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Effect pipes material on water hammer
M. Kandil, A.M. Kamal, T.A. El-Sayed
PII: S0308-0161(19)30385-0
DOI: https://doi.org/10.1016/j.ijpvp.2019.103996
Reference: IPVP 103996
To appear in: International Journal of Pressure Vessels and Piping
Received Date: 8 September 2019
Revised Date: 6 October 2019
Accepted Date: 19 October 2019
Please cite this article as: Kandil M, Kamal AM, El-Sayed TA, Effect pipes material on water
hammer, International Journal of Pressure Vessels and Piping (2019), doi: https://doi.org/10.1016/
j.ijpvp.2019.103996.
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© 2019 Published by Elsevier Ltd.
Effect Pipes Material on Water Hammer
Authors: M. Kandil , A.M. Kamal, T.A. El-Sayed
Affiliations: Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan
University, P.O. Box 11718, Helmeiat - Elzaton, Cairo, Egypt
Contact email: [email protected],
[email protected]
Abstract: Water hammer is a transient flow in pipes that were made by quickly change in
speed in pipes. This phenomenon can cause genuine positive and negative pressures in
pipes and frequently with hazards in pipelines. Generally, water hammer makes by closing
valves quickly and has one of the most dangerous hydrodynamic phenomena in
pressurized pipelines. In this paper, governing equations about water hammer is
numerically fathomed by utilizing MATLAB programing language based on the Method of
Characteristic (MOC), and then the pressure fluctuations have been studied by changing
some effective variables such as pipes Elastic Modulus and Poisson`s Ratio. The numerical
method is based on method of characteristic lines. Results show that the Normalized
Piezometric head is calculated in 4 statuses (pipe’s full length, pipe’s 3/4 length, pipe’s 1/2
length, and pipe’s 1/4 length) direct effect with the change of Elastic Modulus and
Poisson`s Ratio based on the pipe material, The novelty of this study is to show how the
materials with less Elastic Modulus are less likely to occur the water hammer than the high
Elastic Modulus for the same operation condition.
Keywords: Water hammers, MOC, Transient Flow, Fluctuating of Pressure, MATLAB
2
1. Introduction
In hydraulic plants, unsteady flow is often encountered in pipelines, by the
deviations of flow (and consequently of water speed) reason the change of kinetic
energy of water into other forms of energy. However, because water or other liquids are
only marginally compressible, a small flow imbalance can produce great pressure
changes and thus allows a considerable amount of energy to be kept. Quick changes in
water speed can be a reason for the water hammer phenomenon, which can harm
pipelines and other hydraulic plant equipment. The main frequent reasons of water
hammer are the rapid closing (or opening) of a valve and the rapid switching off (or
switching on) of a pump. The modeling of unstable occurrences in pressure systems is
the source for achieving the safe process of a water supply system, and it is the most
challenging part of the design of such a system.
In the end of the 19th century, the development began of models which definite
the differences of speed and of pressure introduced by water hammer [1], and several
programs were made to allow the oscillation of water mass and/or of water hammer
to be replicated.


Figure 1: Water-hammer waves
As performed in the figure (1) a damped sinusoidal wave can appear amid the
transient waves.
As specified by Juokowski [2] when there is rapid change in fluid's flow velocity a
comparing change in pressure intensity is linked as:
a
H V
g
∆ = ∆ (1)

