A model of a centrifugal pump coupled to a windrotor

Citation for published version (APA):

Staasen, A. J. (1988). A model of a centrifugal pump coupled to a windrotor. (TU Eindhoven. Vakgr. Transportfysica : rapport; Vol. R-896-A). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988 Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

A IIlDEL OF A CENfRIfUCAL PUMP ax.JPLID TO A WIIIDROTOR

A.J. Staassen January 1988

R 896-A

Faculty of Physics, Fluid Dynamics, Wind Energy Group, University of Technology Eindhoven

The choice of the type of rotodynamic pump that has to be coupled to a windrotor is motivated. Next. some assumptions about the centrifugal pumps are made and the characteristics following from these assump-tions are calculated. The results are compared with some available measured data. The modelled centrifugal pump is coupled to a wind rotor and some design formulas are derived in order to be able to make a well motivated choice of the pump constants and the transmission ratio. The quality and availability of the optimized system in a wind regime given by the Weibull probability density function are calcula-ted. The sensitivity of the system to the head and the avrage wind speed is calculated. The model of the centrifugal pump is checked by means of some measurements on a pump. Some remarks on the transmission and the safety mechanism are made. Conclusions can be found in the last chapter of this report.

ax.

IEftIS Sununary Contents Symbols Introduction Acknowledgement

CHAPTER 1 CHOICE OF THE TYPE OF ROTODYNAMIC PUMP

CHAPTER 2 THE MODEL OF THE CENTRIFUGAL PUMP

2.1. Starting points

2.2. The characteristics of the pump 2.3. Characteristics for constant head

2.~. Characteristics for static + dynamic head

2.5. Comparison with available data CHAPTER 3 : COUPLING TO A WINDROTOR

3.1. Description of the system 3.2. Design formulas

3.3. Restrictions

3.~. Use of the design formulas

3.5. Output prediction 3.6. Conclusions I page I

III

V

VII

1 7 1 9 12 1~ 15 16 16 18 20 22 23 25 CHAPTER 4 : CALCULATION OF THE QUALITY AND AVAILABILITY OF THE

OPTIMIZED SYSTEM 28

~.l. The quality 28

4.2. The influence of the cut-out wind speed on the quality 31

4.3. The availability 32

4.4. The influence of the cut-out wind speed on the

availabili ty 33

CHAPTER 5 : SENSITIVITY ANALYSIS 5.1. Sensi tivi ty to H

st 5.2. Validity of the results 5.3. Sens it iv ity to

V

CHAPTER 6 : MEASUREMENTS 6.1. Introduction

6.2. Design of an impeller 6.3. Measurements

6.4. The setup of the test rig 6.5. Processing the measured data 6.6. The results

6.7. Conclusions

6.8. Consequences for the model

6.8.1. Consequences for the efficiency

6.8.2. Consequences for the load on the rotor CHAPTER 7 : SOME CENERAL REMARKS ON THE SYSTEM

7.1. The transmission 7.2. The safety mechanism CHAPTER 8 : CONCLUSIONS Literature 36 36 38 38 39 40 40 41 43 45 50 51 52 54 56 57 58 60 Appendices

A Rotational speeds. impeller diameters and hydraulic powers of

some centrifugal pumps

B1-B2 The parabolic H-q curve

C1-C4 Checking the results

01-012: Calculation of the quality and availability

E1-E3 The calculation of an impeller of a centrifugal pump

List of symbols

III

a b c

D.

_10Pt g H

pipeline resistance factor pump constant

pump constant

power coefficient of the rotor maximum C_p

pump constant

optimum impeller diameter gravitational constant head over the pump

static head over the pump transmission ratio

hydraulic power pumpshaft power rotor power

rotor power after transmission flow

design flow

radius of the rotor

optimum specific impeller diameter windspeed

design windspeed rated windspeed

efficiency of the pump maximum

rr

p

[i/m

5] 2 [ m·s ] [s2/m5] [ - ] [ - ] 2 [kg·m ] [ m ] [ m/s2] [ m ] [ m ] [ - ] [ W ]

[

W ]

[ W ]

[

W ]

3 [ m /s] 3 [ m /s] [ m ] [ m ] [ m/s ] [ m/s ] [ m/s ] [ - ] [ - ]

~t efficiency of the transmission ~p+l efficiency of the pump + pipelines A tipspeed ratio of the rotor

Ad,A

opt design tipspeed ratio P a density of air P w density of water W rotational pumpspeed p wpd design w_p wpo

pt pumpspeed at maximum efficiency for a given head maximum W P rotational rotorspeed design w R maximum w R

rotational rotorspeed after transmission

[ - ] [ - ] [ - ] [ - ] 3 [kg/m ] 3 [kg/m ] [radls] [radls] [radls] [radls] [radls] [radls] [radls] [radls] W S specific pumpspeed [ - ]

v

Introduction

This is the final report of the work done for my thesis to obtain the masters degree in mechanical engineering. It has been done at the Wind Energy Group of the Faculty of Physics of the University of Tech-nology Eindhoven. The Wind Energy Group is part of the Consultancy Services Wind Energy in Developing Countries (CWO). Its goal is to do research on and design of water pumping windmills for use in deve-loping countries.

My work was to do research on a waterpumping system for low head and high volume. The water pumping systems that have been designed by the CWO until now mostly make use of piston pumps. The problem with these systems. i f they are designed to pump large amounts of water over a relatively low head. is that the forces in the pumping rod become too large and the efficiency becomes quite small. For this reason the CWO started a research programm to develop systems for low head and high volume making use of other types of pumps. Alternative types of pumps are for example Archemedes' screws and rotodynamic pumps.

An

investi-gation on the performance of Archimedes screws is carried out by L.Linssen at the University of Technology Eindhoven (Not finished yet) and in China systems using Archimedes' screws coupled to a wind rotor are already in use (see [16]).

In this report. the possibility of rotodynamic pumps is investigated. Other systems making use of a centrifugal pump for a high head have been designed by the CWO already (Wind Electric Pumping System.WEPS) [4]. [15]. They make use of an electrical transmission. Other

litera-ture on this subject can be found in [11] .A system with a mechani-cally coupled centrifugal pump has been designed and tested by IWECO [12].[13]. Centrifugal pumps are also used by one of the largest Dutch companies producing water pumping wind mills. Bosman. The design of these kind of systems was done by trial and error.

In this report is tried to derive a general valid model describing the behaviour of centrifugal pumps. Some theory on centrifugal pumps

cou-pled to a windrotor can be found in [5] and [14].

Of special interest is a report by J.Burton [10]. who derived a model similar to that derived in this report.

VII

I wish to thank the following persons for their help in the work done for this report.

Paul Smulders and Jan de Jongh of the Wind Energy Group for their advises and their contribution in completing the work and making this report.

Prof. Vossers and Prof. Schouten of the University of Technology Eindhoven for their time reading and judging my work and this report. SIHI-Maters B.V. Beverwijk for putting a pump for testing purposes at my disposal.

The personel of the Wind Energy Group of the University of Technology Eindhoven for their cooperation in building the test rig and for the very nice time during my stay at the Group.

Thanks.

