Handout notes gas liquid flow patterns as directed graphs

Gas-liquid pipe flow patterns as directed graphsTheir creation and application

Pablo Adames, Schlumberger

Banff, Canada, June 11, 2014

This document contains the notes to the slides used to present the paper of the same title at the9thNorth American Conference on Multiphase Technology in Banff, Canada on June 11, 2014.

Gas-liquid pipe flow patterns asdirected graphs

(a) Title page

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

(b) Abstract

Figure 1: First two slides presented on June 11, 2014

1 Abstract

A visual representation of the abstract appears in Figure 1b. The main idea across the top from leftto right is that the ideas in this paper come from observations of flow patterns being mapped totraditional control volume models and then being transformed into directed graph representations.

The second idea is that using the directed graphs for simple flow patterns and a consistent set ofrules one can compose the directed graphs of more complex flow patterns.

2 IntroductionTwo common applications of directed graphs are illustrated in Figure 2. The first one is as a graphicrepresentation of an optimization algorithm that sweeps several states of the process flow sheet

1

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


(a) Tearing cycles in process flow sheeting

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


(b) Graphic mass balance in dynamic distillation

Figure 2: Slides 3 and 4 presented on June 11, 2014

while it looks to minimize the cost of each alternative solution path by finding the one that involvesguessing the least number of variables possible.

The second is an example of a graphic representation of the dynamic mass balance of a componentin a distillation column that separates the feed stream into a distilled stream rich in the morevolatile component(s) and the bottoms richer in the less volatile component(s). The edges representaccumulation flow rates and the nodes represent mass balances around sections of the column or thewhole unit (node f).

3 The graph structure

Introduction

Applications of directed graphs

This paper What does it look like?


(a) Aim: to draw directed graphs for flow patterns

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


(b) Definitions


This paper presents directed graphs as a means to understand the distribution of mass and momentumamong the regions of the flow patterns, see Figure 3a. The directed graphs that are produced are

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similar to those used to represent the relations between interrelated sets of equations.

Figure 3b shows the working definition of a directed graph in term of its components: nodes andedges. The nodes in pipe flow directed graphs can represent:

1. Regions of the flow pattern2. Conceptual operations on phases: stream or phase weighting3. Force generation4. Mass sources or sinks

The edges represent connecting equations, they can be of two types:

1. Mass flux2. Force applied on the control volume center of mass

We will see examples of all these kind of components as we develop the directed graphs for flowpatterns.

3.1 A simple flow pattern

The graph structure A simple flow pattern

A well-mixed flow pattern

Dispersed bubble control volume


(a) The control volume


From control volume to DG


(b) The sub graphs


Figure 4a shows a well-mixed flow pattern like dispersed bubble flow in a horizontal pipe with lightmachine oil and air. The name comes from the fact that the gas and liquid phases are intimatelymixed and the turbulence is so high that the dispersed gas phase remains confined to very finebubbles that travel almost at the same average speed of the continuous liquid phase.

Figure 4b shows the superposition of the control volume concept over the actual flow pattern obser-vation.

3.1.1 The mass balance subgraphThe mass flow in and out of the control volume shown in Figure 4b can be mapped to arches a1,2and a2,3. The corresponding input and output nodes become the mass source and sink respectively.And the middle node becomes the slip generator that represents also the mass balance for the wholecontrol volume. Figure 5a represent directed graph for this simple mass balance.

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Figure 5b, shows how the mass flows in and out of the slip generator node have the same referencecross section area, namely, the area for flow of the pipe. This is because node 2 is associated withthe mass balance over the entire volume for a well-mixed flow pattern that occupies all the crosssection of the pipe.


Well-mixedmass balance subgraph

Control volume Mass balance subgraph


(a) 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


(b) Mass balance equation


3.1.2 The force balance subgraph


Well-mixed up flowforce balance

(a) Up flow: control volume

(b) Force

diagram (c) Directed graph


(a) Force balance for up flow


Well-mixed down flowforce balance

control volume diagram(d) Down flow:

(f) Directed graph (e) Force


(b) Force balance for down flow


The momentum balance can be transformed into a static force balance for the case of steady state(no acceleration) and thus the forces acting on the control volume can be pictured through free bodydiagrams. This aids in understanding the direction conventions in the force balance subgraph.Figure 6a shows the case of up flow. In this case the pressure force for flow is the only one actingin the direction of flow, ∆Fpressure, while frictional, kinetic, and gravitational forces drag the controlvolume down by pointing opposite the direction of flow.Figure 6b shows the case of down flow. In this case the pressure and the gravitational forces goin the direction of flow, ∆Fpressure and Fhead, while frictional and kinetic forces, Ffriction and Fkinetic,

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offer resistance to motion and thus act in the direction opposite to flow, and thus point verticallyupwards.

