Gasch R., Twele J. (Eds.) Wind Power Plants: Fundamentals, Design, Construction and Operation
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352 10.4 Design of wind pump systems
Piston pump Centrifugal pump
v v
Ș
WP
Ș
WP
v
in
v
in
Piston pump Centrifugal pump
v v
Ș
WP
Ș
WP
v
in
v
in
Fig. 10-19 Total efficiency of a wind pump with a piston pump and a centrifugal pump
= konst.
v
Q
<
2
HH
1
H
2
= konst.
1
H
const.
const.
= konst.
v
Q
<
2
HH
1
H
2
= konst.
1
H
const.
const.
Fig. 10-20 Flow rate versus wind speed of a wind pump for two different constant geodetic
heads (discharge curves)
10.4 Design of wind pump systems
10.4.1 Design target
The design calculations of wind pump systems aim for the most efficient conver-
sion of wind energy into potential energy of the water pumped into the water
storage, cf. Fig. 10-1. The quality factor
J
characterizes the efficiency of this
energy conversion:
10 Wind pump systems 353
i
3
Rotor
iigeow
Wind
geow
2
1
hvTA
hQTHg
E
VHg
i
¦
¦
U
U
U
J
. (10.18)
It is the ratio of the available potential energy of the discharge volume V to the
energy contained in the wind in the considered period T:
i
3
iRotorWind
2
1
hvTAE ¦
U
(10.19)
In contrast to the efficiency of the wind pump according to equation (10-16), the
quality factor also considers the losses due to friction in the pipes, and it averages
over the considered period. Wind pump systems with a maximized quality factor
have the highest discharge volume possible in the period under consideration. This
period may be a season or a certain period when water supply is critical. In prac-
tice, the design is not based on a maximum quality factor, but the following sim-
plifications are used:
- the rated wind speed v
r
is chosen wisely, and
- the requirement that the wind pump reaches its maximum total efficiency
K
WP
at this rated wind speed.
10.4.2 Selection of the rated wind speed for the wind pump design
For typical European wind conditions, it is common practice to choose the rated
wind speed v
r
in relation to the mean wind speed
v
using a factor
v
r
/ v | 1.4...1.6 [2, 4, 7]. This has the positive effect that at wind speeds of a
high energy density the energy E
i
is converted with high total efficiencies
K
WP
,
Fig. 10-21. If a control mechanism or storm control of the wind turbine is already
activated at wind speeds below triple the mean wind speed (3
v ), the factor v
r
/ v
reduces to a smaller value. A study on the influence of the wind speed of control
start on the determination of the rated wind speed with optimum yield is given in
[7].
In addition considering that the efficiency of the gearbox is not constant, but
decreases in the partial load range below the rated efficiency, the factor vr /
v
moves to higher values v
r
/ v | 1.5…1.95 [11]. These values are valid for a wind
pump system with a centrifugal pump taking into consideration the control of the
wind turbine.
354 10.4 Design of wind pump systems
i
E
3
i
v
0
h
i
5
0
h
i
5
Q
,
0
5
WP
v
v
10
N
10
15
m/s
15
m/s
N
v
10
15
Q
m/s
WP
20
v
i
c
20
v
i
b
20
v
i
a
v
r
v
r
v
r
v
r
v
v
i
E
3
i
v
0
h
i
5
0
h
i
5
Q
,
0
5
WP
v
v
10
N
10
15
m/s
15
m/s
N
v
10
15
Q
m/s
WP
20
v
i
c
20
v
i
b
20
v
i
a
v
r
v
r
v
r
v
r
v
v
Fig. 10-21 Selection of the rated wind speed v
r
| 1,5 v for a wind pump system, a) Discharge
curve and efficiency of the wind pump, b) Energy content of the wind classes, c) Wind histogram
(Frequency distribution of the wind),
v
mean wind speed
The wind speed v
in
at which the discharge starts influences the discharged volume,
too. However, for a given hydraulic system it has a fixed relation to v
N
. Therefore,
it is usually sufficient to dimension the wind pump for the rated wind speed. The
rated total head H
r
is mainly determined by the geodetic head H
geo
, but the pipe
losses have to be considered with an allowance. For low heads in the range of up
to 10m, the rated total head is approximately H
r
= (1.2…1.4) H
geo
. For a higher
rated total head, a common estimation is H
r
d 1.2 H
geo
[3].
10 Wind pump systems 355
The following sections will treat the equations that can be used to determine the
parameters of turbine and pump, and especially the required transmission gear
ratio. Thus, the focused aim cosists in the wind pump system reaching the maximum
total efficiency
K
WP
at the rated wind speed v
r
.
