numerical hydraulics w. kinzelbach with marc wolf and cornel beffa lecture 3: computation of...
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Numerical Hydraulics
W. Kinzelbach withMarc Wolf andCornel Beffa
Lecture 3: Computation of pressure surges
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The phenomenon
Till time t=0: Steady state flow QAt time t = 0: Instantaneous closing of valveObservation: Sudden pressure rise at valve
→ pressure surge (water hammer)
ValveReservoir
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And some typical damages
Sayano-Shushenskaya plant in southern Siberia
Water pipe damage due to pressure surge
San Bruno: PG&E Power Outage and Pressure Surge Preceded Blast
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Where pressure surges occur…
• water distribution systems• waste water transfers• storm water rising mains• power station cooling systems• oil pipelines• RAS (activated sludge) onsite pipelines• hydropower stations• and any fluid system in which the inertia (mass
and velocity) of the fluid is significant.
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The phenomenon
Pressure wave propagates with wave velocity c
If valve closing time is smaller than the run time of the wave to the reflection point and back the surge is called
Joukowski surge
Pressure vs. time at valve
50 100 150 200 250 300-1.5
-1
-0.5
0
0.5
1
1.5x 10
7 Druck am letzten Knoten gegen Zeit
Damping of amplitude throughfriction
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Negative pressure wave
The negative pressure wave cannot become lower than thevapour pressure of the fluid.If the pressure falls below the vapour pressure, a vapour bubble is formed. The water columnseparates from the valve.When the pressure increasesagain the bubble collapses. This phenomenon is calledcavitation.
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Pressure surge with vapour bubble formation
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Computed and observed pressure surge at twoplaces along a pipe
Today the reliable computation of pressure surges is possible
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Measures against pressure surges
Surge shaft
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Measures against pressure surges
Surge vessels (Windkessel)
Special valves
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The equations of unsteady pipe flow
• Continuity
• Momentum equation (per unit volume) ( inclination angle of
pipe)
( ) ( )0 0
( ) ( )
A Av v A Av v
t x t x x A t x
with p and A A p
sin( ) 0R
v v pv g g I
t x x
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Further transformations (1)
• Continuity: As density and cross-sectional area depend on x and t only via the pressure p, the chain rule can be applied.
0
1 10
d p p dA p p vv v
dp t x A dp t x x
d dA p p vv
dp A dp t x x
Using the moduli of elasticity of water EW und of the pipe Epipe
1 1 1 / 1
'W pipe
d dA D e
dp A dp E E E
e is the pipe wall thickness, E‘ is the combined modulus of elasticity of the system
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Further transformations (2)
• Compressibility: Definition
• Using pressure tank formula
/ / 1W
W
dV V d
dp dp E
1 1 /
pipe
dA dA dD D e
A dp A dD dp E
2
2 /
pipe
pipe
p xD xe
D e p
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Further transformations(3)
• Momentum equation
21( )
2
2
R
R
h vI Hydraulics I
x D g
v vTaking into account the flow direction I
Dg
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The equations of unsteady pipe flow
• Continuity
• Momentum equation
10
'
p p vv
E t x x
1sin( ) 0
2
v vv v pv g
t x x D
2 PDE with 2 unknown functions p(x,t) und v(x,t)plus initial and boundary conditions
(1)
(2)
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Boundary conditions
• Pressure boundary condition: p given– e.g. water level in reservoir, controlled pump
• Velocity/Flux boundary condition: v given– e.g. flow controlled (v from Q/A)
• Combination: Relation between pressure and flux given– Z. B. function of pressure reduction valve, characteristic
curve of pump
• Closing of a valve at the end of a pipe– Initially flow Q, then according to closing function reduction
to zero withing closing time of the valve.
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Linearised equations
• Delete all terms in (1) and (2) which are non-linear (for convenience: = 0):
10
'
p v
E t x
1
0v p
t x
General solution by elimination: - Take partial derivative of first equation
with respect to t- Take partial derivative of second
equation with respect to xSubtraction yields:
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Linearised equations
• Wave equation (for p, analogously for v)
which has general solution
• Wave with wave velocity
• Example: Modulus of elasticity of steel = 200‘000 MN/m2, Modulus of elasticity of water = 2‘000 MN/m2, wall thickness e = 0.02 m, D = 1 m, = 1000 kg/m3 yields c = 1333 m/s
2 2
2 2
'0
p E p
t x
'Ec
( , ) ( )f x t f x ct
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Joukowski surge
• Estimate of surge pressure after instantaneous closing of valve (neglecting friction, linearized equations): „Worst case“
• General solution:
0 ( ) ( )p p F x ct f x ct
0
1( ) ( )v v F x ct f x ct
c
Proof by insertion into linearised equations!!
