Steve Back
New York Blower Company
Willowbrook, Illinois
The purpose of this article is to explain the basis and application of rules used to “estimate” fan performance for a fan installed in a duct system moving air when the speed of the fan is changed. With a basic understanding of these rules, the performance of a fan can be quickly calculated for various speed conditions.
System Requirements
The rules governing fan performance are derived from the affinity (similarity) laws, which are also known as the “fan laws” when expressing the performance relationship for fans and “pump laws” when doing the same for pumps. To calculate precise changes in fan flow, static pressure and power requirement with a speed change to the fan, several variables such as the air density, compressibility, Reynolds* and Mach** numbers must be calculated. These variables can be time consuming to calculate. Therefore, the fan laws presented in this article have been condensed, so they may be applied to most industrial fan applications in a convenient method. The following assumptions must be true so the fan law estimations are within +/- 5% of the precise calculations:
- A system is the combination of fan(s) ductwork, hoods, filters, grills, silencers, dampers, etc., through which standard air is distributed.
- These rules are only valid within a system where there has been no change to the location, mechanical, aerodynamic or airflow design and characteristics of the system and fan.
- The system is moving standard air at a pressure so that the change in compressibility, Reynolds or Mach number is small.
- Fan static pressure shall be less than 40 in. wg (9.952 kPa) and the air density remains constant. The "in. wg" is an abbreviation for inches of water gauge.
*Reynolds Number (Re) is a dimensionless quantity in fluid dynamics used to help determine flow patterns.
**Mach Number (M) is a dimensionless quantity in fluid dynamics used to calculate the ratio of the flow velocity to the speed of sound.
Volume and Pressure - System Resistance
The movement of air through a system causes friction between the air molecules and their surroundings (duct walls, filter media, etc.). The faster the air moves, the greater the resistance to flow and the more energy is required to push or pull the air through the system.
The static pressure necessary to cause flow is proportional to the square of the volume flow. In a system, this means that static pressure will vary as the square of the change in volume flow. This is called the system resistance curve and is shown in Figure 1.
This makes it possible to predict all possible combinations of static pressure (Ps) at the corresponding volume flow (Q), given any other static pressure (Ps) and volume flow (Q) for that same system.
For example, an air duct system is calculated to require a static pressure equal to 2 in. wg (498 Pa) at a volume flow of 1000 cfm (0.47 m3/s). If it is desired to increase the flow to 1500 cfm (0.71 m3/s), the required static pressure would be:
\[ (1500 \div 1000)^2 \times 2 \,\text{in. wg} = 4.5 \,\text{in. wg} \] \[ \left( \frac{Q_{\text{new}}}{Q_{\text{current}}} \right)^2 = \frac{Ps_{\text{new}}}{Ps_{\text{Current}}} \]
This calculation will yield results so that an extension of the system resistance curve can be plotted as shown in Figure 1.
The relationship of volume flow to static pressure will not change unless the system itself is altered. (Refer to assumptions addressed in System Requirements.)
Fan Laws
In air movement systems, it is the fan that does the work. In a sense, the fan acts like a shovel. Refer to Figure 2. As it revolves, it discharges the same volume of air with each revolution. Working within a duct system, a fan will discharge the same volume of air regardless of air density. When the air being moved changes temperature, the density of the air changes.
While the fan will still move the same volume with the air temperature change, it will operate at a different static pressure and a different fan power requirement. For example, it may be necessary to reduce the air flow of hot fans via dampers when they are moving cold air so that the fan motor is not operating in an overcurrent situation.
The condensed fan laws included in this article for estimating the change in fan volume, static pressure and power requirement assume the air temperature and density of the fan does not change. The following standards may be referenced for the precise formulas of the fan laws: ANSI/AMCA 99, ANSI/AMCA 210 or ISO 5801.
If the fan speed (N) is changed, the fan will discharge a volume (Q) of air in exact proportion to the change in fan speed. This is the first “fan law.”
