Tyler, page 228
What is the maximum pressure developed in a water pipeline with a pressure "p", if a valve is closed nearly instantly or line are all stope at the same instant. Pipe data is "steel", "d", "sch", "L". The water flow rate is "Q". What is the maximu the valve closes in a time "∆τ" 1. Data, SI (Tyler data in Anex A) Operating pressure and flowrate p= 13.8 bar Q= Pipe data Material: dn =
0.1767
8
#NAME?
3.-Water speed of sound
c: speed of sound (m/s)
m /s
Bulk modulus and density of water K: water bulk modulus (Pa) ρ: water density (kg/m³) k= 20,684 bar ρ= (K / ρ )^0.5 1000 kg/m³ c
in
Steel pipe elasticity module Et = 2,068,428 bar
Steel
sch = 40 L= 1524.0 Valve closing time ∆τ = 5.0 Pipe dimensions di = #NAME? s=
3
Pipe section A= (pi()/4)*d^2 d= #NAME? m A= #NAME? m2 Material data
m s mm mm
2. Fluid velocity v= Q/A Q= 0.1767 A= v=
K=
2.1E+09
ρ= 1000 c= 1438.2 4.- Celerity
m3/s
#NAME? m2 #NAME? m/s
5.- Pressure increment due to water 6.- Maximum pressure developed due 8.- Pressure to increment due to water hammer produced by a sudden shutoff.a sudden shutoff hammer produced by a Not sudden shutoff. The pressure increment can be calculatedpmax = pop = with Joukovsky elasticity theory h= pmax = pmax =
pop + h 13.8 bar #NAME? bar
For a shutoff time greater than the critical time, the Michaud relation can be used.
#NAME? bar
h : pressure increment [mwc] a : wave velocity [m/s] ∆v : speed variation [m/s] vfinal - vinitial ∆v =
7. Critical time
g : acceleration of gravity m/s² The pressure change "h" is
Maximum over- pressure or under- ∆v :speed change pressure are obtained when the g : acceleration of gravity m/s²
h=
(- a * ∆v ) / g
a= ∆v = vf = vi =
#NAME? vf - vi 0 #NAME?
m/s m/s m/s m/s
∆v = g= h=
#NAME? #REF! #NAME?
m/s m/s² mwc
h=
#NAME?
bar
#NAME? psi ∆P: presure increment (mwc) L : pipe length (m)
shutoff time "∆τ", is less or equal to∆τ: shutoff time interval (s) the critical time "τc",
τc = L= a= τc =
2*L/a 1,524 m #NAME? m/s #NAME? s
∆P = h= L=
2 * h * L / (a * ∆τ) #NAME? 1524.0
a= ∆τ = ∆P =
#NAME? 5.0 #NAME?
Anex A Anex A. Tyler data p= 200 dn = 8
psi in
sch = L= Q= τ=
40 5,000 2,800 5
k=
300,000
psi
E=
30,000,000
psi
ft gpm s
Data, SI p= dn =
Tyler results 13.8
8 sch = 40 L= 1,524 Q= 0.177 τ= 5 Bulk modulus of water k= 20,684 k= 2,068 Pipe elasticity module E= 2,068,428 E=
bar in m m3/s s bar Mpa bar
206,843 Mpa
5.- Celerity a= 1287.9 Calculated value a= #NAME? 6.- Pressure increment due to water hammer produced by a sudden shutoff. h= 70.5 Calculated value h= #NAME? 8. Pressure developed due to valve shutoff in the time interval pmax =
g=
9.81
m/s²
di =
#NAME? mm
s=
#NAME? mm
44.7
Calculated value pmax = #NAME?
The bulk modulus K>0 can be formally defin
a valve is closed nearly instantly or pumps discharging into the low rate is "Q". What is the maximum pressure developed if
peed of sound
f sound (m/s)
ulk modulus (Pa) ensity (kg/m³) (K / ρ )^0.5
a : celerity (wave velocity) (m/s) c: speed of sound (m/s) d: inside pipe diameter (mm) s: minimum wall thickness (mm) K: water bulk modulus (bar) c / (1 + (K/Et) * (d/s) )^(0.5) a=
Pa kg/m³ m/s
m/s Pa
2.1E+11
Pa
K=
2.20E+09
d= s= a=
#NAME? #NAME? #NAME?
mm mm m/s
ρ= c=
1000 1483.2
a/c=
#NAME?
∆τ =
a= k=
#NAME?
bar
#NAME?
s
5.0
s
#NAME?
