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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
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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
In this report a mathematical model to describe the behaviour of
centrifugal pumps is derived.
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
4.5. Analysis of the results 34
II
CHAPTER 5 : SENSITIVITY ANALYSIS
5.1. Sensi tivi ty to Hst
5.2. Validity of the results
5.3. Sens i tivity 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
F1-F5 The testing of a centrifugal pump
List of symbols
III
a
b
c
D. t10P
g
H
pipeline resistance factor
pump constant
pump constant
power coefficient of the rotor
maximum Cp
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 rrp
[i/m5]
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,Aopt design tipspeed ratio
Pa density of air
Pw density of water
W rotational pumpspeedp
wpd design wp
wpopt pumpspeed at maximum efficiency for a given head
maximum WP
rotational rotorspeed
design wR
maximum wR
rotational rotorspeed after transmission
IV
[ -]
[ -]
[ -]
[ -]
3[kg/m ]
3[kg/m ]
[radls]
[radls]
[radls]
[radls]
[radls]
[radls]
[radls]
[radls]
WS
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. if 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-
VI
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.
Ton Staassen
1
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]
3qd design flow [m /s]
Hd 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 =0.5s
(J = 1.0s
Figure 1.1
(J = 1.5s
(J =2.5s
(J = 3.5s
2
As guidelines for the selection the following values can be taken [1]:
w < 0.6 radial pumps (also called centrifugal pumps)s
0.4 < w < 3 'mixed-flow' pumpss
2 < w axial pumpss
With the help of the specific pumpspeed it 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, PR,
delivered at the design windspeed Vd by a windrotor with a maximum
powercoefficient CPmax
and radius R is:
1 _3 2P = C .-.p ·V:·'JI"RR Pmax 2 a d
with Pa 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 .qH w d d (1.3)
with p : density of waterw
(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 = ~.~ .pH tpmaxR (1.4)
3
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/WR
) 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 Vd 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 Vd
~ 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 2Pw = 1000 kg/m ,g = 9.81 mls
values of p ,CPm ,~ and ~t are:a ax pmax3
Pa = 1.23 kg/m ,CPmax = 004 , ~pmax = 0.65 , ~t = 0.90
With these values equation (1.8) becomes:
V 5/2-3 dC&ls = 1.2·10 .i·Ad ·---.,;=::,...,..,.....
H 5/4d
(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 Vd , H
and i·Ad . 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 Vd = 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 Vd = 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. Ins
the example mentioned above a centrifugal pump would be the choice.
5
....
10
(; 7 8 9 10532
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/4sopt - iopt 1/2
qd(1.10)
with Diopt :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 rangings
from 0.1 to 0.25 at pumping requirements with Hd 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 • .,
,,
W5
Figure 1.3.
7
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 canp p
be written:
q = q (0. g-H. Pw' v. w)
P = P (0. g-H. Pw' v. w)P p
According to the IT-theorem of Buckingham these equations can be written
with dimensionless numbers as:
2
][ gonw -0q P
3 = f 1 2 2'w -0 w -0 vp p
P 2
][ gonw -0p = f 2P
3 5 2 2 •p -w -0 w -0 vw p p
(2.1)
(2.2)
In these equations the expression w -02/v is the Reynolds number Re.p
This means that. if Reynolds influence is negligible. for a given pump
(0 is constant) in a given situation (p and g are constant). (2.1)w
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 a3
. This has been done before in lit [4].
Assume: at = band a3 = -c. For equation (2.t) this results in:
H n ·2-2- = b - c e (.;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:
3P = deCa)p p
with d depending on the pump only.
(2.6)
9
b
w~ rp
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 ofp
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:
(2.9)
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. Thisp
gives:
I3=Hw t=~2=i)pop (2.11)
with Ca.I t: w at maximum efficiencypop 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
(2.13)
11
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 TJpmax
Equation (2.5) gives the head as function of q for a given w . For anp
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:
Heq3wp
=constant (2.14)
This turns out to be true along all parabolas
12
2H = 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:
2H = 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 staticp p p
head H t only. The q - w curve for constant head can be found directlys p
by applying equation (2.5), substituting Hst
for H. The ~p - wp curve
for H =Hst 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=Cg, =: q·r-JrH
st
~w _. w .~~-p -. P st
With these definitions the dimensionless flow characteristic for con-
stant head becomes:
13
'23-w -2-p
and the dimensionless efficiency curve for constant head becomes
(2.16)
= 3w-p
(2.17)
These characteristics have been drawn in figure 2.3.
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. if Reynolds influence is negligible.
proportional to the square of the flow. The total head then has the
form:
2H =H + a-qst 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 dimensionless 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-Hw st
Pp
IS
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
o
16
Chapter 3: <XlJPLING TO A WIIURJIlIl
3.1 Description of the system
In chapter 2 is stated that the pumpshaft power Pp is proportional to
the cube of the pumpspeed ~ . The power characteristic of a windrotor isp
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 ~;
/. 1\I
/ \• , .\. I • •
u
u
..,
...
Figure 3.1.
1 __"=l 2-p ·y-·...·R2 a
and
wi th PR output power of the rotor at windspeed V [W]
peed V [s-l]rotational speed of the rotor at winds
17
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. 11
t ---- 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 = ~ .pP t R
(3.4)
18
With Pp 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 X will 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 CPmax '
X t' T}tand T} are known. The goal is to be able to determine theop pmax
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 CPm and X tax op
19
The radius of the rotor can be calculated from the power balance at the
design windspeed:
P = 1) -1) - PH tpmaxR
This gives for the radius R:
(3.6)
R =1) -1) .C -p -~-vt 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
2H = 2-c-q (2.15). So. the first design formula becomes:
EtiJ= H22-q
d
(3.8)
The rotor has to run at CPmax
and Aopt . If this is the case, the rotor
speed at the design windspeed, wRopt ' is:
A t.VdopwRopt = -....o..:R:--- (3.9)
20
With i =wp/wR and equation (2.11). the second design formula becomes:
.2 b1 - (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 ther
following restriction is valid:
W -Ri < _ ....pm~ax:-:-__
X-Vr
(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 ).
Also the mathematic model derived in chapter 2 will not be valid in this
21
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.r
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
CPmax = 0.34 • Aopt = 2
-3 3H = 6 m • qd =5-10 m /s • Vd = 4.5 m/s • Vr =10 m/s
11t = 0.8
11 = 0.75 • wpd = 152 radlspmax
W = 377 radlspmax
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 ) 5 2/5---+ c = 1. 20-10 s m
(3.10) -4 2---+ b = 3.90-10 ms
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
0.6-86 = 51.6. This is still to large to realise in one or two steps.
