Fan Laws
It may seem like heresy to many fan engineers to question the validity of the socalled “Fan Laws”. They are in fact approximations albeit, in many well defined situations, very close approximations. As they are so widely used without query or comment, it seems appropriate to look at their derivation.
When considering the performance of a series of fans, it is apparent that they can be made in a geometrically similar range of sizes and that they can be run at an infinite number of rotational speeds. They can also handle gases or air having varying physical properties — temperature, humidity, density, viscosity, and specific heats. For the manufacturer to test under all these varying conditions would be impossible and it is therefore desirable to be able to predict the performance of one fan in a series from tests made on another, perhaps with a variation also in speed and gas conditions.
To develop the Fan Laws requires that we appreciate the concept of similarity and recognize its limitations.
In geometry, we are aware that similar triangles have equal angles and the lengths of sides are in proportion. From this we are able to develop three complementary types of similarity:
• Geometric similarity in which two units have length dimensions in a constant ratio throughout and equivalent angles are equal.
• Dynamic similarity in which acceleration is introduced and the forces at corresponding points in the two machines also bear a constant relationship.
Whilst it might be thought that geometric similarity would be easy to achieve, it should be remembered that if strict adherence is necessary then this would require that metal thicknesses would have to be proportional, along with clearances, weld dimensions, fasteners etc. The exigencies of manufacturing methods and the commercial availability of the required elements dictate that this cannot be the case.
Surface roughness would also need to be proportional with size. Sheet metal roughness is almost constant over a range of thicknesses whilst welding protuberances etc., may well be a function of operator skill and quality control. Shaft diameters and the scantlings of impellers and other items are determined by the mechanical loads imposed such as centrifugal stresses, critical speeds, and fatigue stresses. This may result in the dimensions of such rotating parts diverging from those calculated by strict geometrical similarity.
Fortunately the effect of these differences is usually small and can be ignored in all but the most extreme cases. The relative
Critical dimensions 
% 
Impeller 

Blade tip diameter 
±0.25 
Blade heel diameter 
±0.25 
Blade chord & width 
±0.2 
Blade profile (deviation from template) 
±0.2 
Rim inlet diameter — formed 
±1.0 
Rim inlet diameter — machined 
±1.0 
Rim inlet curvature (deviation from template) 
± 1.0 
Peripheral runout 
±1.0 
Inlet 

Throat curvature (deviation from template) 
± 1.0 
Inlet/impeller rim clearance when running[1] 
±20.0 
Inlet/Impeller setting when running* 
± 10.0 
Housing, inlet box(es), and all accessories 
±0.4 
* Expressed as a percentage of actual clearances 
Table 4.2 Permissible divergences from strict geometrical similarity for a centrifugal fan 
Critical dimensions 
Pitch design 

% 
% 

Impeller 
Fixed 
Variable 
Blade tip diameter 
±0.25 
+ 0.125 0.25 
Hub diameter 
± 0.375 
±0.125 
Blade chord length 
±0.1 
±0.1 
Blade profile 
±0.1 
±0.1 
Blade angle of twist 
+ 2.0° 
± 1.5 ° 
Blade angular setting 
±0.1 ° 
±0.5° 
Blade tip clearance when running* 
±20.0 
±20.0 
Casing 

Impeller casing 
±0.2 
±0.2 
Inlet box, inlet bell and discharge casing 
±0.4 
±0.4 
Angular setting guide vanes 
±2.0° 
±2.0 ° 
Axial setting of guide vanes 
±0.2 
±0.2 
Accessories 
±0.4 
±0.4 
* Expressed as a percentage of actual clearances 
Table 4.3 Permissible divergences from strict geometrical similarity for an axial fan 



















Clearances between different parts of the fan can also vary but these may be of great importance and should be eliminated by both careful design and by quality control at the manufacturing stage.
Figures 4.23 and 4.24 give the terminology and show those dimensions which are critical. These, together with Tables 4.2 and 4.3 have been abstracted from AMCA 802. They give recommendations for maximum divergences of these critical dimensions from strict geometrical similarity without invalidating the “Fan Laws” used in performance prediction, within the stated uncertainties of the method.
One of the requirements of dynamic similarity is that Reynolds numbers be equal at all corresponding points in the two fans — model and predicted. Differing crosssectional areas within the impeller blade passages and into and out of the casing, dictate that Reynolds number vary considerably. It is, therefore, both customary and convenient to refer to a single arbitrary figure based on the impeller tip diameter D and the peripheral velocity at this point tcND together with the air or gas properties at the fan inlet — mass density p and viscosity p.
Thus fan Reynolds number ReF = ^ nND2
Changes in ReF can be the result of varying N or D or both. By altering only N, any size effects that might accompany a change of D can be eliminated. Tests by Phelan suggest that there is a threshold limit for ReF for each and every fan design below which increasing deviations from the fan aerodynamic laws occur.
The approximate threshold limits for various designs are given in Table 4.4. It will be noted that the lowest limiting value is for the paddle fan where, due to its simple design, flow is highly turbulent throughout the flow passages. More sophisticated designs have higher threshold values indicating that flow is in the transitional region, until speeds are reached at which most of the passages are hydraulically rough. Shock losses follow the Fan Laws and are independent of Reynolds number but are less with the increasingly efficient designs.
Type of Fan 
Impeller design 
Rep Threshold Fan Reynolds number 
Centrifugal 
Radial 
0.4 x 106 
Forward curved 
0.8 x 106 