The strange changes in pressure and fluid velocity are a danger to the pipeline system
3
included - valves. Consequently, it is important to study the water hammer possibilities in
hydraulic plants and model them for accomplishing a more secure design of the hydraulic
plants.
The study on the Water-hammer has been complete for more than a hundred years, and
the prototypes and the strategies to achieve water hammer state solutions for flow
characteristics have progressive enormously. Be that as it may, the essentials philosophies
continue as before. The next area respects the basic assumptions, managing equations and
the pressure wave speed concept which form the origin of each model or strategy.
1.1 Several Methods Applied for Water-Hammer Study
In the sequence of recent years, analysts have proposed approaches for explaining
water hammer state problems with many conditions and assumptions. As expressed in the
previous section it was after the coming of the 20th century when frankly precise
mathematical models came into the picture. In 1902 Allievi [3] was presented the
mathematical method in which he solves the differential equations without considering
the effects of friction, despite the result of this technique were inaccurate but this method
used as the basis for future works. In the opinion of Jurkowski's theory, N.R. Gibson [4]
stretched Allievi’s method by consider the non-linear losses friction and called it Gibson
Method, however the results were still inaccurate.
In 1928, Löwy [5] was presented the use of a graphical method, which was created by
Gaspard Monge [6] in 1798, to solve the water-hammer equations. He was starting to
include the friction part into the differential equations. Later. Bergeron [7] strained out
this method to resolve the flow conditions at middle points in the pipe, not just at the
valve or tank. The method was additionally improved by J. Parmakian [8] in the 1950s.
This method solved the problem consider the quasi-steady friction model but for consider
the dynamic effects a "correction term" was additional in solution. Despite the method bent
fast results but the method was inaccurate, it just gave exact values for the initial
wave-period for late stages the friction term was not satisfactory.
In the 1950s, with the attendance of PCs, the study became directed towards gating
methods for computerized research of water hammers phenomenon. The first was Gray
[9] who planned a calculation to execute a method for the method of characteristics to give a
computational method of Water-hammer. His method was later corrected by the works of V.L.
Streeter and C.Lai [10]. In this method, the important governing equations which are
hyperbolic and partial first order in nature are changed over into ordinary first order, along
lines named the characteristic lines. This method results quite accurate when the unsteady
friction considered with it. It is, thus the most widely accepted model to date. Be that as it
may, the boundaries of stability and convergence [11] of the finite-difference method
pose a limit to the range of applicability of this method. In the present condition, the
frequency domain study of the water hammer has approached. In 1989, E.B. Wylie along
with L. Suo [12] published a paper exhibiting the impulse response method, which includes
frequency-dependent friction and wave speed. This method includes the use of the inverse
4
Fourier transform (IFT) to solve the flow conditions. This method is noticeably quicker than
the method of characteristics, though there is a loss of accuracy then it linearizes the friction
which is not suitable for each situation.
2. MATERIALS AND METHODS
2.1 Classical Assumptions
The present method for the water hammer flows in pipelines is drawn on the following
assumptions: [13] [11].
The flow is one dimensional, low compressible that is, it elastically deformation
with high pressures with negligible relative changes in density
The liquid flow velocity is very small as compared to the pressure water hammer wave
velocity
3. THEORY AND MODELING
3.1 Method of Characteristics Equations for Water Hammer
The water hammer problems are represented by the pressure and velocity in the pipeline
are calculated using the continuity equation, Equation (2), and momentum Equation (3) [14].
2 2 2
0
H a Q c x
t gA x gE t
∂ ∂ υ ∂σ
+ − =
∂ ∂ ∂
(2)
0
2
Q H f A g Q Q
t x D A
∂ ∂
+ + =
∂ ∂
(3)
Where g is the acceleration, t describe time, H describes the piezometric head at the
valve, A describes the cross-sectional area of the pipe, Q is the flow rate, x is the position
in the axial direction. In the equations, it is assumed that the cross-sectional area and the
wave speed, a, are constant and that a >> v which means that the convective terms can be
ignored.
The 2 x
gE t
υ ∂σ
∂
, where σ x
describe the axial stress in the pipe, is neglected if not
considering axial stress or strain, an assumption that generally is complete. The following
method is completed under that assumption, however will later be changed [15].
5
The pressure wave speed as describe in Equation (4).
( )
1
2
2
1 1 K K D bm bm a
Et
µ
ρ
   
= + −        

(4)
Where ρ the density of the fluid, Kbm is the bulk modulus of the fluid flow, E is
modulus of elasticity of the pipe, D is the internal pipe diameter, t describe the pipe wall
thickness, and Poisson’s ratio µ .
By using the method of characteristic (MOC) used to solve the set of Partial
Differential Equations (PDEs).
The method of characteristic transforms the set of PDEs into the four Ordinary
Differential Equations (ODEs) seen in Equations (5) and (6).


(5)

(6)

Figure 2:
Characteristic lines
in the x-t plane
The equation
(5) can be solved
1 1 0
2
:
g dH dQ fQ Q
a dt A dt A D
dx
a
dt
C
+ + =
+
= +






