Fluids Handling
Simplified Analysis of Water Hammer Alejandro Anaya Durand Mauricio Marquez Lucero Maria del Carmen Rojas Ocampo Carlos David Ramos Vilchis Gonzalez Vargas Maria de Lourdes National Autonomous University of Mexico
Use this graphical method to quickly and reliably determine the main data — wave celerity, critical time, maximum head developed in the maximum pressure time and the minimum head developed in the critical time — produced by water hammer.
W
ater hammer is generally defined as a pressure surge or wave caused by the kinetic energy of a fluid in motion when it is forced to stop or change direction suddenly, such as in the slow or abrupt startup or shutdown of a pump system. It can also occur because of other operating conditions, such as turbine failure, pipe breakage or electric power interruption to the pump’s motor. Today, informatics methods are commonly used to perform complex water hammer calculations. However, the lack of information about certain physical flow properties, as well as the intricacy in handling complicated equations make this phenomena analysis difficult. The objective of this paper is to provide a practical and simplified methodology to calculate four main phenomenon parameters of water hammer: • velocity of the pressure wave or celerity • phenomena critical time • maximum head developed in the maximum pressure time • minimum head developed in the critical time.
Equations and basic considerations One of the most basic equations for the maximum pressure gradient calculation is Joukowsky’s equation: hw max = (a)(v)/g
(1)
where: hw max is the maximum fluid elevation head to water hammer, ft; a is the wave celerity, ft/s; v is the flow veloci40
www.aiche.org/cep
December 2006
CEP
ty, ft/s; and g is the gravitational acceleration constant, ft/s2. The volumetric flow is given as: Q = vA
(2)
where A is the flow area, ft2. A can be defined as: A = πDi2/4
(3)
where Di is the internal diameter of the pipe, in. Substituting Eqs. 2 and 3 into Eq. 1 and solving for a, yields: 2
2 ⎛ g π ⎞ ⎛ hw maxε ⎞ ⎛ Di ⎞ a=⎜ ⎝ 4 ⎟⎠ ⎜⎝ Q ⎟⎠ ⎜⎝ ε ⎟⎠ where ε is the pipe thickness, in. Eq. 4 can then be rewritten as: 1
⎞ ⎛ ⎟ ⎜ 1 ⎟ a=⎜ ⎜ ⎛ ρ ⎞ ⎛ 1 ⎛ D ⎞ ⎛ C1 ⎞ ⎞ ⎟ ⎜⎝ ⎜⎝ g ⎟⎠ ⎜⎝ K + ⎜⎝ ε ⎟⎠ ⎜⎝ E ⎟⎠ ⎟⎠ ⎟⎠
(4 )
2
(5 )
where ρ is the density, lb/ft3; K is the liquid compressibility volume factor, lb/in2; C1 is Poisson’s ratio; and E is the maximum yield stress, lb/in2. These equations were used to generate Figures 1–3. When the valve that stops the fluid flow is closed in a time slower than the critical time (this reduces the effect of water hammer), the Allievi equation is used:
ho 2 C ± C (4 + C2 2
(
)
0.5
)
