effect of water hammer on pipes containing a crack...

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:18 No:03 25

181003-4242-IJMME-IJENS © June 2018 IJENS I J E N S

Abstract— Water hammer pressure wave may occur in

pipelines transporting fluids when a sudden change in flow occurs.

To model this phenomenon, the method of characteristic technic is

used to solve mathematical equations describing the transient flow

in the pipe considering the transportation of four different fluids.

The results of pressure variation are then used in a proposed

analytical model for the calculation of the stress intensity factor

based on Raju and Newman model.

This model is calibrated against results obtained from a

numerical model composed of a pipeline that contains a

longitudinal semi-elliptical crack defect. A simple program using

the calibrated analytical model is validated and permits to predict

fatigue life of cracked pipe subjected to static and dynamic loads.

A parametric study conducted on pipe with external and

internal crack of different geometries allowed to conclude on the

harmfulness of the crack defect when subjected to water hammer

phenomenon.

Index Term— Fatigue, Pipeline, Semi-elliptical, Crack, Water

Hammer, Dynamic load, Method of characteristics, Leak, Break

I. INTRODUCTION

Pipelines constitute the most used mode for transporting oil,

gas, water and various petroleum products. Aging, impact with

foreign bodies and corrosive environment may lead to

formation of various defects such as crack, dent, gouge, and

corrosion [1]-[3]. Mechanical components in a presence of a

defect experiencing loading with variable amplitude are prone

to fail due to fatigue, thus their fatigue life prediction has

become a focal research issue.

Fatigue life prediction problems of pipes subjected to dynamic

loads is a complex problem in engineering practices compared

to cyclic loading or constant amplitude loading.

In a pipeline network, pump failure, pipe ruptures or sudden

change of state of the valve (opening or closing) creates

transient flows that generates a pressure pulse which could burst

the pipe and can generate pipeline vibrations [4], [5]. This

phenomenon is called water hammer. When occurring, pipe

motion itself generates water hammer, thus invoking fluid-

structure interaction. Those generated waves can be modelled

with an exponential decaying amplitude [6] (typical case of

pressure surge oscillations) with a frequency defined as the

number of combined start cycles and stop cycles (Fig. 1).

The modeling of this phenomenon is one of the most important

problems related to fatigue life prediction of structures

subjected to dynamic loads.

Fig. 1. Internal pressure range of the pipeline.

The dynamic fatigue consideration requires accurate

assessment of crack depth and remaining fatigue life of

pipelines with crack defect. This is vital for pipeline’s structural

integrity, inspection interval, management, and maintenance to

avoid economic and environmental disaster.

In industrial activities, fast and accurate approaches are needed

when there is a cracked pipe to avoid unnecessary

unavailability. During pipeline inspections, the operator must

take the decision between intervening immediately if the depth

of the crack is critical and scheduling the repair in the future if

the crack is not yet critical. Therefore, an evaluation tool is

required, in order to make accurate and quick evaluations of

these cracked configurations. Some tools are available in the

literature [7] to assess crack propagation in cylindrical shells.

Studies and tools that account for the effect of water hammer

still a lack in the literature [8].

Accordingly, this paper proposes a simplified tool that

considers the dynamic behavior of the material to analyze the

cases mentioned above. In this case, the fatigue life is calculated

using the stress intensity factor (SIF) which is an essential

Linear Elastic Fracture Mechanics (LEFM) parameter that can

characterize clearly the fatigue crack growth and thus assess

structural integrity of cracked structures.

Effect of Water Hammer on Pipes Containing

a Crack Defect

Z. MIGHOUAR, L. ZAHIRI, H. KHATIB, K. MANSOURI, Z. EL MAJID SSDIA Laboratory, Hassan II University of Casablanca, ENSET of Mohammedia, Post Box No. 159,

Mohammedia, Morocco

Corresponding author mail: [email protected]



The estimation of stress intensity factors can be done by

analytical or numerical techniques. Normally, the analytical

ones are more complex to calculate; however they have some

advantages, because an analytical solution can be applied for a

range of crack lengths. The numerical techniques require the

calculation of stress or strain field for each crack length and

therefore for each value of the SIF.