23
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 = 1 the dimensionless flow ~ is-p
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. wp is pro
portional to V. Also w is proportional to w . So. for w can also be-p p-p
written VlVd 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. ~ according to (2.17)p
can be used. If the system is optimally matched, the value of Cp can be
taken equal to CPmax
' The efficiency of the transmission was assumed to
24
be constant. The overall efficiency then becomes:
J1)tot = C -1) -1) -Pmax t pmax
23-(VlVd ) - 2
(VlVd
)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) - ~ curve can be transformed in a 1)p -p ~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 CPmax
' 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
VlVd for a piston pump coupled to a wind rotor:
Ad ]--) -CA Pmaxmax(3.16)
25
If the efficiency of the piston pump Tlpp and the efficiency of the
transmission Tltp 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 /Ad = 2. Tlt = Tlt and Tl =max p pp
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-(VlVd)2 - 2
2 - (VdlV)2
(3.18)
This quotient is tabulated in the following table for several values of
VlVd
Tltot VlVd
Tltot VlVd
Tltot VlVd
TltotTltotp 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 lAd =2.max
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
VlVd 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 CPmax
at every windspeed. The disadvantage of
the centrifugal pump is that the efficiency is very much depending on
27
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 inay
that windregime. It can be calculated with:
co
qay =Jq(V)-W(V) dV
o(4.2)
29
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' =11
- - --:
I; I l' I I•• I---- -; - - -~"-- ----'1
I I I I...~~~-----t------t-'-·t--~
: I+-f---++---+--- ----++-~;:t_________t_I- --_JI
Va-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 · Vdln
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 ther
delivery at the rated windspeed. q. In heavy storm condi tions ther
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 theco
time being. So the output of the system is given by: .
q = 0
q = qd-J 3-(VlVd
)2 - 2
q = qr = qd-J 3-(Vr lVd )2 - 2
for V < ..'/2/3 ~Vd
for J 2/3 -Vd
< V < Vr
for V > Vr
30
The average yearly output of the system is then:
CIO
- W(V) dV + qr-J W(V) dV
Vr
(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 = p -g-H -qay w st ay (4.5)
The output power of a given system at the average windspeed V. if the
design wind speed Vd is equal to the average wind speed Vis given by:
(4.6)
31
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.(VrlVd ) - 2.J W(V) dV
Vr
(4.8)
With this integral the quality factor a is a function of the wind speeds
V, Vd , 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 xd and xr .
For k = 1.5, k = 2 and k = 4 the qual! ty factor is calculated as a
function of xd 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 theco co
number of variables on which a and P depend. Similar to the definitions
of xd and x , x can be defined as x = V /Y.r co co co
The influence of x on the quality a can be investigated by replacingco
the upper boundary of the second integral in equation (4.8) by V . Thisco
32
results in the fact that in the integrals 12 , 14 and 16 the upper boun
dary ~ must be replaced by x . The new values of a including the influco
ence of x ,a ,become:co co
(4.9)
with aco the quality if x is not neglectedco
a the quality as calculated in appendix 0
Calculating several values of aco for some realistic values of xd ' x r
and x learns that the influence of x at k = 2 and k = 4 is onlyco co
minor. For k = 1.5 the influence of x can be significant. For exampleco
at xd = 1.2 ,x = 1.6 and x = 3 the difference between a and a isr co co
0.037. However, there is hardly any change in the value of xd at which
the quality is maximum (for a given x ) if x is not neglected, as longr 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 output flow. The point at which the output flow is 10% of
33
the design flow is determined by:
(4.10)
So. the wind speed VO. 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
~ = JW(V) dV
VO•1
(4.12)
In this equation V is not taken into account also. If it is desirableco
to take V into account. the upper boundary of the integration CIO mustco
be replaced by V . In appendices D11 and D12 ~ is calculated and drawnco
in a figure for several values of xd 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 thisco
results in values that hardly differ from the calculated values of ~ for
x larger than approximately 3. For k = 1.5 the influence can beco
34
significant. This influence can be calculated by substracting
4- O.3030-xcoe
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 isr
rather big. If xr is chosen 1.4 the best xd from the quality point of
view is 1.25; if xr is chosen 2.5 the best xd is 1.7.
Figure 2 shows that, if k = 2. taking x larger than 2 has hardly anyr
effect on the quality. The optimum xd
doesn't vary quite as much with xr
as for k = 1.5. For x = 1.4 the optimum xd is 1.20; for x = 2 ther r
optimum xd is 1.35.
Figure 3 shows that the influence of x on the quality and the optimumr
xd is even smaller. Taking xr larger than 1.8 is not useful. The optimum
xd varies between 0.95 and 1.05 for 1.0 < xr <1.8.
If the system is designed for high quality. it can be seen from figure 4
on appendix 012 that xd should be chosen as small as possible. This is
trivial because of the fact that the smaller Vd (and so xd ) is chosen
the sooner the system starts to deliver. Taking xd very small however
results in a low quality.
In real situations a compromise between quality and availability should
35
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,
storage tanks can be used to increase availability.
36
(])apter 5 Sensitiyity analysis
If xd 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 Vand Hst ' 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 Vand Hst is investigated.
5.1 Sensitivity to H ts
In order to investigate the sensitivity to Hst ' the head for which the
system was designed HO 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 HO' 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 HO 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 wpl = WpO and b remaining unchanged. the quotient of the dimension-
less pump speeds is:
(5.1)
And with ~p = VlVd (see paragraph 3.5) the system operating at the head
37
HI is identical to a system optimized for Vd = 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 VdO and qdO by their new values Vd1 and qd1·
Figure 5.1 is shows what happens if for a system designed for Vd = 4 mls
and a head of HO
= 4 m the head changes to 1. 2. 6 or 8 m.
~_IIIas
Figure 5.1V [mls]
Quali ty and avallabili ty of the system operating at the new head can be
found in the figures in appendix D at the new value'of xd = VdllV.
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.HO.
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 xd 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 xd = VdlV by the correct one. The effect can be seen
In the fIgures of appendix D.
39
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, CPmax::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 xd = 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 it 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.J1 the
head H2 and the flow q2 at another rotational speed c.J2 are given by:
c.J 2 c.J
H H (_2_) and (.......£.)2 = 1· c.J1
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
the maximum efficiency at the design rotational speed as long as that
41
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%)
For the calculated accuracies see appendices F1 and F2.
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
Figure 6.1: The test rig
43
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:
(Pm specific gravity of mercury).