10.4.3 Design of a wind pump system with a piston pump
By equating wind turbine power and (mechanical) piston pump power at the rated
operating point (Fig. 10-22)
volm
rPHubw
RotorP.opt
3
r
2
KK
U
U
HnVg
Acv
(10.20)
a relation is obtained that is useful to determine the required transmission gear ra-
tio i of wind turbine’s rotational speed n
WT
to pump’s rotational speed n
P
P
WT
n
n
i
. (10.21)
Additionally, the optimum tip speed ratio
O
opt
at which the wind turbine reaches
its optimum power has to be considered
O
opt
=
r
WTWT
v
dn
S
. (10.22)
This gives the transmission gear ratio
2
r
3
WTP.opt
2
opt
volm
rw
Hub
1
8
vdc
Hg
Vi
US
O
KK
U
.
(10.23)
356 10.4 Design of wind pump systems
.
.
.
.
n in rpm
P in kW
.
.
.
.
.
.
.
.
n in rpm
P in kW
Fig. 10-22 Wind pump system with a piston pump designed for a rated wind speed of v
r
= 6 m/s
Basically, the goal is to do without a gearbox, i.e. i = 1. For larger Western mills
(
O
opt
| 1, d
WT
t 5 m), this is achieved by selecting a suitable stroke volume
V
Hub
=
S
/4 d
K
2
2 r
K
or a suitable crank radius. For smaller systems of this type,
down-gearing is necessary.
Wind pump systems with a piston pump show a particular operating behaviour
during start-up, Fig. 10-23. The wind turbine only starts to rotate when, due to a
sufficiently high wind speed, the rotor torque provides, via the eccentric lever, a
piston force that is larger than the force from the weight of the water column
above the piston area. The static friction of the system is neglected in this consid-
eration. So, for starting the maximum torque M
max
of the oscillating curve of the
pump torque (cf. Fig. 10-16) has to be overcome. In Fig. 10-23, the wind pump
starts to operate at the start-up wind speed v
in
= 5 m/s. Moreover, the simplifying
assumption is made that the wind speed remains constant during start-up until the
mean pump torque
M
is reached.
During operation, the rotor acts as a flywheel. According to the relations dis-
cussed in section 10.3.2, the operating behaviour can be described by the mean
torque
M
as the rotational speed of the wind turbine does not follow the oscillat-
ing torque of the piston pump, in a first approximation. If the wind speed falls below
v
in
, the rotor will keep turning due to its inertia, and the discharge will continue.
Only if the wind speed falls below the minimum discharge wind speed v
min
where the rotor torque is smaller than the mean torque
M
, does the system stop.
In Fig. 10-23, this happens at v
min
= 2.8 m/s.
The wind pump system with a piston pump has therefore a characterisic hystere-
sis in the discharge curve, Fig. 10-24, in the range between start-up wind speed
v
in
and the minimum discharge wind speed v
min
. This means that within the
hysteresis range, a discharge is only possible if v
in
is exceeded often enough, i.e.
10 Wind pump systems 357
during a day with relatively low wind speeds, there are often enough gusts to start
up the wind pump system.
Equating the individual operating point the mean torque
M
of the piston pump,
from equation (10.12) and (10.14), and the torque of the wind turbine:
O
S
U
M
2
WT
2
WT
WT
242
cv
dd
M
(10.24)
gives the characteristic wind speeds of the hysteresis range.
Fig. 10-23 Start-up and shut-down behaviour (hysteresis) in the torque diagram of a wind pump
system with a piston pump
Fig. 10-24 Flow rate versus wind speed of a wind pump system with piston pump, showing the
hysteresis in the starting range of the discharge curve
358 10.4 Design of wind pump systems
The minimum discharge wind speed v
min
, is calculated from the torque equilib-
rium of the mean pump torque
M
and the torque of the wind turbine for the point
of maximum torque moment coefficient
max.M
c :
mM.max
3
WT
K
2
Kw
min
4
KSU
U
cd
rdHg
v
(10.25)
The start-up wind speed v
in
results from the torque equilibrium of the required
maximum pump torque M
max
and the wind turbine torque at standstill (
O
= 0) with
the torque moment coefficient c
M.0
mM.0
3
WT
K
2
Kw
4
KSU
US
cd
rdHg
v
in
. (10.26)
This results in a hysteresis range of
M.0
M.max
min
in
c
c
v
v
S
. (10.27)
This wind speed ratio strongly depends on the characteristic torque curve of the
wind turbine. For a wind turbine with an extremely low tip speed ratio the two
torque moment coefficients are nearly identical, c
M.0
§ c
M.max
, therefore
minin
vv
S
| . (10.28)
In order to achieve a load-free or load-reduced start-up of the piston pump, load-
relieving measures have to be included for smoothing the torque curve, cf. section
10.2, and [4, 8]. These load-relieving measures allow the wind pump to start the
discharge at lower wind speeds or even the installation of a bigger pump with a
higher hydraulic power which may increase the discharge volume significantly.
For the yield calculation of a wind pump system with a piston pump, the start-
ing behaviour and the hysteresis range of the discharge curve have to be taken into
account. This procedure is suggested e.g. in [9]. It is based on a corrected dis-
charge curve using the probabilities of wind speeds in the hysteresis range which
can be determined by the wind speed distribution on site. This curve is then used
to determine the yield according to section 10.3.2, equation (10.17).
