Hydropower Plants - UniTrentorighetti/lezioni HPP/HPP/HPP_Gabl_Part1...Hydropower Plants Inlet...
Transcript of Hydropower Plants - UniTrentorighetti/lezioni HPP/HPP/HPP_Gabl_Part1...Hydropower Plants Inlet...
Hydropower Plants
Inlet
Reservoir
Surge tank
Power
station
Penstock
- Water hammer- Surge tank
1
Gallery / headrace tunnel
Maurizio RIGHETTI, Roman GABL
Environmental Fluid Mechanics / Hydropower Plants – 2015/16
What is … ???
2
= Pressure wave
every (!!!) change in the operation
Surge Tan
k
= Free surface -> Reflexion of the WH -> Limitation of the influence
= volume potential <-> kinetic energy
3
Differences between water hammer and mass oscillation
m1
m2
manometer upper m1 long period (100-500 sec)lower m2 short period in the range of seconds
Pressure wave
Slow changes <- inertia
Why is WH a problem?
4
!!! Pressure > resistance !!!
crack(Chaudhry 2014)
5
Why is WH a problem?
!!! Pressure < resistance !!!
(Chaudhry 2014)
What to do?
6(Chaudhry 2014)
- surge tank
- air chamber
- pressure-regulating valves
- slower changes->
WH basic equations
7
Frictionless pipeWalls are rigid, constant cross section AInitial pressure ≈upstream head H0
Steady conditions v0
-> closing valve suddenly at t=0
before after
Travels as a wavefrontwith a wave velocity a
material parameter of the pipe
=
Change of the reference system ->observer travels with the wave
(Chaudhry 2014)
8
Change of the reference system ->observer travels with the wave
(Chaudhry 2014)
WH basic equations
inflow outflow velocity
Resulting force F on the CV
Equal to the change of momentum
Range of 10 m/s 1000m/sFor wave front moving downstream
Pressure head increases for a reduction in velocity
Wave Propagation
9(Chaudhry 2014)
Time t = e > 0
Time t = L/a
(Chaudhry 2014)
10
Wave Propagation
(Chaudhry 2014)
Time t = L/a + e
Time t = 2*L/a
11
Wave Propagation
(Chaudhry 2014)
Time t = 2*L/a + e
Time t = 3*L/a
12
Wave Propagation
(Chaudhry 2014)
Time t = 3*L/a + e
Time t = 4*L/a
(Chaudhry 2014)
13
Wave Propagation
with
(Giesecke and Heimerl 2014)
14(Chaudhry 2014)
Wave Propagation
Refection at the reservoir
Refection at the valve
No friction!
Wave Propagation … with friction
15
Refection at the reservoir
Refection at the valve
Lower !
(Chaudhry 2014)
Reservoir <-> Dead end
16
Incident pressure wave F Reflected pressure wave f
with r = f/F
r = -1
r = 1 !!! Increasing !!!
(Chaudhry 2014)
Change of the cross section / Junction
17
Incident pressure wave Reflected pressure wave
A1 > A2
v1 < v2
(Chaudhry 2014)
Wave speed
18
water at 10°C:
= (Fluid Elastic Modulus/ Fluid Density)^0.5
… elastic pipe
Pipe Elastic Modulus and diameter / thickness
(Giesecke and Heimerl 2014)
Wave speed
19
thick pipe
concrete and steal
with
with
only concrete only rock
(Giesecke and Heimerl 2014)
How big is the effect of the WH?
• Joukowski
• Allievi
• Method of characteristics (MoC)
• Software
20
Joukowski‘s theory
21
[m/s] * [m/s][m/s^2]
!!Max!!!
needed when closing time tS < TR
wave speed change of the velocity
gravitational acceleration ≈9.81 [m/s2]
22
Joukowski‘s theory including closing time
(Giesecke and Heimerl 2014)
Reality:
-> Allievi
How big is the effect of the WH?
