Performance matching and optimization of wind powered water pumping systems

Energl Contcrsion. Vol. 19. pp. 33 to 39

0013-7480 79:0401-0033502.CX%0

~i Pergamon Press Ltd 1979. Printed in Great Brilain

PERFORMANCE MATCHING A N D OPTIMIZATION OF W I N D POWERED WATER P U M P I N G SYSTEMS G. M. BRAGG and W. L. SCHMIDT Department of Mechanical Engineering, University of Waterloo, Waterloo N2L 3G1, Ontario, Canada

(Received 18 July 1978) Abstract A procedure is presented which allows optimum selection of pumps and windmills for a given water pumping situation. When information on the wind, pump characteristics and windmill characteristics is available, the best pump and windmill for the application may be selected, and the design and off-design performance of the complete system may be predicted. Wind power Pumping Optimization Design Matching

Generally, most of the pumps and all of the wind turbines available for wind systems have optimum characteristics within a given operating range. If a pump or windmill is allowed or forced to operate outside of this performance envelope, its efficiency is reduced considerably and power that would otherwise be available for water pumping is wasted. Therefore, it is important to ensure that the system components are properly matched and that the system itself is matched to the operating conditions. Previous work on the matching of pumps and windmills has been done by Vadot [1] and Banas and Sullivan I-2]. Vadot has presented some specific examples for wind pumping systems to illustrate how an optimization procedure is effective in maximizing system output. Banas and Sullivan have given a more generalized presentation that illustrates how operating points may be predicted for wind systems driving speed-dependent and constant-speed loads.

LIST OF SYMBOLS A

Cp Dp Dr g

GR H ht Np Nr P~ Q R V A 7 2 r/p /7 p

Swept area Coefficientof power Pump diameter Windmill diameter Gravitational constant Gear ratio Head Height of rotor Pump rotational speed (rad/sec) Windmill rotational speed Shaft power Flow rate Geometry correction factor Wind velocity Specific diameter Specific weight of water Tip speed ratio Pump efficiency Power specific tip speed Air density Specific speed

SYSTEM PARAMETERS AND PERFORMANCE

INTRODUCTION This paper outlines a systematic approach for matching windmills and waterpnmps to give optimum wind pumping system performance. Dimensional analysis techniques are used to predict the most suitable type of pump for the water lifting applications, the best type of wind machine for the design wind speed, and the final configuration to maximize total system performance. The following input information must be available for the analysis: (1) Wind speed frequency and duration information (2) Pumping parameters (3) Windmill and pump characteristics. The procedure is similar in many ways to the matching that allows the system components in turbomachines, such as gas turbines, turbo-prop engines and turbo chargers, to operate together efficiently. E.C.

19/I--<"

Each wind power system must be designed for the conditions that exist at the selected site. The important parameters for consideration are: (i) Wind characteristics: duration, frequency and peak values of velocity (ii) pumping head (iii) water delivery required (iv) component efficiency.

Wind characteristics The wind regime at the selected site is the factor that determines the power available to the system. Therefore several characteristics must be known in some detail. The power available from the wind may be calculated from the usual power equation: Power = Ce~pA V 3, 33

(1)

34

BRAGG AND SCHMIDT:

~[

PERFORMANCE OF WATER PUMPING SYSTEMS

I~ading Interval

/ ~\....~

3s,~c

Anemometerat Anemometerat Anemometerat

....

4,30'?." 60"

The maximum expected wind speed is always beyond the design range of the wind turbine. It must, however, be known so the structural design for the system will be adequate to ensure continued operation when the wind speed returns to the operating range. Wimt machines

5

10

15

20 25 3O VELOCITY (mph)

35

Fig. 1. Velocity frequency curves for one site [4]. where Cp p A V

= = = =

coefficient of performance or efficiency density of air swept area velocity.

The value obtained from equation (1) is an instantaneous value, so the time dependency of the wind must also be known. The velocity frequency data for the selected site will relate wind speed and time for the period when the measurements were taken. Figure 1 is an example of this type of data at a particular site. The variation of wind speed with height as well is shown here. Unfortunately, information of this type is generally not available for more than a few locations, none of which are prospective water pumping sites. Furthermore, it is difficult to obtain reliable data for power predictions in a short period of time. Other characteristics that may influence power output are wind gustiness and wind direction rate of change, length of calm periods and maximum expected wind speeds. The gustiness and rate of direction change will influence the total efficiency of the wind machine depending on its axis, horizontal or vertical, and rotational inertia. Some machines will be able to respond to gusts quickly and will be able to use the energy available from them. The duration of calm periods will determine the storage capacity that is required to ensure a constant supply of water.