1 1 0
2
:
g dH dQ fQ Q
a dt A dt A D
dx
a
dt
C
− + + =
−
= −






6
along the characteristic lines with the slope determined by Equation (6), as is illustrated in
Figure (2). By use of the characteristic lines, points P can be discovered using point A and
B. This is ended by the positive characteristic line, C+
, consistent to a positive a and the
negative characteristic line,C
−
consistent to a negative a. For this situation, the
characteristic lines are linear since it is assumed that a is constant.
Equation (4) with finite differences and integrating, a pipe of length “L” is separated into
“N” number of components, giving N + 1 number of nodes. For every time step∆t , the
pressure and velocity are computed in every node. The time step is determined by the pipe
length and the wave speed according to: ∆ = ∆ t x a/ , along the positive and negative
characteristic lines produce Equations (7) and (8), where CP
and C m
are defined with
Equations (9) and (10), respectively and a
B
gA
= .
: C H C B Q i P P i
+
= −
(7)
: C H C B Q i M M i
−
= +
(8)
2
2
P
f x B B
gDA
∆
= +
(9)
B 2
2
M
f x B
gDA
∆
= +
(10)
C H BQ P i i = +− − 1 1
(11)
C H BQ m i i = −+ + 1 1
(12)
The main compatibility equation is valid along the characteristic line ∆ = ∆ x c t . ,
getting CP
and BP
at the distance i −1 from the point of interest, i.e. at the previous
7
time stept t − ∆ . The other one is valid along ∆ = − x c , with C M
and B M
at the
distance. The referenced quantities can be calculated through Equation (9)-(12).
Using the expressions above in combination with Equation (7) and (8), the pressure
head and velocity can be calculated for each meddle points as indicated by Equation (13)
and
Equation (14), respectively [14].
P M M P
i
P M
C B C B H
B B
+
=
+

(13)
QP can then be calculated with Equation (13).
P M
i
P M
C C Q
B B
−
=
−

(14)
A mesh is picked over the less computationally concentrated diamond grid to enhance
the plotting of the pressure pattern.
4. BOUNDARY CONDITIONS
To get the pressure head and flow at the boundary nodes, there are different
boundary conditions that can be applied depending on which element the pipe is
appended to. Which equation to use depends on how the pipe is connected to the
element i.e. if the boundary condition is the upstream or downstream end of the pipe.
Having a downstream boundary condition the C
+
equation is utilized, and for upstream
boundary condition the C
−
equation is utilized [16].
8
Figure 3: x-t Diagram
4.1 Two Pipes Connected In a Series
When two pipes are connected in a series as illustrated in Figure 4, the conservation
of mass gives that the flow toward the finish of the primary pipe, i.e. in the last node, and
the flow in the start of another pipe, i.e. the first node, will be the same.
Q Q pipe N pipe 1, 1 2,1 +
=
(15)
Figure 4: Pipes connected in a series
Similar, the conservation of energy gives
H H pipe N pipe 1, 1 2,1 + = (16)
The unknown pressure head is obtain by Equation (17).
1 2
1, 1
1 2
1 1
P M
pipe pipe
pipe N
pipe pipe
C C B B
H
B B
+
+
=
+
(17)
And the flow is compute by using Equation (7).
4.2 Tanks
9
Figure 5: Boundary condition: reservoir
The pressure head in the node in contact with the tank is assumed to be equal to the
pressure head of the tank. Having a tank at the upstream end of the pipe this yields:
H H pipe res up ,1 ,
= (18)
And with a tank at the downstream end, the pressure at the last node in the
adjacent pipe is set to
H H pipe res dwn ,N 1 , +
= (19)
The flow can then be calculated by using Equation (8) for the first case, and Equation
(7) for the latter. This get the following equations:
1
1
M
M
H C Q
B
−
= (20)
1
1
P N
N
P
C H Q
B
+
+
−
= (21)
4.3 Valves
Figure 6: Boundary condition: Valve
The modeling of two pipes connected through a valve can be simplified as an abrupt
change of area of the pipe. The pressure drop over the valve is compute through the
following equation:
10
1
2 ( ) up down i
valve
H H H Q Qi
gA t
∆ = − = (22)
Replacing H
up with Equation 7, and Hdown with Equation 8, Qi
for the point
of interest can be solved
1
( ) ( )
2 ( ) M M i P P i i i
valve
C B Q C B Q Q Q
gA t
+ − − =
(23)
( ) ( )( ) ( )
2
(2 ( )) 2 ( )
2 2
M P valve M P valve
i P M
B B gA t B B gA t
Q C C
+ +  
= − ± − −    
(24)
The flow will be equal in and out through valve, giving
Q Q Q i up down = = (25)
The pressure head can then be computed with Equation (7) and (8) for the
previous pipe respective the next pipe.
If the Constant Head Tank is present at the upstream end of the pipe, then
Hp can be assumed equal to the Reservoir head and then Qp can be calculated as
P m
P
H C Q
B
−
=
(26)
Otherwise, if the Constant Head Reservoir is present at the downstream end of
the pipe, then Hp can be assumed equal to the Reservoir head and then
n 1 Q P
+
can be calculated as
1 1
1
1
n n
n P m
P n
H C Q
B
+ +
+
+
−
= (27)
4.4 Free Discharge Valve at an End
If the valve is present at downstream end of the pipe, then
11
1 1 1
2
n n n v
P P P
K
H Q Q
g
+ + +
= (28)
Where Kv
is valve-loss coefficient, Further, Kv
depends on the Coefficient of
Discharge (Cd
) as K C v d = varies with valve-opening and the type of valve Also,
n n n n 1 1 1 1 H C B Q P P P P
+ + + +
= − (29)
5. DISCUSSION
In this part reaction of water hammer for a system consisting of a pipe with different
materials will be examined. For this purpose, a code in MATLAB language has been written
that the parameters are allowed to be replaced and plotted. Method to solve the
governing equations is the characteristics method.
Elasticity modulus and Poisson`s ratio have a remarkable effect on pressure wave
propagation in 1D pipes. The equation from Streeter, Wylie [17] provides the speed of the
pressure wave a in 1D elastic pipe filled with single-phase fluid, Equation No. (4).
Figure (6) shows the speed of the pressure wave as a function of the elasticity
modulus. Constant values used in Equation (17) to plot the curve in Figure (6) were: 0
a
=1490 m/s, D = 1.905 cm, t = 1.588 mm (pipe from Simpson’s [18] experiment), and
variable elasticity modulus E .
12
Figure 7: Speed of sound - dependency on Elasticity modulus
The pressure waves propagate along the pipe length with the speed of sound
modified by the pipe elasticity. If the elasticity modulus is infinite (stiff pipe) then the
propagation speed of the pressure wave is equal to the speed of sound. Figure (7) shows
the need to introduce the pipe elasticity into the basic equations for the given pipe. If the
speed of sound in the stiff pipe of a given diameter and wall thickness is approximately
1490 m/s, we can see in Figure (7) that reduced speed ( = 1.25 N/m2 copper) is about
1360 m/s. The difference is significant and it is approximately 9 %.
Figures (8-11) show the change of the Normalized Piezometric head at the valve for
different pipes marital for four materials were studied PVC, concrete, ductile iron, and
steel. The other variables were taken as constant fluid density ( ρ ) =1000 kg/m3 (fluid is
water), liquid bulk modulus (Kbm) = 2.15 Gpa, pipe length is 2500 m, the pipe diameter is
50 cm, 0
a is 1300 m/s and the pipe wall thickness for all pipe materials was taken 2 cm.
The used materials data is shown in table (1) and were adopted from Jones and
Bosserman [19, 20], Richard and Svindland [21] and Sharp and Sharp [22].

