stability of direct spring operated pressure relief valves from...

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



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)




Summary

Table of Contents

1 Introduction, motivation

2 ModellingCFD1D model

3 Qualitative stability analysis

4 Bifurcations of impacting periodic orbits

5 Summary




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).




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




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




Summary

Measurement

video 1




Summary

CFD1D model

Table of Contents





5 Summary




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.




Summary

CFD1D model

CFD




Summary

CFD1D model

CFD

video 2video 3




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)




Summary

CFD1D model

Simulation results

pipe length: 0.5mpipe length: 1.1mpipe length: 1.5m




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]




Summary

Table of Contents





5 Summary




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




Summary

Quarter-wave instability




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




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.




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).




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)




Summary


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),


Y =−1

1− ω2p

B +O(κ)




Summary



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)




Summary



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)




Summary



B ′′+

(π/2γ

)2

B = −konst.α

γσd

dτ

(Y√

P0 +x0√P0

+ B

)+O(β),


Y =−1

1− ω2p

B +O(κ)




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.




Summary

CFD (red/blue) vs. 1D model (black) vs. QWM analytical(blue)




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




Summary

Table of Contents





5 Summary




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




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




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




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).




Summary



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).




Summary



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).




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.




Summary


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.




Summary


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.




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




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)




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




Summary

Table of Contents





5 Summary




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.




Summary

Thank you for you attention!




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




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[-]




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


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