KOREA INSTITUTE OF MACHINERY & MATERIALS

High-Order Perturbation Solutions To A LH2 Spreading Model With

Continuous Spill

Sep 13, 2011

Myungbae Kim

ICHS 2011

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IntroductionIntroduction

FormulationFormulation

Results & DiscussionResults & Discussion

ConclusionsConclusions

Contents

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Introduction - 1

Leak of flammable liquid

Most common undesirable event in several plants, especially offshore production installations

Initiating event in almost all Event Tree Analysis Top event of almost all Fault Tree Analysis

Makes a pool or a puddle

The pool radius is important for determining a fire size.

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Introduction - 2

Leak model Instantaneous Release Continuous Release

Spread model Simple Physical Model Shallow Layer Model Full Navier-Stokes Equation

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Simple Physical Model - Briscoe & Shaw

R(t)

H(t)Hydrostatic Pressure

Spill source rate, Evaporation, E

Radius always tends to increase Neglecting viscosity and surface tension

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Formulation

▣ Governing Equations

gwhereHdT

dR2,

2

2

R

VH

REdT

dV

▣ Initial Conditions

0R 0Vand

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Previous Results

The 1st order solution for the pool volume

0 20 40 60 80 1000

5

10

15

20

25

30

Vo

lum

e(m

3 )

Time(sec)

Continuous release

with = 1 m3/s Runge-Kutta Perturbation

23

3

4TE

dT

dV

25

15

8TETV

due to the non-uniform expansion by secular terms as time increases

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Non-dimensionalization

▣ Characteristic Time & Length

,5/1

3

5/12

L

▣ Dimensionless Variables

T

tL

Hh

L

Rr

L

Vv ,,,

3

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Dimensionless Governing Equations

L

Ewherer

dt

dv

,1 2

hdt

dr

2r

vh

with 00,0,0 tathrv

Dimensionless evaporation rate ; an unique governing parameter

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3rd Order Expansion for Solutions

33

22

1 vvvvv o

33

22

1 rrrrr o

33

22

1 hhhhh o

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Perturbation Equations

0

33

0

22

0

110

0

2,

2,

2,

h

h

dt

dr

h

h

dt

dr

h

h

dt

drh

dt

dr

202

13

1022

010 2,2,,

1rrr

dt

dvrr

dt

dvr

dt

dv

dt

dv

20

332

0

222

0

112

0

00 ,,,

r

vh

r

vh

r

vh

r

vh

Initial conditions;

All variables is zero at t=0

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Perturbation Solutions

4/214/534/154/324/94/14/34/1

280665

364

2025

38

135

381

3

2ttttr

2113422/5

13365

64

135

8

15

81ttttv

432/522/1

4455

16

45

2

5

2

4

3tttth

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Results and Discussion -1

LH2 Spill on wax E=4.210-4 m/s (Verfondern & Dienhart)

0 50 100 150 200 250 3000

5

10

15

20

25

3rd order

2nd order

Dim

ensi

onle

ss v

olum

e, v

Dimensionless time, t

m3/s Runge-Kutta

1st order

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Results and Discussion -2

LH2 Spill on concrete E=4.210-4 m/s

0 50 100 150 200 250 3000

10

20

30

40

50D

imen

sion

less

rad

ius,

r

Dimensionless time, t

m3/s Runge-Kutta

1st order

2nd & 3rd ordernearly identical

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Conclusions

The unique governing parameter by non-dimensionalization.

For the pool volume, improvement of the 3rd-order solutions is shown.

For the pool radius, the 3rd-order and 2nd-order solutions are nearly identical, and the improvement by high order solutions is not so large compared to 1st-order solutions.

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Thank youfor your attention!!