For wind machines whose characteristics are known, the coefficient of performance is normally plotted against tip speed ratio, 2, as shown in Fig. 2. This non-dimensional presentation allows the properties of geometrically similar machines of the same family to be plotted as a single line assuming independence of Reynolds number. The result is independent of wind speed and machine size over a wide range, The parameters used are: 2 = DTNT/2 V (2) Cv = Ps/~ pA V 3.

Ps = shaft power, DT = rotor diameter, N r = rotational speed (rad/sec),

). = tip speed ratio. From the values presented for several types of wind machines, general observations may be made about their performance. These include: (i) each type of wind machine has a peak performance at a particular tip speed ratio; (ii) rotors having a high efficiency for a wide range of tip speed ratios will provide good off-design performance. (iii) the comparative starting torques of various wind machines may be predicted by the slope of the efficiency curve at zero tip speed ratio. The greater the slope, the higher the starting torque; (iv) the solidity ratio of the rotors (area of blades/ swept area) with the best performance generally increases as the tip speed ratio decreases.

Eflici~cv U~mit.__

iIII/~

oI 0.5

/

0.4

/

/

/

Cp 0,3

~ 0,2

~ X ~

~

Filippini Rotor

/

\/

\

Four-bladed Horizontal Rotor

0.1

2

3

4

(3)

where

5

TIp SPEED RATIO

6

Fig. 2. Summary of C v )o curves.

7

8

9

BRAGG AND SCHMIDT: PERFORMANCE OF WATER PUMPING SYSTEMS

35

I0: ......

I

. . . . . . . .

,

. . . . . . .

101

IO

r I I I I 1 ~ 1 I0 z

10

I IIII

I I11

I0 z 0 (USGPM)

?Os

lO 4

Fig. 3. Pump operating ranges [5]. Pumps Individual pumps operate most efficiently within a range of heads of flow rates. Figure 3 outlines the operating envelopes for several types of pumps. It has been found for families of similar pumps that within each operating range there is a point at which all the similar pumps have a maximum efficiency. This is determined by calculating the dimensionless specific speed, fL of the pump: f ~ - NpQ r2 (H0)3/4

opt.,

(4)

where = specific speed, Np = rotational speed of pump, H = head, Q = flow rate. I00

I

All quantities are measured at the most efficient operating point of the pump. Figure 4 illustrates how specific speed, efficiency, discharge and pump configuration are related for rotodynamic pumps. Generally, a pump of low specific speed has a relatively low rpm, low flow rate and high head at its best operating point. Positive displacement pumps have very low specific speeds but no optimum operating point. Pumps with high specific speed are designed to provide large quantities of water against low heads and operate at high rpms. Axial flow and propellor pumps are classed in this category. It is not enough, however, to know just the best range of specific speeds for a given family of pumps. It is possible, without further guidelines, to design 2 pumps with similar specific speeds for the same job and find that the diameters and efficiencies may be I

I

~

80

'

~

1

~

3000

Opm"

60

" 1 . 0 US i l ~

5~

,la 0.2

I 0.5 I

Radial Flow

Centrifugal

I 1.0 J~. $11~CIFIC SPEED

FfMcis

- OoN3 I / s

I 2.0

Mixed Row

5.0

Pn)p4dlot

Fig. 4. Pump efficiency vs specific speed and configuration [6].

36

BRAGG AND SCHMIDT:

PERFORMANCE OF WATER PUMPING SYSTEMS

quite different. To indicate optimum diameters for best performance, the concept of specific diameter, A, was introduced by Cordier I-3], and is defined as:

A - Dp(H'q)I/4 QZ/2 opt., where A Dp H Q

= : = =

(5)

specific diameter, diameter of pump, head flow rate.