With:
the equations above yield:
p - p + P egeH = (p - p )egeh2 1 w m w m
Ihe difference between the energy pressure in the inlet and in the
outlet of the pump is
1 2 2APe = P2 - PI + pwegeH + 2ePwe(V2 - vI)
(VI and v2
: velocities at inlet and outlet)
So. if h in the manometer is measured. the energy pressure can bem
calculated with:
1 2 2Ap = (p - p )egeh + -dp e(v - V )e m w m 2w 21
The velocities v2 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:
20.00302 m
20.00189 m
The net hydraulic output energy is:
3(q : the measured flow through the pump in m Is)
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)
45
The efficiency of the pump is:
1) = Ph d /P.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 2are given. The best fitting parabolas of the type H = b·~ - c·q were
calculated by means of regression. The calculated values of band care
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 2and 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. [W]1n ~o 900 r.p.m.
ocPo ++ + 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 .... + 00o +++ DO 0 )()(Xx)()( 600 r.p.m.
+ + 00 ~+<~o 0 ~ ~ OCDCD 540 r.p.m.o ~x OCDxXa:P 0o xxxxx 420
XX X ++x ++++ 360
900
1000
Figure 6.3: The measured input power
1 2 3 4 5 6 7 8 q (lis)
-47
Hl' ~:::::::::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
and some parabolas
P3Cal
48
151413 Q ...~~o
o~~12 G "......~'2[0
<>~+ --11 D+v. 1J
tJr:p~· ~10 . c tJ 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. Lmanufactur~r6X 2900 r.p.m. I data
1 2 3 4 5 6 7 8 9 10.-9-
Cal
3Figure 6.5: P/Cal as function of q/Cal (Measured points)
P -7 -5 CIBest fitting line: 3= 5.6-10' + 9.7-10 -~
Cal
with q in l/s
Cal in r.p.m.
P in W
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~ X11 I ~o..a -=l.L
0·" .. fI! f""-a H ~.2 1l7.~...
w ~ 0
9 • ~~~~
8 " +,a'\IIi
7
6
5
4
3
2
1
49
1 2 3 4 5 6 7 8q/w
9 10
Figure 6.6: Hlw2 as function of q/w
(Measured points and best fitting parabola)
H -5 -7 a. 2Best fitting parabola: 2: 1.21-10 - 7.47-10 -(~)fa)
with q in lIs
H in m.w.c.
fa) in r.p.m.
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 3Pin- 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-
23'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
10%. Because of the fact that the most important part of the H-q curve
51
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 2Pin = 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 ~ =J3b. w·c
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 ap
simple way. An analysis of the way the calculated value of 1) changesp
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:
= bd + e.(~ - .-)
w 'l3·c
(6.-4)
Dividing (6.-4) by (6.2) yields:
1= -------:----e ~ b
1 + d (w - ~-3.-c-)
(6.5)
53
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
)( modelT 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 t and atop
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 it 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.
55
900 +
p [W] +x+'11.
800 +'11.'II.
model+'JI.
'II. 't'oI.
700 measurements -t>c.+ ..'II.
lie~
600 ,.;l
500lIM
'll.xt
'JI.~'11.+
400 'JI.'JI.+'II. +
>ex +
300lIex +
>e+lie
x
+200
tt"
100......
100 200 0300 400 500 600 700 BOO 900 1000fa) [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 tor
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. if 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.
57
7.2 The safety mechanism
Several safety mechanisms for water pumping windmills have been designed
by the CWO. These safety mechanisms are mainly applied to water pumping
systems that make use of piston pumps. These safety mechanisms are all
besed on the principle of turning the rotor out of the wind at high wind
speeds. This motion of the rotor round a vertical axis (i.e. the shaft
going down from the top of the tower to the pump) is called yawing. In
these systems the shaft going down from the top of the tower to the
piston pump is in reciprocating motion. The shaft of the system using a
rotodynamic pump is in a rotating motion. The torque in this shaft
causes the head of the mill to yaw. Dependent on the orientation of the
torque in the shaft. it will help or work against the safety ~chanism.
Because of the fact that the torque in the shaft is proportional to the
square of the wind speed (according to the model). the driving torque
can even be used as a safety mechanism. Special care should then be
given to avoid the pump from running dry.
No further attention is given to the safety mechanism in this report.
58
The model of the H-q curve of a centrifugal pump can reasonably be
represented by a parabola as done in this report. The input power was
assumed to be a cubic function of the pump speed only. The measurements
however show that the input power is also dependent of the flow going
through the pump. Instead of the function P. = d_Ca)3 it is better to useln
3 2P. = f-Ca) + e-q-Ca) . If accurate calculations of the performance of theln2pump are wanted it is better to use a power series of the H!Ca) -q/Ca) curve
that extends to higher powers. This has been done by J.Burton in [10]. a
draft report finished just before this report was finalised. Disadvan-
tage of these higher power polynomes is that it is not possible to give
simple efficiency curves. calculate the maximum efficiency and to deter-
mine the parameters in this polynome in a simple way by calculating the
quality and the availability of the system. The model derived in this
report can be used as a first approximation of the desired pump specifi-
cations and to calculate quality and availability of a wind driven water
pumping system using a centrifugal pump.
The assumption that the system runs at constant A seems very reasonable.
The best design windspeed for reaching a high quality is approximately
equal to the average wind speed at sites with a narrow wind distribution
and apprOXimately 1.5 times the average wind speed at sites with a wide
wind distribution.
The main problem in mechanically coupling a centrifugal pump wi th a wind
rotor is the transmission ratio. that becomes quite large. specially
with slow running wind rotors. A centrifugal pump. when driven by a slow
59
running rotor at a site with a low average windspeed. is a good alterna
tive for very small heads. less than about 1 m.w.c .• (more exact data
about this can be found in figure 1.2 of this report) or to pump water
over a static head equal to zero against pipe resistance only. For
example: pumping water over a dyke. Also. they can be used in combina
tion with a wind rotor supplying electrical energy. because of the fact
that in this case there is no problem with a large transmission.
60
Literature
[1] F.Weber. Arbeitsmaschinen. YEB Verlag. 1962.
[2] H.Lameris. Roterende Stromingsmachines. Faculty of Mechanical
Engineering TU Eindhoven. Dictaatno.4400.
[3] A.Stepanoff. Radial- und Axialpumpen. Springer Verlag. 1959.
[4] F.Coezinne. Performance of WESP. Internal note. Wind Energy
Croup. Faculty of Mechanical Engineering TU Twente. 1986.