From these free-body diagrams the following convention is used to map forces to edges on the forcebalance subgraph: a force that aids flow acts in the direction of flow in the free-body diagram andgoes in the direction from force generator to slip nodes. On the contrary, a force opposing motion,acts in the direction opposite to flow in the free-body diagram and corresponds to an edge connectingthe slip with the force nodes.

Figure 7a shows the force balance equation around the slip node, node 2, for the case of up verticalflow. Each force has been expressed in terms of variables that can be used to compute its value.

• Fpressure = A∆P , where A =cross section area and ∆P = net pressure change.

• Ffriction = τW S∆l, where τW = wall shear stress, S =wetted perimeter, and ∆l = lengthchange.

• Fhead = Aρmix g∆l sinθ, where ρmix = mixture density, g = acceleration of gravity, and sinθ =sine of the inclination angle.

• Fkinetic = Aρmix vmix ∆vmix, with ∆vmix = change in mixture velocity.

3.1.3 The combined mass and force graph

The complete directed graph for a well-mixed pipe flow pattern can be obtained from overlaying themass and force balance subgraphs as shown in Figure 7b. The slip generator is the only node sharedby both subgraphs.


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


(a) Force balance formulation


Well-mixed completedirected graph

Mass balance Force balance

Combined mass and force balances


(b) Mass and force balance in well mixed flow


3.2 The slip generatorThe nature of the slip node is to generate slippage between the phases due to the net balance offorces and mass across it. Figure 8a shows how slippage would manifest as a change in the inputand equilibrium area fractions, cL and El, repectively. As a consequence of this slip differential theaverage liquid and gas velocities would be change from input to output, this is what is also know asthe holdup phenomenon.

5

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


(a) Slip generation

More complex flow patterns

Separated flow patternsStratified wavy Annular mist


(b) Two separated flow patterns


4 More complex flow patternsSeparated flow patterns are those where the phases flow co-currently in their own relatively well-defined paths. Figure 8b shows two common separated flow patterns: stratified wavy and annularmist flows.4.1 Separated flow pattern graphFigure 9a shows the observation of annular mist flow and the diagram representation for it. Thereare two well defined flow paths: the liquid film and the gas core. In annular mist flow inertial forcesdominate over gravitational and this determines that the lighter gas core travels in and around theaxis of the pipe where the velocity is highest while the heavier phase moves along the paths of lowervelocities near the walls. In stratified flow gravitational forces control and hence the heavier liquidphase flows at the bottom while the lighter gas flows at the top of the pipe.

More complex flow patterns Separated flow pattern graph

Separated flow patternAnnular mist control volume

1

6

4

3

FG-L fric

−FG-L fric

Gas core

vL

vG

Liquid film

vG vL


(a) Separated flow pattern


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


(b) Separated directed graph


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4.1.1 The directed graph for separated flow

The corresponding directed graph for separated flow can be seen in Figure 9b. Gas core and liquidfilm are modelled as well mixed regions in the control volume and as such they are assigned slipnodes 3 and 4 respectively.

Two interesting features appear in this graph: the splitting and mixing nodes that maintain massbalance between source and sink, and the force exchanged between the two slip nodes.

4.1.2 The mass balance equations

A straight forward mass balance around node 2 looks more interesting after the introduction of theinput split parameter, cF , which is a measure of the fraction of area for film relative to the total areaof the cross section of the pipe at input (no slip) conditions. Figure 10a shows those equations andthe velocities going into each slip node as a consequence of the mass splitting node.

Figure 10b shows that the simplification of no entrainment of the opposite phase into the dominantphase in each flow path reduces cF to cL for the flow pattern and the mass balance around the splitnode to the expression for the input mixture density for the whole control volume.


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.


(a) Mass balance around splitter


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.


(b) When there is no entrainment


A similar analysis can be made around node 5. Figure 11a shows that for the case of of no entrapmentof gas bubbles in the liquid film or liquid droplets in the gas core, the resulting expression is the slipmixture density for the whole flow pattern, which uses EL for phase density weighting

4.2 The force balance equationsThe force balances on the film and gas core nodes are equivalent to the ones formulated aroundwell-mixed control volumes. However there is a new force balance node in this graph, node 11. Thisnode is due to the interfacial shear between gas and liquid phases. The balance for this node can beseen in Figure 11b.

If force accounts for all energy exchanges between major flow paths in this flow pattern, the effectof that energy exchange through their interface can be seen as drag exerted by the film on the gas,or equivalently as the pull of the gas on the film.

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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.


