Parameters affecting water ham

Parameters affecting water hammer in a high-head
hydropower plant with Pelton turbines

U Karadžic1, A Bergant2, P Vukoslavcevic1
1 Faculty of Mechanical Engineering, 81000 Podgorica, Montenegro
2 Litostroj E.I. d.o.o., 1000 Ljubljana, Slovenia

ABSTRACT
This paper investigates water hammer effects in a high-head hydropower plant Perucica, Montenegro. During its first phase of modernisation and revitalisation new distributors (needle valves) have been installed on the first two Pelton turbine units. Numerical results using the standard quasi-steady friction model and the convolution based unsteady friction model for different needle closing laws are compared with results of measurements. The numerical model with the actual two-speed closing law gives the best match between computed and measured results for the case of emergency shut-down of the turbine unit. Inclusion of unsteady friction effects into the numerical models slightly improves numerical results.
1 NOTATION
The following symbols are used in this paper:
A pipe area
Am nozzle area
a pressure wave speed
bov width of the overflow weir
D pipe diameter, diameter
Db shaft bearing diameter
Dk wheel diameter
Dz spherical valve diameter
dm nozzle diameter
e pipe wall thickness
Fh jet hydraulic force
f Darcy-Weisbach friction factor
g gravitational acceleration
H piezometric head (head)
Hd head downstream the nozzle
Hu head upstream the nozzle
J polar moment of inertia
K constant equal to 4?/D2
KQ nozzle discharge coefficient
kair fluid (air) damping coefficient
L pipe length, length
Mair fluid (air) damping torque
Mfr shaft bearing friction torque
Mh hydraulic torque
Mr rated torque
m dimensionless torque (m = M/Mr)
mk, nk exponential sum coefficients
N number of computational reaches
Nk number of exponential terms
n turbine rotational speed
P power
Q discharge
Qm discharge to the turbine wheel
Qu nozzle discharge
Rb resultant force in the bearing
Re Reynolds number (Re = VD/?)
s needle stroke
Ta mechanical starting time
t time
tc needle closing time
to needle opening time
tdef jet deflector operating time
u peripheral velocity
V average flow velocity
Vm jet velocity
v flow velocity
x distance along the pipe
yk weighting function component
z water level, elevation
? momentum correction factor
?t time step
?x reach length
? absolute pipe roughness
? relative speed change
? water mass density
?b shaft bearing friction coefficient
?ov overflow discharge coefficient
? kinematic viscosity
? dimensionless nozzle opening
? angular velocity
?in surge tank inflow coefficient
?out surge tank outflow coefficient

Subscripts:
def deflector
max maximum
q quasi-steady part
R reservoir
r rated conditions
T tunnel
u unsteady part
0 initial or steady state conditions
I,II,III penstock number

Abbreviations:
QSF quasi-steady friction
CBM convolution-based model
HPP hydropower plant
MOC method of characteristics

2 INTRODUCTION

Planning of new or refurbishment and modernisation of existing hydropower plants (HPPs) requires detailed water hammer analysis in order to get maximum and minimum pressures as the most important parameters in the design process of the plant components. Water hammer induces pressure rise or drop in hydraulic systems, rotational speed variation in hydraulic turbomachinery (pumps and water turbines) and level fluctuation in surge tanks and air chambers (1). In hydropower plants water hammer is caused by turbine load acceptance and reduction, load rejection under governor control, emergency shut-down and unwanted runaway, and closure and opening of the safety shutoff valve. Unsteady pipe flow equations are used to describe water hammer phenomena. Boundary conditions for reaction type water turbines (Francis, Kaplan, bulb) are well defined in the literature (2), (3), (4). A little is published about detailed modelling of impulse type Pelton turbine (5), (6). This turbine is usually represented as an end-valve boundary condition.