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Joukowski surge
• After t = 0 only the backward running wave F(x+ct) is found in the upstream
• v at the valve is 0• Maximum p is given by:
• Solution:
0 ( )p p p F x ct
0
1( )v F x ct
c
0p cv
Example continued: c=1333 m/s, Q0=1 m3/s, L=100 m yields: p=1.7E6 N/m2
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Numerical solution of the complete equations
• Difference method– Discretisation of space and time– x and t
• Difference equations for time step t, t+t• Problem: Discretisation „softens“ pressure
front numerically• Way out: Method of characteristics
– Follows the pressure signal in moving coordinate system
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• Normal difference method– Softening of pressure front
• Method of characteristics– Grid is adapted to frontal velocity (feasible, as
v<<c, c+v ≈ c-v ≈ c)
Method of characteristics
Front ofpressure wave
Front ofpressure wave ct = x
ct < x
x
x
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Method of characteristics
• Replacing equations (1) and (2) by 2 linear combinations
yields:
1( ) ( ) sin( ) 0
2
v vv v p pv c v c g
t x c t x D
1( ) ( ) sin( ) 0
2
v vv v p pv c v c g
t x c t x D
(1) (2) (1) (2)c and c
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Method of characteristics
• With total derivative along x(t)
the equations have the form:
1sin( ) 0
2
v vDv Dp dxg with v c c
Dt c Dt D dt
1sin( ) 0
2
v vDv Dp dxg with v c c
Dt c Dt D dt
Forward characteristic
Backward characteristic
D dx
Dt t dt x
(c is actually relative wave velocity with respect to average water movement.)
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Difference scheme
Chose time step such that x = c t
In every time step there are two unknowns at each of the N+1 nodes:
Divide pipe of length L in N sections, length of one section x = L/N
Node 1
section 1
Node N+1
section Nx
1 1,j ji iv p
To determine these unknowns 2N+2 equations are required. From quantities at time j quantities at time j+1 are computed.The new times j+1 become the old times j of the next time step.
Upper index time step, lower index node
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Difference scheme
x
t
x = c t
ii-1 i+1
j
j +13
1
2
1 2
3
1j j
i i
j
p pp
x x
1j ji i
i
p pp
t t
1 1
1 1
, ,
.j j j j
i i i i
i forward i backward
p p p pDp Dpviz
Dt t Dt t
Total derivative or derivative along characteristic line
space
time
Using c + v ≈ c - v ≈ cnode i communicates within time intervalt with node i-1 via the forward characteristic and with node i+1via thebackward characteristic
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Difference form of equations
• Equations for nodes 2 to N: 2N-2 equations
1 11 11 11
sin( ) 02
j jj j j ji ii i i i
v vv v p pg
t c t D
forward characteristic
backward characteristic
1 11 11 11
sin( ) 02
j jj j j ji ii i i i
v vv v p pg
t c t D
The pressure loss term is linearised by evaluating it at the old time jEquations can be solved for 1 1,j j
i iv p
The further equations are determined by the boundary conditions and the one characteristic which can be used at the respective boundary
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Method of characteristics• Example: Reservoir with pipe which is closed instantaneously at t=0• 2 further equations from boundary conditions
In the example:
• 2 further equations from characteristic equationsIn the example:
111
sin( ) 02
j jj j jN NN N N
v vv p pg
t c t D
From forward characteristic for i=N+1
From backward characteristic for i=1
12 21 2 2 11
sin( ) 02
j jj j jB
v vv v p pg
t c t D
1 11 10 ,j j
N Bv after closing of valve p reservoir pressure p
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Simplified case for basic Matlab-Program
=0, friction neglected, equations nodes 2 to N
1 11 11 11
sin( ) 02
j jj j j ji ii i i i
v vv v p pg
t c t D
1 11 11 11
sin( ) 02
j jj j j ji ii i i i
v vv v p pg
t c t D
forward characteristic
backward characteristic
= 0
= 0
Solution by subtracting resp. adding the two equations
)(5.0)(5.0
)(5.0
)(5.0
11111
11111
ji
ji
ji
ji
ji
ji
ji
ji
ji
ji
ppvvcp
ppc
vvv
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Simplified case for basic Matlab-Program
2 equations from boundary conditions1 11 10 ,j j
N Bv after closing of valve p reservoir pressure p
2 equations from characteristics for i = 1 and i = N+1
111
sin( ) 02
j jj j jN NN N N
v vv p pg
t c t D
From forward characteristic for i=N+1
From backward characteristic for i=1
12 21 2 2 11
sin( ) 02
j jj j jB
v vv v p pg
t c t D
= 0
= 0
jN
jN
jN cvpp
11
)(1
221
1j
Bjj pp
cvv
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Additions
• Formation of vapour bubble
• Branching pipes
• Closing functions
• Pumps and pressure reduction valves
• ….
• Consistent initial conditions through steady state computation of flow/pressure distribution
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Example (1)
Tank 1Tank 2
connecting pipe
valve
L=500 m
D = 0.2 m, e = 0.01 m, roughness k = 0.0005 m = 1000 kg/m3, Ew = 2000 MN/m2, Epiper = 210000 MN/m2
pressure downstream reservoir 80 mWS, pressure upstream reservoir 90 mWS closing time of valve1 s, Q before closing: 0.2 m3/sloss coefficient valve 2, time of calculation 60 s,number of pipe sections n = 10
Use Program„Hydraulic System“
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Example (2)
Tank 1Tank 2
ValveL=500 m
D = 0.2 m, e = 0.01 m, roughness k = 0.0005 m = 1000 kg/m3, Ew = 2000 MN/m2, Epipe = 210000 MN/m2
pressure of both downstream reservoirs 80 mWS, pressure upstream reservoir 90 mWS closing time 1 s, Q before closing of valve: 0.2 m3/sloss coefficient of valve 2, computation time 60 s,number of pipe sections n = 10
Tank 3
L=500 m