the right side to determine the quotient value tc/L. The value of hw maxε2/Q is determined by observing the value of the vertical curve at which D/ε and the density in LV C= (7) °API meets. With the knowledge of the value of hw gh0t c ε2/Q, the maximum pressure with instant valve closing max 4Q time can be obtained. This value should be added to the v= (8 ) system’s pressure to determine the system overpressure. π Di2 The maximum and the minimum head, with different where: C is a valve constant; h0 is the head pump, ft; L is valves closing can be calculated using Figure 4, which the length, ft; V is the velocity, ft; and tc is the time to close plots the pump head against the maximum pump head to the valve. Substituting Eqs. 7 and 8 into Eq. 6 results in: water hammer. It also shows a family of curves with different LQ/D2tc values. Knowing the values ⎛ ⎛ ⎛ 1 ⎞ ⎛ 4 ⎞ ⎛ LQ ⎞ ⎞ 2 ⎞ for the pump head and LQ/D2tc, the maxi⎜⎜⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟⎟ ± ⎟ mum and minimum pressure that the system h ⎝ gπ ⎠ ⎝ Di t c ⎠ ⎠ ⎟ ho ⎜ ⎝ ⎝ o ⎠ ∆h = ⎜ (9 ) can handle can be determined by drawing a ⎟ 2 0.5 2 ⎜ ⎛ ⎛ 1 ⎞ ⎛ 4 ⎞ ⎛ LQ ⎞ ⎞ ⎛ horizontal line from the point that the pump ⎛ ⎛ 1 ⎞ ⎛ 4 ⎞ ⎛ LQ ⎞ ⎞ ⎞ ⎟ head and LQ/D2tc intersects. ⎜⎜⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟⎟ ⎜4 + ⎜⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟⎟ ⎟ ⎟ ⎜⎝ ⎝ ⎝ ho ⎠ ⎝ gπ ⎠ ⎝ Di t c ⎠ ⎠ ⎜⎝ ⎝ ⎝ ho ⎠ ⎝ gπ ⎠ ⎝ Di t c ⎠ ⎠ ⎟⎠ ⎟⎠ ∆h =
(6 )
Methodology The simplified graphic Nomenclature methodology is based on a = wave celerity, ft/s L = length, ft the simple nomograms A = flow area, ft2 Q = volumetric flow, ft3/s developed for pipeline C = valve constant as defined by the Allievi Eq. t = close time valve, s transportation of hydrocarC = Poisson’s ratio v = flow velocity, ft/s bons (piping API-5L-X52 D = internal diameter, in. V = velocity, ft/s E = maximum yield stress, lb/in2 X = value of the quotient obtained in Figure 4 of carbonated steel). 2 g = gravitational acceleration constant, ft/s Figures 1–3 plot the wave h = pump head, ft Greek Letters celerity on the y-axis and h = fluid elevation head to water hammer, ft µ = pipe thickness, in. the diameter divided by the K = liquid compressibility volume factor, lb/in.2 ρ = density, lb/ft3 pipe thickness in the x-axis for commercial LPG, crude 2,800.0 7.1x10 oil and water, respectively. Also plotted in Figures 1–3 90 °API 7.4x10 2,700.0 are the different values of 80 °API density in degrees (°API) 2,600.0 7.7x10 70 °API and the quotient value, 60 °API hw maxε2/Q. Using Figures 1, 2,500.0 8.0x10 50 °API 2 or 3, the maximum head 8.3x10 2,400.0 40 °API at the instant the valve closes or the pump stops can be 30 °API 8.7x10 2,300.0 calculated. 20 °API Given the values of D/ε 9.5x10 2,200.0 10 °API and the density in °API, the 9.1x10 2,100.0 wave celerity, hw maxε2/Q and tc/L can be determined. 1.0x10 2,000.0 This is done by drawing a 10.0 15.0 20.0 25.0 30.0 35.0 horizontal line from the Diameter (D) / Thickness (ε) intersection to the left side of the figure to determine Figure 1. Wave celerity vs. D/ε vs. valve close time for commercial liquefied propane gas (LPG; volume the wave celerity, and to factor = 67,000 lb/in.2) in carbon steel pipe. c
1 i
o
w
-4
-4
0.80
-4
= 0. 10
-4
hw ε 2 /Q
