Handout notes gas liquid flow patterns as directed graphs

Gas-liquid pipe flow patterns as directed graphs
Their 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 the
9th
North American Conference on Multiphase Technology in Banff, Canada on June 11, 2014.
Gas-liquid pipe flow patterns as
directed graphs
(a) Title page
Abstract
Abstract
From 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 left
to right is that the ideas in this paper come from observations of flow patterns being mapped to
traditional 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 of
rules one can compose the directed graphs of more complex flow patterns.
2 Introduction
Two common applications of directed graphs are illustrated in Figure 2. The first one is as a graphic
representation 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
Dynamic 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 mathematical
models. 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 involves
guessing the least number of variables possible.
The second is an example of a graphic representation of the dynamic mass balance of a component
in a distillation column that separates the feed stream into a distilled stream rich in the more
volatile component(s) and the bottoms richer in the less volatile component(s). The edges represent
accumulation flow rates and the nodes represent mass balances around sections of the column or the
whole unit (node f).
3 The graph structure
Introduction
This paper What does it look like?
(a) Aim: to draw directed graphs for flow patterns
The graph structure
What is a directed graph?
Definition
Abstract representation of interconnected sets
Components
1 Node: a point representing a set
2 Edge: a link connecting two nodes
Refined definition
The collection of all edges {(i, j)} such that i = j
(b) Definitions
This paper presents directed graphs as a means to understand the distribution of mass and momentum
among the regions of the flow patterns, see Figure 3a. The directed graphs that are produced are
2

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 and
edges. The nodes in pipe flow directed graphs can represent:
1. Regions of the flow pattern
2. Conceptual operations on phases: stream or phase weighting
3. Force generation
4. Mass sources or sinks
The edges represent connecting equations, they can be of two types:
1. Mass flux
2. 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 flow
patterns.
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 light
machine oil and air. The name comes from the fact that the gas and liquid phases are intimately
mixed and the turbulence is so high that the dispersed gas phase remains confined to very fine
bubbles 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 subgraph
The mass flow in and out of the control volume shown in Figure 4b can be mapped to arches a1,2
and 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 whole
control volume. Figure 5a represent directed graph for this simple mass balance.
3

Figure 5b, shows how the mass flows in and out of the slip generator node have the same reference
cross section area, namely, the area for flow of the pipe. This is because node 2 is associated with
the mass balance over the entire volume for a well-mixed flow pattern that occupies all the cross
section of the pipe.
Well-mixed
mass balance subgraph
Control volume Mass balance subgraph
(a) Mass balance subgraph
Well-mixed
mass 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 flow
force balance
(a) Up flow:
control volume
(b) Force
diagram
(c) Directed graph
(a) Force balance for up flow
Well-mixed down flow
force 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 body
diagrams. 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 acting
in the direction of flow, ∆Fpressure, while frictional, kinetic, and gravitational forces drag the control
volume 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 go
in the direction of flow, ∆Fpressure and Fhead, while frictional and kinetic forces, Ffriction and Fkinetic,
4

offer resistance to motion and thus act in the direction opposite to flow, and thus point vertically
upwards.
From these free-body diagrams the following convention is used to map forces to edges on the force
balance subgraph: a force that aids flow acts in the direction of flow in the free-body diagram and
goes 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 connecting
the slip with the force nodes.
Figure 7a shows the force balance equation around the slip node, node 2, for the case of up vertical
flow. 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 = length
change.
• Fhead = Aρmix g ∆l sinθ, where ρmix = mixture density, g = acceleration of gravity, and sinθ =
sine of the inclination angle.
• Fkinetic = A ρmix 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 the
mass and force balance subgraphs as shown in Figure 7b. The slip generator is the only node shared
by 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 complete
directed graph
Mass balance Force balance
Combined mass
and force balances
(b) Mass and force balance in well mixed flow
3.2 The slip generator
The nature of the slip node is to generate slippage between the phases due to the net balance of
forces and mass across it. Figure 8a shows how slippage would manifest as a change in the input
and equilibrium area fractions, cL and El, repectively. As a consequence of this slip differential the
average liquid and gas velocities would be change from input to output, this is what is also know as
the 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 = vsG
1−c L,1
− vsL
c L,1
(1)
vslip,2 = v G,2 − v L,2 = vsG
1−E L,2
− vsL
E 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 patterns
Stratified wavy Annular mist
(b) Two separated flow patterns
4 More complex flow patterns
Separated 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 annular
mist flows.
4.1 Separated flow pattern graph
Figure 9a shows the observation of annular mist flow and the diagram representation for it. There
are two well defined flow paths: the liquid film and the gas core. In annular mist flow inertial forces
dominate over gravitational and this determines that the lighter gas core travels in and around the
axis of the pipe where the velocity is highest while the heavier phase moves along the paths of lower
velocities near the walls. In stratified flow gravitational forces control and hence the heavier liquid
phase 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 pattern
Annular 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 flow
directed graph
1
vmix,2 ρ2
6
vmix,5ρ5
2
(1−
cF)v
m
ix,3
ρ
in
3
cFv
m
ix,4
ρ
in4
511
4
E
Fv
m
ix,4
ρ
out4
3
(1−
E
F)v
m
ix,3
ρ
out3
7
8 9
10
12
13 14
15
Gas core
Film
Ffriction
Fhead Facceleration
∆Fpressure
slip
node
source sink
Ffriction
Fhead Facceleration
∆Fpressureslip
node
input
split
eqilibrium
mix
FG-L fric
(b) Separated directed graph
6