5
pmax =
pop + h
pop =
13.8 #NAME?
bar bar
#NAME?
bar
#NAME?
psi
s
e increment (mwc)
2 * h * L / (a * ∆τ) bar m m/s s bar
h= pmax = pmax =
1000 2.06E+11 543
a=
#NAME?
bar
∆τ =
c / ( 1 + (k/Et) * (d/s))^0.5 2.20E+09
ρw = Et = d=
h *( τc / ∆τ )
9. Pressure developed due to valve shutoff in the time interval ∆τ > τc
time interval (s)
Seawater 2.34 t= 10 (K / ρ )^(1/2) c=
1438.2 2.1E+09
e, the Michaud relation can∆P =
ation of gravity m/s²
Water 2.2×109 Pa (value increases at highe Water 2.15
c= K= Et =
re increment due to water ∆P = produced by a Not sudden h = τc =
off time greater than the
In a fluid, the bulk modulus speed of sound c (pressure waves), accord Newton-Laplace formula
1,483.24
m/s
m/s re increment due to water roduced by a sudden shutoff. bar
bar e developed due to valve he time interval ∆τ > τc bar bar
Not e1
Bulk modulus of water k= 22000
bar
Pipe elasticity module E= 2068428 bar Note 1. Tyler error 484 + 200 = 684 (psi) = 47.16 (bar)
odulus K>0 can be formally defined by the equation:
he bulk modulus K and the density ρ determine the ound c (pressure waves), according to the place formula
109 Pa (value increases at higher pressures
ºC (K / ρ )^(1/2) Pa kg/m³ m/s c / ( 1 + (k/Et) * (d/s))^0.5 Pa kg/m3 Pa mm m/s
###
1000 2.06E+11 543
Pehmco water-hammer [3], page 7.21 1. Data Operating pressure and flowrate pop = 15 mwc pop = Q= Q= Pipe data Material: dn =
#REF! 100 0.100
bar l/s 3
m /s
PECC 63, 80, 100 280
mm
PN 10 bar L= 1000 m Valve closing time ∆τ = < Tc 2. Pipe dimensions and section de = 280 mm s= di =
Pipe section A= d=
5.-Water speed of sound (pi()/4)*d^2 0.2292
A= 0.0413 3. Material data
m m2 c: speed of sound (m/s)
Bulk modulus and density of water k= #REF! bar ρ= 1000 kg/m³ Ppipe elasticity module Ep = #REF!
bar
4. Fluid velocity v= Q/A Q= 0.1000
m3/s
K=
A=
0.0413
m2
v=
2.42
m/s
25.4 229.2
mm
7.- Pressure increment due to water hammer produced by a sudden shutoff. For this case, the pressure increment can be calculated with Joukovsky elasticity theory
8.- Maximum pressure developed due to a sudden shutoff (Joukovsky) Pmax_Jouk =
pop + hJouk
pop =
#REF!
bar
hJouk =
#REF!
bar
Pmax_Jouk =
#REF!
bar
hJouk : pressure increment, Joukovsky a : wave velocity [m/s] ∆v : speed variation [m/s] vfinal - vinitial ∆v =
#REF! 9. Critical time
g : acceleration of gravity m/s² The pressure change "h" is hJouk = (- a * ∆v ) / g m
Maximum over- pressure or underpressures are obtained when the shutoff time "∆τ", is less or equal to the critical time "τc",
a= ∆v = vf = vi =
#REF! vf - vi 0 2.42
m/s m/s m/s m/s
∆v = g= hJouk =
-2.42 #REF!
m/s m/s²
τc =
#REF!
mwc
hJouk =
#REF!
bar
a= τc =
L=
2*L/a 1,000
K: water bulk modulus (Pa) ρ: water density (kg/m³) c
m
#REF!
m/s
#REF!
s
ρ= c= 6.- Celerity
Approximate bulk modulus Water 2.2×109 Pa (value increases at higher pressures) K= 2.20E+09 Pa K= 22,000 bar Air 1.42×105 Pa (adiabatic bulk modulus) Air 1.01×105 Pa (constant temperature bulk modulus)
g=
9.81
m/s²
Módulo de compresión del agua Agua K=
2.06E+04
kp/cm²
K=
#REF!
Pa
K=
#REF!
bar
Water speed of sound
a : celerity (wave velocity) (m/s) c: speed of sound (m/s) d: inside pipe diameter (mm)
peed of sound (m/s)
water bulk modulus (Pa) water density (kg/m³) (K / ρ )^0.5 #REF!
Pa
1000 #REF!
kg/m³ m/s
s: minimum wall thickness (mm) K: water bulk modulus (bar) c / (1 + (K/Ep) * (d/s) )^(0.5) a= c= K= Ep =
#REF! #REF!
m/s Pa
#REF!
Pa
d= s= a=
229.2 25.4 #REF!
mm mm m/s
Steel elasticity modulus E= 2.95E+07 E= 2.03E+11 E= 2.03E+06
psi Pa bar
Módulo de elasticidad del HDPE PEEC Ep = 8,000 kp/cm² Ep =
#REF!
Pa
Ep =
#REF!
bar
[2]
Power generation calculations reference Tyler G. Hicks., P.E., Editor The McGraw-Hill Engineering reference guide series 1985 Water-hammer in liquid pipelines. Page 228
[3]
Productos PECC Tehmco S.A. Example page 7.21
[4]
Heat ans mass transfer Anthony F. Mills Irwin, 1995
[5]
Heat transfer J. P. Holman McGraw-Hill, 1989
[6]
Water Hammer by Robert Pelikan April 1, 2005