• Joukowski
• Allievi
• Method of characteristics (MoC)
• Software
23
Allievi
24
Allievi‘s approach
25
Assumption
• No friction (IE = 0)
• Constant pipe (sin α = 0) with a fixed boundary condition (big reservoir)
• v << a
0D2
vvλsinαg
x
p
ρ
1
x
vv
t
v
= 0= 0= 0
= 0
0t
p
a²ρ
1
x
v
0x
hg
t
v
0t
h
a²
g
x
v
and
0x
p
ρ
1
t
v
0x
pv
t
p
²aρ
1
x
v
Continuity equation
Momentum equation
Very good explanation in Jaeger (1977)
x
a
26
Allievi‘s approach
Initial value
current valueDepending on time t and location x
(Chaudhry 2014)
a
xLtf
a
xLtfhtxh 210),(
a
xLtf
a
xLtf
a
gvtxv 210),(
Idea: two functions f1 and f2 are defined, which - are assumed as being complicated a priori … “and we don’t want it to know” - depend on the boundary conditions - independent of time t - f1+f2= pressure wave
27(Chaudhry 2014)
A
Reservoir = constant -> Δh have to be 0 -> f1=-f2A
Allievi‘s approach
a
xLtf
a
xLtf 21
… the primary wave f1 is fully reflected at A total reflection with reversal of the sign
iii
i
tfa
Ltf
a
L
a
Ltf
a
Ltf
aLtt
21112
/
for
… the function f2(ti) is equal to –f1(ti-2L/a) one period TR earlier
28
Allievi‘s approach
A
ALL r
2L t n T n n = 1, 2, 3, ... für x = L
a
B
B
Only at the main periods and at B
(Chaudhry 2014)
2 1 f (t ) f (t 2L / a)
n 0 1,n 1,n 1 h h f f
n 0 1,n 1,n 1
gv v f f
a
-> only f1
a
xLtf
a
xLtf
a
gvtxv 210),(
a
xLtf
a
xLtfhtxh 210),(
29
Allievi‘s approach
n 0 1,n 1,n 1 h h f f
n 0 1,n 1,n 1
gv v f f
a
n n 1 0 1,n 1,n 2 h h 2h f f
n n 1 1,n 1,n 2
gv v f f
a
Initial value
current valueat time= n*2L/a
n 1 0 1,n 1 1,n 2 h h f f
n 1 0 1,n 1 1,n 2
gv v f f
a
previous valueat time= n*2L/a
n n 1 0 n 1 n
a h h 2h v v
g
equal
n n 1 0 n 1 n h f (h ,h ,v ,v )
hn is now a function depending on only known values
30
0n n 1 0 n 1 n 1 n n
0
a v h h 2h h h
g h
Allievi‘s approach
rn
s
rn
s
T1 n
T
Tn
T
… closing
… opening
(Chaudhry 2014)
n n 1 0 n 1 n
a h h 2h v v
g
1st equation
A
A
nn 0 n
0
hv v
h
2nd equation neededfor examples:
B
B
31
Allievi‘s approach 0n n 1 0 n 1 n 1 n n
0
a v h h 2h h h
g h
0
0r
hg
va2
0
n2
nh
h
0
0n2
nh
hh1
with
… pressure ration
… pipeline characteristic
… ratio of excess to steady pressure
nn1n1nr
2
1n
2
n 22
Allievi‘s equation (classical form)
nn1n1nr
2
1n
2
n 22 … n-th reflection tn=n ·Tr
2211r
2
1
2
2 22 … 2nd reflection t2=2 Tr
1100r
2
0
2
1 22 … 1st reflection t1= Tr
Connecting Allievi with Joukowski
32
1h
h
0
02
0 r0
s
T1 0 1
T
1100r
2
0
2
1 22 … 1st reflection t1= Tr
Abrupt closure
11r
2
1 121
r
2
1 21
0
0
0
01
hg
va
h
hh
0v
g
ah
a/L2TT rS
11 1 0 by t1= 1
(Jaeger 1977)
… pipeline characteristic0
0r
hg
va2
0
0n2
nh
hh1
… ratio of excess to steady pressure
Remember:
Slow closing and opening
33
rS TT
t
TR
(t)
2TR
3TR
4TR
5T =TR s
1
1
0
0
32
4 t
TR
(t)
2TR
3TR
4TR
5T =TR s
1
1
0
0
3
2
45
1T/T rS
rS TT
t
TR
0vg
ah
0vg
ah
hn
2TR 3TR 4TR 5TR 6TR
h0
t
TR
0vg
ah
0vg
ah
hn
2TR 3TR 4TR 5TR 6TR
