture. The practical rule: correct the fan's pressure and power to actual density before you size the motor, or you undersize for cold dense air and oversize for hot thin air. The CFM stays put. The pressure and the horsepower do not.
Brake horsepower and motor sizing: the cube cuts both ways
Brake horsepower is the power the fan actually demands at the shaft, and the cube law governs how it changes with speed. This is where the fan laws turn into a motor decision, because the motor has to deliver the brake horsepower at the operating point without running into overload.
The trap is speeding a fan up. A sheave change or a VFD bump that raises speed 20 percent raises brake horsepower by 1.2 cubed, about 73 percent. The 5 horsepower motor that was comfortable can be pulling well past its rating at the new speed, tripping overloads or cooking slowly. Before any speed increase, calculate the new brake horsepower with the cube law and check it against the motor nameplate and the service factor.
Motors are commonly selected so the brake horsepower at the worst-case operating point stays at or below the nameplate rating, with the service factor as margin, not as the plan. Sizing the motor for non-overloading across the whole fan curve is the conservative habit, because a fan can wander to a higher-power point if the system resistance comes in lower than design. The cube law is unforgiving here. A little more speed is a lot more load.
Commissioning use: set design CFM, record RPM and amps
The clean way to commission a fan is to set it to deliver the design airflow, then record the RPM, amps, and power that produced it. Design CFM is the target. The speed and the draw are the evidence you hit it and the baseline anyone can check later.
Adjust the fan speed, by sheave or VFD, until the measured airflow matches design at the operating static. Then write down the final RPM, the motor amps on each phase, the static across the unit, and the brake horsepower those imply. That set of numbers is the fan's fingerprint. A year later, if airflow complaints come in, the tech compares today's RPM and amps to the commissioning record and knows in minutes whether the fan drifted, the duct fouled, or nothing changed.
Set total air first, then balance the branches, which is the sequence the balancing guide walks through. The fan-law numbers belong in that report. They are what lets the next person tell a fan problem from a duct problem without re-deriving the whole job.
Fan walls and EC fan arrays
Fan walls and EC fan arrays are the fan laws applied many times over. Instead of one large fan, an array of small direct-drive EC plug fans shares the load, and the controls run only as many as the airflow needs at the speed it needs. Because each fan rides the cube law, dropping array speed at part load saves the same way a single VFD fan does, and staging fans off saves further.
Data centers pushed this hard, because they run enormous, steady airflow and the energy is the whole game. An array also gives redundancy: lose one fan in a wall of twenty and you lose a slice of capacity, not the unit, and the survivors speed up a little to cover, paying a small cube-law power increase for it. The thermal-management standards for data centers, the ASHRAE TC 9.9 guidance among them, set the temperature envelope these systems hold, and the fan array is how the airflow gets trimmed to hold it efficiently.
The math is the same fan laws you use on a single rooftop unit. There are just more fans sharing the curve, and a controller deciding how many run and how fast.
What to document
Write down the speed change and what the laws say it did, so the next person can check your work without re-deriving it. A fan-law change is three numbers off one ratio, and the value of recording it is that anyone can reproduce it.
Capture the old and new RPM or frequency, the speed ratio, and the resulting airflow, static, and brake horsepower with the law that produced each. Note the motor amps before and after, because amps are what catch a cube-law overload. If you changed a sheave, record the old and new sheave and the belt. If you set a VFD, record the final hertz and the speed reference. A speed change touches air, pressure, and power at once, and the record has to show all three or it does not tell the story.
| Parameter | Law | Ratio applied |
|---|---|---|
| Airflow (CFM) | Varies directly with speed | CFM2 = CFM1 x (RPM2 / RPM1) |
| Static pressure | Varies with square of speed | SP2 = SP1 x (RPM2 / RPM1) squared |
| Brake horsepower | Varies with cube of speed | HP2 = HP1 x (RPM2 / RPM1) cubed |
| Motor amps | Track power, check after change | Rises roughly with the cube of speed |
| Air density | Pressure and power scale with it | Correct to actual density before sizing |
Common mistakes
- Speeding a fan up without recalculating brake horsepower, then overloading the motor on the cube law.
- Applying the fan laws across a duct change, when they only hold for the same system at the same density.
- Reading rising static and falling airflow as a fan problem, when a closing damper or dirty filter moved the operating point up the curve.
- Oversizing the fan and throttling it down with a damper instead of slowing it, throwing away the cube-law savings as heat and noise.
- Selecting a fan far out on the curve or inside the surge region instead of near its best efficiency point.
- Forgetting the density correction at altitude or on hot air, so the motor is sized for the wrong power.
- Changing a sheave during balancing and never clamping the amps afterward.
- Comparing a total-efficiency number on one fan to a static-efficiency number on another.
Field checklist
Want this checklist to run itself on every job — with photo proof and a signed record crews can hand the customer? That's FieldOS.
Standards and references
The fan laws themselves are physics, exact ratios that hold as long as the fan, the system, and the density stay constant, so no code grants or limits them. What the standards govern is how fan performance gets rated and how the air system gets designed and tested around the laws.