0.15
0.20
0.25
0.30
0.40
-4
0.50
0.70 0.60
Wave Celerity (a), ft/s
-4
Valve Close Time (tc) / Length (L), s/ft
-4
-4
-3
CEP
December 2006
www.aiche.org/cep
41
Fluids Handling
Sample calculation A 12-in. pipe used to transport hydrocarbons 5.0x10 4,000.0 (commercial LPG with a 3,900.0 5.1x10 90 °API density of 51.4 lb/ft3) with 80 °API 3,800.0 5.3x10 40°API has a length of 12 70 °API km and a flow of 25,000 5.4x10 3,700.0 60 °API bbl/d. The pipe is con5.6x10 3,600.0 structed from carbonated 50 °API 3,500.0 5.7x10 steel API-5L-X-52 and has 40 °API 3,400.0 5.9x10 a constant thickness of 30 °API 0.406 in. The pump’s dis3,300.0 6.1x10 20 °API charge pressure is 400 6.3x10 3,200.0 lb/in.2 and the valve section 10 °API 6.5x10 3,100.0 closing time is 60 s. 3,000.0 6.7x10 Calculate the: maximum 10.0 15.0 20.0 25.0 30.0 40.0 35.0 pressure; maximum presDiameter (D) / Thickness (ε) sure critical time; maximum pressure with a valve closing time of 30 s; and 2 Figure 2. Wave celerity vs. D/ε vs. valve close time for crude oil (volume factor = 150,000 lb/in. ) in overpressure. carbon steel pipe. Calculate the maximum pressure. First, determine D/ε. Start by 5,700.0 calculating the internal 3.51x10 diameter: 5,600.0 3.57x10 4.9x10-4 -4
-4
-4
-4
-4
-4
-4
-4
-4
Valve Close Time (tc) / Length (L), s/ft
= 0. 10
hw ε 2 /Q
0.15
0.20
0.25
0.30
0.40
0.50
0.60
Wave Celerity (a), ft/s
0.80 0.70
4,100.0
-4
-4
0.10
hw ε 2 /Q =
3.64x10-4
3.77x10
3.85x10-4
90 °API
3.92x10-4
5,100.0 80 °API
4.00x10-4
5,000.0 70 °API
4.08x10-4
4,900.0 60 °API
4,800.0
4.17x10-4
50 °API
4,700.0 4,600.0
4.26x10-4 4.35x10-4
40 °API
4.44x10-4
4,500.0 30 °API
4.55x10-4
4,400.0 20 °API
4,300.0 4,200.0
4.65x10-4
4.88x10-4
4,000.0
5.00x10-4 5.13x10-4
17.0
22.0
27.0
32.0
37.0
Diameter (D) / Thickness (ε)
Figure 3. Wave celerity vs. D/ε vs. valve close time for water (volume factor = 300,000 lb/in.2) in carbon steel pipe.
42
www.aiche.org/cep
December 2006
CEP
Therefore: D/ε = (11.18 in.)/(0.406 in.) = 27.53 The maximum system pressure is: System pressure = (400 lb/in.2)(144)/(51.4 lb/ft3) = 1,120.6 ft
4.76x10-4
10 °API
4,100.0
3,900.0 12.0
D = 12 in. – ((2)(0.406 in.)) = 11.18 in. = 0.93 ft
3.70x10-4
Valve Close Time (tc) / Length (L), s/ft
5,200.0
Wave Celerity (a), ft/s
-4
-4
0.20
5,300.0
0.25
0.30
0.40
0.50
0.70
5,400.0
0.60
5,500.0
0.80
-4
Calculate the maximum pressure critical time. Using Figure 1, starting at D/ε = 27.53, draw a vertical line until it intercepts the 40°API line. At this intersection, the following information is obtained:
Fluid Elevation Head, ft
a = 2.389 ft/s hw max ε2/Q = 0.124 tc/L = 0.00083 s/ft
100
1,100
tc= (tc/L)L = (0.00083 s/ft)(39,370 ft) = 32.6 s
8,000 6,000
Pressure
200 4,000 100
2,000
0
Subpressure
1,120.6 ft + 192.8 ft = 1,313.4 ft Calculate the maximum pressure critical time:
Maximum Elevation Head to Water Hammer, ft
The maximum overpressure is:
2,000
-100
4,000 6,000
-200 8,000 10,000
-300
Figure 4. Maximum fluid elevation head to water hammer.