Backward inclined 
1.0×10® 

Backward curved 
1.5 x 10® 

Backward aerofoil 
2.0×10® 

Mixed flow 
Compound curvature 
2.0 x 106 
Meridional acceleration 
2.5 x 10® 

Axial 
High hub/tip ratio 
2.5 x 10® 
Low hub/tip ratio 
3.0 x 10® 
Table 4.4 Approximate threshold fan Reynolds numbers for different types of fan 
For dynamic similarity Mach numbers in the test and predicted fan must be the same, which is unlikely unless they develop the same pressure. When operating at high pressures, above say 2.0 kPa, the air or gas may no longer be considered incompressible and a compressibility coefficient has to be introduced into the simplified form of the Fan Laws. This coefficient is a function of the polytropic exponent n and the absolute pressures at fan inlet and outlet.
Imation of what actually happens inside the fan. It is, however, adequate for predictive purposes.
To simplify any analysis, it is again convenient to specify a single fan Mach number based on the peripheral velocity of the impeller blade tips when compared with the speed of sound C as defined by the air or gas density at the fan inlet. Thus:
,, tcND 71ND MaF = —
Where
C = speed of sound (m/s)
R = gas constant (287 J/kg.°K)
T = absolute gas temperature (°K)
From compressibility effects, variations in MaF produce no deviation from the simple fan laws unless they approach a value of around 0.3.
This value may appear lower than anticipated, but it should be recognised may well indicate a local value within the blade passages approaching 1.0. Critical conditions can then develop resulting in a “choking” effect where there is a limitation on the flowrate. It is not usually a problem unless the blade passage is highly obstructed. Figure 4.25, also abstracted from AMCA 802 gives allowable variations in MaF.
Figure 4.25 Allowable variations in fan Mach numbers 
The capacity of a fan “Q” is dependent on:
Capacity Q (m3/s)
Fan size D (m)
Fan speed N (rev/s)
Gas density p (kg/m3)
Gas viscosity n (Pa. s)
Thus:
Q ce fn (D, N, p, n)
Or
Q x Da Nb pc nd
If we assign to each of the physical properties detailed above the fundamental units of mass M, length L and time T we then have:
L3T1 ccfn (l, T1,ML"3ML1T1) 
Or 
The assumption of a polytropic process between the fan connections as defined by total pressures is in itself only an approx
Or c = — d or a = d + 3c + 3 or a = 32d or b = 1d 
ND" 
Without affect 
Rep — 
Q 
Friction factor length of duct air/gas velocity mean hydraulic depth cross — section area 
Or a = 22d or b = 2d 
Equ 4.2 
L3 T’1 ocLa Tb IVP L~3c IVf L~d T Equating indices we have:

