# 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:

$c=\sqrt{\frac{K}{\rho }}$

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

K - volume elastic modulus - [Pa]

ρ - destiny - [Kg/m3]

Volume elastic modulus:

$K=\frac{1}{\beta }$

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.

$t\ge \frac{2L}{a}$

t - valve closing time - [s]

L - pipe length - [m]

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

$P=\frac{\rho *L*v}{t}$

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.

$t={t}_{r}*{c}_{ef}$

t - valve closing time - [s]

tr - closing time - [s]

cef - Effective closing time factor - []

$t\ge \frac{2L}{a}$

t - valve closing time - [s]

L - pipe length - [m]

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

$P=\frac{\rho *L*v}{t}$

P - water hammer - [Pa]

ρ - destiny - [Kg/m3]

L - pipe length - [m]

v - pipeline speed - [m/s]

t - valve closing time - [s]

Pipeline speed:

$v=\frac{Q}{\frac{\pi *{D}^{2}}{4}}$

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 Sef. 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 cef [] from of the proportional flow characteristics of the knife valve Fig.2

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

Pressure parameter:

$p=\frac{\Delta h}{{h}_{0}}$

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:

$\Delta h=\frac{{v}_{0}^{2}}{2g}*\left(\xi +1\right)$

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:

${v}_{0}=\frac{Q}{\frac{\pi *{D}_{0}^{2}}{4}}$

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

$K=\frac{1}{\beta }=\frac{1}{477,1*{10}^{-12}}$ $=2,096*{10}^{9}\left[Pa\right]$

- sound speed in liquid

$c=\sqrt{\frac{K}{\rho }}=\sqrt{\frac{2,096*{10}^{9}}{998,3}}$ $=1448,989\left[m/s\right]$

- speed pressure waves in the pipe

$a=\frac{c}{\sqrt{1+\frac{D}{e}*\frac{K}{E}}}$ $=\frac{1448,989}{\sqrt{1+\frac{600}{10}*\frac{2,096*{10}^{9}}{2*{10}^{11}}}}$ $=1135,3\left[m/s\right]$
$t\ge \frac{2L}{a}\to \frac{2*12000}{1135,2}=21,14\left[s\right]$ $\le 200\left[s\right]\to OK$

- pipeline speed

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

- water hammer

$P=\frac{\rho *L*v}{t}=\frac{998,3*12000*1,11}{200}=$ 66486,78[Pa]

Water hammer (for nonlinear closure):

- valve speed

${v}_{0}=\frac{Q}{\frac{\pi *{D}_{0}^{2}}{4}}=\frac{0,314}{\frac{\pi *{0,3}^{2}}{4}}=4,44\left[m/s\right]$

- theoretical pressure in the closure at full opening

$\Delta h=\frac{{v}_{0}^{2}}{2g}*\left(\xi +1\right)=\frac{{4,44}^{2}}{2*9,81}*\left(0,01+1\right)$ $=1,015\left[m\right]$

- pressure parameter

$p=\frac{\Delta h}{{h}_{0}}=\frac{1,015}{33}=0,03\left[\right]$ $\to {c}_{ef}=0,2\left[\right]$

this value is determined from the table using interpolation.

- valve closed time water hammer calculation

$t={t}_{r}*{c}_{ef}=200*0,2=40\left[s\right]$

- volume elastic modulus

$K=\frac{1}{\beta }=\frac{1}{477,1*{10}^{-12}}$ $=2,096*{10}^{9}\left[Pa\right]$

- sound speed in liquid

$c=\sqrt{\frac{K}{\rho }}=\sqrt{\frac{2,096*{10}^{9}}{998,3}}$ $=1448,989\left[m/s\right]$

- speed pressure waves in the pipe

$a=\frac{c}{\sqrt{1+\frac{D}{e}*\frac{K}{E}}}$ $=\frac{1448,989}{\sqrt{1+\frac{600}{10}*\frac{2,096*{10}^{9}}{2*{10}^{11}}}}$ $=1135,3\left[m/s\right]$
$t\ge \frac{2L}{a}\to \frac{2*12000}{1135,2}=21,14\left[s\right]$ $\le 40\left[s\right]\to OK$

- pipeline speed

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

- water hammer

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