stability of direct spring operated pressure relief valves from...
TRANSCRIPT
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Stability of Direct Spring Operated Pressure ReliefValves – from CFD to spreadsheet
Csaba Hős (BME, Dept. of Hydrodynamic Systems)
12th September 2017
HDR
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
... with major contributions byProf. Alan Champneys (University of Bristol, Dept. of Eng.Mathematics)Dr. Csaba Bazsó (BME HDS)István Erdei (BME HDS)Paul Kenneth, Mike McNelly (Pentair, Houston, TX)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Table of Contents
1 Introduction, motivation
2 ModellingCFD1D model
3 Qualitative stability analysis
4 Bifurcations of impacting periodic orbits
5 Summary
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
DSOPRV
Direct Spring Operated Pressure ReliefValves: safety device to limit systempressure - last line of defence.
Valve opens @ set pressure pset
Valve closes @ reseat pressure prs 6= pset
Capacity: mass flow rate @ p = 1.1pset
and x = xmax (full lift)
Challenges
Valve chatter
API code: 3% rule based on upstreampipe pressure loss – is that sufficient?
Lack of measurements (in the opendomain).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Example of stable opening
0 20 40 60 80 1000
20
40
60
80
lift,
%
data file: 2J3 ON LIQUID tank Test 47.csv
0 20 40 60 80 10080
100
120
140
160
PH
, psi
g
0 20 40 60 80 10080
100
120
140
160
TK
, psi
g
0 50 100 150 200 250 300
10−5
100
ampl
. lift
f valve: 46.8 Hz (black)
0 50 100 150 200 250 300
10−5
100
ampl
. PH
f pipe: 233.4464 Hz (red)
0 50 100 150 200 250 300
10−5
100
ampl
. TK
f QW: 58.3616 Hz (blue)
frequency, Hz
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Example of unstable opening
20 25 30 350
100
200
300
lift,
%
data file: 2J3 ON LIQUID tank Test 22.csv
20 25 30 350
200
400
PH
, psi
g
20 25 30 350
200
400
TK
, psi
g
0 50 100 150 200 250 300
100
ampl
. lift
f valve: 47 Hz (black)
0 50 100 150 200 250 300
100
ampl
. PH
f pipe: 243 Hz (red)
0 50 100 150 200 250 300
100
ampl
. TK
f QW: 61 Hz (blue)
frequency, Hz
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Measurement
video 1
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
Table of Contents
1 Introduction, motivation
2 ModellingCFD1D model
3 Qualitative stability analysis
4 Bifurcations of impacting periodic orbits
5 Summary
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
Computational Fluid Dynamics
ANSYS CFX + Icem
Deforming mesh + automatic remeshing
Upstream pipe + simple valve model
Axisymmetric
Valve disc as rigid body
High-resolution, lots of information butslow and no qualitative understanding
Stable and unstable behaviourreproduced.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
CFD
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
CFD
video 2video 3
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
1D model for liquid service
valve: 1DoF oscillatormxv + kxv + s(x0 + xv ) = Flift,Flift = Aeff(xv )(pv − p0)
reservoir pressure dynamics:mr ,in − mr ,out =
Va2 pr
1D unsteady pipeline dynamics:∂p∂t +ρa2 ∂v
∂ξ+v ∂p∂ξ=0
∂v∂t +v ∂v
∂ξ=−1ρ∂p∂ξ+
λ2Dpipe
v |v |
BC @ res. side:pt = p(0, t) + ρ
2 (v(0, t))2
