- Email: [email protected]
Effect of pipematerials on water hammer
Journal Pre-proof 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. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 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:
∆H =
a ∆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].
∂H a 2 ∂Q 2c 2υ ∂ σ x + − =0 gA ∂ x gE ∂ t ∂t
∂Q ∂H f + Ag + Q Q =0 ∂t ∂x 2DA
(2)
(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
2υ ∂σ x gE ∂t
, 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
K 2 K D a = bm 1 + (1 − µ 2 ) bm Et ρ
(4) Where ρ the density of the fluid, K bm 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).
C
C
+
−
g dH a dt :
+
g dH − a dt :
Figure 2: Characteristic lines in the x-t plane The equation (5) can be solved
1 dQ 1 fQ Q + =0 A dt A 2 D dx =+a dt
+
(5)
1 dQ 1 fQ Q + =0 A dt A 2 D
(6) dx =−a dt
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 C P and C m are defined with Equations (9) and (10), respectively and B =
a . gA
C+ : Hi =CP −BPQi (7)
C− : Hi =CM +BMQi (8)
BP = B +
f ∆x 2 gDA 2
(9)
BM = B +
f ∆x 2 gDA 2
(10)
CP = Hi −1 +BQi −1 (11)
Cm = Hi +1 −BQi +1 (12) The main compatibility equation is valid along the characteristic line ∆x = c .∆t , getting C P and B P at the distance i − 1 from the point of interest, i.e. at the previous
7
time step t − ∆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].
Hi =
CPBM +CM BP BP +BM
(13) QP can then be calculated with Equation (13).
Qi =
CP −CM BP −BM
(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 pipe 1, N +1 = Q pipe 2,1 (15)
Figure 4: Pipes connected in a series
Similar, the conservation of energy gives
H pipe 1, N +1 = H pipe 2,1
(16)
The unknown pressure head is obtain by Equation (17).
H pipe 1,N +1 =
CP 1
B pipe 1 B pipe 1
+C M +1
B pipe 2 (17)
B pipe 2
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 pipe ,1 = H res ,up
(18)
And with a tank at the downstream end, the pressure at the last node in the adjacent pipe is set to
H pipe ,N +1 = H res ,dwn
(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:
Q1 =
H 1 −C M BM
Q N +1 =
(20)
C P − H N +1 BP
(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
∆H = H up − H down =
Replacing
H up
1 2 gAvalve (t )
Q i Qi
(22)
with Equation 7, and H down with Equation 8,
Qi
for the point
of interest can be solved
(C M + B M Q i ) − (C P − B PQ i ) =
1 2 gAvalve (t )
Qi Qi
(23)
Qi
(B =− M
+ B P ) (2 gAvalve (t )) 2
( B + B P )( 2 gAvalve (t ) ) ± M − (C P − C M 2 2
)
(24)
The flow will be equal in and out through valve, giving
Q i = Qup = Q 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
QP =
H P −C m 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
QP
n +1
can be calculated as
QP
n+1
HPn+1 −Cmn+1 = B n+1
4.4 Free Discharge Valve at an End If the valve is present at downstream end of the pipe, then
(27)
11
H P n +1 =
Kv n +1 n +1 QP QP 2g
(28)
Where K v is valve-loss coefficient, Further, K v depends on the Coefficient of Discharge (C d ) as K v = C d varies with valve-opening and the type of valve Also,
H P n +1 = C P n +1 − B P n +1QP n +1
(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: =1490 m/s,
a0
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, 50 cm,
a0 is 1300 m/s
pipe length is 2500 m, the pipe diameter is
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
References 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21.
22.
