Mathematical tools
Most fan duties can be met by selecting an appropriate fan size from a standard range. It is just a case of choosing the best size and type from those which are available or predesigned. Theoretically, if it could be run fast enough, a single fan could meet all duties. Whilst much to be desired by managing directors, this would be at the expense of less than optimum efficiency, noise and outlet velocity. It might also be required to be constructed in exotic materials with high strength and ductility.
Fan selection begins with the customer specifying what he wants and ends with the supplier putting forward a solution, having evaluated many alternatives. Of the many types and sizes of fan that may be capable of producing the required flowrate and pressure, the best selection is the one that is the most economic in terms of capital, energy, maintenance and disposal costs (see Chapter 19). Low noise levels may also be of importance.
A customer’s fan specification should give as much information as possible regarding the required performance, design life, mechanical arrangement etc, so that the best possible solution can be put forward.
If the supplier received all the information given in Table 20.2 he should be highly delighted, but undoubtedly dumbfounded! Invariably bad selections are the result of insufficient information being available for the best selection.
It is also desirable to have a schematic drawing of the ducting to and from the fan with their sizes. This will enable the system effect factors to be evaluated. The customer needs to be aware how these have been included in his assessment. Inclusion of the duct sizes will enable any diffuser/reducer regains to be assessed.
Users should remember that virtually all information published by manufacturers is based on the fans handling standard air. Whilst this has varied by a small amount over the years it is currently specified as air having a density of 1.2 kg/m3. Reference to Chapter 4, Section 4.6 and the Fan Laws, showed that whilst
Information 
Units 

General (see Chapter 1 ) 
No. of fans Aerodynamic type Size Installation category (see Chapter 4) 

Physical data (see Chapter 9) 
No. of inlets Type of drive Arrangement No. Direction of entry for inlet boxes Rotation, discharge angle and motor position 

Gas compositions and conditions (see Chapter 2) 
Gas analysis and /or name Molecular weight or specific gravity referred to out Ambient barometric pressure Temperature at fan inlet Relative humidity Dust loading 
KPa °C % Kg dust/kg air 
Flowrate per fan at inlet conditions (see Chapter 4) 
Design MCR (maximum continuous rating) NER (normal economic rating) Minimum 
M3/s 
Fan pressure at each flowrate (see Chapter 5) 
Fan (total) pressure Total pressure at outlet Total pressure at inlet Fan velocity pressure Fan static pressure 
Pa or kPa 
Constructional Features 
Ancillaries (see Chapter 16) Special materials (see Chapter 7) Bearing type (see Chapter 10) 