[5] W.Janssen/P.Smulders. Matching Centrifugal Pumps and Windmills.
Internal note. Wind Energy Croup. TU Twente. 1980
[6] B.Roorda. Eindopdracht HTS Enschede. internal note. Wind Energy
Croup TU Twente. 1978.
[7] F.Fuchslocher/J.Schulz. Die Pumpen. 12th edition. Springer
Verlag. 1967
[8] E.Lysen. Introduction to Wind Energy. 2nd. edition. 1983.
Publication SWD 82-1. SWD c/o DHV Consulting Engineers B.V .•
P.O.Box 85. Amersfoort. The Netherlands.
[9] L.Bianchi/P.BUstraan/J.Stolk. Pompano 11th edition. Stam
Technische Boaken. 1976
[10] J.Burton. The Mechanical Coupling of Wind Turbines to Low Lift
Rotodynamic Water Pumps. Report Department of Engineering of the
University of Reading (U.K.). 1987
[11] M.Falchetta/D.Prischich/E.Dal Pane. Wind Pumping with Electri
cal Transmission. Article taken from Wind Pumping Applications.
Food and Agriculture Organisation. 1986
61
[12] T.Dekker. The Design of a Waterlifting Windmill Coupled to a
Centrifugal Pump. IWECO Report 5165018-80-1. 1980
[13] B.Westgeest. Veldmetingen aan Proto-type van Windmolen/Centrifu
gaalpomp Combinatie. IWECO Report 5165018-80-2. 1980
[14] H.BoslL.Janssen. Matching Centrifugal Pumps and Windmills.
University of Technology Twente. Faculty of Mechanical
Engineering. Vakgroep Ontwerp- en Constructieleer.
Reportnr. WMOll.
[15] F.Coezinne. Performance of Wind Electric Pumoing Systems.
Article taken from Proceedings of the Conference and Exhibition.
Rome 7-9 Oct 1986. Volume 2 pA31-436. European Wind Energy
Association
[16] D.Shen/J.Shi/J.Wei/Z.Zheng. The National Research Progranme
about Wind Power-Screw Pump Unit. Article taken from Proceedings
of the Conference and Exhibition. Rome 7-9 Oct 1986. Volume 2.
p.259-265. European Wind Energy Association
Appendix A
Rotational speeds n [r.p.m]. impeller diameters D [cm] and hydraulic
powers P [W]of some centrifugal pumps
C&l = 0.1 D =9.5s sH rml 1 3 5 10
3D Pqd [m /s] n D P n D P n D P n
10~ 529 5 1 1206 -i 3 1769 -i 5 297-i 3 10
10-3 167 17 10 381 13 30 559 11 50 9-i1 10 100
10-2 53 5-i 100 121 -i1 300 177 36 500 297 30 1000
10-1 17 170 1000 38 129 3000 56 1~ 5000 9-i 96 10-i
C&l =0.15 D =7s sH rml 1 3 5 10
3qd [m /s] n D P n D P n D P n D P
10--i 79-i -i 1 1809 3 3 265-i 3 5 ~1 2 10
10-3 251 13 10 571 10 30 839 8 50 H21 7 100
10-2 79 -iO 100 181 30 300 265 27 500 ~ 22 1000
10-1 25 125 1000 57 95 3000 8-i 8-i 5000 1-i1 70 10-i
C&l =0.20 D =-i.5s sH rml 1 3 5 10
3 n D P n D P n D P n D Pqd [m /s]
10~ 1058 3 1 2-i12 2 3 3538 2 5 59-i8 1 10
10-3 3:M 8 10 762 6 30 1118 5 50 1882 5 100
10-2 106 26 100 2-i1 19 300 354 17 500 595 1-i 1000
10-133 81 1000 76 61 3000 112 5-i 5000 188 -is 10-i
C&l = 0.25 D = 3.5s sH rml 1 3 5 10
3 n D P n D P n D P n D Pqd [m /s]
10--i 1323 2 1 3015 2 3 ~23 1 5 7-i35 1 10
10-3 -i18 6 10 953 5 30 1398 -i 50 2353 -i 100
10-2 132 20 100 302 15 300 ~2 13 500 7~ 11 1000
10-1 -i2 63 1000 95 -i8 3000 1-iO -i2 5000 235 35 10-i
Appendix Bl
The parabolic H - q curve
In this appendix the parabolic H - q curve is compared with some data
given by manufacturers of centrifugal pumps. These curves are usually
given for only one pump speed, the design pumpspeed wpd. The parabolic
characteristic can then be written as :
2H = b' - c-q 2with b' =b-wpd.
In the following table values for b' and c of several pumps are given.
They have been determined by finding the best fitting parabola over the
given characteristic. On the next page these characteristics and the
points calculated with the parabolic function are given. These points
are marked with X. In the table the deviations are also given between
the given and the calculated values for HO =H(q =0), ~ =q(H =0) and
in between : H ~ HO/2.
,2 5 Deviation in :
number b [m] c [s /m ] HOH
O/2
~
100-105 11.3 6.3 _103 6% 3% -t%
100-65 7.0 7.8 _103 9% 6% 0%
80-85 8.9 1.3 _1O-t< 3% -t% 2%
5O-1-t5 16.0 1.3 -lOS 10 % H% 7%
50-65 6.6 7.9 _10-t 0% -t% 6%
50-30 6.0 1.5 _lOS 0% -t% 0%
MT 80-60 6.0 1.6 _10-t 0% 2% ?
100-60 5.9 7.6 _103 0% 1 % ?
65-125 6.5 3.6 _10-t 0% 2% ?LM
u
UMT
UMT
HMT
HMT
HMT
HMT
Pump
HMT
Appendix B2
10o
..
.... (m)r·...;...,.--,....T'""-r---,-,--....,...--------.
III
-
.~ ..HIIT~'46 NW50
r--......~
HIIT~11O '"r----.. .... \..""""~.~ \
HMT~75
~ -~ i\.~~\HUT - r--......~
~~,\ '\•4
..~~"~' ,.
a ~
"~.\ 1\0
0 10 20 .....'" •
) .; " ; 1 ~. I t- :J- t t i"""+- Uta·.I' • --,,. H-- -- .-+- .. ~ . I I - .....
, ! --- KIOI T .. -.'"
• lM'~lUI. .;.-+ t-I• ~ No.- I';1 -~...
.- - ...~r toolr- ~--r-
~--f- f- .. --- ~ . --I.AI 1'3 3
..
I.J.• 1.- ~
'-'1
I1 Ib 5 30 ~. 4D alio¥
••
10
12It -.