Chapter 1: (]I)IQ: OF THE TYPE OF IIJIUJY1QJIIC PUIIP

Rotodynamic pumps can be classified by a dimensionless number, the specific pumpspeed ( J ,

s defined for the point of operation for which the efficiency is maximum: the design point. By definition [1]:

in which _(Jpd design pump speed [lis]

3

qd design flow [m /s]

H

d the design head. over the pump [m.w.c]

g gravity constant 9.8 [m/s2]

(1.1)

A certain value of the specific pump speed correspOnds to a specific type of pump: figure 1.1.

(J = O. 25 .--~--L+L-=_.., s (J

=

0.5 s (J

=

1.0 s Figure 1.1 (J = 1.5 s (J

=

2.5 s (J = 3.5 s

2

As guidelines for the selection the following values can be taken [1]:

w_s

<

0.6 radial pumps (also called centrifugal pumps) 0.4

<

w_s

<

3 'mixed-flow' pumps

2

<

w_s axial pumps

With the help of the specific pumpspeed i t is possible to select the most suitable type of rotodynamic pump for a given duty.

If a rotodynamic pump is coupled to a windrotor via a transmission, the balance of power will be valid at the point of operation. The power, P

R, delivered at the design windspeed Vd by a windrotor with a maximum powercoefficient C

Pmax and radius R is:

1 _3 2 P = C .-.p ·V:·'JI"R

R

Pmax 2 a d with P a density of air (1.2)

The net hydraulic power output, PH' delivered by the pump at the point of operation is:

P

=

p

·g·H .q

H w d d (1.3)

with p : density of water w

(only waterpumping is considered in this report) Assume an efficiency of the transmission ~t and a maximum efficiency of the pump ~pmax' The power balance of the system will then be:

P = ~.~ .p

Equations (1.2) and (1.3) substituted in (1.4) give for qd:

(l.S)

If the design tipspeed ratio Ad of the rotor is given •the rotational

speed of the rotor at the design windspeed will be:

(1.6)

and if the transmission ratio i (= Wp/W

R) is given. the rotational speed

of the pump at the point of operation and at design windspeed. wpd' will

be:

(1.1)

Substituting (1.5) and (1.1) in (1.1) gives:

(1.8)

constants system parameters site specifications

In this equation V

d is not really a site parameter. but an appropriate

value of Vd is directly related to the average windspeed V. which is

site specific. In fact V

d ~ 1 - 1.S-V. A definite value of vdlV will follow from considerations of optimizing the system.

Typical

The values of the constants in (1.8) are:

3 2 P w= 1000 kg/m ,g = 9.81 mls values of p ,C Pm ,~ and ~t are: a ax pmax 3 Pa = 1.23 kg/m ,C Pmax = 004 , ~pmax = 0.65 , ~t = 0.90

With these values equation (1.8) becomes:

V

5/2 -3 d C&ls = 1.2·10 .i·A d·---.,;=::,...,..,...H5/4 d (l.9)

With the help of figure 1.2, that is based on' equation (1.9), it is

possible to select a type of rotodynamic pump for given values of V_d, H

and i·A

d. On vertical axis of the upper part of this diagram the areas

of pumptypes are numbered from I to III. These areas are:

I radial pumps

II 'mixed-flow' pumps

I I I axial pumps

As an example, the diagram shows what the choice would be for the

situ-ation that V

d = 4 mls , H = 3.5 m.w.c. and i·Ad = 40.

A horizontal line starting from the value H

=

3.5 m.w.c. in the lower

part of the diagram is drawn. From the intersection of this line with

the line V_d

=

4 mls a vertical line is drawn. From the intersection of

this line with the line of the value i·A

=

40 in the upper part of the

diagram a horizontal line is drawn. On the vertical axis of the upper

part of the diagram the corresponding value of C&l can then be read. In s

....

10 (; 7 8 9 10 5 3 2 10 100

:

_{1j~.llllllilllllli~IIII~llilllllllllillliil; I;ii~}

..

'-~, 1

~~,~.~

I

~

1

I"-l=;

~

Figure 1.2

6

Closer examination of the diagram shows that. in the case of rotodynamic pumps coupled to a windrotor. centrifugal pumps have the widest

applica-tion. If slow running rotors are to be used and i is not chosen very

large. the chosen type of pump will usually be a centrifugal pump. In

the rest of this report only centrifugal pumps will be considered.

In figure 1.3. taken from lit.[2]. it is shown in which way the optimum specific diameter of the pump depends on the specific pumpspeed. The optimum specific impeller diameter is defined as :

6 -D e (geH)I/4

sopt - iopt 1/2

qd

(1.10)

with

D

_iopt :optimum diameter of the impeller Em]

In appendix A some values of the rotational speed. impeller diameter

and net hydraulic power are given for centrifugal pumps with w ranging

s

from 0.1 to 0.25 at pumping requirements with

H

_d ranging from 1 to 10 m

and qd from 10-4 to 10-1 m3/s.

t

...

I-

"

/'

--

--

/

CI

--

---

_...

_f..~

..

7 41 I ₆

/

~ sopt / 1'\ 5

.,

,

/ If

--

- __u ,/"

_'-

r-3

--~I - - -

--=-

CO' ~ z _{j

----

,....

'~

,CJ

CJ

\.

~

.'to-

~ :t"""I

r

t

,

'u

IS

,,,

..

• II I III

•

.,

,

,

W 5 Figure 1.3.

Chapter 2: 1lIE JIlDEI.. OF 1lIE CElffRIf1JGAL IUIP

2.1 Starting points

In literature dealing with rotodynamic pumps. rules of similarity have been derived in the following way [3].

For a given pump working at a given point of operation O. Ap =(pw-g-H).

v and ware known. So. for the flow q and the pump shaft power P can

p p

be written:

q = q (0. g-H. Pw' v. w)

P_P

=

P (0. g-H. Pw' v. w)

p

According to the IT-theorem of Buckingham these equations can be written with dimensionless numbers as:

2

]

[ gon

w -0 q _P 3

=

f 1 2 2 ' w -0 w -0 v p p P 2

]

[ gon

w -0 p

=

f 2

P

3 5 _{2 2 •} p -w -0 w -0 v w p p (2.1) (2.2)

In these equations the expression w_p-02/v is the Reynolds number Re.

This means that. if Reynolds influence is negligible. for a given pump

8

gives:

-+=

C(+)

Ca)p p

or expanding in power series:

(2.3)

(2.-4)

with at' a₂, a₃... :constants

For many centrifugal pumps it seems to be reasonable to assume that all a's are zero, except at and a

3. This has been done before in lit [4].

Assume: at

=

band a

3

=

-c. For equation (2.t) this results in:

H n ·2

- 2 - = b - ce_(.;L.)