Ghidaoui, M.S., et al., A review of water hammer theory and practice. Applied Mechanics Reviews, 2005. 58(1): p. 49-76. Joukowski, N., Memoirs of the Imperial Academy Society of St. Petersburg. Proceedings of the American Water Works Association, 1898. 24: p. 341-424. Allievi, L., Teoria generale del moto perturbato dell'acqua nei tubi in pressione (colpo d'ariete) memoria dell'ing. LB Allievi. 1903, Unione Cooperativa. Gibson, N.R. The Gibson method and apparatus for measuring the flow of water in closed conduits. in ASME. 1923. Löwy, R., Druckschwankungen in Druckrohrleitungen. Mit 45 Abb. 1928: Springer. Fuamba, M., N. Bouaanani, and C. Marche, Modeling of dam break wave propagation in a partially ice-covered channel. Advances in Water Resources, 2007. 30(12): p. 2499-2510. Bergeron, L., Variations in flow in water conduits. Comptes Redus des Traveux de la Societe-Hydrotechnique de France, 1932. Parmakian, J., Waterhammer Analysis, PrenticeHall, linc., Englewood Cliffs, N. J. 1955. Gray, C. The analysis of the dissipation of energy in water hammer. in Proc. ASCE. 1953. Streeter, V.L. and C. Lai, Water-hammer analysis including fluid friction. Journal of the Hydraulics Division, 1962. 88(3): p. 79-112. Chaudhry, M.H., Applied hydraulic transients. 1979, Springer. Suo, L. and E. Wylie, Impulse response method for frequency-dependent pipeline transients. Journal of fluids engineering, 1989. 111(4): p. 478-483. Wylie, E., Streeter, Fluid Transients. 1978, McGraw-Hill. Thorley, D.A., Fluid transients in pipeline systems. 2004: ASME Press. Bergant, A., et al., Parameters affecting water-hammer wave attenuation, shape and timing—Part 1: Mathematical tools. Journal of Hydraulic Research, 2008. 46(3): p. 373-381. Wylie, E.B., V.L. Streeter, and L. Suo, Fluid transients in systems. Vol. 1. 1993: Prentice Hall Englewood Cliffs, NJ. Wylie, E.B. and V.L. Streeter, Fluid transients. New York, McGraw-Hill International Book Co., 1978. 401 p., 1978. Tiselj, I. and S. Petelin, Modelling of two-phase flow with second-order accurate scheme. Journal of Computational Physics, 1997. 136(2): p. 503-521. Ali, N.A., et al., Analysis of transient flow phenomenon in pressurized pipes system and methods of protection. J. Eng. Sci. Assiut University, 2010. 38(2): p. 323-342. Jones, G.M., et al., Pumping station design. 2006: Gulf Professional Publishing. Svindland, R.C. and S.C. Williams, Predicting the location and duration of transient induced low or negative pressures within a large water distribution system, in Pipelines 2009: Infrastructure's Hidden Assets. 2009. p. 1115-1124. Sharp, B., D. Sharp, and W. Hammer, Practical Solutions. 1996, Butterworth-Heinemann.
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Conflict of Interest and Authorship Conformation Form Please check the following as appropriate:
o
All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.
o
This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.
o
The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript
o
The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:
Author’s name Mostafa Kandil
Ahmed Kamal
Tamer Elsayed
Affiliation Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat - Elzaton, Cairo, Egypt Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat - Elzaton, Cairo, Egypt Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat - Elzaton, Cairo, Egypt
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. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 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:
∆H =
a ∆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].
∂H a 2 ∂Q 2c 2υ ∂ σ x + − =0 gA ∂ x gE ∂ t ∂t
∂Q ∂H f + Ag + Q Q =0 ∂t ∂x 2DA
(2)
(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
2υ ∂σ x gE ∂t
, 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
K 2 K D a = bm 1 + (1 − µ 2 ) bm Et ρ
(4) Where ρ the density of the fluid, K bm 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).
C
C
+
−
g dH a dt :
+
g dH − a dt :
Figure 2: Characteristic lines in the x-t plane The equation (5) can be solved
1 dQ 1 fQ Q + =0 A dt A 2 D dx =+a dt
+
(5)
1 dQ 1 fQ Q + =0 A dt A 2 D
(6) dx =−a dt
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 C P and C m are defined with Equations (9) and (10), respectively and B =
a . gA
C+ : Hi =CP −BPQi (7)
C− : Hi =CM +BMQi (8)
BP = B +
f ∆x 2 gDA 2
(9)
BM = B +
f ∆x 2 gDA 2
(10)
CP = Hi −1 +BQi −1 (11)
Cm = Hi +1 −BQi +1 (12) The main compatibility equation is valid along the characteristic line ∆x = c .∆t , getting C P and B P at the distance i − 1 from the point of interest, i.e. at the previous
7
time step t − ∆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].
Hi =
CPBM +CM BP BP +BM
(13) QP can then be calculated with Equation (13).
Qi =
CP −CM BP −BM
(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 pipe 1, N +1 = Q pipe 2,1 (15)
Figure 4: Pipes connected in a series
Similar, the conservation of energy gives
H pipe 1, N +1 = H pipe 2,1
(16)
The unknown pressure head is obtain by Equation (17).
