Water hammer


Speed pressure waves in the pipe:

a=c1+De*KE

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=Kρ

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

K - volume elastic modulus - [Pa]

ρ - destiny - [Kg/m3]


Volume elastic modulus:

K=1β

K - volume elastic modulus - [Pa]

β - medium compressibility factor - [Pa-1]


Young's modulus for pipe:

Pipe material[Pa]
Steel2*1011
Copper1,17*1011
Cast iron0,7*1011
Glass0,8*1011
Polyvinyl chloride (PVC)3*109
Rubber4,2*106
Reinforced concrete0,21*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]
10°15°20°30°40°50°60°70°80°90°100°
0,1-10520,9502,6492,4482,2477,1468,9457,7457,7463,8471478,1487,3-
10-20501,5484,2469,9459,7450,6444,4437,3433,2435,3447,5459,7477,1822,6
20-30489,3471461,8451,6442,4430,2422421423433,2444,4467,9783,9
30-40475457,7449,5441,4432,2421414,9409,8413,9419430,2454,6745,2
40-50463,8452,6438,3430,2423413,9411,8406,7401,6405,7415,9442,2695,2
50-60446,5438,3426,1419411,8399,6397,6397,6395,5398,6406,7424,1671,8
60-70437,3416,9412,8405,7401,6394,5389,4384,3390,4387,4394,5414,9639,1

Density:

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

Temperature [°C]Pressure [MPa]
0,10,250,511,522,533,544,5567891012,51517,520253035404550607080
999,8999,910001000,31000,61000,81001,11001,31001,61001,81002,11002,31002,81003,31003,81004,31004,810061007,31008,51009,71012,11014,51016,91019,31021,61023,91028,31032,71037
10°999,7999,8999,91000,11000,41000,61000,810011001,31001,61001,810021002,510031003,41003,91004,41005,51006,71007,910091011,31013,61015,710181020,21022,31026,61030,71034,9
20°998,2998,3998,4998,6998,8999,1999,3999,5999,810001000,21000,41000,91001,31001,81002,21002,71003,81004,910061007,21009,31011,41013,61015,71017,81019,91024,11028,11032
30°995,6995,7995,8996996,3996,5996,7996,9997,2997,4997,6997,8998,3998,7999,1999,610001001,11002,21003,21004,31006,51008,61010,61012,81014,71016,81020,81024,71028,5
40°992,2992,3992,4992,7992,9993993,3993,4993,7993,9994,1994,3994,8995,2995,6996,1996,5997,6998,6999,71000,81002,81004,9100710091011101310171020,81024,6
50°988,1988,1988,2988,4988,6988,8989,1989,2989,5989,7989,9990,2990,6991991,5991,9992,3993,3994,4995,5996,5998,61000,71002,71004,71006,81008,71012,61016,41020,2
60°983,2983,3983,4983,6983,9984,1984,3984,5984,6984,9985,1985,3985,8989,2986,6987,1987,5988,5989,6990,7991,7993,7995,8997,9999,91001,91003,81007,81011,51015,3
70°977,8977,8978978,2978,4978,6978,9979,1979,2979,5979,7979,9980,4980,8981,3981,6982,1983,2984,3985,3986,4988,4990,5992,6994,6996,6998,61002,51006,31010,1
80°971,8971,9972972,2972,4972,7972,9973,1973,3973,5973,8974974,5974,9975,3975,7976,2977,2978,4979,4980,5982,6984,7986,8988,8990,9992,9996,81000,71004,4
90°965,3965,3965,7965,7966966,2966,4966,6966,8967,1967,3967,6968968,4968,9969,4969,7970,9972973,1974,2976,4978,5980,6982,7984,7986,8990,8994,6998,5
100°-958,4958,8958,8959959,2959,5959,7960960,2960,4960,6961,1961,5962962,5962,9964965,2966,3967,4969,7971,8974976,1978,2980,3984,3988,3992,3

Water hammer (for linear closure):

For slow and linear closing of the valve.

t2La

t - valve closing time - [s]

L - pipe length - [m]

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

P=ρ*L*vt

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=tr*cef

t - valve closing time - [s]

tr - closing time - [s]

cef - Effective closing time factor - []

t2La

t - valve closing time - [s]

L - pipe length - [m]

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

P=ρ*L*vt

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=Qπ*D24

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

p10,50,20,10,050,01
cef10,730,460,330,240,141

Pressure parameter:

p=Δhh0

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:

Δh=v022g*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:

v0=Qπ*D024

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=1β=1477,1*10-12 =2,096*109Pa

- sound speed in liquid

c=Kρ=2,096*109998,3 =1448,989m/s

- speed pressure waves in the pipe

a=c1+De*KE =1448,9891+60010*2,096*1092*1011 =1135,3m/s
t2La2*120001135,2=21,14s 200sOK

- pipeline speed

v=Qπ*D24=0,314π*0,624=1,11m/s

- water hammer

P=ρ*L*vt=998,3*12000*1,11200= 66486,78[Pa]


Water hammer (for nonlinear closure):

- valve speed

v0=Qπ*D024=0,314π*0,324=4,44m/s

- theoretical pressure in the closure at full opening

Δh=v022g*ξ+1=4,4422*9,81*0,01+1 =1,015m

- pressure parameter

p=Δhh0=1,01533=0,03 cef=0,2

this value is determined from the table using interpolation.

- valve closed time water hammer calculation

t=tr*cef=200*0,2=40s

- volume elastic modulus

K=1β=1477,1*10-12 =2,096*109Pa

- sound speed in liquid

c=Kρ=2,096*109998,3 =1448,989m/s

- speed pressure waves in the pipe

a=c1+De*KE =1448,9891+60010*2,096*1092*1011 =1135,3m/s
t2La2*120001135,2=21,14s 40sOK

- pipeline speed

v=Qπ*D24=0,314π*0,624=1,11m/s

- water hammer

P=ρ*L*vt=998,3*12000*1,1140= 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


Download PDF:

Water hammer.pdf


Social media:

Twitter Scribd Pinterest