Ca) Ca)

p p

(2.5)

In figure 2.t this function is shown. This parabolic function has been checked for several centrifugal pumps. The results are shown in appendi-ces Bt and B2. The parabolic function seems to fit quite well in most

cases. In [5] it is stated that the pumpshaft power P depends on Ca)

p p

only, independent of q and H. This has been measured and confirmed in lit [6]. From (2.2) can then be derived if Reynolds influence is negli-gible:

3 P

=

deCa)

p p

with d depending on the pump only.

b

w~

r p Figure 2.1 -9... w p

2.2 The characteristics of the

pump

If the constants b. c. and d of a given pump are mown. the behaviour of that pump is mown. as long as there is no Reynolds in£1uence. The constants b. c. and d are related to each other via the pump efficiency Tl . This relation can be found in the following way. The balance of

p

power of the pump is:

PH = ~.pp (2.7)

in which PH net hydraulic power ~ efficiency of the pump For PH can be written:

(2.8)

with equations (2.6) and (2.7). (2.8) gives:

10

Eliminating q in (2.9) by means of (2.5) results in:

(2.10)

The maximum efficiency of the pump for a constant head

H

will be found

by differentiating (2.10) with respect to Ca.I keeping H constant. This

p

gives:

I3=H

w_popt=~2=i) (2.11)

with Ca.I t : w at maximum efficiency

pop p

Equation (2.11) substituted in (2.10) gives the maximum efficiency ~Pmax

for the head

H:

p .g.b3/ 2

~ =

..1:.../3

._w _

Pmax 9 d./C (2.12)

The maximum efficiency of a given centrifugal PUmP can not be chosen freely; it can be determined by means of measuring and is usually given by the manufacturer. This means that equation (2.12) gives a relation

between b. c. d and ~ . It can be re-written as:

Pmax

Now. the centrifugal pump is characterised by the two equations (2.5) and (2.6) with d in (2.6) according to (2.13) and by the three parameters b. c and TJ

pmax

Equation (2.5) gives the head as function of q for a given w . For an

p

imaginary pump with the values b

=

10-3 [ms2] and c

=

_{105 [s2/m5] the}

H - q characteristics are given for several values of w in figure 2.2. p

Figure 2.2

It is of interest to know where in this figure the curves lie that connect the points with equal efficiency; the iso-efficiency curves. According to (2.9) along these lines the following equation has to be

valid: Heq 3 w p

=

constant (2.14)

This turns out to be true along all parabolas

12

2

H

=

p.q • with P being some positive constant. In order to find the parabola with the maximum efficiency, the value of p has to be found for which the expression in (2.14) reaches its maximum.

A

small calculation learns that the maximum is reached for p

=

2·c, so the maximum iso-efficiency curve is repre-sented by:

2

H

=

2·c·q

2.3 Characteristics for constant head

(2.15)

With the help of the theory derived above, it is possible to calculate the q - wand the ~ - w curves if the pump is loaded with a static

p p p

head

H

t only. The q - w curve for constant head can be found directly

s p

by applying equation (2.5), substituting

H

_st for

H.

The ~p - w_p curve for

H

=

H

_st is given by equation (2.10).

Both the equations can be made dimensionless by means of the following definitions of the dimensionless flow and pump speed:

I2=C

g, =: q·r-JrH-st ~ w _. w .~~­ -p -. P st

With these definitions the dimensionless flow characteristic for con-stant head becomes:

'2

3-w -2 -p

and the dimensionless efficiency curve for constant head becomes

(2.16)

= ₃

w

-p

(2.17)

These characteristics have been drawn in figure 2.3.

2.4 Characteristics for static + dynamic load

Similar results as in paragraph 2.3 can be derived when the pump is

loaded with a static plus a dynamic load. such as the resistance of a

pipeline. The dynamic load is. i f Reynolds influence is negligible.

proportional to the square of the flow. The total head then has the

form:

2

H

=

H

_st + a-q wi th a the pipeline resistance factor

The same results as in paragraph 2.3 are found when c in all the

equa-tions are replaced by (a+c). The shape of the dimensionless flow charac-teristic remains unchanged as long as c in the definition of the dimen-sionless flow is replaced by (a+c). The dimendimen-sionless pump efficiency

characteristic (2.17) turns into a pump + pipeline efficiency

characte-ristic by replacing ~ /~ by the expression:

p pmax

~p+l

_

J

a : c

~pmax

The efficiency ~p+l is defined as:

~p+l

=

p -g-H

w st

P p

2.5 Comparison with available data

The characteristics as calculated above have been compared with some available data. As stated before the parabolic shape of the H - q curve has been checked for several pumps. The assumption turned out to be very reasonable (appendix BI and B2).

The iso-efficiency curves as calculated however are different from those found in literature. Figure 2.4 shows the iso-efficiency curves as given by Fuchslocher and Schulz [1] together with some parabolas. For small heads there is a difference, but in the neighbourhood of the design point of the pump the difference is small.

The power, flow and efficiency curves have been compared with measurements on a Stork centrifugal pump, executed at the

ur

Twente [6] The results of this comparison are shown on appendiX CI to C4. They show

that the model is very accurate as long as the head over the pump isn't too far away from its design value (In that case ReYnOlds influence is not negligible).

Figure 2.4

16

Chapter 3: <XlJPLING TO AWIIURJIlIl

3.1 Description of the system

In chapter 2 is stated that the pumpshaft power P

p is proportional to

the cube of the pumpspeed ~_p. The power characteristic of a windrotor is

given by the powercoefficient Cp as function of the tipspeed ratio A. Figure 3.1 shows a typical Cp-A curve of a slow running rotor .

-

). ep€ c..,-....

----.UI..--- - - - -c,..-••

/ '

I

_~

/

I

~

;

/

.

I

1\

/

\

•

,

.\. I

•

•

u u

..

,

...

Figure 3.1. 1 __"=l 2 - p₂ ·y-·...·R a and wi th P

R output power of the rotor at windspeed V [W]

peed V [s-l] rotational speed of the rotor at winds

For a given point on the ep-X curve the output power of the rotor is given by :

{3.3}

Figure 3.2 shows schematically a rotor coupled to a centrifugal pump via

a fixed transmission. Rotor - - g.._ _....i. 11t - - - - Transmission - - - - Pumpshaft - - - Pressure pipe - - - Centrifugal pump Suction pipe -Figure 3.2.

The transmission is characterised by the transmission ratio i = (o)p/~

and the transmission efficiency ~t.

For this system the power balance is

P = ~ .p

18

With P

p according to (2.6) and PR to (3.3) and i = wp/wR equation

(3.4) results in:

(3.5)

If T}t is assumed to be constant (This is not generally true, but the

assumption is neccesary in order to keep the model simple.). it can be

seen from (3.5) that the system will always run at one point of the

Cp-X curve. Both _Cp and Xwill then be constant.

3.2 Design formulas

In the previous paragraph i thas been derived that. if a centrifugal

pump is coupled to a windrotor, the rotor will run at a constant _Cp

value. In this paragraph design formulas will be derived in order to be able to design a the system so that it works optimally.

The starting points of the design are the head "d' the required design

flow qd and the design windspeed Yd. It is assumed here that C_Pmax'

X_opt' T}tand T}_pmax are known. The goal is to be able to determine the

rotor radius

R,

the pump parameters band c and the transmission ratio

1.

In order to achieve an optimal match between rotor and pump in a given situation, two criteria must be met:

1. The maximum efficiency of the pump is reached at the design

windspeed Yd.

2. The rotor runs at C

The radius of the rotor can be calculated from the power balance at the design windspeed:

P

=

1) -1) -P

H tpmaxR

This gives for the radius R:

(3.6)

R

=

1) -1) .C -p

-~-v

t pmax Pmax a d

(3.7)

The flow at the design windspeed is now qd. According to the first

criterion. at the design windspeed the efficiency of the pump has to be maximum. The maximum efficiency of the pump lies on the parabola

2

H

=

2-c-q (2.15). So. the first design formula becomes:

EtiJ

=

H2 2-q

d

(3.8)

The rotor has to run at C

Pmax and Aopt. If this is the case, the rotor

speed at the design windspeed, w

Ropt' is:

A_opt.Vd

20

With i

=

wp/w

R and equation (2.11). the second design formula becomes:

.2 b

1 - (3.10)

3.3 Restrictions

With the first design formula (3.8) it is possible to calculate the

required value of c.(Or c+a if pipeline resistance plays a role.)