This paper investigates water hammer effects on operation of high-head Perucica HPP, Montenegro. In the first part of the paper mathematical tools for solving water hammer equations are presented. Water hammer is fully described by two hyperbolic partial differential equations, the equation of continuity and the momentum equation (3), (4). These equations are solved by the method of characteristics (MOC). Friction losses in closed conduits are usually estimated by the quasi-steady friction model. In addition, a convolution based unsteady friction model (7) can be used for calculation of friction factor. Different closing laws for the distributor (needle valve) of the Pelton turbine are presented. A novel model for calculation of Pelton turbine rotational speed change during emergency shut-down is presented in detail. In the second part of the paper comparisons of numerical and experimental results are made for turbine load rejection cases (load rejection under governor control, emergency shut-down) from different initial powers. It is shown that numerical model with actual two-speed closing law gives the best fit with results of measurements for the case of emergency shut-down of the turbine unit. It is also shown that unsteady friction effects have small impact on water hammer events in the Perucica HPP flow-passage system.
3 THEORETICAL MODELLING

Water hammer equations are applied for calculation of the liquid unsteady pipe flow. A simplified form of the equations neglecting the convective terms is used for most engineering applications (3), (4),

(1)

(2)

where H = piezometric head (head), Q = discharge, a = pressure wave speed, D = pipe diameter, A = pipe area, g = gravitational acceleration, f = Darcy-Weisbach friction factor, x = distance, and t = time. The staggered (diamond) grid (4) in applying the method of characteristics (MOC) is used in this paper. At a boundary (reservoir, Pelton turbine), a device-specific equation replaces one of the MOC water hammer compatibility equations.

3.1 Modelling friction losses

Friction losses in penstocks of hydropower plants are traditionally estimated by quasi-steady friction model. This approach does not give good results for fast transients when numerical results are compared with experiments (8). The friction factor f can be expressed as the sum of the quasi-steady part fq and the unsteady part fu,

(3)

The quasi-steady friction factor is dependent on the Reynolds number and relative pipe roughness and it is updated every time step. The explicit Halland equation (9) is used in this paper,

(4)

where Re = Reynolds number (Re = VD/?), V = average flow velocity, ? = kinematic viscosity, and ? = absolute pipe roughness. For evaluating of the unsteady friction factor a convolution based model (CBM) (7) is used in this paper. The unsteady friction factor is expressed as a finite sum of Nk functions yk(t) (10),

(5)

with

(6)

where Nk = number of exponential terms (Nk,max = 10), ?t = time step, and K = constant equal to 4?/D2. Coefficients mk and nk have been developed for Zielke’s (7) and Vardy-Brown’s (11), (12) weighting functions and can be found in Vítkovský et al. (10). In addition, a momentum correction factor ?, defined by Eq. (7), is incorporated into the MOC solution when CBM model is used (13),

(7)

where v = flow velocity.

3.2 Pelton turbine boundary condition

3.2.1 Pelton turbine distributor (needle valve)
Pelton turbine distributor (needle valve) is utilized for regulation of discharge and consequently for regulation of the turbine output. Discharge is controlled by closing or opening the mouth of the nozzle by means of a needle (Fig. 1) and with appropriate position of the jet deflector. The discharge through the nozzle is only dependent on the position of the needle and it is not dependent on the turbine speed (14). Therefore, the water hammer equations and the dynamic equation of the unit rotating parts can be solved separately. In this way the head and discharge through the nozzle during the transient event are calculated by the MOC and these values are used as input in the solution method for the dynamic equation of the unit rotating parts.

Fig. 1 Pelton turbine distributor (needle valve)

The discharge through the nozzle is defined by the following equation,

(8)

where KQ = nozzle discharge coefficient, Am = nozzle area (Am = ?dm2/4), Hu,t = head upstream the nozzle, and Hd = const. = head downstream the nozzle. Typical functional dependency of the discharge coefficient KQ and the ratio of needle stroke s and nozzle diameter dm is shown in Fig. 2.