Calculate the maximum pressure with a valve closing time of 30 s. LQ/Di2tc = (39,370 ft)(1.78 ft3/s) / (0.93 ft)2 (30 s) = 2,700.8 ft2 Calculate the overpressure and subpressure. Knowing the maximum pump head value system, as well as LQ/Di2tc, the overpressure can be determined by using Figure 4. The overpressure is 108.8 ft. Therefore, the total system’s overpressure is 1,120.6 ft + 108.8 ft = 1,229.4 ft. Similarly, using Figure 4, the subpressure is 104.5 ft. Thus, the total system’s subpressure is 1,120.6 ft – 104.5 ft = 1,016.0 ft. CEP
Literature Cited
3. 4. 5.
5,100
4,100
300
hw max = XQ/ε2 = 0.124(1.78ft3/s)/(0.406 in./ 12 in./ft)2 = 192.8 ft
2.
3,100
LQ/Di2Tc(max) = 10,000
Solve for hw max, given Q = 25,000 bbl/d = 1.78 ft3/s and ρ = 51.4 lb/ft3 = 0.029 lb/in.3:
1.
2,100
Perry, J. H., “Chemical Engineers Handbook,” Mc Graw Hill, 4th edition, pp. 6–44. UNAM, “Selection and Operation of Pump System,” National Autonomous University of Mexico (March 1983). American Society of Civil Engineers, “Pipe Line Design for Water and Wastewater,” Pipeline Committee on Pipeline Planning (1975). Douglas, J. F., “Solution of Problems in Fluid Mechanics,” Pitman Paperbacks, pp. 109–117 (1967). Potter, M. C. and D. C. Wiggert, “Mechanics of Fluids,” Prentice Hall, 2nd edition, p. 18 (1997).
ALEJANDRO ANAYA DURAND is professor of chemical engineering at the National Autonomous University of Mexico (UNAM; Parque España 15B Colonia Condesa, México D.F. C.P. 06140, México; Phone and fax: 52110385; E-mail:
[email protected]). He has 40 years of experience as an educator, teaching subjects such as heat transfer, fluid flow and project engineering. After 30 years of holding top-level positions in project engineering, Durand retired from the Instituto Mexicano del Petroleo in 1998. He is also a consultant of several engineering companies. Durand is a Fellow of AIChE and holds an M.S. in chemical engineering from UNAM. He has published more than 200 technical articles in local and international magazines, related to chemical engineering and education. MAURICIO MARQUEZ LUCERO Is project manager for the Delta Project and Development Co. (Sur 73 No 311 bis 1 Colonia Sinatel, Del Iztapalapa, México D.F., C.P. 09470; Phone: 56721237; Fax: 55392883; E-mail:
[email protected]). He is a member of the Instituto Mexicano de Ingenieros Químicos. Lucero has published several technical articles in local and international magazines, in chemical engineering. MARIA DEL CARMEN ROJAS OCAMPO is a chemical engineering student at the National University Autonomous of Mexico (Sur 73 No 311 bis 1 Colonia Sinatel, Del Iztapalapa, México D.F., C.P. 09470; Phone: 56721237; Fax: 55392883; E-mail:
[email protected]). She has published several technical articles on local and international magazines, in chemical engineering. CARLOS DAVID RAMOS VILCHIS is a chemical engineering student at the National University Autonomous of Mexico (Cerrada Canal Nacional #6 col. Tejomulco Nativitas C.P. 16510 Delegacion Xochimilco Mexico. D.F.; Phone: 21578360; E-mail:
[email protected]). Presently, he collaborates in a company dedicated to training groups for work in the chemical industry (pilot plant). GONZALEZ VARGAS MARIA DE LOURDES is a chemical engineering student at the National University Autonomous of Mexico (México Tacuba 1523 edificio Managua departamento 505 colonia Argentina Poniente C.P.: 11230 Delegación Miguel Hidalgo México, D.F.; Phone: 55761849; Fax: 55761855).
CEP
December 2006
www.aiche.org/cep
43