In this paper, a correction factor is introduced in a modified

analytical model of Raju and Newman [9] (RN-M) to take into

consideration the position of the crack (external or internal) and

is determined is such a way to improve the correlation between

the results of SIF obtained from the RN-M with results obtained

from a hereby developed numerical model. The obtained

calibrated analytical model (CRN-M) is then validated to make

it able to replace numerical calculation since they provide

nearly the same results in terms of SIF measured in the crack

front. This model allows analyzing the effect of water hammer

on pipeline made of X52 steel containing longitudinally

oriented cracks and predict the number of cycles to rupture and

the critical crack depth with a simple algorithm.

This program considers the variation of pressure wave

propagation history due to water hammer to compute the stress

intensity factor of semi-elliptical cracks defects present in the

pipe.

To model water hammer phenomenon provoked by

instantaneous valve closure/opening at the end of a pipeline, the

method of characteristic [10] (MOC) technic is applied to

mathematical equations that describes transient flow for four

different fluids transported by a pipe which are considered to

have a linear elastic behavior.

II. NUMERICAL MODELING OF CRACKS IN PIPE

A. Stress intensity factor - Analytical model

Various method are used for analyzing the problem of

longitudinal semi-elliptical surface cracks in the wall of

cylindrical shells [11], [12]. Finite element method, method of

boundary integral equations, weight function method and other

methods are used to estimate the stress intensity factor. The first

solution for semi-elliptical surface cracks in a plate subjected to

uniaxial tension was derived from solutions for an elliptical

plane crack in an infinite three dimensions body. In order to

account for the finite thickness of the surface and the plastic

zone at the crack tip, correction factors were introduced for both

surface sides of the body and for the plastic region at the crack

tip [13]. However, different authors showed rather considerable

disagreement in terms of the obtained solutions. The accuracy

of the solutions presented by various authors were therefore

tested [14], [15] by analyzing the evolution of crack throughout

its fatigue growth. They concluded that the best engineering

estimation of the stress intensity factor for a part through crack

in a plate was provided by Newman’s solution [9].

An adjusted form of this solution for thin-walled shell [16] for

cracks with (a/c) ≤1 is given by the equation (1), which we will

be calling hereby RN-M.

The parameters used in this equation are presented in equations

(2), (3), (4), (5) and (6). Pipe and crack parameters are shown

in Fig.2.

Fig. 2. Pipe and crack parameters

, 1 2 1 sI RN TMK k p a M k a (1)

Where:

2

11

6.4 1 1

TM

i e

a

tMa ct D D t

(2)

1 3

2

iDk k

t

(3)

32

11

s

c

k ak

t

(4)

3 1.65

1.13 0.1

1 1.464

a

cka

c

(5)

3

2 8 a

sc

(6)

In the current study, a correction factor MC (8) is introduced to

take into consideration the position of the crack (internal or

external) and improve results for high pressures, hereby, we call

the model described by the equation (7) the CRN-M.

,I C I RNK M K (7)

21

fCM f a (8)

The factors f1 and f2 used in the CRN-M need to be adapted to

get the best agreement with numerical results obtained from a

finite element model. The pressure range taken into

consideration is the one concerned by the water hammer

phenomenon.

B. Finite element model

A numerical model composed of a pipe with a crack is

performed by ANSYS code [17]. The pipe has an outer

diameter De of 274 mm, an inner diameter Di of 246.6 mm and

a length of 1000 mm.

From studies conducted on similar pipe steels [18], [19], values

of Young Modulus E, Poisson ratio υ yield stress σY, fracture

toughness KIC and Paris law constants C and m of the pipe

material are reported in Table 1.



TABLE I MECHANICAL PROPERTIES OF X52 STEEL

E

(GPa)

σY (MPa) [18] υ

KIC (MPa.m0.5) [18]

Material constants [1]

Static Dynamic Static Dynamic C m

200.02 436 478 0.3 44.7 53.36 3.3e-09 2.74

The boundary conditions consists of restricting:

- Longitudinal displacement of one of the pipe ends;

- Lateral and radial displacements of a line on the outer surface

of the pipe;

- Lateral displacement of a line on the inner surface of the pipe.