HIlT~-
10 - .......... l• N 1IMO 'It•0-..... HllT1..~~ "-- 0.;.:
• "HMi'l~, l.- -I
~ "- \.I
~~, " "...
4
" -"" \. [\3
I
2
-~~ \ \,I\. ~ ,~
t~~\ \a ....."' .......,0 20 40 eo III 1CIGo 10 20 30 40 50 It' 70 .lft·/u 100
~... ...~I"~
'" .... "HIlT'" ~~ "-r--....'f' "'" " t
Pilo.
"" "- i\~
\ r\ \I
i\a ~ r\\ ~\ \
1
~
, ,0
••
7
•
10..•
Appendix C1
Checking the results
At the UT Twente, a centrifugal pump, the Stork KCE-11-4. was tested by
Roorda [7] In this test the caracteristics for constant head were deter-
mined. In this appendix the results of these tests are compared with the
values as calculated with the help of the theory in chapter 2 of this
report.
Fitting a parabola over the H - q characteristic as given by the manu-
facturer yields the following values for band c ':
-4 2b = 1.90-10 [ms] and
5 5c = 2.98-10 [s/m]
According to the data given by Stork, the maximum efficiency of the pump
is 0.60. The results of the testing however show that the efficiency
never exceeds 0.49. This value is used here :
T} = 0.49pmax
With the help of equation (2.13) the constant d can be calculated. This
results in :
With the help of the equations (2.5). (2.6) and (2.10) the characteris-
tics q - w • P - w • T} - w for constant head can be calculated. On thep p p p p
next pages these characteristics are shown. Also in these figures. the
measured values are given.
>
.0I
'"de
g..,<~In...,o..,()
gIn
iI"t
•
•
•• H = 10 m
.. H = 5 m
••
..o Measured at H = 1 m
•
-- Calculated at H = 1. 5. 10 m
o I----+---..a-....----+----....----+-........---+---~--.-----,l_. ---....,I-----~, _.o 12Df) ~1feO 'IBoo 3¥. 3600 WDD
n I R.p. MJ
- Calculated at H = 1. 5. 10 m
>CD
5-....x
....o'1
n§UI
§rotH = 1, 5. 10 m
.)6~ tI...
n l R.p.Ml2800ZOCIO1600
.. H = 5 m
.. H = 10 m
..
..
o Measured at H = 1 m
•
2
1
•
0.1
.kJ Jl~ :J~oo 4000
n 1R.p.MJ
>
Appendix D1
Calculation of the quality and availability
On this appendix the quality and availability of the optimized system
are calculated for the values of the Weibull shape factor k = 1.5. 2 and
4.
Here. the quality is defined as the yearly average output power devided
by the output power at the average windspeed in a given wind regime. In
the text is stated that the expression for the quality is given by the
equation:
v
JJ3 o (VlVd)2 - 2o W(V) dV + J
V.1n
CXI
3o (Vr lVd)2 - 2 0 IW(V) dV
Vr
with V. =J 2/3 -Vd1n
and W(V): the Weibull probability density function.
In this equation V is not taken into account.co
The availability is defined as the fraction of the time that the output
flow is larger than lOX of the design output flow.
According to the text. the equation for the availability is:
CXI
~ = JW(V) dV
..JO.67-Vd
In this equation the influence of V is neglected also.co
Appendix D2
A. Quality
k = 1.5
For k - 1.5 with:
v
Jr 0 5 -m-( Y)1.5
I = J 3-(VlVd)2 - 2_ 1.5-m-V · e V dV1 V1.5
Yin
and
with m := r1.5
(1 + 1~5) = 0.8577
11 and 12 can be rewritten substituting
and x • as:r
t + 2 )3/43 dt
1.5-q-xre
12 is an analytical expression. 11 must be calculated numerically.
The calculated values for a are tabulated in tabel 1. In figure 1. a is
shown as a function of xd with xr as parameter.
Appendix 03
Tabel 1: a for several values of xd and xr for k = 1.5
x'lz 1.00 1.10 1.20 1.30 1.40 1.60 1.80 2.00 2.50 3.00
0.10 409 450 484 512 531 515 602 621 646 655
0.15 441 488 528 562 591 635 666 688 111 128
0.80 461 523 510 610 643 694 130 155 188 800
0.85 481 553 608 654 692 150 191 820 858 811
0.90 499 518 641 693 131 804 851 883 926 940
0.95 502 595 668 129 118 854 901 944 991 1008
1.00 493 603 689 158 815 901 960 1001 1054 1013
1.05 469 602 102 181 846 943 1009 1055 1114 1135
1.10 426 589 106 191 811 980 1054 1104 1110 1193
1.20 215 515 683 805 899 1031 1129 1191 1210 1298
1.30 325 603 112 896 1069 1182 1251 1353 1385
1.40 423 685 853 1013 1211 1302 1416 1454
1.50 512 160 1046 1215 1324 1459 1504
1.60 592 984 1194 1324 1482 1535
1.10 880 1144 1301 1486 1541
1.80 119 1065 1255 1412 1542
1.90 455 953 1181 1441 1521
3tabulated is a-l0
Appendix D4
Figure 1: The qua11 ty a as function of xd
and xr
for k = 1.5
1.5
a
1.0
0.5
o
13.0.- -./~- -....
2.5
~V
1# /V .........
~
k=1.1: ~~V f'"-
2.0
) ~./ ...... .....r--... ~./
~~ " " 1.8
,hV V"" ...........~ r\
~W~V ""'"~ " \I\. 1.6
~V ~ r\ \ 1.04
~[:/ f",. 1\ \ 1.3
~~
" \ ' 1.2
\ 1.1 Xr ~
/
1.0
o 0.5 1.0 1.11 2.0
For k = 2 with
Appendix OS
and
v
Jj 3.(V/Vdl - 2
Yin
11" V- -- - e2y2
co
Jr-----=2~- JTr V - ~ ( ~)23- (V /V) - 2- -- -e V dV
r d 2 y2
Vr
With the substitution t = Tr_2"(3-y2 -2-~) 13 can be reduced to:12-V
The integral in this expression is an incomplete Gamma function and can
be found in tables or calculated numerically.
14 can be calculated by substituting
11" 2
J2 - :rxr
14 = 3-(xr/xd) - 2 -e
In table 2 and in figure 2 values of a for some values of xd and xr are
shown.