AMCA, the Air Movement and Control Association, publishes the ratings and test methods behind the fan curves you read, so a fan rated to AMCA standards has been tested in a defined way you can compare across manufacturers. ASHRAE covers the design and energy side, with the energy standard commonly cited as ASHRAE 90.1 pushing variable-speed control and fan-power limits that exist because of the cube law. For test and balance, the NEBB and AABC procedures govern how the fan gets set and the airflow proven in the field.
The fan curve and the rating tables come from the manufacturer, and they are the controlling document for any specific fan. Selection efficiencies, the surge region, the do-not-select zones, and the exact brake horsepower at a point are manufacturer data, so confirm them against the current published curve for the model, at the right speed and density, before you commit a motor or a sheave.
Units, terms, and conversions
Fan performance shows up in a few units and a few names, and they cross between the air side and the water side.
Airflow is CFM (cubic feet per minute) on the air side and GPM (gallons per minute) on the water side, or cubic meters per second and liters per second in metric. Pressure is inches of water column (in. wg) for fans and feet of head or psi for pumps. Power is brake horsepower at the shaft or kilowatts at the motor. Speed is RPM at the shaft or hertz at a VFD, which track each other on a direct drive. Density is in pounds per cubic foot, referenced to standard air at about 0.075 lb per cubic foot.
- Fan laws / affinity laws
- The ratios linking fan or pump speed to airflow, pressure, and power
- Operating point
- Where the fan curve and the system curve cross, the airflow and pressure the fan actually delivers
- Fan curve
- The manufacturer's plot of pressure against airflow for a fan at a given speed
- System curve
- The duct's square-law plot of pressure required against airflow through that arrangement
- Brake horsepower (BHP)
- The power the fan demands at the shaft, varying with the cube of speed
- VFD
- Variable frequency drive, which changes motor speed electrically by changing the frequency
- ECM
- Electronically commutated motor, a variable-speed motor common in modern air handlers
- Standard air
- Reference density of about 0.075 lb per cubic foot at sea level and 70 degree F
FAQ
What are the three fan laws?
The three fan laws relate speed to output. Airflow varies directly with speed: CFM2 = CFM1 x (RPM2 / RPM1). Static pressure varies with the square of the speed ratio. Power varies with the cube of the speed ratio. The pressure and power laws both follow from the airflow law and the fact that power is flow times pressure.
Why do VFDs save so much energy?
VFDs save energy through the cube law. Shaft power falls with the cube of speed, so slowing a fan to 70 percent speed cuts its power to about a third. Most systems need full airflow only a few hours a year, so a drive that trims speed at part load collects large savings the rest of the time.
What is the operating point of a fan?
The operating point is where the fan curve crosses the system curve, and it is the only airflow and pressure the installed fan can deliver into the installed duct. The fan curve slopes down, the system curve slopes up, and they meet at one point. Change speed or change the duct resistance to move it.
How does fan speed affect power?
Fan power changes with the cube of speed. Raise speed 20 percent and power rises about 73 percent (1.2 cubed). Cut speed 20 percent and power drops to about half (0.8 cubed). Airflow only tracks speed one for one, so small speed changes move power far more than they move air.
Why is my airflow dropping while static pressure rises?
Rising static with falling airflow at constant speed is a restriction, not a fan fault. A dirty filter or closing damper steepens the system curve and slides the operating point up the fan curve, giving less air at more pressure. Clean or open the restriction. The fan never slowed down.
Can I just turn up the fan to get more air?
Only within the fan's curve and its motor rating. Speeding the fan lifts the whole curve, but the cube law raises brake horsepower fast, so check the new power against the motor nameplate first. If the fan is already near its limit, the real fixes are larger or cleaner duct or a bigger fan.
Do pumps follow the fan laws?
Yes. The affinity laws apply to centrifugal pumps the same way. Flow tracks speed, head rises with the square of speed, and power rises with the cube. That is why variable-speed pumping on chilled-water and hydronic systems saves energy at part load exactly like a variable-speed fan does on the air side.
What is the difference between a sheave change and a VFD?
A sheave change sets a belt-drive fan's speed mechanically, once, and is cheap for a fixed airflow. A VFD changes motor speed electrically and can vary it continuously while running, which is what variable-air-volume needs. After either change, clamp the motor amps, because the cube law swings power fast with small speed changes.
How do altitude and temperature change fan power?
A fan moves the same CFM regardless of density, but its pressure and power both scale with density. Thin air at altitude or hot process air develops less static and draws less power than the catalog point at standard air. Correct pressure and brake horsepower to actual density before sizing the motor.
How do I tell where a fan is running on its curve?
Read fan RPM with a tach and motor amps with a clamp meter, then bring both to the manufacturer's fan curve. RPM and static pressure place you on the published table, which gives the expected CFM and brake horsepower. It is triage, not a flow measurement; traverse the duct to prove airflow.
People also ask
Codes cited in this guide
This guide is written and reviewed against the published standards below. Always confirm the current adopted edition with the authority having jurisdiction.