The needle closing law is expressed as follows,

(9)

where ? = dimensionless nozzle opening, and smax = maximum needle stroke.

Fig. 2 Typical discharge coefficient of Pelton turbine nozzle

3.2.2 Rotational speed change during emergency shut-down of the unit
The equation that describes dynamic behaviour of the Pelton turbine unit rotating parts during emergency shut-down is,

(10)

where J = polar moment of inertia of the turbine unit rotating parts, ? = angular velocity, Mh = hydraulic torque, Mfr = shaft bearing friction torque, and Mair = fluid damping torque (ventilation losses in the turbine housing). The turbine is disconnected from the electrical grid followed by simultaneous gradual full-closure of the needle(s) and rapid activation of the jet deflector(s) (deflection of the jet from the wheel).

Let us introduce the relative speed change ?,

(11)

and the mechanical starting time Ta (2), (3),

(12)

into Eq. (10), after rearrangement it follows,

(13)

where n = turbine rotational speed (traditionally in rpm), r = rated, Mr = rated torque, and m = dimensionless torque (m = M/Mr).

The dimensionless hydraulic torque is expressed as follows,

(14)

where Fh = jet hydraulic force (15),

(15)

u = peripheral velocity,

(16)

Dk = wheel diameter, Qm = discharge to the turbine wheel (Qm ? Qu), and Vm = jet velocity (Vm = Qm / Am).

The dimensionless shaft bearing friction torque is,

(17)

where Db = shaft bearing diameter, and ?b = shaft bearing friction coefficient. The resultant forces in the shaft bearings RAb, RBb of the horizontal-shaft unit are due to hydraulic force, weight of the wheels, weight of the shaft and weight of the generator.

The dimensionless fluid damping torque is expressed by,

(18)

where kair = fluid (air) damping coefficient.

The hydraulic torque affects the turbine wheel until the jet deflector deflects all the water into the tailrace (t=tdef). During t ? tdef the hydraulic torque is much greater than the dissipation torques. Neglecting the dissipation torques in Eq. (13) gives the following solution for the speed change during t ? tdef,

(19)

At t > tdef the hydraulic torque is set to zero and the dissipation torques mfr and mair are considered in Eq. (13). Now, the relative speed is given by,

(20)

where ?def = relative speed at t = tdef.

The solution method that describes dynamic behaviour of the Pelton turbine unit rotating parts during load rejection under governor control has been developed in a similar way. The turbine is disconnected from the electrical grid followed by simultaneous gradual closure of the needle(s) to the speed-no load position and controlled manouvre of the jet deflector(s) i.e. rapid activation at the first instant followed by gradual adjustment of the deflector to the speed-no load position.
4 PERUCICA HPP FLOW-PASSAGE SYSTEM

Perucica HPP is a complex system comprised of a concrete tunnel (LT = 3335 m, DT = 4.8m), surge tank and three parallel steel penstocks with horizontal-shaft Pelton turbines built at their downstream ends (Fig. 3). The length of each penstock is about 2000 m whereby penstock I feeds two turbine units (A1 and A2) with rated unit power of 39 MW, penstock II feeds three turbine units (A3, A4 and A5) of 39 MW each and penstock III feeds two units (A6 and A7) of 59 MW each. A new turbine unit (A8) with a rated power of 59 MW is to be installed in the near future. The maximum water level at the intake is 613 m and the minimum one is 602.5 m.

Fig. 3 Layout of Perucica HPP, Montenegro

The surge tank is of cylindrical type with an expansion and overflow (Fig. 4). Width of the overflow weir is bov = 7.98m and the discharge coefficient is ?ov = 0.4. At the surge tank intake there is a non-symmetrical orifice with head loss coefficients ?in = 1.65 and ?out = 2.48 during inflow and outflow, respectively.

Table 1 shows geometrical characteristics of the three penstocks (Fig. 3).

