Slides gas liquid flow patterns as directed graphs

Gas-liquid pipe flow patterns asdirected graphs

Abstract

AbstractFrom observation

From simple

to concept

to complex

Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 2 / 34

Introduction

Applications of directed graphs

Optimun stream to tear Given all possible states

Upadhye, R. and E.A. Grens. An efficient algorithm for optimum decomposition of recycle systems.AIChE Journal, Vol. 18, No. 3, 1972, pp 533-39


Introduction

Applications of directed graphsDynamic distillation column Accumulation in node f

rate of accumulation in edges

Smith, C.L., Pike, R. W., and P. W. Murrill. Formulation and optimization of mathematicalmodels. International Textbook Company, Scranton, Pennsylvania, 1970, p. 420


Introduction

Applications of directed graphs

This paper What does it look like?


The graph structure

What is a directed graph?

DefinitionAbstract representation of interconnected sets

Components1 Node: a point representing a set2 Edge: a link connecting two nodes

Refined definitionThe collection of all edges {(i , j)} such that i 6= j


The graph structure A simple flow pattern

A well-mixed flow pattern

Dispersed bubble Control volume



From control volume to DG



From control volume to DG



Well-mixedmass balance subgraph

Control volume Mass balance subgraph



Well-mixedmass balance subgraph

Mass balance subgraph Balance equations∑ai,2 −

∑a2,j = 0

a1,2 − a2,3 = 0

m1−2 − m2−3 = 0

ρ1 v1 A − ρ2 v2 A = 0

m L,1−2 − m L,2−3 = 0

ρ L,1 vsL A − ρ L,2 vsL A = 0

ρ L,1 v L,1 c L,1 A − ρ L,2 v L,2 E L,2 A = 0



Well-mixed up flowforce balance

(a) Up flow: control volume

(b) Force

diagram (c) Directed graph



Well-mixed down flowforce balance

control volume diagram(d) Down flow:

(f) Directed graph (e) Force



The force balance

Mass balance subgraph Balance equations∑i

ai,2 −∑

j

a2,j = 0

A∆p︸︷︷︸pressure force

− τw S ∆l︸︷︷︸frictional force

−A ρmix g ∆l sin θ︸︷︷︸hydrostatic force

− A ρmix vmix ∆vmix︸︷︷︸kinetic force

= 0



Well-mixed completedirected graph

Mass balance Force balance

Combined mass and force balances


The graph structure The slip generator

Slip generator

Slip at source and sink nodes:vslip,1 = v G,1 − v L,1 =

vsG1−c L,1

− vsLc L,1

(1)vslip,2 = v G,2 − v L,2 =

vsG1−E L,2

− vsLE L,2

(2)

Mass flows through arches, area and density fixed:v L,1 c L,1 = v L,2 E L,2

v G,1 (1− c L,1) = v G,2 (1− E L,2)

Any change in mechanical equilibrium will affect the slip and holdup


More complex flow patterns

Separated flow patterns

Stratified wavy

Annular mist


More complex flow patterns Separated flow pattern graph

Separated flow pattern

Annular mistControl volume

1

6

4

3

FG-L fric

−FG-L fric

Gas core

vL

vG

Liquid film

vG vL



The separated flowdirected graph

1vmix,2 ρ2

6vmix,5ρ5

2

(1−

cF)v

mix

,3ρ

in3

cFv

mix,4ρ

in4

511

4

EFv

mix

,4ρ

out

4

3(1−

EF)v

mix,3ρ

out3

7

8 9

10

12

13 14

15

Gas core

Film

Ffriction

Fhead Facceleration

∆Fpressure

slipnode

source sink

Ffriction

Fhead Facceleration

∆Fpressureslip

node

input

split

eqilibrium

mix

FG-L fric



The mass balance equations

Mass balance around splitter node 2:vmix,2 A2ρmix,2 = v in

s4 A2 ρ4 + v ins3 A2 ρ3

After introducing the input split parameter, cF :v2 ρ2 = cF v in

4 ρin4+ (1− cF) v in

3 ρin3

Mixture velocities into film and gas core:v in

3 and v in4 are relative to the area for flow going into the

slip nodes 3 and 4.



The mass balance equationsIf all liquid in film and all gas in core:

cF = cL

v in3 = vsG,3

v in4 = vsL,4

cL vsL,4 = vsL,2

(1− cL) vsG,3 = vsG,2

And the splitter mass balance would be:ρ2 = cL ρL + (1− cL) ρG

This is the definition of the input mixture density to the controlvolume.




Similarly for the mixer, node 5:EF v out

4 ρout4+ (1− EF) v out

3 ρout3

= v5 ρ5

If no entrainment in the gas core or film:EL ρL + (1− EL) ρG = ρ5

This is the definition of the equilibrium mixture density of the controlvolume.


