Understanding the Basic Fan Laws

Geoff Edwards, technical product engineer at Axair Fans UK Limited, explains the three basic fan laws when applied to warehouse ventilation studies.
The Fan Laws are a group of useful equations for determining the effects of a change in the speed, the diameter of the fan and the density of air in the system. They are most useful for determining the imapct of extrapolating from a known fan performa ce to a desired performance. So in short the basic fan laws are used to express the relationship between fan performance and power.
To start we will consider only the effect of change in the speed of the fan on the flow rate, pressure and power consumption. We will assume that the fan size and air density are to remain constant.
The first three derivations of the Fan Laws are predicated on a couple of assumptions:
·         That there is not an extreme difference in the change of rotational speed of the impeller in question and as such creating significant differences in the density of the air. However, it is unlikely that this would be a problem. You will not be looking at situations beyond the design speed of the impeller. Ignoring special applications, the upper limit for the RPM will be approximately 3600 (60hz supply frequency)
·         That there is no change in the diameter of the fan
The First Fan laws: Volume of Air
The first law of fans is a useful tool when working out the volumetric flow rate supplied by a fan under speed control or conversely working out what the RPM would be to deliver a required volume of air and hence what frequency to set a variable speed drive (VSD) to.
Volumetric flow rate (V, m³/hr) varies directly proportional to the ratio of the rotational speed (RPM) of the impeller.
Volume of Air Example – Industrial Warehouse, Process Machinery.
A Factory of 37500m3 space currently requires five air changes an hour to remove waste heat generated by industrial process machinery. Later additional machines are added to the factory and the required number of air changes per hour increases to 6.1 to maintain the desired maximum air temperature with in the factory. The original air flow rate, V1 is 187500 m3/hr to achieve this. At a pressure loss of 40Pa due to duct work, louvers and other ancillary items. 20 number 630mm 6 pole short cased external rotor motor fans were used. From the manufacturers data sheet we know that to deliver this performance, the RPM (U1) of the fan is 865u/min. V2, calculated by multiplying the space by the new air change requirements, is simply 37500m3 x 6.1 which give a new requirement of 228750 m3/hr. So what is the RPM of the fan required to be to deliver this flow rate increase?
All that then needs to be determined is when under VSD control whether the motor and or impeller is rated to operate at 1055.3u/min. If they are, great, if not either a compromise on the system performance needs to be reached, additional fans added or a reduction in the system pressure needs to be made.
The Second Fan Law: Pressure
This second law describes the relationship between the pressure developed by the fan and its rotational speed. From this equation we can see just how powerful the effect of increasing the rotational speed of the fan is on pressure development, double the speed and you quadruple the pressure development.
Pressure (P, Pa) varies as the square to the ratio of the rotational speed (RPM, u/min) of the impeller.
Eq. 2
Pressure Example
Continuing with our first situation of the industrial process factory which has added machinery and now requires additional air flow to maintain working conditions what will the pressure development of the fans now be?
This derivation of the first of the Fan Laws is predicated on a couple of assumptions:
The Third Fan Law: Power
The third law provides the required power to deliver the performance change that the system designer is looking for. The cubic nature of this relationship between power and the rotational speed shows how even for small performance gains, large amounts of additional power are needed.
Power (P, kW) varies as the square to the ratio of the rotational speed (RPM, u/min) of the impeller.
Power Example
If we continue to look at the situation of the expanding factory as we have done with the previous two examples we can see the effect of the additional air flow on the power consumption of the fan. From the original duty point we know that the power consumption was 2.12kW at 18750m3/hr @ 40Pa. so what will the total additional power consumption be for all 20 fans?
Over 20 fans the total power increase is 34.6kW! So for an air flow increase of a little over 18% the power needed has risen by nearly 45%.
In summary fan laws are essentially about impellers and what happens to their characteristics when they undergo changes in rotational speed, air density, or are scaled in size. They also help with the understanding of ventilation systems and the relationship between volume air flow rate and system total pressure. Although there are many fan selection softwares available in the market place, it is necessary for engineers to have at least a basic understanding of these basic fan laws to aid their overall awareness of how changes within ventilation systems can influence performance.