Table 1. Geometrical characteristics of penstocks

Section

Lenght L (m) Penstock I Penstock II Penstock III Pipe diameter D (mm) Pipe wall thickness e (mm) Pipe diameter D (mm) Pipe wall thickness e (mm) Pipe diameter D (mm) Pipe wall thickness e (mm) T1-T2 75.0 2200 10 2200 10 2650 12 T2-T3 61.0 2200 10 2200 10 2650 12 T3-T4 330.5 2200-2100 10 2200 10-16 2650 12-13 T4-T5 318.0 2100-2000 16-25 2200 17-23 2650 13-21 T5-T6 123.0 2000 26-29 2200 24-27 2650 21-24 T6-T7 672.0 1900 27.5 2200-2100 26 2500 23 T7-T8 238.0 1800 27-39 2100 26-34 2500 24-29 T8-T9 I: 53.0
II: 99.8
III: 146.6 1800 39 2100 34 2500 29

Fig. 4 Perucica HPP surge tank

Basic characteristics of the Pelton turbine units, built at the downstream end of the penstocks, are presented in Table 2.

Table 2. Characteristics of Pelton turbine units
Turbine unit Rated unit power
Pr (MW) Rated net head
Hr (m) Rated speed
n (min-1) A1,A2,A3,A4 39 526 375 A5 39 526 375 A6,A7 59 526 428
Turbine unit Number of runners per turbine unit The polar moment of inertia of the unit rotating parts J (tm2) Number of needles per turbine runner A1,A2,A3,A4 2 168.75 1 A5 2 168.75 1 A6,A7 2 200 2 Turbine unit Stroke of the needle
smax (mm) Closing time of the needle
tc (s) Opening time of the needle
to (s) A1,A2,A3,A4 150 85 30 A5 195 80 30 A6,A7 166 80 50
The runner diameter of turbine units A1 to A5 is Dk= 2400 mm and for turbine units A6 and A7 is Dk = 2100 mm. The turbine inlet spherical valves diameters are Dz = 1000 mm and Dz = 1200 mm, respectively. The valves are equipped with a passive actuator comprised of a hydraulic servomotor. These valves are fully opened during turbine load rejection tests.

The following quantities have been continuously measured during transient regimes: pressure at the upstream and the downstream end of the turbine inlet valve (spherical valve), stroke of the needle, stroke of the jet deflector and turbine rotational speed. Pressures at the upstream and downstream end of the turbine inlet valve are the same. All measurements have been carried out on turbine units A1 and A2. Absolute pressures have been measured by high-pressure piezoelectric transducers Cerabar T PMP 131-A1101A70 Endress+Hauser (pressure range 0 to 100 bar, uncertainty in measurement ?0.5 %). The needle stroke and the stroke of the jet deflector have been measured by discplacement transducers Balluff BTL5-S112-M0175-B-532 and Balluff BTL5-S112-M0275-B-532, respectively. Uncertainty of these sensors is ?0.03 mm. The turbine rotational speed has been measured using inductive sensor Balluff BES M18MI-PSC50B-S04K (uncertainty in measurement ?0.03 %).

5 COMPARISONS OF NUMERICAL AND FIELD TEST RESULTS

Various operating regimes have been performed in the plant during commissioning of the turbine units A1 and A2, including the unit start-up, load acceptance and reduction, load rejection under governor control and emergency shut-down, and closure of turbine safety valve against the discharge. Water hammer model includes intake, tunnel, surge tank and three parallel penstocks with Pelton units at their downstream end (see Fig. 3). Numerical results from standard quasi-steady friction model (QSF) and convolution based unsteady friction model (CBM) for different needle’s closing laws are compared with results of measurements. The following results of measurements and corresponding numerical simulations are presented:
1. Emergency shut-down of turbine unit A1 from initial power P0 = 39.5 MW (Test A1P39.5MW)
2. Simultaneous load rejection under governor control of turbine units A1 and A2 from initial power of P0 = 42 MW each (Test A1&A2P42MW)
Tables below present the main initial parameters for these two tests. Flow in penstock I is turbulent with a large Reynolds number.