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
BC @ valve-end:v(L, t)Apipeρ = Cd(xv )Aft(xv )
√2ρ (p(L, t)− p0)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
1D model for liquid service
valve: 1DoF oscillatormxv + kxv + s(x0 + xv ) = Flift,Flift = Aeff(xv )(pv − p0)
reservoir pressure dynamics:mr ,in − mr ,out =
Va2 pr
1D unsteady pipeline dynamics:∂p∂t +ρa2 ∂v
∂ξ+v ∂p∂ξ=0
∂v∂t +v ∂v
∂ξ=−1ρ∂p∂ξ+
λ2Dpipe
v |v |
BC @ res. side:pt = p(0, t) + ρ
2 (v(0, t))2
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
BC @ valve-end:v(L, t)Apipeρ = Cd(xv )Aft(xv )
√2ρ (p(L, t)− p0)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
1D model for liquid service
valve: 1DoF oscillatormxv + kxv + s(x0 + xv ) = Flift,Flift = Aeff(xv )(pv − p0)
reservoir pressure dynamics:mr ,in − mr ,out =
Va2 pr
1D unsteady pipeline dynamics:∂p∂t +ρa2 ∂v
∂ξ+v ∂p∂ξ=0
∂v∂t +v ∂v
∂ξ=−1ρ∂p∂ξ+
λ2Dpipe
v |v |
BC @ res. side:pt = p(0, t) + ρ
2 (v(0, t))2
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
BC @ valve-end:v(L, t)Apipeρ = Cd(xv )Aft(xv )
√2ρ (p(L, t)− p0)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
1D model for liquid service
valve: 1DoF oscillatormxv + kxv + s(x0 + xv ) = Flift,Flift = Aeff(xv )(pv − p0)
reservoir pressure dynamics:mr ,in − mr ,out =
Va2 pr
1D unsteady pipeline dynamics:∂p∂t +ρa2 ∂v
∂ξ+v ∂p∂ξ=0
∂v∂t +v ∂v
∂ξ=−1ρ∂p∂ξ+
λ2Dpipe
v |v |
BC @ res. side:pt = p(0, t) + ρ
2 (v(0, t))2
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
BC @ valve-end:v(L, t)Apipeρ = Cd(xv )Aft(xv )
√2ρ (p(L, t)− p0)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
1D model for liquid service
valve: 1DoF oscillatormxv + kxv + s(x0 + xv ) = Flift,Flift = Aeff(xv )(pv − p0)
reservoir pressure dynamics:mr ,in − mr ,out =
Va2 pr
1D unsteady pipeline dynamics:∂p∂t +ρa2 ∂v
∂ξ+v ∂p∂ξ=0
∂v∂t +v ∂v
∂ξ=−1ρ∂p∂ξ+
λ2Dpipe
v |v |
BC @ res. side:pt = p(0, t) + ρ
2 (v(0, t))2
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
BC @ valve-end:v(L, t)Apipeρ = Cd(xv )Aft(xv )
√2ρ (p(L, t)− p0)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
1D model for liquid service
valve: 1DoF oscillatormxv + kxv + s(x0 + xv ) = Flift,Flift = Aeff(xv )(pv − p0)
reservoir pressure dynamics:mr ,in − mr ,out =
Va2 pr
1D unsteady pipeline dynamics:∂p∂t +ρa2 ∂v
∂ξ+v ∂p∂ξ=0
∂v∂t +v ∂v
∂ξ=−1ρ∂p∂ξ+
λ2Dpipe
v |v |
BC @ res. side:pt = p(0, t) + ρ
2 (v(0, t))2
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
BC @ valve-end:v(L, t)Apipeρ = Cd(xv )Aft(xv )
√2ρ (p(L, t)− p0)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
Simulation results
pipe length: 0.5mpipe length: 1.1mpipe length: 1.5m
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD1D model
Simulation results – CFD vs. 1D
0 0.5 1 1.5 2 2.5 3 3.53
3.2
t [s]
pt,res[bar]
0 1 2 30
50
100
xv[%
]
0 1 2 30
50
100
0 1 2 33
3.1
3.2
pe[bar]
0 1 2 32.5
3
3.5
0 1 2 32.8
3
3.2
t [s]
pv[bar]
0 1 2 31
3
5
t [s]
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Table of Contents
1 Introduction, motivation
2 ModellingCFD1D model
3 Qualitative stability analysis
4 Bifurcations of impacting periodic orbits
5 Summary
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Primary instability types
Remember, our model is...