Appendix D6
Tabel 2: The quality a for several values of xd and x r for k = 2
X';!{ 1.00 1.10 1.20 1.30 1.-«> 1.60 2.00 Cll»
0.70 -457 500 534 562 585 617 6-46 657
0.80 521 581 629 667 697 7-«> 779 793
0.90 555 638 703 75-4 79-4 849 900 918
0.95 557 655 730 788 83-4 897 954 97-4 .
1.00 5-45 661 749 816 868 939 1003 1026
1.05 516 656 758 835 895 975 10-47 1072
1.10 -465 638 758 847 91-4 1004 108-4 1113
1.15 382 60-4 746 848 925 1026 1115 1146
1.20 550 721 8-«> 927 10-41 11-«> 1173
1.25 469 682 821 920 10-48 1157 119-4
1.30 625 789 903 10-47 1168 1208
1.-«> -429 684 838 1021 1168 1216
1.50 -498 726 963 11-43 1200
1.60 5-48 87-4 1094 1160
1.70 751 102-4 1102
1.80 589 937 1029
1.90 837 9+t
3tabulated Is a-10
Figure 2: a as function of xd and xr for k =2
Appendix J17
1.5
a
1.0
0.5
k= 2 - ........
"// ...-...........
~ l"-I"-....- .......
~V .......~ "- '\ [00r-.
~~~ " I\. '\v ........... \.. 2.0
I.~~ - "'- '\ "........ I\.
~~ '\ \ \ \r-.. 1.6
~ """- " \ \,
1.4V r.... \ 1.3
" \ \\ 1.1 1.2
1.0
Xr ~
I
0.00.0 0.5 1.0 1.5 2.0
Appendix OS
For k = 4
and
j 3.(V IV )2 - 2 •r d
4 1with m := r (1 + 4') = 0.6750
With the same substitution as used for 11 and 12
these integrals become:
and
23·(x /x) - 2r d
4 J2·m·xIS = 9 d t
1/ 2 .(t + 2)
o
m 4 2- -x • (t + 2)
• e 9 d dt
Again. 16 can be calculated analytically and IS must be calculated
numerically. The results of this calculation are shown in table 3 and in
figure 3.
/
Appendix D9
Tabel 3: The quality a for several values of xd and xr for k = 4.
x't-z 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80
0.70 560 6b4 634 653 663 668 670 670 670
0.75 606 659 694 716 727 733 735 736 736
0.80 643 706 748 773 786 793 795 796 796
0.85 671 745 793 822 838 845 848 849 849
0.90 685 772 829 862 880 889 892 893 893
0.95 684 787 852 891 912 921 925 926 927
1.00 664 786 863 907 931 941 946 947 947
1.05 622 769 858 909 936 948 953 955 955
1.10 552 734 838 897 927 941 946 948 949
1.20 263 599 749 827 866 884 890 893 893
1.30 344 593 702 753 776 786 787 788
1.40 362 532 602 631 642 645 646
1.50 328 432 470 483 488 489
3tabulated is a-tO
1.5
a
1.0
O.S
Figure 3: a as function of xd
and xr for k = 4.
L--
.4 ,/, I-
~k'= ...- :-...
~y.- ........ '\ '\~r\.
~./~~ \ \ ~\
V ,1\ i\ ~\ 1.8
\ \ \ \1.4
" 1.1 1.2
1.0
X I.i'r/
Appendix DIO
0.5 1.0 1.15 2.:0
B Availability
For k = 1.5 the availabIlity is given by:
For k = 2 the availabIlity is given by:
Appendix Dll
co
±v{j= --.2y2
O.67.Vd
11" V 2- -(-)4 -e V dV =
And for k = 4 the avilability is given by:
V 4- m·(-)
e V dV =
In figure 4 these functions have been drawn.
Appendix D12
Figure 4: The availability 13 for k = 1.5. 2 and 4 as function of xdo
W
II
II I'fI II
C\t f) IJ!: VI V/,V
'ffr1 L1
! ~V)VI
~V 1//~
V Jr// VI ... ~....
I / V .&
//I /11
"/I,IIfJ
oN
II')
d
od
CJo
Appendix E1
The calculation of an impeller of a centrifugal pump
The design of an impeller of a centrifugal pump is based upon a coside-
ration of the velocity of the medium (i.c. water) at the inlet and the
outlet of the impeller. This theory can be found for example in [9].
In fjgure 1 these velocities together with some dimensions of the impel-
ler are shown.
Figure 1
In this figure the velocities are:
u l ' u2 the velocity of the impeller at the inner and outer diameter
c l ' c2 the velocity of the water at the inner and outer diameter
wl ' w2 the relative (to the impeller) velocity of the water at the
inner and the outer diameter
For the calculation of the dimensions of the impeller the following
equations are used:
(l)
(2)
(3)
crad =u2-sin(a2)-sin(132)
sin(a2 + 132 )
Appendix E2
(4)
b2 = 1. 1- -.".""""_D"...q~--c-2 rad
(5)
(6)
For a theoretical background of these equations see [9]
(7)
Here, a calculation of an impeller for a centrifugal pump is executed
for the windmill CWO 5000 (R = 2.5 m, Ad = 2, CPmax = 0.33), with a
transmission of i =25 working at a total head of H =3 m.
The following efficiencies are assumed:
1}t = 0.8, 1}PmaX = 0.7
The design windspeed is taken 3.5 m/s. This results, according to (1.7),
in a design pump speed of 70 rad/s and according to (1.5) in a design
flow of 0.0033 m3/s.
The factor k-~Ydr in equation (1) is assumed to be 0.7.
The dimensions of a suitable impeller can be calculated with the 7
equations using the values mentioned above. These calculations are
executed with a small computer prograDlD in which a large number of
values of D1, a2
and 132
are assumed. An example of a suitable impeller
is shown in figure 2 on appendix E3.
Appendix E3
Jw
-rf: 50 100
tl30 :
I
200
(All dimensions in millimeters and degrees)
Figure 2
Appendix Fl
The testing of a centrifugal pump
Accuracy of the measurements
The rotational speed
The rotational speed is measured with a magnetic contact at the pump
shaft. To the pupshaft 5· magnets are attached. These magnets pass an
electric switch that closes every time a magnet passes it. The number of
times this switch closes is counted with a puIs counter. So. every
revolution 5 pulses are counted. The measuring time at every measurement
is 10 seconds. The smallest speed that is measured is 360 r.p.m. (or 6
r.p.s.) The minimum number of pulses counted at every measurement is 10
s * 5 pulses/s * 6 r.p.s. = 300 pulses. The accuracy that these pulses
are counted with is +/- 1 puIs. The maximum deviation of the r.p.m. is
then +/- 1*360/300 =+/- 1.2 r.p.m.. This is less than 0.5 X.