A distributed pressure is applied on the inner surface of the pipe.

This pressure is expressed in terms of Maximum Operating

Pressure (MOP) which is equal to 72% of yield strength of the

pipe material.

The mesh of the pipe is performed in three dimensions using

tetrahedral elements. A sphere of influence refines the mesh

around the crack with a size of elements of 0.5 mm and a

diameter of 1.2 times the crack length. The rest of the pipe is

meshed by elements of larger size to reduce computation time.

The longitudinal semi-elliptical crack is inserted using fracture

feature available in ANSYS software.

C. Calibration of the analytical model

Numerical results for external and internal crack are displayed

in Table 2. Since the water hammer phenomenon is more

harmful for the structure at high pressures. The calibration of

the analytical model is made for pressure interval between 70%

and 100% MOP.

Comparing the numerical data in Fig. 3 and Fig. 4 with both the

model prediction results of Raju and Newman (RN-M) and the

calibrated one (CRN-M). The deviation between the RN-M and

numerical model varies from 0.8% to 12.4%. The deviation

between the CRM-M varies from 0.04% to 2.3%.

TABLE II

NUMERICAL RESULTS OF THE STRESS INTENSITY FACTOR

Stress intensity factor: KI (MPa.m0,5)

External crack Internal crack

a/c p

a/t

70%

MOP

85%

MOP

100%

MOP

70%

MOP

85%

MOP

100%

MOP

0,4

0,25 41,72 44,41 46,36 46,09 49,07 51,27

0,4 60,91 64,97 68,30 64,62 68,92 72,46

0,55 89,59 95,69 101,11 92,47 98,76 104,34

0,7 146,70 156,87 166,12 148,29 158,57 167,89

0,6

0,25 32,21 34,22 35,92 35,61 37,84 39,71

0,4 44,57 47,46 49,60 47,30 50,36 52,65

0,55 60,76 64,80 68,11 62,73 66,90 70,31

0,7 91,00 97,19 102,70 92,01 98,27 103,81

0,85 178,97 191,45 202,71 177,95 190,34 201,52

0,8

0,25 25,62 27,14 28,22 28,30 29,99 31,20

0,4 34,25 36,16 37,86 36,33 38,36 40,17

0,55 43,65 46,37 48,42 45,05 47,85 49,96

0,7 60,06 64,05 67,30 60,72 64,74 68,04

0,85 110,29 117,86 124,77 109,64 117,16 124,00

0,9 159,49 170,58 180,75 157,79 168,74 178,72

1

0,25 20,17 21,32 22,65 22,42 23,61 25,09

0,4 26,93 28,56 30,06 28,61 30,34 31,94

0,55 33,38 35,47 37,11 34,49 36,65 38,33

0,7 42,32 45,04 46,99 42,82 45,56 47,52

0,85 70,49 75,22 79,23 70,12 74,81 78,76

0,95 189,90 203,15 215,88 187,04 200,08 212,63

It should be noted that the calibrated analytical model gives

results close to those of the numerical model and therefore can

replace it in further calculation of the stress intensity factor to

gain in flexibility, simplicity and calculation time.

Fig. 3. Comparison of RN-M, CRN-M against numerical results in the

case of pipes containing external cracks.

Fig. 4. Comparison of RN-M, CRN-M against numerical results in the

case of pipes containing internal cracks.

Parameter values used in the calculation of the analytical

models are shown in table 3.

TABLE III

SUMMARY OF THE CONSTANTS USED IN THE ANALYTICAL CALCULATION

a/c s k1 k2 External crack Internal crack

f1 f2 f1 f2

0,4 2,51 13,145 1,28E-03 -0,056 0,052 0,0447 0,00188

0,6 3,73 10,470 5,59E-05 -0,056 0,052 0,0447 0,00188

0,8 6,10 8,320 1,35E-07 -0,056 0,052 0,0447 0,00188

1 10,00 6,668 5,98E-12 -0,056 0,052 0,0447 0,00188



D. Program description

The program was written in C language and has two

subroutines, Fig. 5 shows its flow diagram. The main

subroutine computes the SIF from the input data that are the

initial crack depth and the loading history returned by the

subroutine that converts the curve of pressure loading history to

a table of pressure peaks.