13
Table 1: Properties of used pipe materials
Pipe material Modules of elasticity Poisson`s Ratio
Steel 210 GN/m2
0.3
Ductile iron 165 GN/m2
0.28
Concrete 25 GN/m2
0.15
PVC 3.3 GN/m2
0.42
Figure 8: Normalized Piezometric head at the valve - dependency on Elasticity modulus
and Poisson`s ratio for Steel material
14
Figure 9: Normalized Piezometric head at the valve - dependency on Elasticity modulus
and Poisson`s ratio for Ductile iron
15
Figure 10: Normalized Piezometric head at the valve - dependency on Elasticity modulus
and Poisson`s ratio for Concrete
16
Figure 11: Normalized Piezometric head at the valve - dependency on Elasticity modulus
and Poisson`s ratio for PVC
5. Conclusions
• Within increasing Elasticity Modulus (E) in the pipe, in the areas with negative pressure
wave, would have more pressure reduction and in the areas with positive pressure
wave, would have less pressure increment. The pressure fluctuation range within the
increasing of Elasticity Modulus coefficient would increase slightly.
• The plastic (PVC) which has low Elasticity Modulus 3.3 Gpa will have pressure waves
with low values compere with the steel martial with high Elasticity Modulus 210 Gpa
under the same operating conditions.
• A maximum and minimum of pressure would occur at the end of the pipe, and so that
the ending section of the pipe is a critical zone for design.

17
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Author’s name Affiliation
Mostafa Kandil Department of Mechanical Design,
Faculty of Engineering, Mataria,
Helwan University, P.O. Box 11718,
Helmeiat - Elzaton, Cairo, Egypt
Ahmed Kamal Department of Mechanical Design,
Faculty of Engineering, Mataria,
Helwan University, P.O. Box 11718,
Helmeiat - Elzaton, Cairo, Egypt
Tamer Elsayed Department of Mechanical Design,
Faculty of Engineering, Mataria,
Helwan University, P.O. Box 11718,
Helmeiat - Elzaton, Cairo, Egypt