Water hammer

Speed pressure waves in the pipe:

a = \frac{c}{\sqrt{1 + \frac{D}{e} * \frac{K}{E}}}

a - speed pressure waves in the pipe - [m/s]

c - sound speed in liquid - [m/s]

D - internal pipe diameter - [mm]

e - thickness of the pipe wall - [mm]

K - volume elastic modulus - [Pa]

E - Young's modulus for pipe - [Pa]

Sound speed in liquid:

c = \sqrt{\frac{K}{ρ}}

c - sound speed in liquid - [m/s]

K - volume elastic modulus - [Pa]

ρ - destiny - [Kg/m³]

Volume elastic modulus:

K = \frac{1}{β}

K - volume elastic modulus - [Pa]

β - medium compressibility factor - [Pa^-1]

Young's modulus for pipe:

Pipe material	[Pa]
Steel	2*10¹¹
Copper	1,17*10¹¹
Cast iron	0,7*10¹¹
Glass	0,8*10¹¹
Polyvinyl chloride (PVC)	3*10⁹
Rubber	4,2*10⁶
Reinforced concrete	0,21*10¹¹
Polypropylene (PP)	7*10⁸

Medium compressibility factor:

Medium compressibility factor β*10^-12 [Pa^-1] water depending on pressure and temperature

Pressure [MPa]	Temperature [°C]
	0°	5°	10°	15°	20°	30°	40°	50°	60°	70°	80°	90°	100°
0,1-10	520,9	502,6	492,4	482,2	477,1	468,9	457,7	457,7	463,8	471	478,1	487,3	-
10-20	501,5	484,2	469,9	459,7	450,6	444,4	437,3	433,2	435,3	447,5	459,7	477,1	822,6
20-30	489,3	471	461,8	451,6	442,4	430,2	422	421	423	433,2	444,4	467,9	783,9
30-40	475	457,7	449,5	441,4	432,2	421	414,9	409,8	413,9	419	430,2	454,6	745,2
40-50	463,8	452,6	438,3	430,2	423	413,9	411,8	406,7	401,6	405,7	415,9	442,2	695,2
50-60	446,5	438,3	426,1	419	411,8	399,6	397,6	397,6	395,5	398,6	406,7	424,1	671,8
60-70	437,3	416,9	412,8	405,7	401,6	394,5	389,4	384,3	390,4	387,4	394,5	414,9	639,1

Density:

Density ρ [Kg/m³] water depending on temperature and pressure

Temperature [°C]	Pressure [MPa]
	0,1	0,25	0,5	1	1,5	2	2,5	3	3,5	4	4,5	5	6	7	8	9	10	12,5	15	17,5	20	25	30	35	40	45	50	60	70	80
0°	999,8	999,9	1000	1000,3	1000,6	1000,8	1001,1	1001,3	1001,6	1001,8	1002,1	1002,3	1002,8	1003,3	1003,8	1004,3	1004,8	1006	1007,3	1008,5	1009,7	1012,1	1014,5	1016,9	1019,3	1021,6	1023,9	1028,3	1032,7	1037
10°	999,7	999,8	999,9	1000,1	1000,4	1000,6	1000,8	1001	1001,3	1001,6	1001,8	1002	1002,5	1003	1003,4	1003,9	1004,4	1005,5	1006,7	1007,9	1009	1011,3	1013,6	1015,7	1018	1020,2	1022,3	1026,6	1030,7	1034,9
20°	998,2	998,3	998,4	998,6	998,8	999,1	999,3	999,5	999,8	1000	1000,2	1000,4	1000,9	1001,3	1001,8	1002,2	1002,7	1003,8	1004,9	1006	1007,2	1009,3	1011,4	1013,6	1015,7	1017,8	1019,9	1024,1	1028,1	1032
30°	995,6	995,7	995,8	996	996,3	996,5	996,7	996,9	997,2	997,4	997,6	997,8	998,3	998,7	999,1	999,6	1000	1001,1	1002,2	1003,2	1004,3	1006,5	1008,6	1010,6	1012,8	1014,7	1016,8	1020,8	1024,7	1028,5
40°	992,2	992,3	992,4	992,7	992,9	993	993,3	993,4	993,7	993,9	994,1	994,3	994,8	995,2	995,6	996,1	996,5	997,6	998,6	999,7	1000,8	1002,8	1004,9	1007	1009	1011	1013	1017	1020,8	1024,6
50°	988,1	988,1	988,2	988,4	988,6	988,8	989,1	989,2	989,5	989,7	989,9	990,2	990,6	991	991,5	991,9	992,3	993,3	994,4	995,5	996,5	998,6	1000,7	1002,7	1004,7	1006,8	1008,7	1012,6	1016,4	1020,2
60°	983,2	983,3	983,4	983,6	983,9	984,1	984,3	984,5	984,6	984,9	985,1	985,3	985,8	989,2	986,6	987,1	987,5	988,5	989,6	990,7	991,7	993,7	995,8	997,9	999,9	1001,9	1003,8	1007,8	1011,5	1015,3
70°	977,8	977,8	978	978,2	978,4	978,6	978,9	979,1	979,2	979,5	979,7	979,9	980,4	980,8	981,3	981,6	982,1	983,2	984,3	985,3	986,4	988,4	990,5	992,6	994,6	996,6	998,6	1002,5	1006,3	1010,1
80°	971,8	971,9	972	972,2	972,4	972,7	972,9	973,1	973,3	973,5	973,8	974	974,5	974,9	975,3	975,7	976,2	977,2	978,4	979,4	980,5	982,6	984,7	986,8	988,8	990,9	992,9	996,8	1000,7	1004,4
90°	965,3	965,3	965,7	965,7	966	966,2	966,4	966,6	966,8	967,1	967,3	967,6	968	968,4	968,9	969,4	969,7	970,9	972	973,1	974,2	976,4	978,5	980,6	982,7	984,7	986,8	990,8	994,6	998,5
100°	-	958,4	958,8	958,8	959	959,2	959,5	959,7	960	960,2	960,4	960,6	961,1	961,5	962	962,5	962,9	964	965,2	966,3	967,4	969,7	971,8	974	976,1	978,2	980,3	984,3	988,3	992,3

