Water hammer
Speed pressure waves in the pipe:
$a$ | speed pressure waves in the pipe | $\mathrm{m/s}$ |
$c$ | sound Speed in liquid | $\mathrm{m/s}$ |
$D_p$ | internal pipe diameter | $\mathrm{mm}$ |
$e$ | thickness of the pipe wall | $\mathrm{mm}$ |
$K$ | volume elastic modulus | $\mathrm{Pa}$ |
$E$ | Young's modulus for pipe | $\mathrm{Pa}$ |
Sound Speed in liquid:
$c$ | sound Speed in liquid | $\mathrm{m/s}$ |
$K$ | volume elastic modulus | $\mathrm{Pa}$ |
$ρ$ | density | $\mathrm{kg/m^3}$ |
Volume elastic modulus:
$K$ | volume elastic modulus | $\mathrm{Pa}$ |
$β$ | medium compressibility factor | $\mathrm{Pa^{-1}}$ |
Young's modulus for pipe:
Pipe material | $\mathrm{Pa}$ |
---|---|
Steel | $2\cdot10^{11}$ |
Copper | $1,17\cdot10^{11}$ |
Cast iron | $0,7\cdot10^{11}$ |
Glass | $0,8\cdot10^{11}$ |
Polyvinyl chloride (PVC) | $3\cdot10^{9}$ |
Rubber | $4,2\cdot10^{6}$ |
Reinforced concrete | $0,21\cdot10^{11}$ |
Polypropylene (PP) | $7\cdot10^{8}$ |
Medium compressibility factor:
Medium compressibility factor $\beta \cdot10^{-12}$ $\mathrm{Pa^{-1}}$ water depending on pressure and temperature
Pressure $\mathrm{[MPa]}$ | Temperature $\mathrm{[°C]}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0° | 10° | 20° | 30° | 40° | 50° | 60° | 70° | 80° | 90° | 100° | |||
0.1-10 | 520.9 | 492.4 | 477.1 | 468.9 | 457.7 | 457.7 | 463.8 | 471 | 478.1 | 487.3 | - | ||
10-20 | 501.5 | 469.9 | 450.6 | 444.4 | 437.3 | 433.2 | 435.3 | 447.5 | 459.7 | 477.1 | 822.6 | ||
20-30 | 489.3 | 461.8 | 442.4 | 430.2 | 422 | 421 | 423 | 433.2 | 444.4 | 467.9 | 783.9 | ||
30-40 | 475 | 449.5 | 432.2 | 421 | 414.9 | 409.8 | 413.9 | 419 | 430.2 | 454.6 | 745.2 | ||
40-50 | 463.8 | 438.3 | 423 | 413.9 | 411.8 | 406.7 | 401.6 | 405.7 | 415.9 | 442.2 | 695.2 | ||
50-60 | 446.5 | 426.1 | 411.8 | 399.6 | 397.6 | 397.6 | 395.5 | 398.6 | 406.7 | 424.1 | 671.8 | ||
60-70 | 437.3 | 412.8 | 401.6 | 394.5 | 389.4 | 384.3 | 390.4 | 387.4 | 394.5 | 414.9 | 639.1 |
Density:
Density $\rho\ \mathrm{[kg/m^3]}$ water depending on temperature and pressure
Pressure $\mathrm{[MPa]}$ | Temperature $\mathrm{[°C]}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0° | 10° | 20° | 30° | 40° | 50° | 60° | 70° | 80° | 90° | 100° | |||
0.1 | 999.8 | 999.7 | 998.2 | 995.6 | 992.2 | 988.1 | 983.2 | 977.8 | 971.8 | 965.3 | - | ||
0.25 | 999.9 | 999.8 | 998.3 | 995.7 | 992.3 | 988.1 | 983.3 | 977.8 | 971.9 | 965.3 | 958.4 | ||
0.5 | 1000 | 999.9 | 998.4 | 995.8 | 992.4 | 988.2 | 983.4 | 978 | 972 | 965.5 | 958.5 | ||
1 | 1000.3 | 1000.1 | 998.6 | 996 | 992.7 | 988.4 | 983.6 | 978.2 | 972.2 | 965.7 | 958.8 | ||
1.5 | 1000.6 | 1000.4 | 998.8 | 996.3 | 992.9 | 988.6 | 983.9 | 978.4 | 972.4 | 966 | 959 | ||
2 | 1000.8 | 1000.6 | 999.1 | 996.5 | 993 | 988.8 | 984.1 | 978.6 | 972.7 | 966.2 | 959.2 | ||
