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Modelling valve behaviour in unsteady conditions
J P Ferreira, N M C Martins, D I C Covas
CERIS, Instituto Superior Técnico, Universidade de Lisboa, Portugal
ABSTRACT
The characterization of boundary conditions is essential to accurately describe real
systems when using the numerical models to predict transient pressure variations, to
locate leaks and blockages and to identify pipe stiffness and wall-thickness changes. If
the boundary conditions are not fully understood and correctly described, inaccurate
transient pressures can be calculated that do not allow the correct design and diagnosis.
An existing pipe facility at Instituto Superior Técnico has been used to collect transient
pressure data. The system has a classic reservoir-pipe-valve configuration with a length
of 15.22 m and a pipe inner diameter of 0.02 m. Collected data comprehended steady-
state discharge and transient pressures at upstream and at downstream the valve and at
the pipe mid-length. The downstream valve position was also measured. Three initial
discharges were analysed and each one was tested for three different valve manoeuvres.
The effective closure time of the valve was, at least, ten times lower than the total time
of the valve manoeuvre.
The discharge during valve closure was indirectly estimated and obtained results
discussed. Two mathematical functions (hyperbolic and sigmoidal) were analysed for
describing the discharge time variation during the valve closure for each initial
discharge. The most appropriate function to describe the downstream boundary condition
depends on the assumptions and simplifications of the transient solver. A very good fit
was observed between the numerical results and collected data, when the valve closure is
correctly described. A computational fluid dynamics (CFD) model was used to support
the observed experimental behaviour and to highlight the effect of the pipe length on the
discharge variation.
Keywords: Ball valve, Hydraulic transients, Numerical modelling, Boundary conditions
NOTATION
c = pressure wave speed (ms-1)
D = pipe inner diameter (m)
E = Young's modulus (Pa)
e = pipe-wall thickness (m)
g = gravity acceleration (ms-2)
H = piezometric head (m)
K = Modulus of compressibility (Pa)
L = pipe length (m)
Q = discharge (m3s-1)
Q0 = initial discharge (m3s-1)
Re = Reynolds number (-)
r = inner radius of the pipe-wall (m)
S = pipe cross-sectional area (m2)
t = time (s)
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tc = valve closure time (s)
u = time, in convolution integral (s)
V = fluid mean velocity (ms-1)
W = Zielke (1968) or Vardy and
Brown (2003) weighting function
x = distance along the pipe axis (m)
ν = Poisson's ratio (-)
m = dynamic viscosity (N s m-2)
q = closure angle (º)
r = water specific weight (kg m-3)
w
t = wall shear stress
ws
t= steady wall shear stress
wu
t= unsteady wall shear stress
Underscripts
0 = initial
u = upstream
d = downstream
u,0 = initial upstream
1 INTRODUCTION
Boundary conditions are essential for accurate numerical modelling of real systems in
particular when valves are one of the main components of pressurized pipe systems. As
such, if they are not correctly characterized, models may not be able to describe the
designed/real system leading to different extreme pressures or to different predicted
transient behaviours.
Several studies have been carried out to characterize different types of valves under
steady and unsteady conditions. Tullis (1989), Dickenson (1999) and Nesbitt (2007) have
thoroughly characterized several types of valves under steady state conditions. Lescovich
(1967) was the first to describe valve closures under unsteady state flows, having
consequences on transient analysis and on predicted extreme pressures. Meniconi et al.
(2011), Yang et al. (2016) and Martins et al. (2016) described ball valves closure
manoeuvres under unsteady conditions by using head loss coefficients or by a discharge
variation in time. Good curves fittings were obtained between pressure measurements
and numerical results by Meniconi et al. (2011) due to the energy dissipation due to the
viscoelastic behaviour of the pipe material and by Yang et al. (2016) due to the long
valve closure time. Martins et al. (2016), using a 3D CFD model, obtained a sharp valve
closure manoeuvre when compared with the measure experimental pressure signal, even
though the predicted extreme pressures fitted well the physical measurements.
These studies focused mainly on the pressure variation for different boundary conditions.
However, none has described the influence of the closure time nor the effect of other
elements located at downstream the valve on the valve response. This paper analyses the
effective closure of ball valves and presents experimental valve discharge laws to better
describe the valve dynamic behaviour under unsteady state flows. These laws were
implemented in a numerical transient solver incorporating two unsteady friction models
(Zielke, 1968, Vardy and Brown, 2003). Results are compared with experimental data
collected from an experimental pipe rig at the Laboratory of Hydraulics and
Environment (LHE/IST) of Instituto Superior Técnico, Lisbon, Portugal.