Aim:Systematically isolate instabilitytypes and give design formulae toavoid them.
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Quarter-wave instability
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Valve chatter
experiments theory
valve spring frequency
quarter−wave frequency
self-excited oscillations despite steady-state BCsQuarter-wave frequency of the pipe seems to dominate
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
The Quarter-wave model (QWM)
Simplest case (liquid, one mode only):
Aim: replace the PDEs describing the pipeline dynamics to ODEsthat allow stability analysis.
Ansatz:
p(x , t) = pt(t)−ρ
2v(0, t)2 + B(t) sin
(2π
x
4L
)v(x , t) = v(L, t) + C (t) cos
(2π
x
4L
)where v(L, t) = Cd
Aft(xv )Apipe
√2ρ (p(L, t)− p0)
Solve the above equations for p(L, t) and v(0, t).
Then use 1-point collocation technique (PDE → ODE).
One can perform the same computation for arbitrary wave modes.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
The Quarter-wave model (QWM)
Simplest case (liquid, one mode only):
Aim: replace the PDEs describing the pipeline dynamics to ODEsthat allow stability analysis.
Ansatz:
p(x , t) = pt(t)−ρ
2v(0, t)2 + B(t) sin
(2π
x
4L
)v(x , t) = v(L, t) + C (t) cos
(2π
x
4L
)where v(L, t) = Cd
Aft(xv )Apipe
√2ρ (p(L, t)− p0)
Solve the above equations for p(L, t) and v(0, t).
Then use 1-point collocation technique (PDE → ODE).
One can perform the same computation for arbitrary wave modes.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
The Quarter-wave model (QWM)
Simplest case (liquid, one mode only):
Aim: replace the PDEs describing the pipeline dynamics to ODEsthat allow stability analysis.
Ansatz:
p(x , t) = pt(t)−ρ
2v(0, t)2 + B(t) sin
(2π
x
4L
)v(x , t) = v(L, t) + C (t) cos
(2π
x
4L
)where v(L, t) = Cd
Aft(xv )Apipe
√2ρ (p(L, t)− p0)
Solve the above equations for p(L, t) and v(0, t).
Then use 1-point collocation technique (PDE → ODE).
One can perform the same computation for arbitrary wave modes.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
The Quarter-wave model (QWM)
Simplest case (liquid, one mode only):
Aim: replace the PDEs describing the pipeline dynamics to ODEsthat allow stability analysis.
Ansatz:
p(x , t) = pt(t)−ρ
2v(0, t)2 + B(t) sin
(2π
x
4L
)v(x , t) = v(L, t) + C (t) cos
(2π
x
4L
)where v(L, t) = Cd
Aft(xv )Apipe
√2ρ (p(L, t)− p0)
Solve the above equations for p(L, t) and v(0, t).
Then use 1-point collocation technique (PDE → ODE).
One can perform the same computation for arbitrary wave modes.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
The Quarter-wave model (QWM)
Simplest case (liquid, one mode only):
Aim: replace the PDEs describing the pipeline dynamics to ODEsthat allow stability analysis.
Ansatz:
p(x , t) = pt(t)−ρ
2v(0, t)2 + B(t) sin
(2π
x
4L
)v(x , t) = v(L, t) + C (t) cos
(2π
x
4L
)where v(L, t) = Cd
Aft(xv )Apipe
√2ρ (p(L, t)− p0)
Solve the above equations for p(L, t) and v(0, t).
Then use 1-point collocation technique (PDE → ODE).