The cases that can be analyzed are external and internal part

through-wall longitudinal cracks in pipes under static and

dynamic internal pressure load.

When the SIF corresponding to a crack depth ai exceeds or is

equal to the fracture toughness, the program stop and returns

the number of cycles to failure along with the critical crack

depth. In this case, we consider the pipe to break.

Alternatively, when the crack depth ai reaches the pipe wall

thickness, the program stop and returns the number of cycles to

failure. In this case, we consider the pipe to leak.

The number of cycles to failure is calculated using the Paris

Law using the effective stress to account for the crack closure

[20], [21] that we can write:

2

0.25 0.5 0.25

m

MINMAX MIN

MAX

KdaC K K

dN K

(9)

KMIN: Minimum stress intensity factor (MPa.m0.5)

KMAX: Maximum stress intensity factor (MPa.m0.5)

E. Validation of the program

For the purpose of validation, the study conducted by Bakushi

[7] is considered, in which the critical crack depth was

estimated for leak before break evaluation. To permit

comparison, the stopping condition on wall thickness is

ignored.

The case analyzed is a pipe with wall thickness of the analyzed

pipe is 14.3 mm in presence of semi-elliptical external crack.

The corresponding values of constant k1 is 35.79, k2 is 0.00046,

s is 3 with a load ratio (KMIN/KMAX) of 0.1.

As it can be seen from table 4, the deviation between results

remain acceptable, especially that they lead to the same

conclusion concerning leak before break assessment.

TABLE IV

PREDICTION COMPARISON OF CRITICAL CRACK DEPTH, AND LEAK BEFORE

BREAK ASSESSMENT

Internal

pressure aIC (mm) [7] aIC (mm) Error Leak/Break

70% MOP 16.74 15.63 -6.63% Leak

76% MOP 13.83 13.02 -5.85% Break

84% MOP 11.62 10.98 -5.51% Break

F. Water hammer equations

The analysis of water hammer is done assuming:

Horizontal pipe-slope;

Linear elastic behavior for both the pipe-wall and the

fluid;

Thin-walled pipe ;

One-dimensional flow.

Fig. 5. Flow diagram of the proposed program.

The one-dimensional wave equation set described in equations

(10) and (11) defines the analytic pattern of water hammer

model:

10

2 i

V VV p

t x D

(10)

² 0p V

ct x

(11)

Where:

V: Instantaneous fluid velocity (m/s)

t: Time (s)

ρ: Fluid density (kg/m3)

x: One-dimensional axis (m)

λ: Darcy-Weisbach friction coefficient

C: Sonic velocity in the pipe (m/s)

This system of partial differential equations can easily be solved

using the method of characteristics (MOC), which is

characterized by fast convergence and high accuracy of

calculation results.

The water hammer wave speed C is the speed of sound in the

pipe and is determined by a modified hooks law formula which

takes into account the stiffness of the fluid and the pipe wall.

Halliwell [22] presented an expression for the speed at which

the pressure waves generated by water hammer travel in the

pipe:



(1 )

KC

K

E

(12)

Where K is the bulk modulus of elasticity of the fluid in “Pa”.

The non-dimensional parameter 𝜑 depends on the elastic

properties of the pipe. Since the pipe is assumed to be anchored

against longitudinal movement throughout its length, the

expression in equation (13) is used.