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 liquid
film are modelled as well mixed regions in the control volume and as such they are assigned slip
nodes 3 and 4 respectively.
Two interesting features appear in this graph: the splitting and mixing nodes that maintain mass
balance 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 the
input split parameter, cF , which is a measure of the fraction of area for film relative to the total area
of the cross section of the pipe at input (no slip) conditions. Figure 10a shows those equations and
the 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 dominant
phase in each flow path reduces cF to cL for the flow pattern and the mass balance around the split
node 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 = vin
s4
A2 ρ4 + vin
s3
A2 ρ3
After introducing the input split parameter, cF :
v2 ρ2 = cF vin
4
ρin
4
+ (1 − cF) vin
3
ρin
3
Mixture velocities into film and gas core:
vin
3
and vin
4
are relative to the area for flow going into the
slip nodes 3 and 4.
(a) Mass balance around splitter
If all liquid in film and all gas in core:
cF = cL
vin
3
= vsG,3
vin
4
= 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 control
volume.
(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 entrapment
of gas bubbles in the liquid film or liquid droplets in the gas core, the resulting expression is the slip
mixture density for the whole flow pattern, which uses EL for phase density weighting
4.2 The force balance equations
The force balances on the film and gas core nodes are equivalent to the ones formulated around
well-mixed control volumes. However there is a new force balance node in this graph, node 11. This
node is due to the interfacial shear between gas and liquid phases. The balance for this node can be
seen in Figure 11b.
If force accounts for all energy exchanges between major flow paths in this flow pattern, the effect
of 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.
7

Similarly for the mixer, node 5:
EF vout
4
ρout
4
+ (1 − EF) vout
3
ρout
3
= 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 control
volume.
(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
2
fI ˆρ |vR| vR = 0
The interfacial shear force is equal to the friction force due to a rough
interface
(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 plugs
or slugs. Commonly referred to as slug flow, Figure 12a shows an experimental observation next
to a graphical representation of the control volume associated with the slug unit that repeats itself
indefinitely.
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 flow
directed 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
) v L,3
ρG
c
LU v
L,5 ρ
L
γ v sGS,4
ρG
(1
−
γ)vsG
D,6
ρG
γ
v
sL
S,4ρ
L
(1
−
γ) v
sLD,6 ρ
L
v mix,4
ρmix,4
v
mix,6 ρ
mix,6
v mix,10
ρmix,10
c
Fv
m
,7ρ
m
,7
(1
−
cF
)vm
,7
ρm
,7
(1
−
E
F)v
m
,7ρ
m
,9
EF
vm
,7
ρm
,8
v
mix,11 ρ
mix,11
β
vsG
S,15
ρG
β v
sLS,13 ρ
L
(1
−
β)v
sL
D,13ρ
L
(1
−
β) v sGD,15
ρG
E
LU v
LU,16 ρ
L
(1
−
E LU
) v GU,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 slug
like a well-mixed flow pattern allows one to use the analogy of the directed graphs for those steady
state gas-liquid flow patterns to build the directed graph for this intermittent train of bubbles and
slugs. Figure 12b shows the slide where the directed graph for this flow pattern was presented.
8

