Prediction of Pump as Turbine's (PAT) Best Efficiency Point (BEP) by
Theoretical Method
A.Nourbakhsh1 , Sh.Derakhshan2
1. Professor, Department of Mechanical Engineering, University of Tehran, Tehran 14147, Iran
2. M.S. Student, Department of Mechanical Engineering, University of Tehran, Tehran, Iran
Abstract: Because the behavior of pump when rotates as turbine is change, the prediction of
performance of PAT is difficult. Up to now all efforts to perdition of BEP of PAT are done with test
and experimental. And any empirical relations give the BEP of PAT with any errors. But there is not
existed a certain theoretical method to calculation and prediction of BEP of PAT. Therefore a great
interest for research in this field is to investigate a theoretical way for predicting a pump's behavior
operating as a turbine.
In this present paper, we have predicted the BEP of PAT by calculation of PAT hydraulic
characteristics and then we have given the BEP point of the PAT. Finally our result has compared with
the other methods and test results.
Keywords: turbine-pump-volute-efficiency-head-discharge
1 Nomenclature
Ht: turbine actual head
Hp :pump actual head
H"t: turbine Euler head
Q"nt: turbine shock less discharge
Qt: turbine actual discharge
Qlt: turbine leakage discharge
Qlp: pump leakage discharge
Pmt: pump mechanical losses
Pdp: pump disk friction losses
Plt: turbine leakage losses
Pelt: turbine exit losses
Pvt: turbine volute hydraulic losses
Pit: turbine impeller hydraulic losses
Pmt: turbine mechanical losses
Pdt: turbine disk friction losses
Pnt: turbine exit power
Np: pump rotational speed
Nt: pump rotational speed
hp: pump total efficiency
hmp: pump mechanical efficiency
hdp: pump disk friction efficiency
hqp: pump leakage efficiency
hvp: pump volute efficiency
hvt: turbine volute efficiency
hip: pump impeller efficiency
ht: turbine total efficiency
hit: turbine impeller efficiency
e: turbine utilization factor
b'1: pump impeller inlet angle
b'2: pump impeller outlet angle
D1: pump impeller inlet diameter
D2: pump impeller outlet diameter
z: quantity of pump impeller blade
e: pump impeller blade thickness
b1: pump impeller inlet passage width
b2 : pump impeller outlet passage width
b3: pump volute inlet passage width
av:pump volute angle
a2 : pump impeller outlet absolute velocity angle
b3 : turbine impeller outlet relative velocity angle
a1: pump impeller inlet area
a2 pump impeller outlet area
U2: pump impeller outlet peripheral velocity
U3: turbine impeller outlet peripheral velocity
Vu2 : pump impeller outlet absolute tangential
velocity
Vu3 :turbine impeller outlet absolute tangential
velocity
2 Introduction
Small hydroelectric power stations became
attractive for generating electrical energy after the
oil price crisis of the seventies. However cost per
KW energy produced by these stations is higher
than the hydroelectric power plants with large
capacity. Numerous publications in recent years
emphasize the importance of using simple turbine
in order to reduce the cost of produced electrical
energy.
There is need for installation small hydroelectric
power stations many developing countries.
We considered the idea of using pumps as
hydraulic turbines an attractive and important
alternative. Pumps are relatively simple machine,
are easy to maintain and are readily available in
most developing countries. From the economical
point of view, it is often stated that pumps working
as turbines in the range of 5 to 500 KW allow
capital payback periods of tow years or less which
is considerably less than that of a conventional
turbine.
Because the behavior of pump when rotates as
turbine is change, the prediction of performance of
PAT is difficult. Up to now all efforts to perdition
of BEP of PAT are done with test and
experimental. And any empirical relations give the
BEP of PAT with any errors. The many researchers
such as Stepanof[2], Wong[5], Sharma[6],
Williams[7], gantar[4], Ramos, H., Borga[9],
Alatorre-Frenk[8],
Nourbakhsh-Derakhshan[6]
have presented relations that predicate some of
pumps with special characteristics. And they are
not suitable for all of pumps. But there is not
existed a certain theoretical method to calculation
and prediction of BEP of PAT.
