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Instantaneous acceleration based model
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PRASHANTH R HANMAIAHGARIPROF. SILVA ARAYA WALTER
MOHAMMED ELKHOLYPROF. M HANIF CHAUDHRY
Instantaneous Acceleration based model coefficients using genetic algorithms
Steady State Friction Formula
Darcy Weisbach Equation
f = Darcy friction factor. f is calculated using formula depends on flow is
laminar, transition between laminar and turbulent flow, fully turbulent flow in smooth pipes, fully turbulent flow in rough pipes
g
V
D
Lfh f 2
..2
Steady state friction formula
Haaland equation (1983) is used to determine friction factor f for full flowing circular pipe.
The basic governing equations for one dimensional unsteady pipe liquid flow are
Re
9.6
7.3log8.1
111.1
10D
f
g
V
DfJJ
t
V
gx
H
x
V
g
a
t
H
2.
1. ;0
1 ;0
22
Steady state friction formula
MOC implementation of water hammer equations
PCE:
NCE:
papp HCCQ
panp HCCQ
DA
fR
a
gAC
QtQRHa
gAQC
QtQRHa
gAQC
a
BBBBn
AAAAp
2;
Limitations of Steady state friction formula
Traditionally steady friction terms are incorporated in water hammer equations. This assumption is satisfactory only for slow transients
Experimental validation of steady friction models for rapid transients shown significant discrepancies in dissipation and phase shift of the computed pressures as compared to the measurements
Limitations of Steady state friction formula
Figure from Dr Silva thesis
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20
-10
0
10
20
30
40
50
60
70
Time (s)
Pre
ssure
(m
)
computed
experimental
Other unsteady friction models
1. Convolution based models (Zilke, 1968, Vardy and Hwang, 1991, Vardy and Brown 2004, 2007)
2. Two dimensional models (Silva-Araya and Chaudhry (1997), Pezzinga, 1999)
However these methods are computationally intensive and not been extended beyond simple pipe systems.
Instantaneous acceleration based models
Looking for simplicity and computational efficiency, much effort has been put into a third category, named Instantaneous Acceleration Based models
IAB models ( Brunone et al. 1991, Ramos et al. 2003)
MIAB models (Vitkovsky et al. 2006)
Instantaneous acceleration based models
Head losses per unit length due to steady state and unsteady state friction can be written as
LimitationsAbove eq. does not produce acceptable results when
the transient is caused by a sudden closure of an upstream valve.
Vítkovský et al. (2006) showed that Brunone’s IAB model included in the exact MOC produces no attenuation and decreases the wave speed by a factor of (1+k).
x
Va
t
V
g
kJ
JJJ
U
US
MIAB model (Vitkovsky 2006)
LimitationsAuthors demonstrated that, when the exact
MOC solution is used, the model fails to reproduce damping during valve opening events.
Interpolation must be used in any case due to changes in the wave speed introduced by the additional terms.
x
Va
x
VVsign
t
V
g
kJU
IAB model using two coefficients
The IAB model was given more flexibility to reproduce damping during transient events by adding a second coefficient to the unsteady friction term (Vítkovsky et al., 2001).
Laurerio and Ramos (2003) selected this
model for testing dissipative effects in polyethylene (HDPE) pipes.
IAB model using two coefficients
The two coefficient model is
where Kut and Kux are two decay coefficients; related to the local and convective accelerations respectively.
It has been verified numerically that the term Kut∂V/∂t affects the phase shift of transient pressure waves and Kux∂V/∂x the damping. (Ramos et al. 2003).
x
VaVK
t
VK
g
kJ uxutU sin
Objectives of the present research
The number of parameters could be one or two depending on the chosen model.
The empirical coefficients must be supplied for application of the IAB unsteady friction models.
Three formulas have been proposed to estimate the coefficient for one-parameter formulations;
the first was proposed by Vardy and Hwang (1995, 1996), the second by Ghidaoui et al., 2001; and, the third by Bouazza and Brunelle (2004).
Presently the only way to obtain the parameters for two-coefficient models is by trial and error.
Objectives of the present research
This paper presents a new methodology for estimation of the decay coefficients for IAB models using Genetic Algorithms.
The adjustment requires a time history of
pressure oscillations which could be obtained experimentally or by using more advanced unsteady friction models.
The results are compared with the available
formulas.
Two decay coefficient model
Final MOC equationsPCE:
NCE:
ux
pp
jijijiutjijiuxjiuxpp
jijijiajip
jiapji
K
CC
QQQsignKQQKQKCC
QRQHCQC
HCCQ
1
1
'"
1,11,1,12,11,11,1'
1,11,11,11,1
,'"
,
a
gAC
K
CC
K
CC
QQQsignKQQKQKCC
QRQHCQC
HCCQ
aux
aa
ux
nn
jijijiutjijiuxjiuxnn
jijijiajin
jianji
;1
;1
1
''
"
1,11,1,12,11,11,'
1,11,11,11,1
,'"
,
Genetic algorithms
A genetic algorithm (GA) is a search technique used in computing to find exact or approximate solutions to optimization and search problems. Genetic algorithms are categorized as global search heuristics.
The GA is a random search algorithm based on the concept
of natural selection inherent in natural genetics, presents a robust method for search for the optimum solution to the complex problems.
The artificial survival of better solution in GA search technique is achieved with genetic operators: inheritance, mutation, selection and crossover based on evolutionary biology.
NRMSD (Normalized root mean square deviation)
The objective of the GA is usually formulated to minimize the fitness function, which may be achieved by the following equation.
NRMSD is the RMSD divided by the range of measured values.
RMSD is a frequently-used measure of the differences between values predicted by a model and the values actually measured from the thing being modeled.
*min
*max
1
2*
HH
HH
NRSMD
Ni
iii
Results
Lab experiment
0 1 2 3 4 5 6-2
0
2
4
6
8
10
12
14
16
18
Time (s)
Pre
ssure
(m
)
computed
experimental
Results
Experimental data obtained from Pezzinga
0 1 2 3 4 5 635
36
37
38
39
40
41
Time (s)
Pre
ssure
(m
)
computed
experimental
Results
Slow closure (Dr. Silva thesis)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20
-10
0
10
20
30
40
50
60
70at Valve
Time (s)
Pre
ssure
(m
)
computed
experimental
Conclusions
1. In Ramos method Kut effects phase and shape of the oscillation and Kux effects only damping.
2. Ramos two coefficient method gives better comparison than the Vitkovsky single coefficient method except for PVC pipelines.
3. In most of the cases Kux > Kut but in few cases Kut > Kux.
4. In majority of cases Kux ≤ 5 Kut
References
H. Prashanth Reddy, Silva‐Araya, W. F., and Chaudhry, M. H., (2012) “Estimation of Decay Coefficients for Unsteady Friction for Instantaneous Acceleration Based Models,” Journal of Hydraulic Engineering, ASCE, 138(3), 260‐271.