Water hammer (for linear closure):

For slow and linear closing of the valve.

t \geq \frac{2 L}{a}

t - valve closing time - [s]

L - pipe length - [m]

a - speed pressure waves in the pipe - [m/s]

P = \frac{ρ * L * v}{t}

P - water hammer - [Pa]

ρ - destiny - [Kg/m³]

L - pipe length - [m]

v - pipeline speed - [m/s]

t - valve closing time - [s]

Water hammer (for nonlinear closure):

For slow closing of the valve.

t = t_{r} * c_{e f}

t - valve closing time - [s]

t_r - closing time - [s]

c_ef - Effective closing time factor - []

t \geq \frac{2 L}{a}

t - valve closing time - [s]

L - pipe length - [m]

a - speed pressure waves in the pipe - [m/s]

P = \frac{ρ * L * v}{t}

P - water hammer - [Pa]

ρ - destiny - [Kg/m³]

L - pipe length - [m]

v - pipeline speed - [m/s]

t - valve closing time - [s]

Pipeline speed:

v = \frac{Q}{\frac{π * D^{2}}{4}}

v - pipeline speed - [m/s]

Q - flow - [m³/s]

D - internal pipe diameter - [mm]

Effective closing time factor:

Fig.1

The water hammer formula is valid assuming linear flow characteristics (at even closure - the linear relationship between the flow and the position of the closure valve). This assumption is difficult to accomplish with most valves without pre-treatment (modification of structural characteristics).

If we calculate the proportional flow rate by a valve for several valve positions, we can graphically represent the relationship between the proportional flow and the stroke (or turn) of the closure valve. This dependence is shown in Figure 1 by line ˮaˮ. The line ˮa₁ˮ shows the linear relationship between the flow and the valve of the closure. It can be seen from the figure that only a partial part of the total stroke influences the substantial flow limitation.

To the decreasing line ˮaˮ we can build a tangent line ˮtˮ, which on the horizontal line ˮQˮ determine the effective stroke S_ef. We assume that only the effective stroke has an effect on the flow limitation and, moreover, that in its range the relationship between flow and stroke linear.