One can perform the same computation for arbitrary wave modes.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
The simplest QWM
x ′v = vv
v ′v = −κvv − (xv + δ) + Aeff (p + B)
p′ = β (q − µ(vend + C ))
B ′ =π
2α
γC −√
2p′ + φ
(C 2√
2+ 2Cvend +
√2v2
end
)C ′ = −π
21αγ
B −√
2v ′end
vend = σx√
p + B
... and one can also add pipe friction, convective terms, moremodes (however, you might want to use a computer algebrasystem).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
The simplest QWM
x ′v = vv
v ′v = −κvv − (xv + δ) + Aeff (p + B)
p′ = β (q − µ(vend + C ))
B ′ =π
2α
γC −√
2p′ + φ
(C 2√
2+ 2Cvend +
√2v2
end
)C ′ = −π
21αγ
B −√
2v ′end
vend = σx√
p + B
... and one can also add pipe friction, convective terms, moremodes (however, you might want to use a computer algebrasystem).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria
Assume large reservoir (β ≈ 0) → y3 ≈ konstans.
The pipeline dynamics is
B ′′+
(π/2γ
)2
B = −konst.α
γσd
dτ
(Y√
P0 +x0√P0
+ B
)+O(β),
and the valve dynamics is Y ′′ = −Y + B − κY ′.Close to the stability boundary: B(τ ; τ2) = A(τ2) cos(ωpτ),with which the valve displacement becomes:
Y =−1
1− ω2p
B +O(κ)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria
Assume large reservoir (β ≈ 0) → y3 ≈ konstans.The pipeline dynamics is
B ′′+
(π/2γ
)2
B = −konst.α
γσd
dτ
(Y√
P0 +x0√P0
+ B
)+O(β),
and the valve dynamics is Y ′′ = −Y + B − κY ′.Close to the stability boundary: B(τ ; τ2) = A(τ2) cos(ωpτ),with which the valve displacement becomes:
Y =−1
1− ω2p
B +O(κ)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria
Assume large reservoir (β ≈ 0) → y3 ≈ konstans.The pipeline dynamics is
B ′′+
(π/2γ
)2
B = −konst.α
γσd
dτ
(Y√
P0 +x0√P0
+ B
)+O(β),
and the valve dynamics is Y ′′ = −Y + B − κY ′.
Close to the stability boundary: B(τ ; τ2) = A(τ2) cos(ωpτ),with which the valve displacement becomes:
Y =−1
1− ω2p
B +O(κ)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria
Assume large reservoir (β ≈ 0) → y3 ≈ konstans.The pipeline dynamics is
B ′′+
(π/2γ
)2
B = −konst.α
γσd
dτ
(Y√
P0 +x0√P0
+ B
)+O(β),
and the valve dynamics is Y ′′ = −Y + B − κY ′.Close to the stability boundary: B(τ ; τ2) = A(τ2) cos(ωpτ),
with which the valve displacement becomes:
Y =−1
1− ω2p
B +O(κ)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria
Assume large reservoir (β ≈ 0) → y3 ≈ konstans.The pipeline dynamics is
B ′′+
(π/2γ
)2
B = −konst.α
γσd
dτ
(Y√
P0 +x0√P0
+ B
)+O(β),
and the valve dynamics is Y ′′ = −Y + B − κY ′.Close to the stability boundary: B(τ ; τ2) = A(τ2) cos(ωpτ),with which the valve displacement becomes:
Y =−1
1− ω2p
B +O(κ)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria (cont’d)
The pipe dynamics is:
B ′′ +
(π/2γ
)2
B = −konst.α
γσ
(X0
2√P0−√P0
ω2p − 1
)︸ ︷︷ ︸
!>0
B ′
For small q values, the equilibrium (X0,P0) can be expandedinto Taylor series and given in closed form.
After some algebra, one can arrive ar q ≥ 2 δ3/2
µσ((ωp(L))2−1) ,which is straightforward to implement even in a spreadsheetsoftware.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria (cont’d)
The pipe dynamics is:
B ′′ +
(π/2γ
)2
B = −konst.α
γσ
(X0
2√P0−√P0
ω2p − 1
)︸ ︷︷ ︸
!>0
B ′
For small q values, the equilibrium (X0,P0) can be expandedinto Taylor series and given in closed form.