The formula can be altered to Q oc ND3
Ing its validity as 7i is a constant, and if we note that tcND = fan tip speed u then it will be seen that the term in brackets has the
Form p — i. e. some sort of Reynolds number.
This is a dimensionless quantity. For reasonable variations in this fan Reynolds number, its effects will be small. ISO 5801 requires that the test condition is within the range 0.7 to 1.4 times the fan Reynolds number for the specified duty.
Provided that these limits are met then:
Q oc ND3 Equ 4.1
It is anticipated that this “Law” would be accurate to at least the catalogue tolerances of IS013348. In general if the test fan Reynolds number is lower than the specified fan Reynolds number, then the law will be pessimistic, whilst if the test number is higher than the duty number the results of the calculation will be optimistic.
At very “high” numbers (test and duty) i. e. above the socalled threshold number for a particular design (see Table 4.4), the effects may be ignored but the dangers of predicting the performance of a small and /or highspeed fan are apparent. These effects have been noted as being especially serious with high efficiency fans, e. g. aerofoil bladed centrifugals.
In like manner we can calculate the fan pressure (static or total).
The pressure of a fan p is dependent on the same quantities and thus :
P oc fn (D, N, p, n)
Or
P oc Da Nb pc
Pressure has the dimensions of force (mass x acceleration) per unit area and using dimensional analysis we have:
ML1 T2 oc fn (L, T1, ML3, ML1 T1)
Or
ML1 T2 oc La T b Mc L’3c Md L_d T d Equating indices we have:
M : 1 = c + d or c = 1d
Or a = 3c + d1
L : 1 = a3cd
Or
A = 33d + d1 T : 2 = — bd Thus:
P oc D22d N2d p1d
Or:
ND
P oc N2D2 p
Or:
P cc N2D2 p (7tp
Again the function in brackets is in the form of the fan Reynolds number and with the same provisos we may say that:
P ocpN2D2 Equ 4.3
The fan power absorbed W is proportional to Q x p and therefore:
P ocND3 x pN2D2
Or
PocpN3D5 Equ 4.4
Note: Capital P is for power whilst small p is for pressure.
It must be emphasised that these simplified laws apply to a specific duty point of Q, p and P. AsP oc Qxp, the efficiency of the unit will remain unchanged. When the fan is applied to a system we cannot change the speed N without altering all the quantities.
Just as fans have laws, which govern their behaviour, so have systems. The usual fan system consists of a number of fittings such as bends, grilles, transformation pieces, junctions, etc. Between these will be lengths of straight pipe or ducting.
The pressure loss in fittings, assuming a constant friction loss factor k:
Oc velocity pressure
X v^ ocQ2
As v =
Cross — sectional area In like manner the pressure loss in straight ducting fLv2
M
Where:
F
L
V
M
Of duct
Perimeter
Unfortunately the friction factor is never a constant over the complete fan characteristic. For many ventilation systems we are in the transitional zone between laminar and fully turbulent flow. The index for v may be nearer 1.8 even at the design flow rate. It will fall to 1.0 at zero flow. However, this would upset all those people who for years have been declaring that, on a given system, as Q oc v, we may say that the loss in straight ducting and fittings is also oc Q2. Thus overall p x Q2 and a system line may be plotted on the fan characteristic accordingly, see Figure 4.26. This is only strictly correct for flows varying by about 20% from design (see Chapter 5 and 6).
A change in fan speed alters the point of operation from A to B i. e. along the system curve. This is because, as previously shown in the Fan Laws, for a given fan and system Q oc N, p oc N2 and therefore p oc Q2 for the fan as well, but only if f remains constant, or nearly so. It should be repeated that this system
P =p x<^ 2 1 IN, 
N, 
02=0, x^x[^ 
3 
Fan tip speed velocity of sound 
Equ 4.14 Equ 4.15 Equ 4.16 
• Changes of Reynolds number are maintained within the limits shown. • Relative roughness of fan parts remain unchanged with variation in size. If all these effects were included in our dimensional analysis additional variables would be introduced and the mathematics complicated accordingly. The overall fan laws would then become: Q oc ND3 (ReF)a (MaF)b kpc Ad F, ‘P 
P oc N2D2 (ReF)e(MaF)r kp9 Ah 
< 025 , say (see Figure 4.21 ) 
P oc N3D5 (ReF)j (MaF)k kp1 Am where: RcpND2 
ReF fan Reynolds number: 
2 + (z — where: 
D(r — D Z = (yr YQp R = absolute pressure ratio across fan Y = ratio of specific heats (1.4 for air) R = gas constant (287 J/Kg. °K) T = absolute gas temperature (°K) A = relative roughness _ absolute roughness of component impeller diameter The calculation of r is dependent on whether the fan is ducted on the inlet and/or outlet. The velocity of sound in air at sea level and 20°C (293°K) = 344 m/s. Care must be taken to use N in rev/s in the calculation of fan Reynolds and Mach numbers. 
MaF fan Mach number = 
JiND 
VyRt _ fan tip speed velocity of sound Compressibility coefficient ^ 2 + 2 z(r — 1) 
Equ 4.5 Equ 4.6 
Q oc N i. e. 
Q2 =Q1 
P oc N2 
Law is only valid for speed changes of about 20%. Over this value the divergence in the value of f becomes too great. Thus if a fan is applied to a system and its speed is changed from N1 to N2. N2 N1 _ I Nol2 P2 ~ Pi 
N, 
Q2 =Q, x 
P oc D2 i. e. 
P2 — Pi 
P oc D5 i. e. 
An increase of 10% in fan rotational speed will therefore increase volume flow Q by 10%, pressure developed p by 21% but power absorbed P by 33%, assuming air/gas density is unchanged. Unless large motor margins over the absorbed power are available, therefore, the possibilities of increasing flow by speed increase are usually limited. At the same speed and gas density, a fan of a different size, but geometrically similar, will have a performance as given below: ,3 
In a range of fans to ISO 13351, where the size ratio averages 1.12, the approximate increase per size will therefore be 40% on capacity, 25% on pressure, and 76% on power. At the same tip speed and gas density, N1, D2 will equal N2D2 
Equ 4.8 Equ 4.9 Equ 4.10 
P>=p’i% 
Q oc D3 i. e. 


























Relative roughness should not normally be of interest except when predicting the performance of a very small fan from tests on a larger unit, or where impeller scantlings are varied substantially.
Further information on the above is given in a number of advanced textbooks, e. g Cranfield Series on Turbomachinery. It is important to note however that the exponents a, b, c, etc are peculiar to a given design of fan and probably a given duty point. Work is being carried out in many research establishments to establish them. Usually they only need to be known when it is important to achieve the duty within very close tolerances i. e. within 2%.
Approximate Reynolds numbers and absolute roughness effects are typically combined in manufacturers data. Those for a medium pressure backward inclined centrifugal fan are shown in Figure 4.27. The effect of fan Reynolds numbers on the peak static efficiency is shown in Figures 4.28.
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