2.5 | 1001.1 | 1000.8 | 999.3 | 996.7 | 993.3 | 989.1 | 984.3 | 978.9 | 972.9 | 966.4 | 959.5 | ||
3 | 1001.3 | 1001 | 999.5 | 996.9 | 993.4 | 989.2 | 984.5 | 979.1 | 973.1 | 966.6 | 959.7 | ||
3.5 | 1001.6 | 1001.3 | 999.8 | 997.2 | 993.7 | 989.5 | 984.6 | 979.2 | 973.3 | 966.8 | 960 | ||
4 | 1001.8 | 1001.6 | 1000 | 997.4 | 993.9 | 989.7 | 984.9 | 979.5 | 973.5 | 967.1 | 960.2 | ||
4.5 | 1002.1 | 1001.8 | 1000.2 | 997.6 | 994.1 | 989.9 | 985.1 | 979.7 | 973.8 | 967.3 | 960.4 | ||
5 | 1002.3 | 1002 | 1000.4 | 997.8 | 994.3 | 990.2 | 985.3 | 979.9 | 974 | 967.6 | 960.6 | ||
6 | 1002.8 | 1002.5 | 1000.9 | 998.3 | 994.8 | 990.6 | 985.8 | 980.4 | 974.5 | 968 | 961.1 | ||
7 | 1003.3 | 1003 | 1001.3 | 998.7 | 995.2 | 991 | 989.2 | 980.8 | 974.9 | 968.4 | 961.5 | ||
8 | 1003.8 | 1003.4 | 1001.8 | 999.1 | 995.6 | 991.5 | 986.6 | 981.3 | 975.3 | 968.9 | 962 | ||
9 | 1004.3 | 1003.9 | 1002.2 | 999.6 | 996.1 | 991.9 | 987.1 | 981.6 | 975.7 | 969.4 | 962.5 | ||
10 | 1004.8 | 1004.4 | 1002.7 | 1000 | 996.5 | 992.3 | 987.5 | 982.1 | 976.2 | 969.7 | 962.9 | ||
12.5 | 1006 | 1005.5 | 1003.8 | 1001.1 | 997.6 | 993.3 | 988.5 | 983.2 | 977.2 | 970.9 | 964 | ||
15 | 1007.3 | 1006.7 | 1004.9 | 1002.2 | 998.6 | 994.4 | 989.6 | 984.3 | 978.4 | 972 | 965.2 | ||
17.5 | 1008.5 | 1007.9 | 1006 | 1003.2 | 999.7 | 995.5 | 990.7 | 985.3 | 979.4 | 973.1 | 966.3 | ||
20 | 1009.7 | 1009 | 1007.2 | 1004.3 | 1000.8 | 996.5 | 991.7 | 986.4 | 980.5 | 974.2 | 967.4 | ||
25 | 1012.1 | 1011.3 | 1009.3 | 1006.5 | 1002.8 | 998.6 | 993.7 | 988.4 | 982.6 | 976.4 | 969.7 | ||
30 | 1014.5 | 1013.6 | 1011.4 | 1008.6 | 1004.9 | 1000.7 | 995.8 | 990.5 | 984.7 | 978.5 | 971.8 | ||
35 | 1016.9 | 1015.7 | 1013.6 | 1010.6 | 1007 | 1002.7 | 997.9 | 992.6 | 986.8 | 980.6 | 974 | ||
40 | 1019.3 | 1018 | 1015.7 | 1012.8 | 1009 | 1004.7 | 999.9 | 994.6 | 988.8 | 982.7 | 976.1 | ||
45 | 1021.6 | 1020.2 | 1017.8 | 1014.7 | 1011 | 1006.8 | 1001.9 | 996.6 | 990.9 | 984.7 | 978.2 | ||
50 | 1023.9 | 1022.3 | 1019.9 | 1016.8 | 1013 | 1008.7 | 1003.8 | 998.6 | 992.9 | 986.8 | 980.3 | ||
60 | 1028.3 | 1026.6 | 1024.1 | 1020.8 | 1017 | 1012.6 | 1007.8 | 1002.5 | 996.8 | 990.8 | 984.3 | ||
70 | 1032.7 | 1030.7 | 1028.1 | 1024.7 | 1020.8 | 1016.4 | 1011.5 | 1006.3 | 1000.7 | 994.6 | 988.3 | ||
80 | 1037 | 1034.9 | 1032 | 1028.5 | 1024.6 | 1020.2 | 1015.3 | 1010.1 | 1004.4 | 998.5 | 992.3 |
Water hammer (for linear closure):
$t$ | valve closing time | $\mathrm{s}$ |
$L$ | pipe length | $\mathrm{m}$ |
$a$ | speed pressure waves in the pipe | $\mathrm{m/s}$ |
$P$ | water hammer | $\mathrm{Pa}$ |
$ρ$ | destiny | $\mathrm{kg/m^3}$ |
$L$ | pipe length | $\mathrm{m}$ |
$v$ | pipeline speed | $\mathrm{m/s}$ |
$t$ | valve closing time | $\mathrm{s}$ |