All quantities are, again, measured at the most efficient operating point of the machine. Cordier found that an empirical relationship exists between specific speed and specific diameter for highly efficient machines. A Cordier diagram is presented as Fig. 5. Using this information it is possible to approximate the best configuration for the machine required. For example, in a typical example the head and flow rate are assumed to be known. If a suitable rotational speed is chosen, then fl is known and the best type of pump is selected from the specific speed chart. Then, using f~ as an input parameter again, this time to the Cordier diagram, A, the specific diameter is determined, and from this the diameter of the pump may be calculated. The task that remains is to match the components to produce a system that will maximize the output performance for a given set of input parameters.

i

r

~ ~tr~r]

J

~

I

t xtlr[

~

I

ial Fan (78%)

ILZI'.,.,

~

£ial Water Turbine 195.5%1 evatrifugal Fan 180-90%)

~lh'ltri~gal Centrifugal Water Turbine 191%2

The optimization of pumping components alone is a relatively straightforward job with the aid of the Cordier diagram. The appropriate rotational speed and diameter are calculated for the given head and flow rate and the selected type of pump. The concepts of specific speed and specific diameter are, unfortunately not so versatile for wind turbine design. It is possible to show that the tip speed ratio used for wind machines, is, in fact, a specific speed, and as Fig. 5 shows, wind turbines may also be plotted on the Cordier diagram. However, this information is not really useful as it is the power delivered by the turbine that is of importance. Using dimensional analysis, it is possible to arrive at a power parameter that includes wind turbine rotational speed instead of the diameter. This parameter, which is equivalent to the power specific speed for hydraulic turbines, will be referred to as the power specific tip speed, H, for wind turbines. The appropriate non-dimensional grouping of terms may be written as:

P~NT2 H - (1/2)P VSRGR(s,rl

(6)

and H = Cp2 2 for

GR = 1,

(7)

This is easily proven to be dimensionless and consistent with equations (2) and (3). The geometry correction factor, R, is used to accommodate circular, square and rectangular rotors into the equation. For circular rotors R = n, for square rotors R = 4 and for rectangular rotors R = 4(ht/DT) where h, is the height of the rotor. Using the value presented in equation (7), it is possible to replot the s(andard presentation of power coefficient vs tip speed ratio, Fig. 2, into a graph of power specific tip speed vs tip speed ratio, Fig. 6. The advantage of this presentation is that, using this figure, it is possible to determine the best wind rotor to use when rotational speed and power required are the only input parameters. The figure may also be used to determine the effects of different wind velocities on the system. The effects of a gear ratio used to match a pump and a windmill may be determined as well.

n dmill (74%)

~

Selection of components

where Pc = shaft power, R = geometry correction factor, GR = gear ratio.

,J ~ i p Propeller (76%)

~

SYSTEM M A T C H I N G

Fan 180%)

191%)

~ P u m l p

Matching windmills to a given pump I

I

t

I i t t 11 0.5

1.0

i

I

I

I ~ Ilil

5,0 A - SPIFCIFIC DIAMETER

~ I0

Fig. 5. Cordier diagram [7].

i

150

To find the windmill that will be the best match for a known pump, or pumping requirement, the power needed to drive the pump must be calculated.

BRAGG AND SCHMIDT:

PERFORMANCE OF WATER PUMPING SYSTEMS

/

37

/

/

---,°--.,..-7°>9.--

-_

"E ..,,..°.,,.....***°'"*" . .......

//

• 5.0

......

'

/

........

,

~ t.o

/

o.5

.-"

~

Savonius Rotor

~3) Multi-blotkKI Horizontal Rotor (4) Four-bladed Horizontal Rotor ($) Three-bladed Darrieus (~1 l'~lmde(l Horizontal Rotor o.1

II//1 1 /

2

I

3

I

4

/

5

I

6

f

7

I

8

TIP £~EED RATIO

Fig. 6. Power specific tip speed. This is determined from the pump input power requirements: P~ -