(a) The mass balance around the mixer node

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


(b) Force balance for separated flow


4.3 Intermittent flow pattern graph

A more complex flow pattern involves alternating series of large gas bubbles and well-mixed plugsor slugs. Commonly referred to as slug flow, Figure 12a shows an experimental observation nextto a graphical representation of the control volume associated with the slug unit that repeats itselfindefinitely.

More complex flow patterns Intermittent flow pattern graph

Intermittent flow pattern

Slug flow control volume

10

8

9

Gas core (SC)

Liquid film(SF)

Liquid slug (D)


(a) The slug flow patterns


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


(b) The slug flow directed graph


4.3.1 The intermittent directed graph

The fact that the gas bubble section looks like a separated flow pattern region and the liquid sluglike a well-mixed flow pattern allows one to use the analogy of the directed graphs for those steadystate gas-liquid flow patterns to build the directed graph for this intermittent train of bubbles andslugs. Figure 12b shows the slide where the directed graph for this flow pattern was presented.

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4.3.2 The mass balance equations

The main features of the directed graph for slug flow are presented in Figure 13a. The long bubblesare called the S region (for separated) while the well-mixed slug is called the D region (for dispersed).The appearance of more complex splitting and mixing nodes comes accompanied by the introductionof two key parameters: γ and β, the input and the slip intermittencies.


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


(a) Characteristics of slug flow patterns



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


(b) Features of the input section


Figure 13b points to how the mass flux splitting going from the source node towards the slip nodeshappens in three splitting nodes and two mixing nodes. The key parameters are the input liquidfraction to the flow pattern, cLU, and the input intermittency, γ. They define the base line slipbetween gas and liquid.



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


(a) Features of the equilibrium section

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


(b) The slug flow mass balance equations


Figure 14a highlights the left–right symmetry of the mass balance subgraph because the right,equilibrium, side mirrors the left, input section. Each slip section, S and D, have their respective sinknode on the equilibrium side and from there they blend their mass fluxes through two splitting andthree mixing nodes. The parameters used are the equilibrium intermittency, β, and the equilibriumslug unit liquid fraction, ELU.

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4.4 The force balance equationsAs manifested in Figure 14b the structure of the force balance equations for the slip nodes in S andD is the same of the independent flow patterns they are modelled after. However only the structureis the same because the mathematical expressions for some of these forces may be different, v.g. theinterfacial force in a long bubble may differ slightly from the one in stable stratified or annular flow.There was a complete momentum transfer disconnection between the two slip generating regionsof the unit cell slug flow models published until the introduction of the Zhang et. al. unified slugflow model in 2003. These researchers developed a term for the momentum exchanged between theliquid film and the liquid slug, effectively creating a force node and arches between node 8 and node10 of Figure 12b.

4.5 Similarities between separated and slug flow

An analysis of the directed graphs obtained for separated and slug flows shows a similar structure inthe mass balance equations that express the distribution of the main flow paths. Figure 15a showsthe structure of the liquid volumetric flux balances in the equlibrium section of the directed graphs.The key slip distribution parameters are film fraction, EF , for separated flow, and intermittency, β,for slug flow. Their role is equivalent as flux distribution parameters in space for EF , and in timeand space, for β.

5 Consequences of the graph structureFigures 15b and 16a highlight the main consequences of using directed graphs for steady stategas-liquid flow patterns in pipes. The main flow paths in separated flows can be mapped to the slipnodes representing liquid film and gas core. Similarly, the main flow paths in slug flow can be mappedto separated and liquid slug regions. The separated region of slug flow reuses the full representation ofthat independent flow pattern. This highlights the recursive nature of this conceptual representationand potentially of the way in which nature reuses the same structures.

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


(a) Similarities between separated and slug flows

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


(b) Consequences of the directed graphs


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Another consequence is the potential for easier introduction of refinements to the existing models.The idea is that new model refinement can be mapped from specific regions of the control volumeto new graph nodes for the introduction of more slip, or for the introduction of new flow paths.

6 ConclusionsFigures 16b and 17a show the conclusions for this paper. A consistent directed graph representationwas possible for all major steady state gas-liquid pipe flow patterns. This representation also confirmsunequivocally that the traditional mass balances for flow patterns in pipes preserve mass continuityeven in the case of intermittent flow by always providing a single source and sink.

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


(a) Consequences of the directed graphs

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


(b) Conclusions I


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


(a) Conclusions II

conclusions

Thank you


(b) Thanks


The directed graph for well-mixed flow patterns was reused in separated flow and the directed graphfor separated and well-mixed flow patterns were reused again to construct the slug flow directedgraph. Finally the equivalent role of film fraction and intermittency as flow distributors was shown.

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