2 (1 )iD

t (13)

G. Resolution by the method of characteristics

The method of characteristics is applied to transform the system

of partial differential equations (10) and (11) into a system of

ordinary differential equations that can be integrated

numerically without difficulty (Equation 14 and 15) [23]. One

can get for i ϵ [0, n] and j ϵ [0, m]:

, 1, 1 1, 1 1, 1 1, 1

1 ρC

2 2i j i j i j i j i jp Vp p V

2

-1, -1 -1, -1 1, -1 1, -1

ρ C- -

4i j i j i

ij i jV V V

Dx V

(14)

, 1, 1 1, 1 1, 1 1, 1

1 1

2ρC 2i j i j i j i j i jV p p VV

1, 1 1, 1 1, 1 1, 1

C

4 i j i j i j i j

i

V V VVD

x

(15)

Where : x C t (16)

The parameters used in this method are explained in Fig. 6.

Fig. 6. Characteristic lines in x-t plane

Initial and boundary conditions used in the simulation are:

0, 0 56.05 jp p MPa (17)

1, 1 0n j nV V (18)

2

0, 0 3

21i j i

i

Vp p p i x

D

(19)

, 0 1.2 /i jV V m s (20)

III. PARAMETRIC ANALYSIS

The analysis concerning the pressure range generated by the

water hammer phenomenon was performed for four different

fluids. Their properties are regrouped in Table 5. The wave

celerity is calculated using equation (12).

TABLE V

HYDRAULIC CHARACTERISTIC OF FLUIDS

Fluid ρ (kg/m3) Viscosity (cSt) K (GPa) C (m/s)

Water 998,2 1,347 2,198 1366,03

Water-Glycol 1080 4,88 2,147 1300,26

Light Oil 813 3,7 2,03 1463,21

Heavy Oil 962 513 1,30 1105,13

The pipeline dimensions used for the water hammer modelling

are similar to those used in the finite element analysis except

for the pipe length parameter which is considered equal to 150

m. The friction coefficient λ is equal to 0.019. The time

increment Δt is set to 0.01s.

In order to highlight the phenomenon of crack propagation, the

locations of the semi-elliptical longitudinal cracks are

considered to be in the internal and external surface of the pipe.

The selected cracks have a/c ratio of 0.4, 0.6, 0.8 and 1.

As a matter of comparison, the initial crack depth is the same

for all studied cases and is equal to 3.25mm.

IV. RESULTS AND DISCUSSION

After resolving the equations using the method of characteristic,

the results are plotted for the section of the pipeline in the

position x=0 (Fig. 7, 8, 9 and 10).

From these results, pressure evolve and converge to final steady

state values p0= 56.05 MPa after 14.58, 14.65, 13.41 and 14.13s

for respectively the pipe filled with water, water-glycol, light

oil and heavy oil.

Examination of the results shows that the pipeline is subjected

to pressure peaks which build up from the initial pressure of

56.05 MPa to a maximum pressure of 58.03, 58.07, 58.42 and

58.7 MPa for respectively the pipe filled with water, water-

glycol, light oil and heavy oil.

Fig. 7. Water hammer pressure wave in a pipeline transporting water.



Fig. 8. Water hammer pressure wave in a pipeline transporting water-

glycol.

Fig. 9. Water hammer pressure wave in a pipeline transporting light oil.

Fig. 10. Water hammer pressure wave in a pipeline transporting heavy oil.

Those pressure histories are used as an input in the proposed

program to predict the number of cycles to failure along with

the critical crack depth of pipelines with longitudinal semi-

elliptic cracks. The results obtained from this analysis are

presented in table 6. Hereby, the crack depth is limited to the

thickness of the pipe in order to conclude whether the break of

the failure of the structure will occur first.

TABLE VI

CRITICAL CRACK DEPTH, NUMBER OF CYCLES TO FAILURE, AND LEAK BEFORE

BREAK ASSESSMENT PREDICTED USING THE PROPOSED TOOL

External crack Internal crack

Fluid a/c

aIC

(mm)

N

(Cycles)

Leak /

Break

aIC

(mm)

N

(Cycles)