Table 3. Initial discharges through tunnel and penstocks
Test QI (m3/s) ReI ? 106 QII (m3/s) QIII (m3/s) QT (m3/s) A1P39.5MW 8.4 5.5 0 22.0 30.4 A1&A2P42MW 18.6 12.1 0 6.4 25.0
Table 4. Steady friction factors and momentum correction factors
Test Penstock I Penstock II Penstock III Tunnel f0 ? f0 ? f0 ? f0 ? A1P39.5MW 0.0126 1.0123 0.0125 1.0122 0.0126 1.0123 0.0146 1.0142 A1&A2P42MW 0.0125 1.0122 0.0125 1.0122 0.0129 1.0126 0.0146 1.0142
Table 5. Closure time, intake level and initial opening of the nozzle
Test zR (m) tc (s) s0 (mm) tdef (s) A1P39.5MW 605.9 55.3 117 1.6 A1&A2P42MW 608.5 82.7 146 2.0
Pressure wave speeds are as follows, aT = 1345 m/s , aI = 1000 m/s, aII = 983 m/s and aIII=1007 m/s.

5.1 Comparison of numerical and measured head at the turbine inlet

Transient head and discharge have been computed using a staggered grid MOC code. Basic time step was ?t = 0.040 s. Computed and measured results are shown in Figs. 5 and 6.

Numerical and measured head at the turbine inlet (HI) for emergency shut-down of the unit A1 are compared in Fig. 5 (Test A1P39.5MW). The computed and the measured total needle closure time are the same (tc = 55.3 s – see Fig. 5a). The closure time is much larger than the water hammer reflection time of 2LI/aI = 3.84 s. Maximum measured head of 557.7 m occurs when the nozzle is fully closed. Head rise for this case is 24.5 m. Maximum calculated heads obtained by QSF are 555.9 m (one-speed closure) and 556.4 m (two-speed closure; the cushioning stroke is 2.5 %), and by CBM are 556.4 m (one-speed closure) and 556.8 m (two-speed closure). All maximum calculated heads match the measured one. Calculated and measured heads are much lower than the maximum permissible head of 602 m. Numerical results agree well with measured results during the needle closure period. After this, a phase shift for all numerical models is evident from the third pressure pulse on. A model with two-speed closure better attenuates pressure pulses compared to the model with one-speed closure. It means that at the end of the closure the needle actually moves a little bit slower (natural damping). Friction losses are described slightly better by the CBM model (Fig. 5d). The Perucica flow-passage system is not unsteady friction dominant system during water hammer events.

Fig. 5 Comparison of needle stroke (s) and head at the turbine inlet (HI; datum level z = 65.8 m; time step ?t=0.040 s). Emergency shut-down of A1 from P0 = 39.5 MW

Numerical and measured heads at the turbine inlet for simultaneous load rejection under governor control of turbine units A1 and A2 are compared in Fig. 6. Test A1&A2P42MW induces maximum measured head of 573.9 m with head rise of 57.0 m. The maximum calculated heads obtained by QSF are 571.5 m (one-speed closure) and 572.7 m (two-speed closure), and by CBM are 572.4 m (one-speed closure) and 571.9 m (two-speed closure). In this case the needle is closed to its speed-no load position (3.8 %). Process is governed by the turbine control system. Up to this time all numerical models show good agreement with results of measurement and all maximum heads are well below the maximum permissible head of 602 m. All models produce similar results after the closure period. In this case there is no cushioning effect for the two-speed closure because the needle speed-no load position is larger (3.8 %) than the cushioning one (2.1 %). Consequently, the two-speed closure is actually the one-speed closure with slightly shorter closure time.