For the choice of a set of values for i and b. the second design formula

can be used. This formula still gives some freedom in selecting i and b.

There are however some restrictions in this selection.

The first restriction is that the pumpspeed may not exceed the maximum allowable pumpspeed at high windspeed. The maximum pumpspeed is usually given by the manufacturer and is fixed by the maximum allowable torque

at the pumpshaft. If the windspeed. at which the rotor runs at its

maximum speed. is the rated windspeed V • this means that for i the

r

following restriction is valid:

W -R

i

< _ ....

pm~ax:-:-__

X-V

r

(3.11 )

The second restriction is that the pump should not work too far away from its design speed wpd. This means that. for example. if the head is too small for the chosen pump. that pump will work in the area in which

the maximum efficiency is low ( I.e. the lower part of figure 2.4 ).

area. The best choice would be to match the rotor and the pump in such a way that the pump runs at its design speed 6)pd when the windspeed is Yd. This choice leads to:

(3.12)

It might be the case that it is impossible to choose the transmission ratio i with equation (3.12) without breaking the maximum speed rule (3.13). In that case i must be chosen smaller. Attention must be payed not to choose the transmission ratio to small. because of the decreasing maximum efficiency of the pump at lower Pumpspeeds. For the Stork pump measured at the UT Twente the maximum efficiency decreases rapidly for

pumpspeeds less than about 60% of the design pumpspeed. With this. the

second restriction becomes:

(3.13)

If it is not possible to comply with both the restrictions (3.11) and

(3.13). another pump with a higher value of 6)~6)pd should be chosen.

or the safety mechanism of the mill should be changed so that the rated windspeed V becomes smaller.

22

3.4 Use of the design formulas

In this paragraph it is shown how to use the design formulas derived in the previous paragraph by way of an example.

Assume the following values are given:

(npd

=

1450 r.p.m) • (n = 3600 r.p.m) pmax Rotor Design specs Transmission Pump C_Pmax

=

_{0.34 • Aopt}

=

2 -3 3 H

=

6 m • qd

=

5-10 m /s • V d

=

4.5 m/s • Vr

=

10 m/s 11_t

=

0.8 11_pmax = 0.75 • wpd = 152 radls W

=

377 radls pmax

With the design formulas and the restrictions now can be calculated:

(3.7 ) ---+ R

=

5.1 m (3.12) ---+ i

=

86.0 (3.11) ---+ check that i

<96

(3.8 ) ---+ c

=

1.20-105 s2/5m (3.10) ---+ b

=

3.90-10-4 ms2

O.K.

The results show that the transmission ratio in this case is quite

large. It could be chosen smaller with equation (3.13). Then .i would be

This means that the speed of the chosen pump is too large. or the tip speed ratio of the chosen rotor is too small.

The design formulas can also be used to find a suitable pump and match it with a given rotor in a given situation. In this case R is given. the design flow qd can be calculated with (3.7) and b. c and i can be deter-mined.

3.5 Output prediction

When the parameters b. c and i are chosen via the procedure explained above. the output flow and the overall efficiency of the system can be calculated as a function of the windspeed V.

The output flow can be determined using equation (2.16). with band c

according to the design formulas. For w_-p

=

1 the dimensionless flow ~ is

equal to 1 and the efficiency of the pump is maximum. For the optimal

matched system this point lies at V

=

Yd' The real flow at this point is

qd' Because of the fact that the system runs at constant A. w

p is

pro-portional to V. Also w_-p is proportional to w ._{p - p}So. for w can also be

written VlV_d and for ~ can be written q/qd'

The output flow of the optimal system then becomes:

(3.14)

For the calculation of the overall efficiency. ~_p according to (2.17)

can be used. If the system is optimally matched, the value of Cp can be taken equal to C

24

be constant. The overall efficiency then becomes:

J

1)tot

=

C_Pmax-1) -1)_{t pmax}

-2 3-(VlV d) - 2 (VlV d)3 (3.15)

The curves represented by (3.15) and (3.16) are similar to the dimen-sionless curves as drawn in figure 2.3. By way of simple rescaling the dimensionless

same way the - V curve.

~ - ~ curve can be transformed in a q - V curve. In the

-p

dimensionless 1)_p - ~_-p curve can be transformed in a 1)

~t

3.6 Comparison with the piston pump

The main difference between a centrifugal pump and a piston pump coupled to a wind rotor is the fact that the piston pump runs approximately at a constant torque independent of the rotational speed while the torque-ro-tational speed characteristic of the centrifugal pump is approximately a parabola. This results in the fact that in the case of a piston pump the rotor doesn't always run at C

Pmax' However, while the piston pump

appro-ximately has a constant efficiency, the efficiency of the centrifugal

pump depends on its rotational speed.

In [8] the following equation is given to describe Cp as a function of

VlV_d for a piston pump coupled to a wind rotor:

Ad

]

- - ) -C

A Pmax

max

If the efficiency of the piston pump Tl

pp and the efficiency of the transmission Tl

tp are assumed to be constant. the total efficiency of the system with the piston pump is:

n - n -n -C

"totp - "tp "pp P (3.17)

The efficiency of the system with the centrifugal pump is given by equation (3.15). For the assumptions A_max/Ad = 2. Tl_tp

=

_{Tlt and Tlpp}

=

Tlpmax' the total efficiency of the system with the centrifugal pump and the efficiency of the system with the piston pump' is given by:

J

3-(VlV

d)2 - 2

2 - (V

dlV)2

(3.18)

This quotient is tabulated in the following table for several values of

VlV d Tltot VlV d Tltot VlV d Tltot VlV d Tltot

Tltotp Tltotp Tltotp Tltotp

1.00 1.00 1.50 0.93 2.00 0.90 2.50 0.89 1.10 0.99 1.60 0.93 2.10 0.90 2.60 0.89 1.20 0.97 1.70 0.92 2.20 0.90 2.70 0.89 1.30 0.96 1.80 0.91 2.30 0.89 2.80 0.88 1.40 0.94 1.90 0.91 2.40 0.89 2.90 0.88 Table 5.2

26

This table shows that the efficiency of the system with the centrifugal pump is slightly less than the efficiency of the system with the piston pump for A_maxlAd

=

2.

A great advantage however of the centrifugal pump is that there is no starting problem. A general comparison of the two systems for values of VlV

d smaller than 1 is not possible because of the fact that equation

(3.16) is then not valid.

A disadvantage is that a rotating transmission is necessary when centri-fugal pumps are applied.

3.6 Conclusions

In this chapter it was assumed that the head over the pump is static only. The results however can also be used if the head is static plus

dynamic, like a pipeline resistance. As stated in chapter 2 in all

formulas c then has to be replaced by c+a, if a is the pipe resistance

factor. In the second design formula (3.10) the left term will then be

a+c. In equation (3.16) describing the predicted overall efficiency,

also a has to be replaced by c+a. The overall efficiency is then the efficiency of the whole system, including the pipelines.