10 Wind pump systems 359
10.4.4 Design of a wind pump system with a centrifugal pump
For the rated operating point, the power balance of wind turbine and centrifugal
pump is calculated assuming that both wind turbine and centrifugal pump operate
at their maximum efficiency point which is illustrated by Fig. 10-25. For a
mechanical coupling of wind turbine and centrifugal pump with a gearbox
(mechanical efficiency
K
G
) the power balance is then
opt
rrW
3
r
2
WT
GoptP,
42
K
U
S
U
K
HQg
v
d
c . (10.29)
In the following, the rated flow rate Q
r
in equation (10.29) is substituted by spe-
cific values of the pump, in order to illustrate the relation between the main char-
acteristic values of the centrifugal pump and other system parameters. Similar to
the description of types of wind-driven pumps in section 10.2, the specific speed
n
q
(equation 10.8) is used since it is characteristic of the corresponding type of
centrifugal pump.
Using the model laws based on the laws of similarity, the head of the centrifugal
pump can be expressed by the dimensionless pressure number
\
[4]:
2
2
2
dn
Hg
S
< . (10.30)
The impeller diameter of the centrifugal pump is d
2
, cf. Fig. 10-8. For the best-
efficiency point the rated rotational speed of the pump can be calculated with the
corresponding pressure number
\
opt
:
S
2
12
d
Hg
n
opt
r
r
<
. (10.31)
In the power balance of equation (10.29), the rated flow rate Q
r
is now substituted
by the specific speed n
q
of equation (10.8). Moreover, equation (10.31) is taken
into account, giving
3/2
S.r
3
r
GP.opt
2
W
2/3
q
q
W
opt
opt
2
2
2
q
2
4
H
v
c
d
H
Q
dn
K
S
UU
K
S
<
. (10.32)
At the rated operating point, the rated total head H
S.r
of the hydraulic system is
equal to the rated total head H
r
of the pump, cf. section 10.3.2.
360 10.4 Design of wind pump systems
WT
WT.N
WT
Wind turbine Gearbox Centrifugal pump
Cu
r
v
e
o
f
r
a
t
e
d
o
p
e
r
a
t
i
n
g
p
o
i
n
t
wi
t
h
t
h
e
i
n
v
e
s
t
i
g
a
t
e
d
p
i
p
i
n
g
s
y
s
t
e
m
Head H
Flow rate Q
Curve of the
optimum points
Curve of the
operating points
Power P
W
of the wind turbine
Rotational speed n
W
of the wind turbine
Wind speed v
r
Efficiency Ș
WP
of the wind pump
WT
WT.N
WT
WT
WT.N
WT
Wind turbine Gearbox Centrifugal pump
Cu
r
v
e
o
f
r
a
t
e
d
o
p
e
r
a
t
i
n
g
p
o
i
n
t
wi
t
h
t
h
e
i
n
v
e
s
t
i
g
a
t
e
d
p
i
p
i
n
g
s
y
s
t
e
m
Head H
Flow rate Q
Curve of the
optimum points
Curve of the
operating points
Power P
W
of the wind turbine
Rotational speed n
W
of the wind turbine
Wind speed v
r
Efficiency Ș
WP
of the wind pump
Fig. 10-25 Rated operating point of a wind pump system with a mechanically coupled centrifugal
pump
10 Wind pump systems 361
.
..
d
WT
= 5 m
d
2
= 400 mm
Power P
WT
of the wind turbine
Rotational speed n
WT
of the wind turbine
curve of
operating points
rpm
.
..
.
..
d
WT
= 5 m
d
2
= 400 mm
Power P
WT
of the wind turbine
Rotational speed n
WT
of the wind turbine
curve of
operating points
rpm
Fig. 10-26 Influence of the transmission gear ratio on the curve of operating points in the power
characteristics of a wind turbine, wind turbine diameter d
WT
= 5 m, tip speed ratio
O
opt
= 4, impeller
diameter of the centrifugal pump d
2
= 400 mm, rated wind speed v
r
= 8 m/s
It is considered that a centrifugal pump, suitable for given site conditions and a
given wind turbine, is chosen from a product line of similar pumps, scaled accord-
ing to the model laws. For all these pumps the specific speed n
q
, the optimum
pressure number
\
opt
, and the efficiency
K
opt
in the best point are the same.
Therefore, only the impeller radius d
2
has to be determined from equation (10.32):
23
S.r
3
r
GP.opt
2
WT
q
23
q
opt
opt
wq
2
4
11
H
v
c
d
Q
H
n
d
K
S
U
\
K
US
. (10.33)
Considering the design tip speed ratio
O
opt
of the wind turbine and equation
(10.31) gives the required transmission gear ratio i for the rated rotational speed n
r