H pipe 1,N +1 =
CP 1
B pipe 1 B pipe 1
+C M +1
B pipe 2 (17)
B pipe 2
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 pipe ,1 = H res ,up
(18)
And with a tank at the downstream end, the pressure at the last node in the adjacent pipe is set to
H pipe ,N +1 = H res ,dwn
(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:
Q1 =
H 1 −C M BM
Q N +1 =
(20)
C P − H N +1 BP
(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
∆H = H up − H down =
Replacing
H up
1 2 gAvalve (t )
Q i Qi
(22)
with Equation 7, and H down with Equation 8,
Qi
for the point
of interest can be solved
(C M + B M Q i ) − (C P − B PQ i ) =
1 2 gAvalve (t )
Qi Qi
(23)
Qi
(B =− M
+ B P ) (2 gAvalve (t )) 2
( B + B P )( 2 gAvalve (t ) ) ± M − (C P − C M 2 2
)
(24)
The flow will be equal in and out through valve, giving
Q i = Qup = Q 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
QP =
H P −C m 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
QP
n +1
can be calculated as
QP
n+1
HPn+1 −Cmn+1 = B n+1
4.4 Free Discharge Valve at an End If the valve is present at downstream end of the pipe, then
(27)
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H P n +1 =
Kv n +1 n +1 QP QP 2g
(28)
Where K v is valve-loss coefficient, Further, K v depends on the Coefficient of Discharge (C d ) as K v = C d varies with valve-opening and the type of valve Also,
H P n +1 = C P n +1 − B P n +1QP n +1
(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: =1490 m/s,
a0
D = 1.905 cm, t = 1.588 mm (pipe from Simpson’s [18] experiment), and
variable elasticity modulus
E
.
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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, 50 cm,
a0 is 1300 m/s
pipe length is 2500 m, the pipe diameter is
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].
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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
References 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21.
22.
Ghidaoui, M.S., et al., A review of water hammer theory and practice. Applied Mechanics Reviews, 2005. 58(1): p. 49-76. Joukowski, N., Memoirs of the Imperial Academy Society of St. Petersburg. Proceedings of the American Water Works Association, 1898. 24: p. 341-424. Allievi, L., Teoria generale del moto perturbato dell'acqua nei tubi in pressione (colpo d'ariete) memoria dell'ing. LB Allievi. 1903, Unione Cooperativa. Gibson, N.R. The Gibson method and apparatus for measuring the flow of water in closed conduits. in ASME. 1923. Löwy, R., Druckschwankungen in Druckrohrleitungen. Mit 45 Abb. 1928: Springer. Fuamba, M., N. Bouaanani, and C. Marche, Modeling of dam break wave propagation in a partially ice-covered channel. Advances in Water Resources, 2007. 30(12): p. 2499-2510. Bergeron, L., Variations in flow in water conduits. Comptes Redus des Traveux de la Societe-Hydrotechnique de France, 1932. Parmakian, J., Waterhammer Analysis, PrenticeHall, linc., Englewood Cliffs, N. J. 1955. Gray, C. The analysis of the dissipation of energy in water hammer. in Proc. ASCE. 1953. Streeter, V.L. and C. Lai, Water-hammer analysis including fluid friction. Journal of the Hydraulics Division, 1962. 88(3): p. 79-112. Chaudhry, M.H., Applied hydraulic transients. 1979, Springer. Suo, L. and E. Wylie, Impulse response method for frequency-dependent pipeline transients. Journal of fluids engineering, 1989. 111(4): p. 478-483. Wylie, E., Streeter, Fluid Transients. 1978, McGraw-Hill. Thorley, D.A., Fluid transients in pipeline systems. 2004: ASME Press. Bergant, A., et al., Parameters affecting water-hammer wave attenuation, shape and timing—Part 1: Mathematical tools. Journal of Hydraulic Research, 2008. 46(3): p. 373-381. Wylie, E.B., V.L. Streeter, and L. Suo, Fluid transients in systems. Vol. 1. 1993: Prentice Hall Englewood Cliffs, NJ. Wylie, E.B. and V.L. Streeter, Fluid transients. New York, McGraw-Hill International Book Co., 1978. 401 p., 1978. Tiselj, I. and S. Petelin, Modelling of two-phase flow with second-order accurate scheme. Journal of Computational Physics, 1997. 136(2): p. 503-521. Ali, N.A., et al., Analysis of transient flow phenomenon in pressurized pipes system and methods of protection. J. Eng. Sci. Assiut University, 2010. 38(2): p. 323-342. Jones, G.M., et al., Pumping station design. 2006: Gulf Professional Publishing. Svindland, R.C. and S.C. Williams, Predicting the location and duration of transient induced low or negative pressures within a large water distribution system, in Pipelines 2009: Infrastructure's Hidden Assets. 2009. p. 1115-1124. Sharp, B., D. Sharp, and W. Hammer, Practical Solutions. 1996, Butterworth-Heinemann.
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o
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Author’s name Mostafa Kandil
Ahmed Kamal
Tamer Elsayed
Affiliation Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat - Elzaton, Cairo, Egypt Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat - Elzaton, Cairo, Egypt Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat - Elzaton, Cairo, Egypt