7QH

(8)

where 7 = specific weight of water, ~/p = pump efficiency. If the head and flow rate are known, equation (8) requires that an ~/p must be assumed. Figure 4 or similar information may be used as a guideline to estimate this value. This value may be checked when the final operating conditions are known. Equation (6) is then used to calculate the power specific tip speed for a given wind velocity and shaft speed. This value from equation (6) is entered in Fig. 6 as a straight horizontal line. For example, a line having a value of 5.0 would correspond to the input from a pump operating with 90~o efficiency at 180 rpm and requiring 368 watts of power from a 6.7m/sec wind. This line passes through several curves., All of these points are possible matches for the pump provided other criteria such as starting may be met. The best choice for the job is shown by the first intersection point on the left. This is the rotor with the highest efficiency, lowest tip speed ratio and hence lowest swept area and material requirements for the direct drive arrangement. The diameter of the chosen windmill type is now immediately calculated from:

Dr = 22V/NT,

(9)

since ,i may be read off the figure as well. If a point on a curve for a wind turbine is below the horizontal line drawn on Fig. 6 and a match with the wind machine is desired at that design point, then a speed up gear ratio is required. The value of this ratio is determined from equations (6) and (7). The value for P~, Cp, 22 , V and Np are known and the

gear ratio, GRts/r ) may be calculated. It is important to note the (S/T) subscript as this indicates that it is the gear ratio between the shaft speed and turbine speed that is derived. A number greater than 1.0 indicates that the pump shaft must be driven faster than the rotor and vice versa for a number less than 1.0. In summary, the system component selection when the pumping requirements are known is quite straightforward. The optimum rotational speed is chosen for the pump and the equivalent power specific tip speed for the windmill is calculated. Then the match is made by selecting the intersection point of the input data line and the most efficient windmill line for the direct drive combination. The rotor configuration is then calculated directly as the tip speed ratio is shown on the graph. For other combinations, the required gear ratios and rotor geometries may also be determined in a similar manner.

Matching pumps to a given windmill The procedure for matching a pump to a selected type of wind machine is a bit more involved. A general algebraic relationship is presented so that only the design wind speed and type of windmill and estimated pump efficiency must be selected to begin the process. Other parameters such as pump and windmill rotor diameters, head, flow rate and shaft speed remain unknown until a specific case is to be considered. The general analysis starts off by choosing the appropriate value of power specific tip speed for the type of windmill from Fig. 6. From equations (6) and (7) when V and Cp22 are known and GR is equal to 1.0 for direct drive (Nr = Np = N), it follows that:

Ps N2 = (Cp2 2) (½pVSRGR 2) =C1

BRAGG AND SCHMIDT:

38

PERFORMANCE OF WATER PUMPING SYSTEMS

Table 1. Matching pumps to a wind turbine Pump Positive displ, Centrifugal Centrifugal Mixed flow Propeller

est.

tq

A

Head (fl)

Q (cfs)

N (rad/sec)

Dr (fl)

DI [ill

0.007 0.15 0.50 1.8 4.0

-25 5.5 2.2 1.7

155' 2.8' 1.1' 0.4' 0.2'

-0.758 2.08 5.88 10.33

5.03 5.03 5.03 5.03 5.03

6.56' 3.25' 2.81' 3.43'

6.56' 6.56' 6.56' 6.56' 6.56'

All systems are direct drive. Design wind velocity 22 ft/sec. Wind machine Filippini. since all values on the right hand side are assumed given. F r o m equation (8), Ps may also be evaluated

In summary, the following relationships are known :

(1)

as:

Ps-

(2)

yQH %

Dv =

f(H) Dr

Q = f(H)Dr

(3) N r = const/Dr

For a given efficiency

(4)

Np = N r for direct drive.

7QHN 2 -- P s N 2 = C 1 % Q N 2 _ C2 H'

(10)

where

C2-

Clair 7

Now from equation (4) specific speed is : N Q 1/2 CC~.~ 1 C4 - (Hg)314 - ~ / ~ H 3/4 - H5/4

(11)

Now, using these general relationships, once the size of wind turbine and type of pump are chosen, all other system parameters may be derived. An example is summarized in Table 1. As Table 1 shows, there are several possible matches for various types of pumps to a given type of wind turbine. When gear ratios are included the possibilities are even greater. Hence the efficiencies and system characteristics must be evaluated over the entire operating range to determine the most suitable combination.

where

OFF-DESIGN AND STARTING PERFORMANCE

C2

C3 -- g3/4

C4 = C 1/2. Since spqcific speed and specific diameter are related at the optimum design point, by the Cordier diagram, specific diameter may be expressed as A = fi(f~) = f x ( C , H - S I * ) .