2 EXPERIMENTAL FACILITY
A pipe-rig at the Laboratory of Hydraulics and Environment (LHE) of Instituto Superior
Técnico (IST), Portugal, was used to collect steady-state discharge and pressure head
data for the ball valve characterization. The copper pipe was 15 m length, L, with an
inner diameter, D, of 0.02 m and a pipe wall thickness, e, of 0.001 m. The pipe was
anchored throughout its length with supports equally-spaced 0.60 m. The anchoring
distance was reduced to 0.18 m near the valves to avoid any axial movement during the
730 © BHR Group 2018 Pressure Surges 13
transient events. The system was supplied by a centrifugal pump, with a nominal
discharge of 3.6 m3h-1 and a maximum head of 46 m, installed at the upstream end of the
pipe. A 60 litres air vessel was installed in derivation after the pump to control the inlet
pressure.
Two quarter-turn full-bore ball valves, with 0.02 m of nominal diameter, were installed
at the downstream end of the pipe system (Figure 1). These valves were used to carry out
the experimental tests. The first (V1) had an actuator which was triggered by a solenoid
valve with a working 2-10 bars air pressure to generate transient events (the valve
position was obtained with a potentiometer). The second valve, manual, was used merely
to establish the steady-state discharge. After the set of valves, there was a return pipe
hose with a higher diameter and with ca. 20 m length.
Figure 1. Experimental facility schematic
The data acquisition system (DAS) was composed of a desktop computer, an
oscilloscope, a trigger-synchronizer (which emits a 0-5V signal to initiate the data
acquisition from all sensors at once during a 5 s time-span), an electromagnetic
flowmeter (with a 0.4% accuracy and a low flow cut-off at 15 Lh-1 flow rate) and three
strain-gauge type pressure transducers (25 bars nominal pressure and 0.25% full-scale
span accuracy). All signals were measured with the oscilloscope and furtherly treated.
Pressure transducers 1 and 2 were at the downstream and at the upstream end of the ball
pneumatically actuated valve respectively; the third transducer was at the pipe mid-
length. Valve V1 was subjected to different operating air pressures to achieve different
closures times. The air pressure in the solenoid valve V1 actuator was controlled by a
ServiceJunior pressure transducer with an operating pressure between -1 to 16 bar and a
0.5% full-scale accuracy.
The pressure wave speed, c, was estimated in 1,250 m/s based on the travelling time of
the transient pressure wave between two consecutive transducers (Ferràs et al., 2016,
Soares et al., 2017). All tests were carried out for totally developed initial discharges and
all pressure measurements were collected with a 1 kHz frequency.
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3 NUMERICAL MODEL DEVELOPMENT
A 1D numerical transient solver, developed based on the Method of the Characteristics
(MOC), was used to analyse the transients generated by the valve closure. The model is
based on mass conservation and momentum equations (Almeida and Koelle, 1992, Wylie
and Streeter, 1993, Chaudhry, 2014).
2
0
H c Q
t gS x
¶ ¶
+¶=
¶ (1)
10
4
w
H Q
x gS t gD t
r
¶ ¶ =
¶ ¶
+ + (2)
where H is the piezometric head, Q is the discharge, tand x are the time and distance
coordinates in the MOC grid, respectively, Sis the area of the pipe cross-section and w
t
is the wall shear stress and ris the water specific weight. Wall shear stress is usually
divided into two different components: the steady state, ws
t, and the unsteady state,
wu
t, components. These are commonly summed, resulting in the total wall shear
stress, w ws wu
t t t= + . The former is calculated by steady-state friction formulas (e.g.
Colebrook-White), whereas the latter is calculated by unsteady state friction
formulations. Since the copper pipe has a smooth wall, the steady-state component of
wall shear stress is estimated by means of Blasius and Hagen Poiseuille friction
formulations for smooth wall turbulent and laminar flows, respectively. Unsteady wall
shear stress is very important for fast transient events (Bergant et al., 2008, Brito et al.,
2014, Martins et al., 2016). The convolution-based formulations of Zielke (1968) and of
Vardy and Brown (2003) were implemented in the transient solver and used herein to
estimate unsteady-state friction for laminar and turbulent flows, respectively. This
component of the wall shear stress is obtained according to:
0
4( ) ( )
t
wu u W t u d
Vu
D t
m
t-
¶
=¶
ò (3)
where m stands for the water viscosity and Wto the weighting function as defined by
Zielke (1968) and by Vardy and Brown (2003) for laminar or smooth-wall turbulent
conditions, respectively. These formulations are those that lead to the best fitting of the
numerical results with collected data in terms of pressure wave amplitude, phase and
shape.