Fig. 6 Comparison of needle stroke (s) and head at the turbine inlet (HI; datum level z = 65.8 m; time step ?t=0.040 s). Simultaneous load rejection under governor control of A1 and A2 from P0 = 42 MW

5.1.1 Statement on convergence and stability
Convergence relates to the behaviour of the solution as ?x and ?t approach zero, whereas stability is concerned with the growth of round-off error. Analysis of computational results obtained with different numbers of reaches (see Table 6) reveals that the magnitude and timing of the pressure pulses predicted by QSF and CBM converge to practically the same solution. As an example, Fig. 7 shows heads at the turbine inlet (HI) for the case of the emergency shut-down of turbine unit A1 from initial power P0 = 39.5 MW (Test A1P39.5MW). The QSF model (Fig. 7a) and the CBM model (Fig. 7b) results are compared for two different time steps ?t = {0.040; 0.005} s.

Table 6. Number of reaches for tunnel and penstocks
N (?t=0.04 s) N (?t=0.02 s) N (?t=0.01 s) N (?t=0.005 s) Tunnel 62 124 248 496 Penstock I 48 96 192 384 Penstock II 50 100 200 400 Penstock III 50 100 200 400
Fig. 7 Comparison of head at the turbine inlet (HI; datum level z = 65.8 m) for different time steps. Emergency shut-down of A1 from P0 = 39.5 MW

5.2 Comparison of computed and measured turbine rotational speed

Turbine rotational speed during (1) emergency shut-down of turbine unit A1 from initial power P0 = 39.5 MW (Test A1P39.5MW) and (2) during simultaneous load rejection under governor control of turbine units A1 and A2 from initial power of P0 = 42 MW each (Test A1&A2P42MW) has been calculated using adequate solution method of the dynamic equation of the unit rotating parts (Eq. (10)). The input head and discharge through the nozzle during the transient event have been previously calculated by the MOC (see Section 5.1). Fig. 8 shows comparison between computed and measured rotational speed for both case studies.

Fig. 8 Rotational speed change (n0 = 375 min-1)

The maximum measured turbine speed rise for Test A1P39.5MW of 8.1 % occurs at time t = tdef. The maximum computed turbine speed rise of 8.0 % agrees well with measured one (Fig. 8a). After jet deflector deflects all the water into the tailrace, the computed turbine speed decrease reasonably agree with measured one. There is a good agreement between the maximum measured and calculated turbine speed rise for Test A1&A2P42MW of 11.2 % and 11.1 %, respectively (Fig. 8b). There are some discrepancies in the phase of speed decrease due to complex flow behaviour in the turbine housing. In both investigated cases the maximum speed rise is well below the permissible speed rise of 25%.
6 CONCLUSIONS

Computed results using the standard quasi-steady friction model (QSF) and the convolution based unsteady friction model (CBM) for different distributor (needle valve) closing laws are compared with results of measurements performed in Perucica HPP, Montenegro. Both the QSF model and the CBM model produce practically the same results. The CBM model only slightly better captures measured data during the decay period of the transient event. The effects of unsteady friction on water hammer events in Perucica flow-passage system are indeed small; however, they might have a strong influence on behaviour close to resonance and this is a subject of authors’ future studies. The two-speed closing law considers natural damping effect in the hydraulic servomotor close to its fully closed position. That is why the computed results using the two-speed closing law for the case of the unit emergency shut-down fit better the results of measurement compared with the computed results using the one-speed i.e. linear closing law. For the case of load rejection under governor control the needle is closed to its speed-no load position; then the two-speed closing law is actually one-speed law with slightly shorter closure time. The turbine rotational speed change is calculated separately by a novel model of Pelton turbine rotational speed change. There is a reasonable agreement between the computed and measured results.
7 ACKNOWLEDGMENTS

The authors wish to thank ARRS (Slovenian Research Agency) for their generous support of research on fluid transients. The support of research by ZAMTES (Montenegrian Bureau for International Scientific, Educational, Cultural and Technical Cooperation) and The Ministry of Education and Science of Montenegro are gratefully acknowledged as well.
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