A great advantage of the centrifugal pumps, cOmPared with the piston

pumps, that are now used by the CWO, is that the power characteristic

fits much better to a windrotor. Piston pumps have approximately a

constant torque characteristic. This means that at varying windspeeds the rotor will not always run at its maximum Cp - value. The centrifugal

pump has a parabolic torque characteristic. That is why the rotor, if

matched optimally, runs at C

Pmax at every windspeed. The disadvantage of

the pumpspeed. The efficiency of the piston pump is almost constant at

every speed. In all individual cases. comparison of the two types is

neccesary to decide which pump gives the best performance.

A great disadvantage of centrifugal pumps is that they are usually

manufactured to run at relatively high speeds. This means that in combi-nation with the CWO rotors a rather large transmission ratio will have

to be used. The largest transmission ratio that can be realised in one

step is for gears about 5. If the transmission ratio calculated with the design formulas is larger than about 25. the transmission has to consist

of three steps. This is disadvantageous from the efficiency point of

view. In the selection care should be taken to choose pumps with rather small design speeds. Also fast-running rotors are preferred.

Another disadvantage of the system with the centrifugal pump is that the efficiency of rotating transmissions is depending on the momentary speed of the wheels. This is not taken into account in this report. In assu-ming an efficiency of the transmission a not too high value should be taken. because of this effect.

Chapter 4: CALaJLATI(Jf OF TIlE QUALIlY AND AVAlLABILIlY

OF TIlE OPfIJuzm SYSTEJI

28

4. 1 The quali ty

The pattern of the wind distribution. the so-called windregime. in many areas in the world is best represented by the Weibull ditribution. The probability density function of the Weibull distribution is given by:

(4.1 )

with k: the dimensionless shape factor

V:

the average windspeed

f: the Gamma function

In figure 4.1 taken from lit [8] this function has been drawn for

seve-ral values of k. The shape factor k says something about the width of

the distribution. If k is small the windspeed varies in a relatively

wide range. if k is large the wind speed varies in a relatively narrow

range. for example in areas where trade winds blow. The shape factor

usually lies between 1.5 and 4.

With the help of the wind regime given by the Weibull distribution it is

possible to calculate the average yearly output q of a given system in

ay that windregime. It can be calculated with:

co

qay

=

J

q(V)-W(V) dV

o

t.

:---r-.--

---k.:

r- --

"r----l -

-j

-1

"\-1-

t --

~----\---

---t----

I

--~

I , I I i I I :

"., f·

i---r-

i- --

-t- -

t----+-I

'

=1

1 -- - - :

I;

_{•• I---- - ;}

I

l'