h0
rS TT0
Fast closing
Slow closing
34
TR 2TR 3TR 4TR 5TR
TS
12n
12
m
TR 2TR 3TR 4TR 5TR
TS
12n
12
m
1T/T rS rS TT
Asymptotic behaviour
S
rn
T
Tn1η
S
r1n
T
T1n1η
S
rn1n
T
Tηη
mnm1n
mnm1n hhhh
m1nn
m … asymptotes / final value
nn1n1nr
2
1n
2
n 22
n1nmr
2
m 222
S
rmr
2
mT
T222
1T
T
2T
T
2
2
S
rr
S
rrm
mnm1nr
2
m
2
m 22
Exercise to Allievi
35
L, d, s, Q0
Pipe:Dout= 1,4 mThickness 20 mmE(steel)=2,1*1011N/m2
E(water)=2,06*109N/m2
L= 1400 mH=120 mQ0= 2,6 m³/s Valve:ζ = 380420·x-2,25 + 1,15 (x … position of the valve; 100 = open, 0 = closed)
???a
Joukowskihn(t)
How big is the effect of the WH?
• Joukowski
• Allievi
• Method of characteristics (MoC)
• Software
36
Calculation MoC
37
Continuity equation
Momentum equation
describing transient flow in closed conduits= hyperbolic partial differential equations
? solve ?
Method of characteristics (MoC)
L1 + unknown multiplier * L2 =
withf….friction factor
Calculation MoC
38
total derivative
havewant
Eliminate the independent variable x ->Ordinary differential equation depending on t
if, this is satisfied then
Calculation MoC
39
partial differential equation ordinary differential equation
for
valid everywhere in the x-t plane valid only on a straight line, if a is constant
= characteristic lines
(Chaudhry 2014)
Example MoC
40
Region I:depending only on the initial conditions
back
furt
her
on
in t
ime
Region II:imposed by the downstream condition
excitations on both sides:
(Chaudhry 2014)
Calculation MoC
41
for
known
wanted
friction losses (depending on QP)
first order approximation:(Q is constant from A to P for this term)
satisfactory results /short computational interval needed
(Chaudhry 2014)
Calculation MoC
42
with
positive characteristic equation
negative characteristic equation
two equations = two unknowns: HP and QP
HP based on one characteristic equation
boundary conditionsfurther advanced methods
-> Chaudhry (2014)
(Chaudhry 2014)
How big is the effect of the WH?
• Joukowski
• Allievi
• Method of characteristics (MoC)
• Software
43
Software
44
Hydraulic System – EPFL
AFT ImpulseWater Hammer and Mass Oscillation(WHAMO) 3.0
… more information: Ghidaoui et al. (2005)
Example WANDA
45
WANDA
46
workflow
Most of the time!
WANDA
47
References
• Chaudhry, M. Hanif: Applied Hydraulic Transients. Springer, New York Heidelberg Dordrecht London, 2014. DOI: 10.1007/978-1-4614-8538-4
• Ghidaoui MS, Zhao M, McInnis DA, Axworthy DH. A Review of Water Hammer Theory and Practice. ASME. Appl. Mech. Rev. 2005;58(1):49-76. DOI:10.1115/1.1828050.
• Giesecke, J.; Heimerl S.: Wasserkraftanlagen – Planung, Bau, Betrieb. Springer, Berlin Heidelberg, 2014.DOI: 10.1007/978-3-642-53871-1
• Jaeger, Charles: Fluid Transients in Hydro-Electric Engineering Practice. Blackie, Glasgow London, 1977.
• WANDA User manual 4.3, Deltares, Delft Hydraulics, 2014.
48(Ghidaoui et al 2005)