More complex flow patterns The force balance equations

The force balance equations

Force balance around node 11:a11,3 − a11,4 = 0

τI S ∆l − 1

2fI ρ |vR| vR = 0

The interfacial shear force is equal to the friction force due to a roughinterface


More complex flow patterns Intermittent flow pattern graph

Intermittent flow pattern

Slug flowControl volume

10

8

9

Gas core (SC)

Liquid film(SF)

Liquid slug (D)



The intermittent flowdirected graph

1 172 16

5

6

4

3

14

15

10

12

13

7 1122

9

8

18

19 20

21

23

24 25

26

27

28 29

30

Film (F)

Bubble (S)

Liquid slug (D)

vmix,2 ρmix,2

(1−

c LU) vL,3

ρG

cLU v

L,5 ρL

γv sG

S,4ρG

(1−

γ)v

sGD

,6ρ

G

γv sL

S,4

ρ L

(1−

γ) vsLD,6 ρ

L

vmix,4ρmix,4

vmix,6 ρ

mix,6vmix,10

ρmix,10

c Fv m

,7ρ m

,7

(1−

cF )v

m,7 ρ

m,7

(1−

EF)v

m,7ρ m

,9

EF v

m,7 ρ

m,8

vmix,11 ρ

mix,11

βv

sGS,15

ρG

βvsLS,13 ρ

L

(1−

β)v sL

D,1

3ρ L

(1−

β) v sGD,15

ρG

ELU v

LU,16 ρL

(1−

E LU) vGU,16

ρG

vmix,17 ρU,17



Description of intermittentdirected graph

1 A long bubble section, noted as S2 A dispersed flow section or slug, noted as D3 A periodic slug unit, S+D4 A timed-averaged ratio called intermittency, β = lS

lS+lD

5 An input intermittency, γ =l inS

l inS +l in

D




Input section Features

1 172 16

5

6

4

3

14

15

10

12

13

7 1122

9

8

18

19 20

21

23

24 25

26

27

28 29

30

Film (F)

Bubble (S)

Liquid slug (D)

vmix,2 ρmix,2

(1−

c LU) vL,3

ρG

cLU v

L,5 ρL

γv sG

S,4ρG

(1−

γ)v

sGD

,6ρ

G

γv sL

S,4

ρ L

(1−

γ) vsLD,6 ρ

L

vmix,4ρmix,4

vmix,6 ρ

mix,6vmix,10

ρmix,10

c Fv m

,7ρ m

,7

(1−

cF )v

m,7 ρ

m,7

(1−

EF)v

m,7ρ m

,9

EF v

m,7 ρ

m,8

vmix,11 ρ

mix,11

βv

sGS,15

ρG

βvsLS,13 ρ

L

(1−

β)v sL

D,1

3ρ L

(1−

β) v sGD,15

ρG

ELU v

LU,16 ρL

(1−

E LU) vGU,16

ρG

vmix,17 ρU,17

1 Node 2 uses the input liquidfraction CLU

2 Nodes 3 and 5 use the inputintermittency, γ

3 Node 7 is the source node forthe long bubble, S

4 Node 6 is the source node forthe well-mixed region, D




Equilibrium section Features

1 172 16

5

6

4

3

14

15

10

12

13

7 1122

9

8

18

19 20

21

23

24 25

26

27

28 29

30

Film (F)

Bubble (S)

Liquid slug (D)

vmix,2 ρmix,2

(1−

c LU) vL,3

ρG

cLU v

L,5 ρL

γv sG

S,4ρG

(1−

γ)v

sGD

,6ρ

G

γv sL

S,4

ρ L

(1−

γ) vsLD,6 ρ

L

vmix,4ρmix,4

vmix,6 ρ

mix,6vmix,10

ρmix,10

c Fv m

,7ρ m

,7

(1−

cF )v

m,7 ρ

m,7

(1−

EF)v

m,7ρ m

,9

EF v

m,7 ρ

m,8

vmix,11 ρ

mix,11

βv

sGS,15

ρG

βvsLS,13 ρ

L

(1−

β)v sL

D,1

3ρ L

(1−

β) v sGD,15

ρG

ELU v

LU,16 ρL

(1−

E LU) vGU,16

ρG

vmix,17 ρU,17

1 Node 11 is the sink node forthe long bubble, S

2 Node 14 is the sink node forthe well-mixed region, D

3 Nodes 12 and 14 use theintermittency, β

4 Node 16 uses the equilibriumliquid fraction ELU


More complex flow patterns The force balance equations

The force balance equations

1 The force balances of the S and D regions are doneindependently

2 These are exact replicas of the ones done forseparated and well-mixed

3 The Tulsa Unified model, as an example, adds a forceterm from 8 to 10


More complex flow patterns Similarities between separated and slug flow

Similarities betweenseparated and slug flow

The liquid phase is distributed according to a keyparameter:

vsL = vLEL = EF vLFELF︸︷︷︸liquid in film region

+ (1− EF) vLCELC︸︷︷︸liquid in gas core region

vsL = vLEL = β vLSELS︸︷︷︸liquid in separated region

+ (1− β) vLDELD︸︷︷︸liquid in dispersed region


Consequences of graph structure

Consequences of thedirected graphs

1 In separated flow there are paths for film and gas core

2 The forces controlling the degree of separation aregravitational and inertial

3 In intermittent flow the paths are distributed in timethrough regions S and D

4 Further paths exist in the separated region S ofintermittent flow


Consequences of graph structure

Consequences of thedirected graphs

5 There is a recursive nature in this representation withwell-mixed regions as the primary level

6 Model refinements can be visualized and modelledusing directed graphs for flow patterns:

1 Back mixing regions2 Additional mass flow paths3 Additional momentum exchange


conclusions

Conclusions

1 All major steady state flow pattern types can berepresented as directed graphs

2 The mass balance directed graph always has oneinput and one output

3 This in itself proves that mass is conserved when theyare used to solve discretized pipeline models


conclusions

Conclusions

4 Directed graphs for simple flow patterns can bereused to build more complex ones

5 Intermittency in slug flow plays an equivalent role asfilm fraction in separated flow


conclusions

Thank you


Technology

Slides gas liquid flow patterns as directed graphs