_{- -} I

I

-~"-- ----'1 I

I

I

I

...

~~~---t---t-' -·t--~

: I+-f---++---+---

----++-~;:t_________t_I

-

--_J

I -V

a--

V Figure 4.1 In the previous chapter it was derived that the output q of the optimized system is given by:

(4.3) From this equation can be seen that the system starts to deliver at the windspeed V.

=

&3 ·

V

d

ln

Usually the system is protected against damage at very high wind speeds. Here is assumed that the safety mechanism works in such a way that above

the rated windspeed V the delivery remains constant and equal to the

r

delivery at the rated windspeed. q . In heavy storm condi tions the

r

system is completely stopped and the delivery is equal to zero. This happens at the cut out wind speed V . This fact is neglected for the

co

time being. So the output of the system is given by: . q = 0 q

=

qd-J 3-(VlV d)2 - 2 q

=

qr

=

qd-

J

3-(V rlVd)2 - 2 for V

< ..

'/2/3 ~Vd for

J

2/3 -V d

<

V

<

Vr for V

>

V r

30

The average yearly output of the system is then:

CIO

- W(V) dV + qr-J W(V) dV

V r

(4.4)

With the help of the average yearly output it is possible to calculate

the quality factor of the system. The quality factor says something

about the functioning of the system in a given wind regime. The higher

this factor is. the better the system works in the wind regime. It can

be used in comparing two systems in order to determine which one is the best from the output point of view. The quality factor can be defined in

several different ways. When comparing two systems by means of the

quality factor. care should be taken that for both the systems the same

definition is used. Here it is defined as the average yearly output

power divided by the output power at the average windspeed if the design

wind speed is equal to

V

(Only in that case ~

=

~ at

V).

The

ave-P PmaX rage yearly output power is given by:

P_ay

=

p_w-g-H_st-q_ay (4.5)

The output power of a given system at the average windspeed

V.

if the

design wind speed V

d is equal to the average wind speed

V

is given by:

With the design output qd being:

c

.~.~

.1-

p .T.R2.~

Pmax t pmax 2 a d

pw·g·Hst

and equations (4.4), (4.5) and (4.6) the quality factor a becomes:

(4.7) co 3.(V r

lV

d) -

2.J

W(V) dV V r

(4.8)

With this integral the quality factor a is a function of the wind speeds

V,

V

d, Vr and the Weibull shape factor k. With the definitions:

xd := _{Vd/Y , xr} = Vr/Y

,the quality factor in a given wind regime ( k is given) can be written

as a function of x

d and xr.

For k

=

1.5, k

=

2 and k

=

4 the qual! ty factor is calculated as a

function of x

d wi th xr as parameter. This has been done on the

appen-dices D1 to D10.

4.2 The influence of the cut out wind speed on the quality

In the previous paragraph is assumed that there is no cut out wind speed

V , or in other words V

=

co. This was done in order to reduce the

co co

number of variables on which a and

P

depend. Similar to the definitions

of x

d and x , xr co can be defined as xco = V /Y.co

The influence of x on the quality a can be investigated by replacing

co

32

results in the fact that in the integrals 1

2, 14 and 16 the upper

boun-dary ~ must be replaced by x . The new values of a including the

influ-co

ence of x , a ,become:

co co

(4.9)

with a

co the quality if xco is not neglected

a the quality as calculated in appendix 0

Calculating several values of aco for some realistic values of xd ' xr

and x learns that the influence of x at k = 2 and k = 4 is only

co co

minor. For k

=

1.5 the influence of x can be significant. For example

co at x

d

=

1.2 , xr

=

1.6 and xco

=

3 the difference between a and aco is

0.037. However, there is hardly any change in the value of x

d at which

the quality is maximum (for a given x ) if x is not neglected, as long

r co

as x is not taken smaller than approximately 3.

co

4.3 The availability

The availability can also be defined in several different ways. Here,

the follOWing definition is used: the availability is the fraction of

the total time that the output flow is equal to or larger than 10% of

the design flow is determined by:

( 4.10)

So. the wind speed V

O.1 at which the flow is 10% of the design flow is:

VO. 1 =

J

_{0.67 • Vd} (4.11)

In a windregime characterised by the Weibull distribution the

availabi-lity ~ can be found with:

CIO

~

=

J

W(V) dV

VO• 1

(4.12)

In this equation

V

is not taken into account also. If it is desirable

co

to take

V

into account. the upper boundary of the integration CIO must

co

be replaced by V . In appendices D11 and D12 ~ is calculated and drawn

co

in a figure for several values of x

d and xr for k

=

1.5. 2 and 4.

4.4 The influence of the cut out wind speed on the availability

As stated in the previous paragraph. the influence of the cut out wind

speed on the avallabll ity can be determined by replacing the upper

boundary of the integral (4.12) CIO by V . For k = 2 and k = 4 this

co

results in values that hardly differ from the calculated values of ~ for

34

significant. This influence can be calculated by substracting

4 - O.3030-x

co e

from the calculated values on appendix O.

4.5 Analysis of the results

If the system is designed for maximum quality, the results shown in

figure 1. 2 and 3 on the appendices 04. 07 and 010 can be used.

Figure 1 shows that the influence of x on the quality for k

=

1.5 is

r

rather big. If x

r is chosen 1.4 the best xd from the quality point of

view is 1.25; if x

r is chosen 2.5 the best xd is 1.7.

Figure 2 shows that, if k

=

2. taking x larger than 2 has hardly any

r effect on the quality. The optimum x

d doesn't vary quite as much with xr

as for k

=

1.5. For x

=

1.4 the optimum x

d is 1.20; for x

=

2 the

r r

optimum x

d is 1.35.

Figure 3 shows that the influence of x on the quality and the optimum

r

x_d is even smaller. Taking x

r larger than 1.8 is not useful. The optimum

x_d varies between 0.95 and 1.05 for 1.0 < x

r <1.8.

If the system is designed for high quality. it can be seen from figure 4 on appendix 012 that x

d should be chosen as small as possible. This is

trivial because of the fact that the smaller V

d (and so xd) is chosen

the sooner the system starts to deliver. Taking x

d very small however

results in a low quality.

be made. The way this compromise is achieved depends on local demands as, for example, whether the system is designed to deliver water in a critical period or whether the system is designed for high yearly output

etc. If high quality and high availability are both very important,

36

(])apter 5 Sensitiyity analysis

If x

d is chosen it is possible to design an optimal system for a given

situation with the help of the design formulas derived in chapter 3.

Starting points are the site specifications

V

and H

st' If the system has

been designed the rotor radius R. the pump parameters b. c and d and the

transmission ratio i are known. In this chapter the sensitivity of the

system to

V

and H

st is investigated.

5.1 Sensitivity to H t

s

In order to investigate the sensitivity to H

st' the head for which the

system was designed H

O is replaced by another head HI'

In chapter 2 it was stated that the pump parameter d is independent of the head. This results in the fact that the match between the rotor and

the pump. once chosen optimally. remains optimal for any head HI not

differing much from the head H

O' The rotor also runs at CPmax and Ad for

the new head HI' With the transmission ratio unchanged the pump speed at

a given wind speed is the same for both H_O and HI: wpl(V)

=

wpO(V).

The dimensionless efficiency and the flow of the centrifugal pump as function of the dimensionless pump speed ware given in figure 2.3.

-p

With w_pl

=

WpO and b remaining unchanged. the quotient of the

dimension-less pump speeds is:

(5.1)

And with ~p

=

VlV

HI is identical to a system optimized for V_d = _{Vdl with:}

(5.2)

The flow at the new design wind speed of the system with the head HI fol lows from equation (3.8):

(5.3)

The flow and the efficiency of the system at the new head HI can now be

determined as a function of the windspeed wi th the help of equations (3.14) and (3.15) replacing V

dO and qdO by their new values Vd1 and qd1·

Figure 5.1 is shows what happens i f _{for a system designed for Vd}

=

4 mls

and a head of H

O

=

4 m the head changes to 1. 2. 6 or 8 m.

~_III

as

Figure 5.1

V

[mls]

Quali ty and avallabili ty of the system operating at the new head can be

38

5.2 Validity of the results

Figure 1 shows that the maximum efficiency of the pump is independent of the head. This is not exactly true. Actually the maximum efficiency of the pump decreases when the pump is used for other heads than the head the pump was designed for (See appendiX C4). The only available data about the efficiency of a centrifugal pump running at a different head

is the Stork pump mentioned before. Its maximum efficiency decreases

rapidly for pump speeds less than about 60% of its design pump speed.