Power evaluation (see Chapter 19) 
Expected (design) life Expected operation at each rating Power tariff Demand charges 
Years Hr/year Pence/kWh Ј/kW 
Motor data (see Chapter 13) 
Electrical characteristics 
Volts Phases Frequency 
Table 20.2 Information required for optimizing fan selection 
I itSE mm m» ««fl [ BPW1 
The flowrate is virtually unaffected by air/gas density (qv cc ND3), the fan pressure is directly proportional to this density. Thus if a fan is handling a hot gas with a density of 0.6 kg/m3, then the actual pressure developed will be halved.
It is a relief to know that the resistance of most systems will also be nearly halved, apart from the second order effect of a change in air/gas viscosity (see Chapter 2). Thus the volumetric flowrate in a given system will be almost unchanged, although the mass flowrate qm will of course be halved along with the power absorbed by the impeller. For selection purposes, therefore, where non ambient air is being handled, it is customary to calculate the fan “apparent” pressure (static or total) so that standard multirating tables or the various types of curve can then be used.
Thus:
Fan “apparent” pressure:
Pf. app=Pf. ac, Equ 20.2
Pact
The early history of fan catalogues
For many years the dissemination by companies of information concerning the performance of their products was shrouded in mystery. When the author entered the industry in the early 1950s, this was still very much the case. Baseline characteristic curves were kept under lock and key, with only senior engineers having access to them.
As in so many subjects, the break with the past, started in the USA. A renowned fan engineer H. F. Hagen, then with the B. F.
Figure 20.1 Hagen Chart for selection of aerofoil bladed centrifugal fans
Sturtevant Company of Massachusetts, was the first to devise an ingenious graphical method under US Patent No 1358107, (see Chapter 1). An example of such a “Hagen Chart” is shown in Figure 20.1. It will be noted that a number of such graphs were necessary to cover all the sizes in even one range.
In the UK however, Sturtevant’s associated company, Sturtevant Engineering Company Ltd, continued with the use of multirating tables. Again, an example is shown as Table 20.3, albeit updated for modern SI units.
The actual data from which these tables were produced was in fact surprisingly small. A small number of original test points were simply magnified by the use of the Fan Laws. (See Chapter 4.) Thus the figures at one particular pressure could be calculated for another pressure.
PF oc N2D2 at constant gas density Equ 20.3
Or pF oc(7tND)2 introducing constant n
OrpF oc peripheral speed2 In like manner:
Qv °cND3
Or qv octiND xD2
Or qv oc peripheral speed xD2
Thus at constant pressure (i. e. peripheral speed), flowrate
Qv ccD2
For constant peripheral speed
Table 20.3 Typical multirating table for paddle bladed centrifugal fans 
All these relationships assume constant efficiency so that absorbed power can also be calculated.
It soon became apparent that these tables in their original form were inadequate, so that by the 1960s they had been expanded considerably. The manufacturers simply expanded the number of flowrates and pressures into virtual books, stating the size of fan involved to achieve the duty, together with its speed and absorbed power.
Matthews & Yates Ltd together with Keith Blackman Ltd were prime exponents of this approach.
It will have been noted that use of the Fan Laws is central to all methods of mathematical fan selection. By the use of these laws, fan performance may be reduced to some convenient dimensionless or standard performance.
To repeat:
Qv ocND3
JD 12 J3 12 
Pf = kt 
<Ps 
Pfs — Ks 
<Ps 
Here again it became customary to take the speed of rotation in thousand of rev/min and d in feet. Also relative density ps was used for convenience instead of air density p. The constant was known as ks or kt according to whether comparison was being made for fan static pressure or fan (total) pressure. Thus: 2 / p. 2 
Equ 20.5 
N 
1000 N 1000 
Equ 20.6 
= Kq : 
Where: N = rotational speed (rev/min) D = impeller diameter (in) Q = flowrate (cu. ft/min) Similarly, pressure developed: Pf 
1000 
N 









































When the author was a young apprentice at the Sturtevant Engineering Company, specialist engineers in the fan design department used Master curves. These were jealously guarded as knowledge is of course power! Performance curves were considered to be important sources of knowledge. They were rarely let out of their sight. Certainly, the customer was very privileged to get such information. Nowadays of course everybody asks for a performance curve. Even buyers request them
— whetherthey understand them, or just lock them in theirfiles is another matter.
The Master curves were based upon what are known as R, C and E values. From these curves the characteristics of any fan in the series at any speed can be calculated. They were also useful to bring into the general knowledge base any spot tests taken on installations from which the speed, flowrate, pressure and absorbed power were returned.
To prepare a master curve a full test was made on a fan of reasonable size in the series, and from the results the values of R, C and E were calculated.
The velocity pressure was the velocity equivalent of fan static pressure i. e. in Imperial units:
VSP = 3970Jp~s tip speed ft/m
Conditions (static non delivery) or SND and fully open (free inlet and outlet) or FIO. Curves of the nature of Figure 20.4 resulted. Efficiency was assumed constant over a number of sizes without serious error.
By using logarithmic graph paper (both x and y ordinates) a considerable simplification in the mathematics of the selection procedure can be made.
Consider the drawing in Figure 20.5.
Figure 20.5 Basis for a background chart of y = f(x) 
R = 
X1 = 9 cT 
Pf — Ct 
VSP flowrate ft3/min
C =
Rotational speed rev/min xD
E = fan static efficiency % where D is in feet.
It will be noted that all these parameters are dimensionless so that they can just as easily be converted to SI units provided consistent units are used.
Thus:
VSP = = 1.29^/p^ for standard air
V P
Tip speed m/s
VSP
Flowrate m /’s
C =
Rotational speed rev/secxD
E = fan static efficiency %
Where D is in metres
Values of R, C and E were usually taken at 8 to 10 points over the complete characteristic. These included values at closed
To multiply Ax by N we add log N
Ay by M we add log M
The two operations may be done together and the same operation may be carried out on point B. The line AB will remain a constant length.
If we now plot fan pressures against flowrates, an identical situation is apparent. See Figure 20.6.
Figure 20.6 Background chart of flowrates, pressures, speeds and sizes 
Thus the mathematics become:
CQ = flowrate (m3/s) of 1000 mm fan at 1000 rev/min
CT =fan (total) pressure (Pa) at 1000 mm fan at 1000 rev/min
To find the q„ and pF of a 2000 mm fan at 1500 rev/min
1500V2000 ,
Qv=cQi^^ll^] = 12 c,
Q
1000A1000
1500W2000
1000J Ijooo
On the diagram which has logarithmic scales, the values of qV2ooo and pF2ooo are found by adding lengths of log 12 and log 9 respectively. This would apply to any number of points and to a curve such as the fan characteristic.
00106 08 0/ 09 OS Of 0E 02 01 6 B Z 9 S (• E 2 „ _ 0 1808010 9 0 SO 
Intake capacity mVs 0 3 04 0 5 06 0/ 080910 2 3 « 5 8 7 8 9 10 20 30 «0 SO 60 70 60 90100 