Fan Law 1: Volume varies in direct proportion to fan speed
\[ Q(\text{new}) = \frac{N^{(\text{new})}}{N^{(\text{old})}} \times Q(\text{existing}) \]
As shown earlier, the static pressure in a system varies as the square of the change in volume. Since volume varies directly with speed, speed can be substituted for volume in the system resistance equation. Therefore, static pressure varies as the square of the change in speed. This is the second “fan law.”
Fan Law 2: Static pressure varies in proportion to the change in speed squared N2.
\[ Ps(\text{new}) = \left( \frac{N^{(\text{new})}}{N^{(\text{existing})}} \right)^2 \times Ps(\text{existing}) \]
The power requirement of a fan is the function of its aerodynamic design and point of operation on the static pressure vs. volume fan performance curve. (See Figure 3.) Thus, the fan power requirement varies proportionally as the cube of the change in speed. This is the third “fan law.”
Fan Law 3: Power H varies in proportion to the change in speed cubed N3.
\[ H(\text{new}) = \left( \frac{N^{(\text{new})}}{N^{(\text{existing})}} \right)^3 \times H(\text{existing}) \]
It is important to remember that each of these “fan law” relationships takes place simultaneously and cannot be considered independently.
Fan Performance Curve and System Resistance Curve
The relative shape of the fan curve will not change, regardless of fan speed.
Because the fan and system only operate somewhere on their own respective curves, a fan used on an air duct system can only have one point of operation. The point of operation, as shown in Figure 3, is the intersection of the system resistance curve and the static pressure vs. fan volume curve.
If the fan speed is increased or decreased, the point of operation will move up or down the existing system resistance curve. This is shown in Figure 4.
The following example shows how to use the fan laws to calculate the new flow, static pressure and power requirement of a fan with a new fan speed.
Example: A fan was installed to deliver a volume of 35,530 cfm (16.7 m3/s) at a static pressure of 8 in. wg (1,991 Pa). The fan runs at 1230 rpm and requires 61 hp (45 kW).
After installation, the plant wants to increase the volume flow by 20%. At what speed must the fan run? What static pressure will be developed? What fan power is required?
Fan Law 1: Volume varies with speed
N (new) = Q (new) / Q (existing) x N (existing)
(16.7 m3/s × 1.20) / 16.7 m3/s × 1230 rpm = 1476 rpm **
Fan Law 2: Static pressure varies with speed squared
Ps (new) = ( N (new) / N (existing) )2 x Ps (existing)
(1476 / 1230)2 x 1991 Pa = 2867 Pa**
Fan Law 3: Fan power varies with speed cubed
H (new) = (N (new) / N (existing) ) 3 × H (existing)
(1476 / 1230)3 (61 hp) = 105 hp **
** Ensure the fan, system components and the foundation have been mechanically designed to operate at the higher conditions. The new volume flow requirement of 20% requires a 20% increase in speed, 44% increase in static pressure and a 73% increase in power requirement. The same caution must be considered for the design of the electrical components, i.e., motors, VFDs, starters, wiring, breakers, etc.
Conclusion
Use of the “fan laws” and a system resistance curve equation is based on not modifying the fan or air duct system from the existing operating condition to the new operating condition. Adding or deleting system components such as dampers, silencers or incurring density changes will create completely new system resistance curves. Changing fan accessories such as inlet boxes, evases (expanding duct), diffusors or inlet dampers, or fan temperature and density will change the fan’s performance curve. If such changes are made, the “fan laws” would no longer apply.
An in-situ (installed) fan and air duct system may be able to handle higher capacities aerodynamically, but not mechanically and electronically. If the fan and air duct system have not been designed and manufactured for higher capacities, the fan or the air duct system may experience failures. Also, a fan generates higher noise levels when the fan speed is increased. Fan and duct sound insulation (abatement) along with any airstream silencers may have to be enhanced to stay with in acceptable noise levels.
This document is New York Blower’s Engineering Letter EL-02 revised and updated by Steven F. Back, PE, for nomenclature/clarity consistency with AMCA/ANSI Standard 99, SI units and ease of use. 08/04/2025
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