Fig.2 Proportional flow characteristic of the knife valve - ξ=0,01

The value of the effective closing time factor c_ef [] from of the proportional flow characteristics of the knife valve Fig.2

p	1	0,5	0,2	0,1	0,05	0,01
c_ef	1	0,73	0,46	0,33	0,24	0,141

Pressure parameter:

p = \frac{Δ h}{h_{0}}

p - pressure parameter - []

Δh - theoretical pressure in the closure at full opening - [m]

h₀ - rated net head - [m]

Theoretical pressure in the closure at full opening:

Δ h = \frac{v_{0}^{2}}{2 g} * (ξ + 1)

Δh - theoretical pressure in the closure at full opening - [m]

v₀ - valve speed - [m/s]

g - gravitational acceleration - [m/s²]

ξ - local loss factor for open valve - []

valve speed:

v_{0} = \frac{Q}{\frac{π * D_{0}^{2}}{4}}

v₀ - valve speed - [m/s]

Q - flow - [m³/s]

D₀ - valve diameter - [mm]

Example:

We have to determine the water hammer size for the linear and nonlinear shut-off of the DN300 knife valve with the following parameters:

L = 12000m; steel pipe D = 600mm; thickness of the pipe wall e = 10mm; h₀ = 33m; Q = 0,314 m³/s; density water ρ = 998,3Kg/m³; medium compressibility factor β = 477,1*10^-12; closing time 200s

Water hammer (for linear closure):

- volume elastic modulus

K = \frac{1}{β} = \frac{1}{477,1 * 10^{- 12}}

= 2,096 * 10^{9} [P a]

- sound speed in liquid

c = \sqrt{\frac{K}{ρ}} = \sqrt{\frac{2,096 * 10^{9}}{998,3}}

= 1448,989 [m / s]

- speed pressure waves in the pipe

a = \frac{c}{\sqrt{1 + \frac{D}{e} * \frac{K}{E}}}

= \frac{1448,989}{\sqrt{1 + \frac{600}{10} * \frac{2,096 * 10^{9}}{2 * 10^{11}}}}

= 1135,3 [m / s]

t \geq \frac{2 L}{a} \to \frac{2 * 12000}{1135,2} = 21,14 [s]

\leq 200 [s] \to O K

- pipeline speed

v = \frac{Q}{\frac{π * D^{2}}{4}} = \frac{0,314}{\frac{π * {0,6}^{2}}{4}} = 1,11 [m / s]

- water hammer

P = \frac{ρ * L * v}{t} = \frac{998,3 * 12000 * 1,11}{200} =

66486,78[Pa]

Water hammer (for nonlinear closure):

- valve speed

v_{0} = \frac{Q}{\frac{π * D_{0}^{2}}{4}} = \frac{0,314}{\frac{π * {0,3}^{2}}{4}} = 4,44 [m / s]

- theoretical pressure in the closure at full opening

Δ h = \frac{v_{0}^{2}}{2 g} * (ξ + 1) = \frac{{4,44}^{2}}{2 * 9,81} * (0,01 + 1)

= 1,015 [m]

- pressure parameter

p = \frac{Δ h}{h_{0}} = \frac{1,015}{33} = 0,03 []

\to c_{e f} = 0,2 []

this value is determined from the table using interpolation.

- valve closed time water hammer calculation

t = t_{r} * c_{e f} = 200 * 0,2 = 40 [s]

- volume elastic modulus

K = \frac{1}{β} = \frac{1}{477,1 * 10^{- 12}}

= 2,096 * 10^{9} [P a]

- sound speed in liquid

c = \sqrt{\frac{K}{ρ}} = \sqrt{\frac{2,096 * 10^{9}}{998,3}}

= 1448,989 [m / s]

- speed pressure waves in the pipe

a = \frac{c}{\sqrt{1 + \frac{D}{e} * \frac{K}{E}}}

= \frac{1448,989}{\sqrt{1 + \frac{600}{10} * \frac{2,096 * 10^{9}}{2 * 10^{11}}}}

= 1135,3 [m / s]

t \geq \frac{2 L}{a} \to \frac{2 * 12000}{1135,2} = 21,14 [s]

\leq 40 [s] \to O K

- pipeline speed

v = \frac{Q}{\frac{π * D^{2}}{4}} = \frac{0,314}{\frac{π * {0,6}^{2}}{4}} = 1,11 [m / s]

- water hammer

P = \frac{ρ * L * v}{t} = \frac{998,3 * 12000 * 1,11}{40} =

332433,9[Pa]

Water hammer for nonlinear closure is 5x bigger, than for linear closure.

Literature:

Ing. J. Kvasnička: Určení doby otevření nebo uzavření uzávěru. Vodní hospodářství 6/1969

V. Kolář, S. Vinopal: Hydraulika průmyslových armatur. SNTL 1964

Wikipedia: Water hammer. en.wikipedia.org/wiki/Water_hammer

R. Mareš: Tabulky termodynamických vlastností vody a vodní páry

ČSN EN 13480-3: Simplified static analysis of rapid valve closure

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