Blower Noise and the Fan Laws: Why Speed Is the Loud Number
Sound power scales with tip speed to the fifth: 50 log of the speed ratio, worked examples, sound power versus pressure.
Technical note
Fan laws, without the hand-waving
Three proportionalities explain most conversations about speed change, most of the energy bill, and most of the noise complaints. Here they are with the numbers carried through, including the part everyone forgets: absorbed power goes as the cube.
The laws
What the fan laws say
The affinity laws describe one geometrically fixed wheel moving one gas. Change the rotational speed and everything else follows from three proportionalities. They are exact for a dynamically similar machine, and close enough to be the right first calculation on any industrial fan.
- Airflow
- Q is proportional to speed (N)
- Pressure
- Total or static pressure is proportional to density times speed squared
- Shaft power
- Power is proportional to density times speed cubed
- Diameter swap
- At the same speed, Q goes as D cubed, pressure as D squared, power as D to the fifth
Worked example
Ten per cent more airflow at a cement mill
A bag filter fan is delivering 60,000 m3/hr against 1,800 mmWC and the plant wants 10 per cent more air to clear a filter that is blinding earlier than the maintenance plan allows. The wheel is the same wheel, the gas is the same gas, so only speed moves.
- Speed ratio
- 1.10
- New airflow
- 60,000 x 1.10 = 66,000 m3/hr
- Pressure at the same system
- 1,800 x 1.10 squared = 2,178 mmWC, which is what the fan now makes at the old flow – the system curve is steeper still
- Power
- 1.10 cubed = 1.331, so 33 per cent more absorbed power
- Noise
- 50 x log(1.10) = about 2 dB on sound power
- Bearing load
- Rising with the cube, and this is the number that bites
On the actual system curve, static pressure rises roughly with the square of airflow because most of the resistance is duct and filter loss, so the fan climbs to a point higher than 2,178 mmWC on the old curve and the power rise is at least the cubic one. Three things have to be checked before the speed change is authorised: the motor nameplate, the belt and bearing catalogue, and the critical speed margin. We have seen all three fail a tidy-looking 10 per cent.
The density form
Constant flow, changing gas
There is a second form of the laws that matters more on hot duty. Keep volume flow constant and the pressure a fan produces is simply proportional to gas density, and so is the power needed to produce it. Efficiency is a property of the wheel, so it barely moves.
- At constant Q and N
- Pressure is proportional to density, power is proportional to density
- At constant Q and constant required pressure
- Speed must rise as the square root of the density ratio
- Then power
- Rises as the square root of the density ratio, not the ratio itself
Work the second line through. Standard air is 1.2 kg/m3. Flue gas at 650 C at atmospheric pressure is 101,325 divided by 287.05 x 923.15, which is about 0.38 kg/m3. A machine that makes its pressure on cold air has to run sqrt(1.2 / 0.38), roughly 1.77 times faster, to make the same pressure on that gas – and the absorbed power at the duty point rises by the same factor of about 1.77. Not 3.14, and not 0.32 either. Getting this square-root right is the difference between a motor that runs warm and a motor that never starts.
Limits
Where the laws stop being true
They assume the same wheel
Inlet cones, inlet boxes, silencers and a plenum chamber all change what a machine does at a given speed. A diameter change is not a speed change: power goes as D to the fifth, which is why substituting one frame size for another is a works decision, not a site decision.
They assume similar Reynolds and Mach conditions
On a small wheel at high speed, or a very large one at low speed, the assumption degrades. For the sizes we build the error is small compared with duct leakage, but it is not zero and it is not measured unless a test is called for.
They assume one gas
Compressibility is negligible below roughly 30 kPa rise. Above that, on high pressure blower duty, the isothermal and adiabatic corrections matter and the published curve has to be read as total pressure with the correct density basis.
Practical
What we do with this on an enquiry
When a duty arrives with a speed change already in it, we re-scale to standard air, select on the family curve, then re-scale back to the gas and check absorbed power at the far end of the curve rather than at the design point. If the motor only fits at the design point, it does not fit.
The four numbers we ask for
- – Volume airflow, and at which density basis it was measured
- Static or total pressure, and whether the system loss was calculated or observed
- Gas temperature and composition
- Whether the machine is to run at fixed speed or be modulated
Questions
Fan law questions
Is the fan law the same as the affinity law?
Yes, in fan practice the two names describe the same set of proportionalities. Affinity laws is the older term carried over from turbomachinery; fan laws is what the ventilation world calls them. Both assume one geometric family and one gas.
If I double the speed, do I double the pressure?
No. Double the speed doubles airflow and quadruples pressure, and demands eight times the power. Tripling speed means nine times the pressure and twenty-seven times the power, which is why speed-up projects fail on the motor long before they fail on the wheel.
Does a variable frequency drive save the energy the fan laws promise?
A drive gives you the cubic saving on the shaft, but only against a system that does not need the pressure. If a damper was open and the duty genuinely needs less air at the same static pressure, the fan is now working further along a flatter part of its curve and the saving is smaller than the arithmetic. VFD retrofits on blowers should be selected, not estimated.
Enquiry
Send us the speed change before you make it
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