After some algebra, one can arrive ar q ≥ 2 δ3/2
µσ((ωp(L))2−1) ,which is straightforward to implement even in a spreadsheetsoftware.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Analytical stability criteria (cont’d)
The pipe dynamics is:
B ′′ +
(π/2γ
)2
B = −konst.α
γσ
(X0
2√P0−√P0
ω2p − 1
)︸ ︷︷ ︸
!>0
B ′
For small q values, the equilibrium (X0,P0) can be expandedinto Taylor series and given in closed form.
After some algebra, one can arrive ar q ≥ 2 δ3/2
µσ((ωp(L))2−1) ,which is straightforward to implement even in a spreadsheetsoftware.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
CFD (red/blue) vs. 1D model (black) vs. QWM analytical(blue)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Stability diagram - QWM vs. meas. 2J3 valve
0 10 20 30 40 50 60 70 80 90 100 1100
5
10
15
flow, percent of capacity
pipe
leng
th, f
oot
2J3, ring: −5 and −25
3% rule
QWM prediction
3% rule
QWM prediction
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Table of Contents
1 Introduction, motivation
2 ModellingCFD1D model
3 Qualitative stability analysis
4 Bifurcations of impacting periodic orbits
5 Summary
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Simplified model without pipe
Close-coupled valveSmall reservoirThe resulting model is:
y ′1 = y2
y ′2 = −κy2 − (y1 + δ) + y3
y ′3 = β (q − y1√y3)
... with the impact law aty1 = 0:
(y1, y2, y3)T → (y1,−ry2, y3)
T
s k
V, pr
x0
D, A, L
mr,in
mr,out
m
.
.
mv.
Aft(xv)
ξ
xv
vp(ξ,t), pp(ξ,t)
pv
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Bifurcation diagram
0
m6=
0.5
m5=
1.0 2
m4=
3.0 4 5 6
m3=
6.5
m2=
7.1
m1=
8 9 100
2
4
6
8
10
12
stabilegyensúlyi
helyzet
periodikuspálya,nincs
ütközés
ütközésesperiodikus
pályák
Hopfbifurkáció
grazingbifurkáció
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Some orbits:
-10 0 10
y2
0
2
4
6
y 1
-10 0 10 20
y2
0
2
4
6
8
10
y 1-10 0 10 20
y2
0
2
4
6
8
y 1
-5 0 5
y2
0
0.5
1
1.5
2
2.5
y 1
-5 0 5
y2
0
0.2
0.4
0.6
0.8
1
y 1
-5 0 5
y2
0
0.2
0.4
0.6
0.8
y 1
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Continuation strategy
Formulate the problem as a BVP, i.e. y ′ = TF (y) with
y1(0) = 0y1(1) = 0
−ry2(0) = y2(1)y3(0) = y3(1)
Use pseudo-arclength cont. to track periodic orbits (+1 BC).Stability: solve variational equation to compute monodromymatrix and apply correction at the impact (see Bernardo, M.,Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-smoothDynamical Systems: Theory and Applications, Springer, 2008, ISBN978-1-84628-039-9).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Continuation strategy
Formulate the problem as a BVP, i.e. y ′ = TF (y) with
y1(0) = 0y1(1) = 0
−ry2(0) = y2(1)y3(0) = y3(1)
Use pseudo-arclength cont. to track periodic orbits (+1 BC).
Stability: solve variational equation to compute monodromymatrix and apply correction at the impact (see Bernardo, M.,Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-smoothDynamical Systems: Theory and Applications, Springer, 2008, ISBN978-1-84628-039-9).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Continuation strategy
Formulate the problem as a BVP, i.e. y ′ = TF (y) with
y1(0) = 0y1(1) = 0
−ry2(0) = y2(1)y3(0) = y3(1)
Use pseudo-arclength cont. to track periodic orbits (+1 BC).Stability: solve variational equation to compute monodromymatrix and apply correction at the impact (see Bernardo, M.,Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-smoothDynamical Systems: Theory and Applications, Springer, 2008, ISBN978-1-84628-039-9).