With equation 5.2 this means that the minimum head for which the above derived results may be used for this pump will be' HI

=

O.36.H_O.

The efficiency of the pump will also decrease if it is used for larger

heads than the head the Pump was designed for. However. this decrease is

only small and not very significant.

5.3 Sensitivity to

V

The influence of

V

on the system can be found by investigating the

influence of x

d on the system. An incorrect chosen value of

V

has no

influence on the calculated output of the system at a given windspeed:

the parameter used In the output prediction is not

V

but Vd. The

Influ-ence on the qualIty and the availability can be found by replacing the

incorrect value of x

d

=

VdlV by the correct one. The effect can be seen

6.1 Introduction

The last part of this thesis was to try to apply the theory in a real

situation. It was decided to try to adapt the windmill CWO 5000 (R = 2.5

m, C_Pmax::O.35 at Ad = 2) for lifting large amounts of water over a

relatively small head. The assumed site specifications are the

following:

Static head: H

=

3 m.

Average wind speed:

V=

3.5 mls.

Wind distribution: Weibull with k = 2.

The transmission ratio i is assumed to lie between 5 (one stage) and 25 (two stages). With (3.12) and x

d

=

l,the following design pump speeds

are found:

i

=

5 ---+ (,Jpd

=

14 rad/s or npd

=

134 r.p.m.

i

=

25 ---+ (,Jpd

=

70 rad/s or npd

=

668 r.p.m.

An

examination of pump manufacturer's data shows that available

centri-fugal pumps usually run between 1500 and 3000 r.p.m. Increasing i is

disadvantageous because of the negative effect on the efficiency when

three or more stages of transmission are used. Increasing the design

wind speed Is possible but has a negative influence on the availability. The options are then:

1. Designing a centrifugal pump with a low design pump speed.

2. Using available pumps at a much lower r.p.m. than they have been designed for.

6.2 Design of an impeller

An example of how the main dimensions of an impeller for a centrifugal

pump are calculated is given in the appendices El to E3. From the

equa-tions used in the design process i t can be seen that decreasing the

transmission ratio results in an impeller with a larger diameter and a smaller width. This leads to a higher resistance and a lower efficiency.

So. if the pump is to be driven by a slow running rotor a large

trans-mission ratio is unavoidable.

Building a prototype and testing is neccesary in order to find out the real performance of the designed impeller.

6.3 Measurements

The most suitable pump (b.c and c.Jpd) can be calculated by using the

design formulas derived in chapter 3. Finding a suitable pump from

manufacturer's data is possible by using equation 2.3. If the

manufac-turer gives the specifications HI and ql at the rotational speed c.J

1 the head H

2 and the flow q2 at another rotational speed c.J2 are given by:

c.J 2 c.J

H_{2 =} H_{1 ·} (_2_)_c.J and (...£.)

1 q2 = ql· c.J1

In this way a suitable pump can be chosen when the design rotational

speed is known.

A problem is that manufacturers don' t give the efficiency at other

rotational speeds. According to the model derived in this report the

maximum efficiency of the pump at another rotational speed is equal to

speed does not differ too much from the design speed. In order to find out how much the efficiency decreases with lower speeds. some measure-ments have been executed on a centrifugal pump at the University. This

pump. the NOWA5026. was put at our disposal by a Dutch company

SIHI-Maters B.V. Beverwijk. The setup and the results of these tests are given in the next paragraphs.

6.4 The setup of the test rig

For the testing of the pump a testrig was built at the University. The centrifugal pump is driven by an electric DC machine. The head over the pump is controlled by a valve in the outlet. Figure 1 shows the setup of the test rig.

The items that have been measured are:

1. The rotational speed of the pump. measured with a magnetic contact

and a pulse counter (accuracy: 0.5%).

2. The torque on the pump shaft. measured with a torque measuring device (British Hovercraft Corporation. Transducer type TT2.4.BBS) between the electric motor and the pump (accuracy: 3%).

3. The static pressure over the pump. measured with a manometer filled

with mercury between inlet and outlet of the pump (accuracy: 2.5%).

4. The flow through the pump. measured with a flow meter (Flowtech

Variomag. type Discomag OMI 6531) in the outlet of the pump (accuracy 1%)

42 1 Pump 2 Motor 3 Flow meter 4 Valve 5 Torque meter 6 Manometer

1 Inlet pressure measuring point

8 Outlet pressure measuring point

9 Storage tank

10 Electronic motor control

and measuring equipment

6.5 Processing the measured data

In figure 1 can be seen that there is a difference between the height of the point where the inlet pressure and the point where the outlet pressure is measured. The lines between the pressure measuring points and the manometer as well as the volume in the manometer above the mercury are filled with water. The pressure that is measured in the manometer can be calculated in the following way (see figure 2):

The pressures in the legs of the manometer above the mercury are: and

Because of the equilibrium between the two legs. the relation between P3 and P4 is:

With:

the equations above yield:

p - p + P egeH

=

(p - p )egeh

2 1 w m w m

Ihe difference between the energy pressure in the inlet and in the outlet of the pump is

1 2 2

APe = P2 - PI + pw egeH +

2

ePwe(V_{2 - vI)}

(VI and v

2: velocities at inlet and outlet)

So. if h_m in the manometer is measured. the energy pressure can be calculated with:

1 2 2

Ap

=

(p - p )egeh + -dp e(v - V )

e m w m 2w 2 1

The velocities v

2 and vI can be calculated by dividing the measured flow by the area of the cross sections of the inlet and the outlet pipe. These areas are:

At the inlet measuring point At the outlet measuring point:

2

0.00302 m 2 0.00189 m

The net hydraulic output energy is:

3 (q : the measured flow through the pump in mIs)

The input power at the pump shaft is:

Pin

=

Ie" (1 the measured torque at the pump shaft in Nm " the rotational speed of the pump shaft in radls)

The efficiency of the pump is:

1)_p

=

Ph d /P._{y r 1n}

The accuracies of the calculated values of the efficiency and the input power are derived from the accuracies of the measuring devices:

In the input power: 1.03

*

1.005 = 1.055 ----+ less than -IX

In the efficiency : 1.03 *1.005

*

1.025

*

1.01 = 1.09 ----+ less than 1%

The deviations given above are the deviations calculated in the worst case. Deviations caused by other phenomena, such as a non-uniform flow at the pressure measuring points etc, are assumed to be much smaller than the deviations given above.

The results of the tests are given in table 1 on appendices F3 and F4 and in the figures in the next paragraph.

6.6 The results

The values taken from table 1 in the appendices F3 and F4 have been .put

in figures 6.3 and 6.4. The measured input power at 480 r.p.m. (see

figure 6.3) deviates from the measured power at other speeds very

strongly. There is no reasonable explanation for this. The measured

values at speeds lower than 480 r.p.m. are quite inaccurate. For these

reasons the measurements at speeds of 480 r.p.m. and lower are not taken

into account in the conclusions.

In figure 6.3 the measured points of the input power are given as a

function of q for several values of ~. The curves connecting the points

for a given speed seem to be straight lines with a positive tangent. In figure 6.4 the measured points of the H-q curves at several speeds

2 2

are given. The best fitting parabolas of the type H

=

b·~ - c·q were

given in table 2 on appendix F5. These parabolas are also given in

figure 6A. Other curves that have been drawn in this figure are curves

connecting points with constant efficiency; the iso-efficiency curves.

and some parabolas of the type H

=

p.q2 • with p arbitrarily chosen.

In order to verify equations 2.1 and 2.2 and to check the influence of the Reynolds number. the measured values have been plotted in figure 6.5

3 2

and 6.6 as Pinlw and Hlw as functions of q/w. In these figures also

some points at pump speeds of 1450 r.p.m. and 2900 r.p.m. are given.

These were calculated from the data supplied by the manufacturer (figure 1 on appendix F5)

480

r.p.m.

r.p.m.

r.p.m.

P.

_1n

[W]

~o 900

r.p.m.

ocP

o

++

+

840 r.p.m.

ocP

0

++'

+

DO

+ + + /

780

r.p.m.

eP

+

0

o

0

++

0

cD

,+*

720

r.p.m.

+ +

0

-tt-

T

+

OdJ

-t+

~

o

++

~

660

r.p.m.

0 0 ....

+

00

o

+

+

+

DO 0 )()(Xx)()(

600

r.p.m.

+

+

00

~+<~

o

0

~ ~

OCDCD

540 r.p.m.

o

~x OCD xX

a:P

0

o

xxxxx

420

XX X

+

+x

++++

360

900

1000

Figure 6.3: The measured input power

H

l'

~:::::::::r----HH-tf~~~----+~r--~

(m.w.c)

R"1'--T-~f+-F~H""':'~~~~--+---I---U

1

2

3

4

5

6

7

₈

_q

_O/s)

•

Figure 6.-4: The H-q curves, the iso-efficiency curves