The torgue
The torque measured at the pumpshaft is measured with a torque
measuring device based upon the deformation of rekstrookjes. The device
was gauged before the measurements. This was done on a testrig specially
constructed for this goal. Torques that were imposed on the device were
O. 10. 20•.....• 90 and 100 Nm. The output voltage of the device in mV
at these torques were measured and the linearity of the relation between
the torque and the voltage was checked. This resulted in the T = -2..485
* m + 0.883 with m the measured voltage in mV and the T the torque in
Nm. with a correlation of 0.99999998. The voltage at zero torque is
Appendix F2
0.355 mV. The minimum voltage measured above 540 r.p.m. is -0.33 mV
representing a torque of 1.70 Nm. The accuracy that the voltage is
measured with is +/- 0.02 mV representing +/- .05 Nm. The maximum
deviation in the measured torque is then +/- 3%.
The pressure
The accuracy that the manometer is read with is +/- 2 mm. The minimum
reading is 84 mm. This means a maximum deviation in the measured
pressure of 2.5%.
The flow
The deviation of the measured flow is according to the manufacturer of
the instrument less than 1%.
Appendix F3
Omega H q Pin Eta Omega H q Pin Etar.p.m. m.w.c. lIs W X r.p.m. Ill.W.C. lIs W X360 1.11 2.47 62.3 43.2 600 2.69 4.60 286.4 42.4360 1.23 2.23 58.6 45.9 600 2.79 4.54 284.9 43.6360 1.46 1.98 55.7 50.8 600 2.88 4.45 284.9 44.1360 1.56 1.63 51.1 48.9 600 2.93 4.29 277.1 44.5360 1.58 1.31 47.3 42.8 600 2.99 4.23 277.1 44.8360 1.50 0.56 37.0 22.3 600 3.06 4.13 270.8 45.8360 1.50 0.00 37.0 0.0 600 3.13 4.06 270.8 46.0
600 3.22 3.97 264.6 47.5Omega H q Pin Eta 600 3.27 3.86 264.6 46.8r.p.m. Ill.W.c. lIs W X 600 3.39 3.78 261.5 48.0420 1.56 2.86 100.0 43.7 600 3.43 3.71 259.9 48.1420 1.70 2.60 94.5 45.8 600 3.55 3.62 259.9 48.6420 1.81 2.36 89.1 47.1 600 3.60 3.48 252.1 48.7420 1.88 2.09 84.7_ 45.4 600 3.68 3.38 250.5 48.7420 1.98 1.66 77.1 41.8 600 3.77 3.26 247.4 48.7420 2.02 1.18 68.3 34.2 600 3.89 3.10 244.3 48.4420 2.04 0.98 62.8 31.2 600 3.90 2.82 230.2 46.9420 2.06 0.75 58.5 25.9 600 3.98 2.70 228.7 46.1420 2.04 0.00 46.5 0.0 600 4.00 2.51 220.9 44.6
600 4.09 2.36 216.2 43.8Omega H q Pin Eta 600 4.18 2.21 213.1 42.5r.p.m. m.w.c. lIs W X 600 4.13 1.87 195.9 38.6480 1.76 3.87 217.9 30.6 600 4.18 1.74 191.2 37.3-+80 1.78 3.74 229.1 28.6 600 4.24 1.39 175.6 32.9480 1.80 3.66 229.1 28.3 600 4.22 1.01 161.6 25.9480 1.88 3.71 230.4 29.8 600 4.19 0.77 152.2 20.8480 1.92 3.54 221.7 30.1 600 4.19 0.52 144.4 14.8480 1.99 3.46 216.7 31.2 600 4.21 0.00 128.8 0.0480 2.06 3.38 215.4 31.8480 2.17 3.19 221.7 30.6 OIIesIa H q Pin Eta480 2.28 2.95 210.4 31.4 r.p••• ••w.c. lIs W X480 2.37 2.72 210.4 30.1 660 3.22 5.04 375.2 42.4480 2.51 2.40 202.9 29.1 660 3.31 4.97 373.4 43.3480 2.64 1.81 185.4 25.2 660 3.36 4.92 371.7 43.6480 2.76 1.11 171.7 17.5 660 3.41 4.87 370.0 44.0480 2.80 0.00 135.5 0.0 660 3.49 4.78 370.0 44.2
660 3.58 4.68 364.9 45.0Qnega H q Pin Eta 660 3.72 4.59 364.9 46.0r.p.m. m.w.c. lIs W X 660 3.74 4.53 363.1 45.8540 2.16 4.11 210.0 41.5 660 3.78 4.47 358.0 46.3540 2.22 4.06 210.0 42.1 660 3.85 4.42 356.3 46.8540 2.36 3.92 207.2 43.9 660 3.90 4.31 351.1 47.0540 2.43 3.71 201.6 43.8 660 4.01 4.22 351.1 47.3540 2.53 3.59 200.2 44.5 660 4.09 4.15 347.7 47.9540 2.61 3.62 200.2 46.3 660 4.20 4.07 346.0 48.5540 2.72 3.44 197.4 46.6 660 4.26 3.98 340.8 48.8
540 2.92 3.14 189.0 47.6 660 4.41 3.85 335.7 49.6540 3.05 2.90 181.9 47.6 660 4.40 3.70 330.5 48.3540 3.08 2.61 170.7 46.2 660 4.57 3.54 321.9 49.3540 3.18 2.41 167.9 44.7 660 4.60 3.42 316.8 48.7540 3.33 2.11 159.5 43.2 660 4.66 3.23 309.9 47.7540 3.34 1.68 144.0 38.2 660 4.71 2.96 294.5 46.5540 3.38 1.32 132.8 33.0 660 4.92 2.46 273.9 43.3540 3.38 1.13 124.3 30.1 660 4.95 2.17 260.1 40.5540 3.40 0.98 120.1 27.2 660 5.08 1.69 237.8 35.4540 3.39 0.56 107.5 17.3 660 5.12 1.43 225.8 31.8540 3.41 0.00 96.3 0.0 660 5.05 1.13 208.6 26.8
