# 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 pre-designed. 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.

Specifying requirements

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.

Fan “apparent” pressure

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 multi-rating 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.

Multi-rating tables

In the UK however, Sturtevant’s associated company, Sturte­vant Engineering Company Ltd, continued with the use of multi-rating 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

 (0 CL -2 Fan static pressure Pa Fan Size No. E E O> A> E (0 Ј 3 IA IA Ј A * O O 0) > E A 2 I O E O A> E 250 375 500 750 1000 1250 1500 A> C A> C O > Rpm KW Rpm KW Rpm KW Rpm KW Rpm KW Rpm KW Rpm KW 92 0.51 1410 0.41 1525 0.50 1650 0.58 1900 0.78 2140 0.99 2350 1.20 2520 1.40 229 130 0.60 1575 0.60 1680 0.70 1780 0.80 2000 1.01 2220 1.25 2410 1.49 2590 1.72 0 172 0.70 1870 0.93 1950 1.08 2130 1.30 2320 1.57 2500 1.85 2680 2.10 222 0.79 2140 1.48 2275 1.70 2430 1.95 2610 2.24 — 100 0.79 1165 0.58 1260 0.70 1360 0.81 1570 1.10 1765 1.39 1935 1.69 2080 1.97 1 279 142 0.94 1320 0.89 1385 0.98 1470 1.13 1652 1.42 1830 1.76 1990 2.09 2140 2.42 187 1.08 1550 1.42 1605 1.52 1755 1.83 1910 2.20 2060 2.60 2210 2.95 239 1.23 1765 2.09 1880 2.39 2000 2.74 2150 3.17 — 97 1.09 990 0.81 1070 0.96 1155 1.13 1335 1.52 1500 1.92 1645 2.33 1765 2.72 326 137 1.29 1120 1.23 1178 1.36 1250 1.55 1410 1.96 1555 2.43 1690 2.89 1815 3.34 2 182 1.49 1315 1.95 1365 2.11 1490 2.54 1625 3.06 1755 3.58 1875 4.10 237 1.69 — — 1500 2.89 1595 3.21 1705 3.80 1830 4.36 — — 97 1.44 860 1.06 928 1.29 1000 1.49 1158 2.01 1300 2.54 1425 3.08 1530 3.60 381 135 1.71 970 1.63 1020 1.80 1080 2.05 1220 2.59 1348 3.21 1469 3.80 1572 4.40 3 182 1.98 — — 1140 2.50 1180 2.59 1292 3.36 1410 4.03 1520 4.74 1628 5.41 234 2.24 — — — — 1300 3.82 1380 4.36 1429 5.02 1585 5.76 — — 97 1.84 760 1.36 820 1.63 889 1.90 1022 2.57 1150 3.24 1260 3.92 1352 4.59 429 135 2.18 860 2.07 905 2.29 960 2.61 1075 3.30 1190 4.10 1295 4.86 1390 5.61 4 182 2.52 — — 1010 3.30 1045 3.54 1140 4.29 1248 5.13 1345 6.04 1440 6.90 234 2.86 — — — — 1150 4.86 1220 5.56 1308 6.41 1400 7.35 — — 110 2.45 660 1.80 714 2.16 771 2.54 890 3.42 1000 4.33 1097 5.23 1178 6.11 483 154 2.91 748 2.76 785 3.06 834 3.49 936 4.40 1037 5.46 1128 6.49 1210 7.46 5 204 3.36 — — 878 4.40 910 4.74 995 5.70 1083 6.85 1170 8.05 1250 9.17 259 3.81 — — — — 1000 6.49 1061 7.42 1138 8.54 1220 9.77 — — 107 2.97 600 2.18 650 2.62 702 3.06 810 4.14 910 5.22 996 6.34 1070 7.38 533 149 3.52 680 3.34 715 3.69 758 4.21 852 5.33 945 6.60 1025 7.83 1100 9.06 6 199 4.06 — — 800 5.33 828 5.73 905 6.90 989 8.28 1065 9.77 1140 11.11 249 4.60 — — 910 7.83 965 9.02 1032 10.33 1110 11.86 — — 107 3.54 550 2.60 595 3.12 645 3.65 744 4.92 835 6.23 915 7.53 982 8.80 584 149 4.20 625 3.99 655 4.40 695 5.03 781 6.34 865 7.83 940 9.32 1010 10.74 7 199 4.84 732 6.34 760 6.82 830 8.20 905 9.84 977 11.63 1045 13.20 249 5.48 835 9.32 888 10.66 950 12.30 1020 14.09 — 102 4.15 508 3.04 550 3.65 595 4.29 685 5.78 770 7.31 844 8.87 907 10.37 8 635 144 4.91 575 4.66 605 5.16 641 5.89 721 7.46 798 9.25 869 10.96 934 12.68 192 5.66 675 7.46 700 8.02 766 9.66 835 11.56 901 13.61 965 15.51 249 6.44 770 10.96 818 12.53 875 14.39 940 16.55 —

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.

Performance coefficients

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

 Pfs — Ks

 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
 Or: Qv = kND3 pF = kN2D2P In order to give reasonable values for these constants, it be­came the “norm” (in the pre SI era within the USA and UK) for these constants to be calculated taking the speed of rotation in thousands of revolutions per minute (per 1000 rev/min) and the impeller diameter in feet. Thus the constant became kq for volu­metric flowrate ie:

 \$

 Equ 20.4

 2000 4000 6000 8000 10000 Volume (low cf. m.

 Figure 20.2 Fan characteristic in terms of cfm, ins. w.g and b/h/p

 400 600 Volume coefficient K® Figure 20.3 Fan characteristic in terms of flow, pressure and power coefficients

 Where: Pf or s = pressure (total or static) ins. water gauge In like manner, the powerabsorbed (then measured in b. h.p.):

 P = Kpx| N P ‘1000

 Equ 20.7



 If fan performance was now plotted in terms of k3, ks, kt and kp instead of volumetric flowrate (then ft3/min) fan static or total pressure (then ins w. g.) and absorbed power (then b. h.p.) a ba­sis of comparison between fans of different series or designs was readily available; the shape of the “standard” characteristic being in every way identical with any other fan of the same se­ries. For axial fans this required that the curve was for the same blade pitch angle. Figures 20.2 and 20.3 show the characteris­tics of a Woods J Type axial flow fan, one of 24 inch diameter (2 feet or 610 mm) running at 1440 rev/min and the other in terms of coefficients kq, ks, kj and Kp. With the change to SI units, the use of these coefficients has largely died out. There were, however, distinct advantages in being able to recognise exactly where a fan was operating on its characteristic. Some attempt at reviving their use has therefore been noted, with certain modifications. Calculation becomes much easier if the coefficients are re-jigged on the basis of a 1000 mm, i. e. 1 m fan at 1000 rev/min. They are designated with a “c” to differenti­ate.

 Pf or fs Cr°rsx| 10oo) X(l000j Xps

 Equ 20.9 Equ 20.10

 P ^jooo) x(ioooJ XPs

 Where:

 Qv = flowrate (m3/s) Pf or fs = fan (total) or static pressure (Pa) P = power absorbed (W)

 Thus:

 D

 Equ 20.8

 11000

 1000

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.

Charts and cursors

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

,3

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 m-Vs 0 3 04 0 5 06 0/ 080910 2 3 « 5 8 7 8 9 10 20 30 «0 SO 60 70 60 90100
 Bd’H ejnssajd oueis uej

 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

Q/3

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 re-plotted 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.

Electronic catalogues

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 de-rating 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!

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