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PRASHANTH R HANMAIAHGARI PROF. SILVA ARAYA WALTER MOHAMMED ELKHOLY PROF. M HANIF CHAUDHRY Instantaneous Acceleration based model coefficients using genetic algorithms

IAB Unsteady Friction

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Instantaneous acceleration based model

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Page 1: IAB Unsteady Friction

PRASHANTH R HANMAIAHGARIPROF. SILVA ARAYA WALTER

MOHAMMED ELKHOLYPROF. M HANIF CHAUDHRY

Instantaneous Acceleration based model coefficients using genetic algorithms

Page 2: IAB Unsteady Friction

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

Page 3: IAB Unsteady Friction

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

Page 4: IAB Unsteady Friction

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;

Page 5: IAB Unsteady Friction

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

Page 6: IAB Unsteady Friction

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

Page 7: IAB Unsteady Friction

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.

Page 8: IAB Unsteady Friction

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)

Page 9: IAB Unsteady Friction

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

Page 10: IAB Unsteady Friction

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

Page 11: IAB Unsteady Friction

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.

Page 12: IAB Unsteady Friction

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

Page 13: IAB Unsteady Friction

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.

Page 14: IAB Unsteady Friction

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.

Page 15: IAB Unsteady Friction

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

,'"

,

Page 16: IAB Unsteady Friction

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.

Page 17: IAB Unsteady Friction

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

Page 18: IAB Unsteady Friction

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

Page 19: IAB Unsteady Friction

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

Page 20: IAB Unsteady Friction

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

Page 21: IAB Unsteady Friction

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

Page 22: IAB Unsteady Friction

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.