Leak /

Break

Sta

tic

load

-

0,4 12,37 23292 B 12,40 22358 B

0,6 13,25 54960 B 13,3 53106 B

0,8 13,70 119900 L 13,70 116582 L

1 13,70 245212 L 13,70 239836 L

Dy

nam

ic l

oad

Water

0,4 4,12 6790 B 3,96 4984 B

0,6 6,05 37322 B 6,00 34782 B

0,8 8,51 105676 B 8,61 102402 B

1 10,72 237011 B 10,82 231897 B

Water-

Glycol

0,4 4,11 6731 B 3,95 4922 B

0,6 6,04 37234 B 5,99 34690 B

0,8 8,50 105558 B 8,60 102283 B

1 10,72 236869 B 10,82 231759 B

Light Oil

0,4 4,09 6530 B 3,93 4711 B

0,6 6,01 36932 B 5,96 34374 B

0,8 8,47 105155 B 8,57 101879 B

1 10,70 236394 B 10,79 231297 B

Heavy Oil

0,4 4,05 6249 B 3,89 4415 B

0,6 5,97 36535 B 5,92 33955 B

0,8 8,43 104672 B 8,52 101389 B

1 10,68 235926 B 10,78 230844 B

To allow the analysis of the effect of water hammer on the

structure integrity, a pipe in the presence of a crack with same

dimensions is subjected to a static cyclic loading ignoring the

effect of water hammer.

Independently from the nature or the fluid and the type of

loading, it is observed that the defect is more harmful when the

crack dimension ratio a/c lowers.

For the static loading, one can observe that the structure might

leak for a/c crack dimension ratio of 0.8 and 1. This leads us to

the conclusion that the probability that the pipe leaks increases

with the a/c ratio of the crack which means that the harmfulness

of the defect decreases. This behavior is no more observed

when the loading is considered dynamic (taking into account

the effect of water hammer) for all studied cases.

Taking as a reference the results obtained for the external

cracks, one can observe that the critical crack depth for pipes

with internal cracks subjected to dynamic loading is lower for

low a/c ratio and higher for more important a/c ratio. If we take

for example the pipelines transporting water, we can observe a

difference in terms of number of cycles to failure of -26.6% for

a/c of 0.4 and a difference of -2.16% for a/c of 1. For static

loading, this difference becomes negligible.

Compared to pipelines subjected to static loading, the number

of cycle to failure for pipe filled with water subjected to

dynamic loading decreases by 77.71% for a/c ratio of 0.4 and

by -3.31% for a/c ratio of 1. For pipe transporting heavy oil, this

parameters decreases by -80.25% for a/c ratio of 0.4 and by

3.75% for a/c of 1. This shows that the crack propagation in a

pipe subjected to dynamic loading is slightly related to the

transported fluid and the crack defect become more harmful

when the transported fluid is a heavy oil.



V. CONCLUSION

An analytical model for the calculation of the stress intensity

factor based on Raju and Newman model has been proposed.

This model is calibrated using a numerical model and then

validated against results of a model encountered in the

literature.

In order to predict the condition of the cracked pipeline during

operation, simulation was carried out considering the pressure

variation of the fluid flow inside of a pipeline containing a crack

defect.

The method of characteristic has been used to solve differential

equations describing the transient flow in the pipe. This

pressure variation is caused by the water hammer phenomenon

when a valve is closed/opened and is considered as an input data

for a program that computes the number of cycles to failure and

the critical crack depth for pipes with different dimensions and

positions of a semi-elliptical longitudinal crack.

According to the obtained results, the harmfulness of the crack

defect is greater when the crack is positioned in the internal

surface of the pipe. This phenomenon is more pronounced when

the pipe is subjected to dynamic loading especially when the

crack dimension a/c ratio is low.

The fatigue life of the cracked pipe drops significantly when

subjected to dynamic loads rather than static loads. Therefore,

it is recommended to carefully adjust the closure time of the

valve to limit the water hammer consequences on the structure

integrity.

The results obtained for the critical crack depth indicate that the

pipe will break before it leaks if it is affected by the water

hammer phenomenon.

Therefore, it is suggested that the water hammer effect need be

taken into account when studying the harmfulness of a crack

defect in pipelines.

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https://doi.org/10.15866/ireme.v11i8.12089

effect of water hammer on pipes containing a crack...

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