In this present paper, we have predicted the BEP of
PAT by calculation of PAT hydraulic
characteristics and then we have given the BEP
point of the PAT. Finally our result has compared
with the other methods and test results.
Figure.1. pump impeller outlet velocity triangle
Figure.2. turbine impeller inlet and outlet velocity
triangles
3 Calculations
For doing of calculations, primary we must define
the important geometrical and Hydraulic
characteristics of pump. In following the important
turbine parameters shall be calculated. In the next
section the method of parameters calculations have
been presented.
3.1 The pump important parameters
For calculation of the BEP of turbine, from pump
geometrical and hydraulic characteristics, these
parameters shall be defined. That is divided two
sections:
- Hydraulic characteristics:
The hydraulic characteristics consists three
parameters: head (H), discharge (Q) and efficiency
(h). Be noted that the obtaining of these parameters
is very easy.
- Geometrical characteristics:
Geometrical parameters consist the impeller inlet
and outlet angels (b'1,b'2), the impeller inlet and
outlet diameters (D1,D2), the impeller blade
numbers (z), the impeller blade thickness (e), the
impeller inlet and outlet passage widths (b1,b2), the
volute outlet width (b3) or volute angle(av).
With tow ways we can define these parameters:
from pump manufactures or with sizing.
3.2 The turbine important parameters
In the following the important parameters of
turbine shall be defined. For obtaining the BEP of
turbine, the bellow parameters shall be calculated:
head (H), discharge (Q) and efficiency (h). For
calculation of these parameters the following
characteristics must be obtained:
1. Hydraulic losses in volute and impeller.
2. Mechanical losses and disc friction losses.
3. Leakage losses.
4. Utilization factor.
5. Euler head.
6. Shock less discharge.
In the following these parameters have been
calculated and finally the BEP of turbine mode has
been obtained.
3.3 Analyzing and calculating of turbine mode
parameters
In this section the inlet and outlet velocity triangles
shall be obtained. The pump inlet and outlet
velocity triangles have been shown in figure 1.
And for turbine mode the inlet and outlet velocity
triangles have been shown in figure 2.
Considerations show that the water inlet angle to
impeller in turbine mode (a2) is equal with volute
angle. In fact the volute performance is same as
guide channel. The water outlet angle (b3) from
impeller in turbine operation is equal with impeller
inlet angle (b'1) too (no whirl at exit). So the Euler
Head is:
(1)
Ht  U 2 Vu 2  U 3 Vu 3
Then:
U 3 Qnt m. cot  v cot 3
U 32


(2)
Ht 
[

]
g
a2
a1
g
That Q"n is shock less discharge and obtained with
relation:
U 2 .a 2
(3)
Qnt 
cot  v  cot 2
In the following with hydraulic losses, the actual
head of turbine operation has been obtained.
The head reaches to impeller after reduction in
volute. So the hydraulic losses in volute shall be
added to the Euler head. But the calculation of
these losses is difficult. So shall be tried to
calculation of them from pump hydraulic and
geometrical characteristics.
The leakage losses for pump operation are obtained
from [1], mechanical losses and disc friction losses
from [2]. Finally with upper efficiencies, the pump
hydraulic efficiency is obtained:
p
(4)
hp 
qp .mp .dp
And with assuming the equal hydraulic losses for
impeller and volute in pump operation, the pump
volute hydraulic efficiency is obtained [7]. Because
in turbine mode the volute is converging, so the
hydraulic losses are less than them in pump mode.
So:
1   vt  0.8(1   vp )
(5)
Then volute hydraulic losses are calculated.
The other losses that shall be gotten together with
Euler head are losses that are missed in outlet of
turbine in form of kinetic energy. Then the
Utilization factor is defined:
Ht
(6)

V u 23
Ht 
2g
Finally the head in BEP of turbine operation is
calculated:
(7)
H t  . vt .Ht
For calculation of turbine BEP discharge, it is
assumed that shock less discharge is BEP
discharge. So we shall calculate the leakage
discharge. With calculate pump leakage discharge
[2], the turbine leakage discharge is obtained [7]:
Ht
Q lt  Q lp .