Water hammer (for nonlinear closure):
$t$ | valve closed time water hammer calculation | $\mathrm{s}$ |
$t_r$ | closing time | $\mathrm{s}$ |
$c_{ef}$ | effective closing time factor | $\mathrm{-}$ |
$t$ | valve closing time | $\mathrm{s}$ |
$L$ | pipe length | $\mathrm{m}$ |
$a$ | speed pressure waves in the pipe | $\mathrm{m/s}$ |
$P$ | water hammer | $\mathrm{Pa}$ |
$ρ$ | destiny | $\mathrm{kg/m^3}$ |
$L$ | pipe length | $\mathrm{m}$ |
$v$ | pipeline speed | $\mathrm{m/s}$ |
$t$ | valve closing time | $\mathrm{s}$ |
Pipeline speed:
$v$ | pipeline speed | $\mathrm{m/s}$ |
$Q$ | flow | $\mathrm{m^3/s}$ |
$D$ | internal pipe diameter | $\mathrm{mm}$ |
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 $a$ . The line $a_1$ 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 $a$ we can build a tangent line $t$ , which on the horizontal line $Q$ 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.
The value of the effective closing time factor $c_{ef}$ from of the proportional flow characteristics of the knife valve Fig. 2
$p$ | $c_{ef}$ |
---|---|
$\mathrm{-}$ | $\mathrm{-}$ |
1 | 1 |
0,5 | 0,73 |
0,2 | 0,46 |
0,1 | 0,33 |
0,05 | 0,24 |
0,01 | 0,141 |
Pressure parameter:
$p$ | pressure parameter | $\mathrm{-}$ |
$∆h$ | theoretical pressure in the closure at full opening | $\mathrm{m}$ |
$h_0$ | rated net head | $\mathrm{m}$ |
Theoretical pressure in the closure at full opening:
$∆h$ | theoretical pressure in the closure at full opening | $\mathrm{m}$ |
$v_0$ | valve speed | $\mathrm{m/s}$ |
$g$ | gravitational acceleration | $\mathrm{m/s^2}$ |
$\xi$ | local loss factor for open valve | $\mathrm{-}$ |
valve speed:
$v_0$ | valve speed | $\mathrm{m/s}$ |
$Q$ | flow | $\mathrm{m^3/s}$ |
$D_0$ | valve diameter | $\mathrm{mm}$ |
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:
$L=12000\ \mathrm{m}$; steel pipe $D=600\ \mathrm{mm}$; thickness of the pipe wall $e=10\ \mathrm{mm}$; $h_0=33\ \mathrm{m}$; $Q=0,314\ \mathrm{m^3/s}$; density water $\rho=998,3\ \mathrm{kg/m^3}$; medium compressibility factor $\beta=477,1\cdot 10^{-12}$; closing time $200\ \mathrm{s}$
Water hammer (for linear closure):
Volume elastic modulus
Sound speed in liquid
Speed pressure waves in the pipe
Pipeline speed
Water hammer
Water hammer (for nonlinear closure):
Valve speed
Theoretical pressure in the closure at full opening
Pressure parameter
this value is determined from the table using interpolation.
Valve closed time water hammer calculation
Volume elastic modulus
Sound speed in liquid
Speed pressure waves in the pipe
Pipeline speed
Water hammer
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
- R. Mareš: Tabulky termodynamických vlastností vody a vodní páry.
- ČSN EN 13480-3: Simplified static analysis of rapid valve closure.