660 5.11 0.76 194.9 19.5660 5.10 0.24 174.3 6.9660 5.13 0.00 169.2 0.0
Table 1: Results of the measurements
Appendix F4
Omega H q Pin Eta Omega H q Pin Etar.p.m. m.w.c. lIs W X r.p.m. m.w.c. lIs W Xno 3.88 5.79 486.1 45.4 840 5.24 6.81 m.o 44.9no 3.98 5.73 486.1 46.0 840 5.53 6.49 757.2 46.5no 4.06 5.65 484.2 46.4 840 5.67 6.37 757.2 46.8no 4.11 5.59 480.4 47.0 840 5.75 6.31 748.4 47.6no 4.18 5.46 476.7 46.9 840 5.90 6.20 744.1 48.3no 4.31 5.37 471.1 48.2 840 5.98 6.13 739.7 48.6no 4.40 5.28 469.2 48.6 840 6.03 6.00 n4.4 49.0no 4.42 5.19 463.6 48.5 840 6.11 5.94 no.o 49.5no 4.51 5.12 459.8 49.3 840 6.23 5.85 717.8 49.8no 4.54 5.06 458.0 49.3 840 6.32 5.77 711.3 50.3no 4.64 5.00 458.0 49.7 840 6.43 5.70 709.1 50.7no 4.74 4.90 456.1 50.0 840 6.49 5.65 706.9 50.9no 4.82 4.76 444.9 50.6 840 6.60 5.63 706.9 51.6no 5.06 4.58 437.~ . 52.0 840 6.67 5.53 698.2 51.8no 5.16 4.39 426.1 52.1 840 6.94 5.23 676.3 52.7no 5.29 4.27 422.4 52.4 840 7.22 4.95 663.2 52.9no 5.33 4.05 407.4 52.0 840 7.48 4.61 643.5 52.6no 5.46 3.91 405.5 51.7 840 7.73 4.25 621.7 51.8no 5.54 3.79 401.8 51.3 840 7.77 3.95 595.5 50.6no 5.70 3.65 399.9 51.0 840 7.96 3.51 569.3 48.1no 5.75 3.24 371.8 49.1 840 8.07 . 3.07 536.5 45.3no 5.84 3.09 368.1 48.1 840 8.15 2.67 508.1 42.0no 5.88 2.82 353.1 46.1 840 8.14 2.23 473.1 37.7no 5.89 2.60 340.0 44.2 840 8.16 1.86 444.7 33.5no 5.90 2.38 326.9 42.1 840 8.24 1.37 416.3 26.6no 5.98 1.95 304.4 37.6 840 8.24 0.99 392.3 20.4no 6.00 1.86 306.3 35.8 840 8.15 0.00 328.9 0.0no 6.01 1.64 291.3 33.2no 6.04 1.36 278.2 29.0 Qllega H q Pin Etano 6.04 1.02 257.6 23.5 r.p••• ..w.c. lIs W xno 6.04 0.79 246.3 19.0 900 5.99 6.95 954.0 42.8no 6.09 0.21 222.0 5.6 900 6.22 6.79 944.7 43.9no 6.05 0.00 210.7 0.0 900 6.28 6.66 928.3 44.2900 6.52 6.52 923.6 45.1
Omega H q Pin Eta 900 6.70 6.32 904.9 45.9r .p.m. m.W.c. lIs W X 900 6.84 6.21 900.2 46.3780 4.58 6.00 617.9 43.7 900 6.96 6.15 895.5 46.9780 4.63 5.85 607.7 43.7 900 7.04 6.08 890.8 47.1780 4.70 5.82 605.7 44.3 900 7.18 6.00 881.5 47.9780 4.82 5.68 599.6 44.8 900 7.33 5.87 874.5 48.3780 4.91 5.58 597.6 45.0 900 7.74 5.51 841.7 49.7780 5. " 5.36 583.4 46.1 900 8.01 5.34 832.3 SO.4780 5.30 5.25 577.3 47.3 900 8.31 5.03 808.9 50.7780 5.45 5.09 565.1 48.2 900 8.54 4.69 783.2 50.2780 5.56 5.00 563.1 48.4 900 8.80 4.10 731.6 48.4780 5.64 4.93 561.1 48.6 900 8.91 3.63 696.5 45.5780 5.78 4.81 557.0 49.0 900 9.16 3.46 687.2 45.3780 5.98 4.59 540.8 49.8 900 9.24 3.11 659.1 42.8780 6.29 4.26 524.5 50.1 900 9.32 2.78 626.3 40.6780 6.49 3.89 504.2 49.1 900 9.41 2.54 605.2 38.7780 6.66 3.69 496.1 48.6 900 9.45 2.04 560.7 33.7780 6.77 3.17 459.6 45.8 900 9.45 1.80 539.7 30.9780 6.85 2.61 425.1 41.3 900 9.43 1.40 502.2 25.8780 7.01 2.20 396.7 38.1 900 9.44 0.83 460.1 16.7780 7.17 1.97 388.6 35.6 900 9.42 0.00 401.6 0.0780 7.11 1.70 3n.4 31.8780 7.13 1.28 344.0 26.0780 7.10 1.01 323.7 21.7780 7.12 0.71 307.4 16.1780 7.11 0.45 291.2 10.8780 7.11 0.00 2n.9 0.0
Table 1: Results of the measurements (continued)
Appendix F5
Omega b cr.p.m. ms**2 m(s/l )**2
360 1.12E-03 6.27E-02420 1.OBE-03 5.84E-02480 1.14E-03 7.36E-02540 1.10E-03 7.39£-02600 1. 14E-03 8.07E-02660 1. 14E-03 8.30E-02no 1.16£-03 7.76£-02780 1.10E-03 7.36E-02840 1.13E-03 7.11E-02900 1.11£-03 7.47E-02
Table 2: The calculated values of band c
H
ft....12""•......,4
3GO
III6051lit
5026JI "I. 40%
i n .2900 1/.."•• JWO SO'll. 55 5i% 60%
tI.S ...... 62%~90 .2(,.5 --10 0235
omlG • Z!5
501"-
.~I
It '155a ;If /
,"'a iZ
p'8522 I;f
lit o~+
•"
0215I
... •l2
•,
•"'l.,
3J1 tIM
Z .16".0
,. -_wu.-..I
a •
p
HM--
HH
JI
It
ft
'1tH5_4 .....3
2
'"
III
til
.. le -I.. i.o -I.. loS-I..~.'5s'e/. 5026
!lI.~ n : 11,SO 1/...,.2155
~Hr..... 60%
.21.5I".i
om 51%
om,
"' ~55""'"021 , ,
0 ,SO"l. I
,
I
105%
,
4 I'264
or;~, P
l • 205I
'235
'225
2 0215'205
1
" 26450-"'l
2
1Il K~
1 lO •Wo-· 1II5
10__ ... a~....... .-.-
20
'"
...
..
p
I;"
Figure 1: Specifications at 1450 and 2900 r.p.m.
(Upper curves ~ 264 are to be taken)