(8)
Hp
t
Then the BEP discharge of turbine is calculated:
(9)
Q t  Qnt  Q lt
For calculation of efficiency shall be the total
losses in turbine mode calculated. With available
mechanical and disk friction losses of pump the
mechanical and disk friction in turbine are
calculated:
N
(10)
Pmt  Pmp  t
Np
N 3t
(11)
N 3p
And the leakage losses and the volute hydraulic
losses of turbine are:
Plt  .Q lt .H t   vt
(12)
Pvt  (1   vt )..Q lt .H t
(13)
The kinetic energy losses in turbine exit are:
Pelt  (1  ).( .Q lt .H t  Pvt  Plt )
(14)
In following the hydraulic losses of impeller in
turbine mode has been calculated:
Pdt  Pdp 
Pit  (1  it ).( .Q lt .H t  Pvt  Plt  Pelt )
(15)
Shall be noted that with defining of impeller
hydraulic efficiency of pump, the impeller
hydraulic efficiency of turbine are obtained (the
impeller in turbine mode is converge):
1  it  0.8(1  ip )
(16)
And the turbine exit power is:
(17)
Pnt  .Q t .H t  Pvt  Plt  Pelt  Pit  Pmt  Pdt
And the turbine maximum efficiency is equal to:
t 
.Q t .H t  Pvt  Plt  Pelt  Pit  Pmt  Pdt
.Q t .H t
(18)
4 Example
The Worthington-Simpson 25WB125 pump has
been used as an example of applying the
theoretical method to predict the turbine
performance. The hydraulic characteristics of
pump are:
That this method gives the BEP of turbine
operation in rotational speed Nt=3100 rpm
The results of theoretical method have been
compared with the other methods results in table 1.
Table1. Comparison of theoretical method result with
the other results.
Stepanof [2]
Sharma[6]
Alatorre-Fre-1nk[8]
Alatorre-Frenk-2[8]
Area Ratio[7]
Theoretical
Method
Test[7]
Ht (m)
Qt (l/s)
ht (%)
44.7
52.3
51.2
67.9
56.5
3.53
4.47
4.68
4.63
4.00
46
46
46
263
45
59.45
4.03
47
59.3
3.93
48
5 Conclusions
The theoretical method calculates the BEP of
turbine mode of pump with considering the
hydraulic and geometrical characteristics of pump
and pump behavior in turbine mode. Primary with
standard relations of pump and using of the other
researcher results, the pump hydraulic parameter
are obtained. Then the hydraulic parameters in
turbine mode are calculated. This method has a
little error and predicts the BEP of turbine mode of
pump well. Table1 show that the theoretical
method gives the good result. The other matter is
that the most of methods assume that the efficiency
of pump and turbine is equal. But in fact these
efficiencies are not equal.
References:
[1] Nourbakhsh, A, and Jahangiri, G.(1992).
Inexpressive small hydropower stations for small
areas of developing countries, pp.313-319.
conference on Advances in Planning, Design and
Management of Irrigation Systems as Related to
Sustainable Land use, Louvain, Belgium.
[2] Stepanoff, A.J. (1957). Centrifugal and axial
flow pumps, John Wiley and Sons, New York.
[3] Williams, A. (1995). Pumps as turbines a user's
guide,
pp.34,
intermediate
Technology
publications, London.
[4] Gantar, M. (1988). Propeller pumps running as
turbines, pp 237-248, conference on Hydraulic
Machinery, Ljubljana, Slovenia.
[5] Wong, W. (1987). Application of centrifugal
pumps for power generation, pp. 381-348, World
Pumps.
[6] Nourbakhsh, A., Derakhshan, S.,(2004).
Prediction of pump as turbine performance in small
hydropower stations, pp. 31, Conference of Iran
Society of Mechanical Engineering, Tarbiat
Modares University, Tehran, Iran.
[7] Williams, A., (1992). Pumps as turbines used
with induction generations of stand-alone microhydroelectric
power
plants,
Nathingham
Polytechnic, Doctor of philosophy mechanical
engineering project thesis.
[8] Richard Allan, R.A. (2001). Modeling of
pumped storage and hydropower potential within
water supply networks, Notingham University
[9] Ramos, H., Borga, A., (1999). Pump as
turbines: an unconventional solution to energy
production, Urban Water pp 261-263.