## Water hammer

Speed pressure waves in the pipe:

$a$ $=\cfrac{c}{\sqrt{1+\cfrac{D_p}{e}\cdot\cfrac{K}{E}}}$

where:
 $a$ 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:

$c$ $=\sqrt{\cfrac{K}{ρ}}$

where:
 $c$ sound Speed in liquid $\mathrm{m/s}$ $K$ volume elastic modulus $\mathrm{Pa}$ $ρ$ density $\mathrm{kg/m^3}$

Volume elastic modulus:

$K$ $=\cfrac{1}{β}$

where:
 $K$ 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):

For slow and linear closing of the valve.
$t\geq\cfrac{2L}{a}$
where:
 $t$ valve closing time $\mathrm{s}$ $L$ pipe length $\mathrm{m}$ $a$ speed pressure waves in the pipe $\mathrm{m/s}$
$P=\cfrac{\rho\cdot L\cdot v}{t}$
where:
 $P$ 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):

For slow closing of the valve.
$t=t_r\cdot c_{ef}$
where:
 $t$ valve closed time water hammer calculation $\mathrm{s}$ $t_r$ closing time $\mathrm{s}$ $c_{ef}$ effective closing time factor $\mathrm{-}$
$t\geq\cfrac{2L}{a}$
where:
 $t$ valve closing time $\mathrm{s}$ $L$ pipe length $\mathrm{m}$ $a$ speed pressure waves in the pipe $\mathrm{m/s}$
$P=\cfrac{\rho\cdot L\cdot v}{t}$
where:
 $P$ 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:

$v=\cfrac{4Q}{\pi\cdot D^2}$
where:
 $v$ pipeline speed $\mathrm{m/s}$ $Q$ flow $\mathrm{m^3/s}$ $D$ internal pipe diameter $\mathrm{mm}$

Effective closing time factor:

-●- $a_1$ -●- $a$ -●- $t$
$s\mathrm{[\%]}$
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 $a$ . 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.

-●- $p=1$ -●- $p=0,5$ -●- $p=0,2$ -●- $p=0,1$ -●- $p=0,05$ -●- $p=0,01$
$s\mathrm{[\%]}$
Fig. 2 - Proportional flow characteristic of the knife valve - $\xi=0,01$

The value of the effective closing time factor $c_{ef}$ from of the proportional flow characteristics of the knife valve Fig. 2

$p$ $c_{ef}$
$\mathrm{-}$ $\mathrm{-}$
1 1
0,5 0,73
0,2 0,46
0,1 0,33
0,05 0,24
0,01 0,141

Pressure parameter:

$p=\cfrac{∆h}{h_0}$
where:
 $p$ 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:

$∆h=\cfrac{v_0^2}{2g}\cdot\left(\xi+1\right)$
where:
 $∆h$ 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:

$v_0=\cfrac{4Q}{\pi\cdot D_0^2}$
where:
 $v_0$ 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

$K=\cfrac{1}{\beta}=\cfrac{1}{477,1\cdot10^{-12}}=2,096\cdot10^9\ \mathrm{Pa}$

Sound speed in liquid

$c=\sqrt{\cfrac{K}{\rho}}=\sqrt{\cfrac{2,096\cdot10^9}{998,3}}=1448,989\ \mathrm{m/s}$

Speed pressure waves in the pipe

$a=\cfrac{c}{\sqrt{1+\cfrac{D}{e}\cdot\cfrac{K}{E}}}=\cfrac{1448,989}{\sqrt{1+\cfrac{600}{10}\cdot\cfrac{2,096\cdot10^9}{2\cdot10^{11}}}}=1135,3\ \mathrm{m/s}$

$t\geq\cfrac{2L}{a}\rightarrow\cfrac{2\cdot12000}{1135,2}=21,14\ \mathrm{s}$

$\rightarrow$ does suit

Pipeline speed

$v=\cfrac{4Q}{\pi\cdot D^2}=\cfrac{4\cdot0,314}{\pi\cdot0,6^2}=1,11\ \mathrm{m/s}$

Water hammer

$P=\cfrac{\rho\cdot L\cdot v}{t}=\cfrac{998,3\cdot12000\cdot1,11}{200}=66486,78\ \mathrm{Pa}$

### Water hammer (for nonlinear closure):

Valve speed

$v_0=\cfrac{4Q}{\pi\cdot D_0^2}=\cfrac{4\cdot0,314}{\pi\cdot0,3^2}=4,44\ \mathrm{m/s}$

Theoretical pressure in the closure at full opening

$∆h=\cfrac{v_0^2}{2g}\cdot\left(\xi+1\right)=\cfrac{4,44^2}{2\cdot9,81}\cdot\left(0,01+1\right)=1,015\ \mathrm{m}$

Pressure parameter

$p=\cfrac{∆h}{h_0}=\cfrac{1,015}{33}=0,03\ \rightarrow c_{ef}=0,2$

this value is determined from the table using interpolation.
Valve closed time water hammer calculation

$t=t_r\cdot c_{ef}=200\cdot0,2=40\ \mathrm{s}$

Volume elastic modulus

$K=\cfrac{1}{\beta}=\cfrac{1}{477,1\cdot10^{-12}}=2,096\cdot10^9\ \mathrm{Pa}$

Sound speed in liquid

$c=\sqrt{\cfrac{K}{\rho}}=\sqrt{\cfrac{2,096\cdot10^9}{998,3}}=1448,989\ \mathrm{m/s}$

Speed pressure waves in the pipe

$a=\cfrac{c}{\sqrt{1+\cfrac{D}{e}\cdot\cfrac{K}{E}}}=\cfrac{1448,989}{\sqrt{1+\cfrac{600}{10}\cdot\cfrac{2,096\cdot10^9}{2\cdot10^{11}}}}=1135,3\ \mathrm{m/s}$
$t\geq\cfrac{2L}{a}\rightarrow\cfrac{2\cdot12000}{1135,2}=21,14\ s\le40\ \mathrm{s}$

$\rightarrow$ does suit

Pipeline speed

$v=\cfrac{4Q}{\pi\cdot D^2}=\cfrac{4\cdot0,314}{\pi\cdot0,6^2}=1,11\ \mathrm{m/s}$

Water hammer

$P=\cfrac{\rho\cdot L\cdot v}{t}=\cfrac{998,3\cdot12000\cdot1,11}{40}=332433,9\ \mathrm{Pa}$

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
- R. Mareš: Tabulky termodynamických vlastností vody a vodní páry.
- ČSN EN 13480-3: Simplified static analysis of rapid valve closure.