The model includes the hydropneumatic tank at the upstream end (as a constant head
reservoir), the copper pipe and the pneumatically actuated valve at the downstream end
(which generates transient events and is considered as the system’s boundary condition).
The return hose pipe system is not included in the model, since the pipe cross-section is
not completely circular (it has an irregular oval-shape) and the pipe material is
viscoelastic (Covas et al., 2004, 2005).
The implemented one-dimensional model is composed of a fixed-grid, which gives the
user full control over the grid (Ghidaoui et al., 2005). As the system is a straight copper
pipe without singularities, no interpolations or grid-adjustments are required and all
calculations can be carried out using a time-step Δt = 5 × 10−5s.
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4 EXPERIMENTAL ANALYSIS AND MODEL VALIDATION
Three initial flow rates were analysed – 450, 200 and 56.5 Lh-1 – corresponding to
turbulent, transitional and laminar flows, respectively. The initial discharge was
measured by an electromagnetic flowmeter. Transient pressure measurements were
collected upstream and downstream of the pneumatically actuated valve to analyse the
valve manoeuvre. The valve closure time was reduced by increasing the air pressure that
supplies the valve actuator. Figure 2 depicts the valve closure percentage, / max
q q (being
max
q= 90º), as a function of the ratio between the time, t, and the total closure time, tc.
The starting pressure was 6.5 bar and was progressively decreased until 1.5 bar. An
approximately linear law was obtained for all air pressures expect for 1.5 bar. This value
is the limit of the solenoid actuator operation range.
Figure 2. Valve closure percentage with non-dimensional valve closure time
The discharge variation was estimated by inverting Joukowsky formulation, described by
Eq. (4). This approach requires pressure measurements at the upstream, Hu, and
downstream, Hu,0, ends of the valve, the pipe cross sectional area, S, the gravitational
acceleration gand the initial discharge, 0
Q, in the pipe.
,0
0 0 0
( )
u u
H H
c Q gS H gS
H Q
gS Q c Q c Q
-
D D
D = D = = (4)
However, Eq. (4) is only valid if the valve closure is a fast manoeuvre (tc<2L/c). The
value of 2L/c is 0.024 s, which is lower than the total closure manoeuvres presented in
Table 1. Figure 3 depicts the non-dimensional discharge variation as a function of the
valve closure times for each initial discharge using the pressure both upstream and
downstream valve V1. The discharge varies in a shorter time than the valve closure time.
The valve actually closes in the effective closure time which corresponds to the time of a
linear discharge variation from the maximum discharge until shut-off with a constant
slope (i.e. equal to the maximum slope of the real discharge variation) presented by
Lescovich (1967):
© BHR Group 2018 Pressure Surges 13 733
,c effec
max
tdQ
dt
Q
=æ ö
÷
ç÷
ç÷
÷
çø
D
è
(5)
Table 1. Description of experimental tests and valve total and effective closure times
for each test
Test Q0 (Lh-1) Re Closure time ( c
t) (s) Effective closure time ( ,c effec
t ) (s)
D1
450 7957
0.043 0.0028
D2 0.054 0.0031
D3 0.061 0.0046
D4
200 3537
0.048 0. 0016
D5 0.053 0. 0023
D6 0.074 0. 0032
D7
56.5 999
0.045 0. 0020
D8 0.051 0. 0044
D9 0.055 0. 0050
Figure 3. Non-dimensional discharge variation and maximum slope in time for each
tested conditions
734 © BHR Group 2018 Pressure Surges 13
The effective closure time of the ball valve varies from 4 to 10 % of the total closure
time. This is not the case for other types of valves due to their geometry and size. Snap-
shots of several instances are shown in Figure 4 to illustrate the velocity inside the valve
using a 3D CFD model (Martins et al., 2016, Ferreira et al., 2018).
28.3º 68.0º
83.2º
87.0º
Figure 4. Velocity vectors at different closure angles for an initial discharge
of 450 Lh-1
The experimental discharge variation must be estimated to introduce in the numerical
model as the downstream boundary condition. The obtained discharge variation as a
function of the closure angle is shown in Figure 5. The discharge is almost constant until
a 70% closure and, afterwards, the discharge varies very fast. The discharge is null
before the valve rotation is completed due to the inexistence of available cross section
for the flow. Such behaviour can be observed in Figure 4.