2/ n^/2 N 3 
Equ20.11 
= 2C, 
= 3c, 
■Q 
= 8cr 
Equ20.12 
In like manner for a 1000 mm fan at 2000 rev/min: ^2000"
Qv=c,
1000
2000
Pf_CtIioooJ “4Ct
Whilst for a 1000 mm fan at 3000 rev/min: ‘3000^
Qv=cc
1000
Taking logarithms throughout we may say that:
2 1 1 — logN = — logpF—logqv
Which is a straight line relationship.
In like manner, for a 2000 mm fan at 1000 rev/min: "2000^°
Qv=cc
1000
SHAPE \* MERGEFORMAT
= 4^ 
PF=cT 
2000 r 1000J
Pf_ N 
Now:
Qv ocND3
PF ocN2D2 at constant gas density
Or:
Pf>2
D oc
Thus:
3 n„/2
Similar points may be plotted for the value of qv and pF corresponding to cQ and cT for any size and speed of fan by using the Fan Laws. At any diameter a straight line may be drawn through the points of differing speed resulting in straight lines of constant diameter. In the same way a straight line may be drawn through the points of differing diameter resulting in straight lines of constant speed.
Since more than one pressure and flowrate results from a fan, the whole range of values of cQ and cT (or pF and qv) may be drawn through each point. To prevent this becoming tedious, the characteristic may be drawn on a separate transparent cursor to the same scale.
A reference point of a single value of cQ and cT is selected from which the lines of constant diameter and speed are drawn. This reference point need not lie on the characteristic, but need only be marked on the cursor. The characteristic of the same type of fan, but of another size and speed may be found by placing the
Cursor so that the reference point lies on the intersection of the straight lines of the desired size and speed.
The background chart of pressure and flowrate may be used for any type of fan — only the cursor being replotted to the new characteristic. The same reference point must be used, since the background chart is plotted to this point.
Where a manufacturer has a number of different widths of centrifugal fan (or different pitch angles of axial fan), it is possible to plot these all on the same cursor. The efficiencies may be marked on the cursor characteristic since their relative positions remain constant.
If the cursor is turned through 180°, it is possible to place the reference point over a desired duty of flowrate and fan pressure when the point where the cursor characteristic intersects a fan size will give the speed and efficiency to achieve that duty.
Figure 20.7 is a universal background chart for a range of SI sized fans in accordance with ISO 13351:1996, whilst Figure
20.8 is the cursor for a range of backward inclined bladed centrifugal fans, of varying widths, to be used with it.
All that has been previously said about fan selection has been rendered superfluous by the introduction of electronic catalogues. A CD is now often provided by the fan manufacturer. Contained within the CD are all the fan laws together with the known range of sizes, maximum speeds and powers, temperature derating factors etc, etc. Even noise data and dimensions, together with prices can be included.
When a required duty is entered, the computer can list all possible selections in order of price, efficiency, noise level or some other parameter. It can produce a dimensioned drawing, quotation and specific performance curve. Unfortunately, or fortunately, according to your viewpoint, specialist companies have now entered this field to produce such programmes. Are we to see a situation develop where even the manufacturers’ representatives do not really know what their products do, but are reliant on a CD? But then, they do save time — and time is money!
Posted in Fans Ventilation A Practical Guide