Off-design performance is most conveniently calculated by using the torque-speed curves for the wind turbine and pump. If only the Cp-2 curve is available for the wind machine the torque curves for various wind speeds may be calculated by:

(12)

Torque = Power/NT

(17)

C T = Cp/)~

(18)

or

Therefore Dv(Hg)ll* - f l ( C , , H - 5/*)

(13)

Dp = f 2 ( H ) Q + 112

(14)

Qll2

where

and

This may be extended by adding the expressions for Q from equation (10) and N from equation (2) (as 2 and V are known) to give : /

g.~ \1/2

= f2(H) H ~ 2

(15)

/C2Dr2\

= S3(H)"

Dr.

(16)

The load curve or torque-rpm relationship for the selected pump is added to the torque-rpm figure and the intersection points of the lines show the operating speeds for the system at that wind speed. When the overall system performance is shown in this manner, it is quite easy to see that the best system performance over all wind speeds is obtained when the load curve follows the locus of maximum turbine power. This point has been made by Banas and Sullivan [2]. If this is not possible due to load character-

BRAGG AND SCHMIDT:

PERFORMANCE OF WATER PUMPING SYSTEMS

istics, then the best match will be determined by starting conditions and the local wind conditions. Examples of torque curves often show the starting torque, i.e. torque of the rotor, at zero rpm. When the load curve for a pump is plotted on these figures the minimum starting or threshold velocity for the system may then be found. If starting torque figures are not available for a given type of wind turbine, it may be quite difficult to predict these values. The actual starting torque values must be measured for each rotor configuration. The reason is that dynamic operating values cannot predict static starting torque values accurately. Once the starting torque requirements have been satisfied and the off-design operating points are known for the system, its performance may be assessed for the local wind conditions.

M A T C H I N G A W I N D - P U M P I N G SYSTEM TO LOCAL WIND CONDITIONS

Presently, the only technique that exists to find the most suitable water pumping system for a wind regime is to iterate the system selection and identify the best match. The system parameters to be considered are windmill and pump size and configuration, gear ratio, cut-in and cut-out wind velocities and storage capacity. The wind parameters which must be available are velocity frequency and duration. When the characteristics of the system and wind regime are known in detail, the performance of various well matched combinations of windmills and pumps may be modelled and water delivery for the wind conditions tabulated. In other words, several likely combinations must be selected and the design and off-design performance calculated to show which of these systems is the best. Generally, the best system will provide the required amount of water with a minimum cost and maximum availability.

39

CONCLUSIONS 1. A systematic methodology has been developed to analyze windmill waterpump systems, given the characteristics of the pump and windmill separately. The procedure allows a number of different design problems to be treated. If the windmill is given, the appropriate pump may be chosen for a given head; or conversely the best windmill for a given pump may be selected in an optimum fashion. 2. The concept of power specific tip speed which has been introduced greatly facilitates the matching of windmills and pumps at the design wind velocity in a general manner for a variety of situations. The power specific tip speed is analogous to the power specific speed used in hydraulic turbine technology. 3. When detailed information on the wind characteristics is available the performance of a pumping system may be modelled for the selected site. Acknowledgement The work reported here was supported by the International Development Research Centre, Ottawa, Canada. REFERENCES

[1] Vadot L., Water Pumping by Windmills, La Houille Blanche No. 4 (1957). [2] Banas J. F. & Sullivan W. N., Engineering of Wind Energy Systems, Sandia Labs Report, SAND 75-0530, Albuquerque N.M. (1976). [3] Cordier B., WiJrme Kraft, No. 4 (1953). [4] Cousins D. M., Wind Monitoring, Internal Report 78-1, Dept. of Mechanical Engineering, University of Waterloo (1978). [5] Streeter V. L., Fluid Mechanics, 5th edn, p. 537. McGraw-Hill. New York (1971). [6] Karassik, Krutzsch, Fraser & Messina (Editors), Pump Handbook, pp. 2- 130. McGraw-Hill, New York (1976). [7] Bouwman H. B., Dimensieloos keuze-diagram voor de bepaling van de bouwgrootte en het toerental van verschillende typen ventilatoren en audere stromingsmachines, De Inyenieur W31 (1956).