These non-dimensional discharge variations can be described by a hyperbolic function,
Eq. (6), of two parameters that depend on the Reynolds number. The same analysis was
carried out for 13 flow rates with Reynolds number varying between 1000 and 9000.
0
) 100
1
10
/
0
(max
Q
Q
e
q q d
æ ö
-÷
ç
= - ÷
ç÷
÷
ç
è ø
´ (6)
in which d and e depend on the initial Reynolds number according to Eqs. (7) and (8).
4.36 0.0002 Red= - (7)
164.85 15.62 ln (Re)e= - (8)
© BHR Group 2018 Pressure Surges 13 735
Using the discharge variation given by Eq. (6), the numerical model was used to describe
the tests D1, D3, D7 and D9. The valve closure percentage, / max
q q , was considered
linear in time. Comparing experimental data with the numerical results for two initial
discharges (450 and 56.5 Lh-1) and two closure times, the extreme pressures and the
pressure decay were correctly estimated by the numerical model. However, the pressure
wave shape was not accurately described, as the valve seemed to close faster in the
numerical model than in the experimental facility. This can be confirmed by the position
of the valve, registered at every instant. This was due to the systems not being
completely described in the numerical model. For this approach to be correct, the pipe
after the pneumatic valve should have been included in the model.
Figure 5. Non-dimensional discharge variation with closure angle for turbulent and
laminar initial flow conditions with in-line valve approach
Instead of modelling the whole system and considering the valve as an in-line valve, the
discharge variation is analysed as a discharge to the atmosphere once the model is not
complete. The discharge variation is calculated using Eq. (4).
736 © BHR Group 2018 Pressure Surges 13
(a)
(b)
(c)
(d)
Figure 6. Comparison between pressure measurements and numerical results using
Eq. (6) for tests with in-line valve approach: (a) D1; (b) D3; (c) D7 and (d) D9
The non-dimensional variation is depicted in Figure 7 as a function of the closure angle.
Until 70% of valve closure, the discharge does not vary as in the previous approach.
However, afterwards, the discharge does not show a common variation, especially in the
last stages of closure. This variation can be described by a sigmoidal function, Eq. (9), of
two parameters that depend also on the Reynolds number. These values were obtained
for the same Reynolds numbers from 1000 and 9000 according to Eqs. (10) and (11).
) 100(( / 95)
0
1
11max
Q
e
Q
h
x q q- ´ -
æ ö
÷
ç
= - ÷
ç÷
ç
è ø
+ (9)
in which the x and e coefficients depend on the initial flow Reynolds number:
6
0.7157 6 10 Rex-
= - ´ (10)
0.716 0.602ln(Re)h= - (11)
© BHR Group 2018 Pressure Surges 13 737
Figure 7. Non-dimensional discharge variation with closure angle for turbulent and
laminar initial flow conditions with discharge to the atmosphere approach
(a)
(b)
(c)
(d)
Figure 8. Comparison between pressure measurements and numerical results using
Eq. (10) for tests with in-line valve approach: (a) D1; (b) D3; (c) D7 and (d) D9
The comparison of numerical results obtained for two initial discharges (450 and 56.5
Lh-1) and two closure times with the obtained pressure measurements is depicted in
Figure 8. A better fitting is obtained in terms of the extreme pressure values and the
738 © BHR Group 2018 Pressure Surges 13
pressure wave decay. Thus, even though the pipe characteristics at downstream the
boundary valve are not completely defined, the pressure wave shape and decay were
accurately calculated.
5 CONCLUSIONS
The dynamic behaviour of a ball valve was analysed and the effective closure time was
estimated. For this valve, the effective closure time was estimated between 4 to 10% of
the total closure time and the valve is totally closed for the 95% the valve rotation.
Two approaches were used to describe the valve boundary condition using a 1D transient
solver. These aim to simplify the system modelling at downstream the valve that
generates the transient event. The only way for the valve manoeuvre not to influence the
transient pressure wave is with an instantaneous closure, which is not physically
possible.
The correct estimation of the valve effective closure time is of major importance for
hydraulic transient analysis in the design stage. If not taken into consideration, the
pressures that the system needs to cope with may be higher than the calculated ones. This
is important as well for the diagnosis of existing pipe systems when using inverse
transient analysis for both leak and blockage detection, as the success of this analysis
strongly depends on the accurate description of the system as well as the correct
prediction of transient pressures in terms of wave amplitude, shape and shift.
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