P

3

_Cal 48

15

14

13

Q ...~~o o~~

12

G "...~'2[0 <>~+

--11

_{tJr:p~· ~}

D+v.

1J

10

.

ctJ

c.o..ti<>

X

+t¢.."'!1<l

9

cflr4

:...L-a

co~ ~

I

0540

r.p.m.

~~

D

600

r.p.nl.

7

%~9.

.

+

660

r.p.m.

6

It)( .0..

720

r.p.m.

x

780

r.p.m.

0840

r.p.m.

'V

900

r.p.m.

T

1450 r.p.m.

L

manufactur~r6

X

2900

r.p.m.

I

data

1

2

3

4

5

6

7

8

9

10

.-9-Cal 3

Figure 6.5: P/Cal as function of q/Cal (Measured points)

P -7 -5 CI

Best fitting line:

3 =

5.6-10' + 9.7-10 -~

Cal

with q in l/s Cal in r.p.m.

00 54<3 r.p.m. • 600 r.p.m. t 660

r.p.m.

• 720 r.p.m • .. TBO r.p.m. ~ 840

r.p.m.

.. 900 r.p.m.

+

1450

r.p.m.

L

manufaoturere

X

2900 r.p.m.1 doic

12~_,,;--;Ilt;;;;"'~.A;~~..o

_·J~o~

+

_X

11

I ~o..a -=l.L 0·" ..fI!

f""-a

₂

H

~. 1l7.~... w ~ 0

9

•

~

_~ ~~

8

"

+,a '\IIi

7

6

5

4

3

2

1

1

2

3

4

5

6

7

8

q/w

9

10

Figure 6.6: Hlw2 as function of q/w

(Measured points and best fitting parabola)

H -5 -7 a . 2

Best fitting parabola:

2 :

1.21-10 - 7.47-10 -(~)

fa)

with q in lIs

H

in m.w.c.

50

6.7 Conclusions

The conclusions that can be drawn from the measurements are the fol-lowing:

1. Rules of similarity

Figures 6.5 and 6.6 show that the rules of similarity are quite

accu-rate. If H. q and Pin are made dimensionless. the H-q curves and

2 3

P_in- q curves can be represented by one H/w - q/w and one Pin/w

-the q/w curve very well. The maximum deviations from these lines are

approxima-2 3 '

tely 1% for the H/w -q/w curve and 8% for the P/w -q/w curve. This means

that the maximum efficiency of the pump TJpmax hardly changes with

changing pump speed. In figure 604 can be seen that the decrease of the

maximum efficiency for decreasing pump speed is only very small indeed. From 900 r.p.m. to 540 r.p.m. the decrease of the maximum efficiency is

approximately 4% (from 52% to 48%). The data supplied by the

manufac-turer show that the maximum efficiency decreases 2% in the range from

2900 r.p.m. to 1450 r.p.m. These data don·t fit in very well in the

measured curves. It is possible that there is a rather large decrease of the efficiency between 1450 and 900 r.p.m. It is also possible that the values of H. q and Pin given by the manufacturer are a little optimis-tic. In a wide range of pump speed it is possible to use the

dimension-less curves to calculate the performance of the pump very well.

2. Deviations of the model

The assumption of the H-q curve being a parabola is only partially valid

(see figure 6.4 and 6.6 ). In the measured range the maximum deviation

from the measured points and the best fitting parabola is approximately

is in the middle. the best thing to do is to try to have the best fit of the parabola there. The largest deviations are then to be expected in the left part of the H-q curve. The system then starts to deliver at a somewhat higher speed than calculated.

The iso-efficiency curves as drawn in figure 6.4 can be represented by parabolas quite well.

A big difference with the model is the fact that the input power at a given speed is not independent of the flow. The assumption in the model

was that the input power is given by P. = d_w3. This would result in a

. In

horizontal line in figure 6.5. The measured points in figure 6.5 seem to fit in a straight line with a positive tangent. So. a better equation for the input power is:

3 2

Pin

=

f-w + e-q-w (6.1)

with e: the (positive) tangent in figure 5 f: the intersection of the line with the

vertical axis

6.8 Consequences for the model

The measurements show a rather small decrease in maximum efficiency for decreasing speed. This means that it is reasonable to take the maximum

efficiency at a low pump speed the same as the maximum efficiency at the

design speed. Care should then be taken. because the maximum efficiency

given by the manufacturer could be a little optimistic. This was also a

conclusion of Roorda [6].

The consequences of the fact that that the input power at a given speed is not independent of the flow are shown in paragraph 6.8.1.

52

6.8.1 Consequences for the efficiency

The function of q/w is, according to the model, given by:

(6.2)

The maximum efficiency is then at_.

~

_{w · c}

=

J3

b

According to the measurements, the efficiency of the pump becomes:

(6.3)

For this equation the value q/w where ~ is maximum, can't be found in a

p

simple way. An analysis of the way the calculated value of 1) changes

p

due to the change in the model can be made as follows: first, the

straight line in the denominator of

(6.3)

is rewritten with the

defini-tion d

=

f +

e·J3~C'

The calculated and the measured value of T)Pmax are

now equal at the point

~

=

J3~C'

Equation

(6.3)

becomes:

=

b d + e.(~ - .-) w 'l3·c (6.-4) Dividing (6.-4) by (6.2) yields: 1 = : -e ~ b 1 +

d

(w - ~-3.-c-) (6.5)

From (6.5) can be seen that the measured efficiency increases at speeds lower than the speed where (6.2) is maximum (;

=

~3~c)

and decreases at speeds larger than this speed. As a result of this the maximum of the measured efficiency as drawn in figure 2.3 shifts to the left. This efficiency curve is to be multiplied by the right term of (6.5). The efficiency curve starts steeper. the maximum shifts to the left and the efficiency decreases more at the right of the maximum as drawn in figure 2.3. In figure 6.7 the efficiencies as a function of w for a constant head of H

=

5 m.w.c. for the tested pump are given. calculated with the model (by means of the best fitting curves) and calculated from the measured data. 1.00 0.75 0.50 0.25 )( model T meosurements x )( 100 200 300 400 500 600 700 800 900 1000 wp [r.p.m.] Figure 6.7

6.8.2 Consequences for the load on the rotor

The centrifugal pump modelled in the way that has been done in this

report caused the rotor to run at constant A. Because of (6.1) this will

not exactly be true. If the system was designed to run at A_opt and at

Tlpmax at the design windspeed. (6.1) has the following consequences:

At higher wind speeds the required input power increases more than with

the cube of the speed and at lower wind speeds i t decreases more than

with the cube of the speed. This causes the rotor to run at lower A at higher wind speeds and at higher A at lower wind speed.

This causes the pump to run at a higher efficiency in a wider range. The overall efficiency of the system however decreases at wind speeds above the design windspeed. because of the fact that the pump runs slower than calculated. The flow at this lower speed is also smaller.

An

analytical examination of the new model is very difficult because of

the cubic functions involved. Exact results can only be found by substi-tuting numerical values for all the parameters involved. As an illustra-tion of the change in the load on the rotor in figure 6.8 the required input power at the pump shaft is given as a function of the pump speed for the tested pump at a constant head of 5 m.W.C. as calculated with the model and according to the measurements.

900 + p

[W]

+x +'11. 800 +'11. 'II.

model

+'JI. 'II. 't'oI. 700

measurements

-t>c. + ..'II. lie ~ 600

,

.;l

500 lIM 'll.xt 'JI.~ '11.+ 400 'JI.'JI.+ 'II. + >ex + 300 lIe x

₊

>e + lie x + 200 tt" 100

...

100 200 0300 400 500 600 700 BOO 900 1000 fa) [r .p.m.] p Figure 6.8

56

Chapter 7: SOME CENERAI. REMARKS Of TIlE SYSTEM

In the previous chapters the main object of study was the pump. In this chapter some general remarks on other parts of the system are made.

7.1 The transmission

In the previous chapters· the importance of limiting the transmission ratio is stated several times. The background of this is the following: A good mechanical transmissions (e.g. gearing wheels or belts) has a high efficiency if it is used at the load and speed it was designed for.

The design of a transmission is usualy based upon considerations of

strength. So. in the calculations for the design of a transmission the

maximum occuring load has to be used. For the water pumping windmill

this means that the load used for strenght calculations occurs at high

wind speeds. If the system consists of a safety mechanism that limits

the load to the load at the rated wind speed V • this load is the one to r

be used in determining the required strength of the transmission. The

system however most of the time runs at lower windspeeds (e.g. the

average wind speed

V).

If. for example. the average wind speed is 50% of

the rated wind speed. the transmitted power is 0.503.100%

=

12.5% of the

maximum occuring load. The efficiency of steel gearing wheels depends strong on the percentage of the maximum power the wheels are

transmit-ting. At 10% of the maximum power the efficiency can decrease to 75%.

while the efficiency at the maximum power can be up to 97%. So. i f the

transmission consists of two stages. the efficiency of the total

transmission can be as low as about 55%. This explains the importance of

limiting the transmission ratio as much as possible. Further investi-gations on transmissions are neccesary.