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Continuation strategy (cont’d)
Continuation of grazing points: y2(0) = 0 (+ 4 + 1 BC)
Continuation of period doublings: one of the characteristicmultipliers is -1. Problems with accuracy!Implemented in Matlab, using bvp5c.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Continuation strategy (cont’d)
Continuation of grazing points: y2(0) = 0 (+ 4 + 1 BC)Continuation of period doublings: one of the characteristicmultipliers is -1. Problems with accuracy!
Implemented in Matlab, using bvp5c.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Continuation strategy (cont’d)
Continuation of grazing points: y2(0) = 0 (+ 4 + 1 BC)Continuation of period doublings: one of the characteristicmultipliers is -1. Problems with accuracy!Implemented in Matlab, using bvp5c.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Qualitative bifurcation diagram
Br2
Br1
y3y3y3y3
Br2
y3
HB
GR1
x
Br3Br4Br5Br6
Br3
Br4
GR2
PD1
PD3
Br5
Br6
GR4GR3 PD4Br7
PD2PD
PD
y1
y2
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Shilnikov-like orbit
-0.1 -0.05 0 0.05
y2
0
0.05
0.1
y 1
(1)
-0.05 0 0.05
y2
0
0.02
0.04
y 1
(2)-0.05 0 0.050
0.02
0.04(3)
-0.1 -0.05 0 0.050
0.02
0.04(4)
-0.2 -0.1 0 0.1-0.01
0
0.01
0.02(5)
-0.2 -0.1 0 0.1-0.01
0
0.01
0.02(6)
-0.1 -0.05 0 0.050
0.05(7)
-0.1 -0.05 0 0.050
0.05(8)
0 0.1 0.20
5
10
15
T (
perió
dus)
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Shilnikov-like orbit
0.05
0
y2
0
0.2
0.01
-0.050.15
0.02
y3
y 1
0.1
0.03
0.05
0.04
-0.10
0.050 0.5 1
-0.1
-0.05
0
0.05
y 1
0 0.5 10
0.05
0.1
0.15
0.2
y 2
0 0.5 1
t
0
0.02
0.04
0.06
y 3
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Table of Contents
1 Introduction, motivation
2 ModellingCFD1D model
3 Qualitative stability analysis
4 Bifurcations of impacting periodic orbits
5 Summary
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Summary
Modeling levelsCFD: few hours ↔ full 3D, transient1D unsteady model: few minutes ↔ 1D, transientQWM: seconds ↔ 1D, only close to equilibrium (nolarge-amplitude oscillations)Analytical: ??? ↔ only around the stability boundary,assumptions need to be checked
Impacting periodic orbitsRelatively new mathematical results.Standard nonlin. dyn. toolkit can be used.Surprisingly rich dynamics.
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Thank you for you attention!
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Effective area
A simple yet accurate estimate for the fluid force is essential:
Ffluid =
∫(A)
p(A)dA + Fimp(m, β)
Define effective areaas
Ffluid = Aeff (x)∆p
β
x
D
valve disc
v
A
Aeff
1
100 % of full lift
f,jv
vf,v
Aft
1
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Effective area
A simple yet accurate estimate for the fluid force is essential:
Ffluid =
∫(A)
p(A)dA + Fimp(m, β)
Define effective areaas
Ffluid = Aeff (x)∆p
β
x
D
valve disc
v
A
Aeff
1
100 % of full lift
f,jv
vf,v
Aft
1
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Effective area – theory vs. CFD
0 20 40 60 80 100 120 140 1601
1.5
2
2.5
3
3.5
x [%]
Aeff/A
pipe[-]
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability
Introduction, motivationModelling
Qualitative stability analysisBifurcations of impacting periodic orbits
Summary
Effective area – two more examples
0 0.05 0.1 0.15 0.20.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
X [−]
Cf[−
]
∆p = 5 bar∆p = 10 barmeasurement
0 0.05 0.1 0.15 0.20.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
X [−]
Cf[−
]
∆p = 5 bar∆p = 10 bar
Csaba Hős (BME, Dept. of Hydrodynamic Systems) PRV stability