4.3.2 The mass balance equations
The main features of the directed graph for slug flow are presented in Figure 13a. The long bubbles
are 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 introduction
of two key parameters: γ and β, the input and the slip intermittencies.
Description of intermittent
directed graph
1 A long bubble section, noted as S
2 A dispersed flow section or slug, noted as D
3 A periodic slug unit, S+D
4 A timed-averaged ratio called intermittency, β = lS
lS+lD
5 An input intermittency, γ =
lin
S
lin
S +lin
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
) v L,3
ρG
c
LU v
L,5 ρ
L
γ v sGS,4
ρG
(1
−
γ)vsG
D,6
ρG
γ
v
sL
S,4ρ
L
(1
−
γ) v
sLD,6 ρ
L
v mix,4
ρmix,4
v
mix,6 ρ
mix,6
v mix,10
ρmix,10
c
Fv
m
,7ρ
m
,7
(1
−
cF
)vm
,7
ρm
,7
(1
−
E
F)v
m
,7ρ
m
,9
EF
vm
,7
ρm
,8
v
mix,11 ρ
mix,11
β
vsG
S,15
ρG
β v
sLS,13 ρ
L
(1
−
β)v
sL
D,13ρ
L
(1
−
β) v sGD,15
ρG
E
LU v
LU
,16 ρ
L
(1
−
E LU
) v GU,16
ρG
vmix,17 ρU,17
1 Node 2 uses the input liquid
fraction CLU
2 Nodes 3 and 5 use the input
intermittency, γ
3 Node 7 is the source node for
the long bubble, S
4 Node 6 is the source node for
the 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 nodes
happens in three splitting nodes and two mixing nodes. The key parameters are the input liquid
fraction to the flow pattern, cLU, and the input intermittency, γ. They define the base line slip
between 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
) v L,3
ρG
c
LU v
L,5 ρ
L
γ v sGS,4
ρG
(1
−
γ)vsG
D,6
ρG
γ
v
sL
S,4ρ
L
(1
−
γ) v
sLD,6 ρ
L
v mix,4
ρmix,4
v
mix,6 ρ
mix,6
v mix,10
ρmix,10
c
Fv
m
,7ρ
m
,7
(1
−
cF
)vm
,7
ρm
,7
(1
−
E
F)v
m
,7ρ
m
,9
EF
vm
,7
ρm
,8
v
mix,11 ρ
mix,11
β
vsG
S,15
ρG
β v
sLS,13 ρ
L
(1
−
β)v
sL
D,13ρ
L
(1
−
β) v sGD,15
ρG
E
LU v
LU
,16 ρ
L
(1
−
E LU
) v GU,16
ρG
vmix,17 ρU,17
1 Node 11 is the sink node for
the long bubble, S
2 Node 14 is the sink node for
the well-mixed region, D
3 Nodes 12 and 14 use the
intermittency, β
4 Node 16 uses the equilibrium
liquid 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 done
independently
2 These are exact replicas of the ones done for
separated and well-mixed
3 The Tulsa Unified model, as an example, adds a force
term 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 sink
node on the equilibrium side and from there they blend their mass fluxes through two splitting and
three mixing nodes. The parameters used are the equilibrium intermittency, β, and the equilibrium
slug unit liquid fraction, ELU.
9

4.4 The force balance equations
As manifested in Figure 14b the structure of the force balance equations for the slip nodes in S and
D is the same of the independent flow patterns they are modelled after. However only the structure
is the same because the mathematical expressions for some of these forces may be different, v.g. the
interfacial 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 regions
of the unit cell slug flow models published until the introduction of the Zhang et. al. unified slug
flow model in 2003. These researchers developed a term for the momentum exchanged between the
liquid film and the liquid slug, effectively creating a force node and arches between node 8 and node
10 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 in
the mass balance equations that express the distribution of the main flow paths. Figure 15a shows
the 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 time
and space, for β.
5 Consequences of the graph structure
Figures 15b and 16a highlight the main consequences of using directed graphs for steady state
gas-liquid flow patterns in pipes. The main flow paths in separated flows can be mapped to the slip
nodes representing liquid film and gas core. Similarly, the main flow paths in slug flow can be mapped
to separated and liquid slug regions. The separated region of slug flow reuses the full representation of
that independent flow pattern. This highlights the recursive nature of this conceptual representation
and potentially of the way in which nature reuses the same structures.
More complex flow patterns Similarities between separated and slug flow
Similarities between
separated and slug flow
The liquid phase is distributed according to a key
parameter:
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 the
directed graphs
1 In separated flow there are paths for film and gas core
2 The forces controlling the degree of separation are
gravitational and inertial
3 In intermittent flow the paths are distributed in time
through regions S and D
4 Further paths exist in the separated region S of
intermittent flow
(b) Consequences of the directed graphs
10

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 volume
to new graph nodes for the introduction of more slip, or for the introduction of new flow paths.
6 Conclusions
Figures 16b and 17a show the conclusions for this paper. A consistent directed graph representation
was possible for all major steady state gas-liquid pipe flow patterns. This representation also confirms
unequivocally that the traditional mass balances for flow patterns in pipes preserve mass continuity
even in the case of intermittent flow by always providing a single source and sink.
Consequences of graph structure
Consequences of the
directed graphs
5 There is a recursive nature in this representation with
well-mixed regions as the primary level
6 Model refinements can be visualized and modelled
using directed graphs for flow patterns:
1 Back mixing regions
2 Additional mass flow paths
3 Additional momentum exchange
(a) Consequences of the directed graphs
conclusions
Conclusions
1 All major steady state flow pattern types can be
represented as directed graphs
2 The mass balance directed graph always has one
input and one output
3 This in itself proves that mass is conserved when they
are used to solve discretized pipeline models
(b) Conclusions I
conclusions
Conclusions
4 Directed graphs for simple flow patterns can be
reused to build more complex ones
5 Intermittency in slug flow plays an equivalent role as
film 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 graph
for separated and well-mixed flow patterns were reused again to construct the slug flow directed
graph. Finally the equivalent role of film fraction and intermittency as flow distributors was shown.
11

Handout notes gas liquid flow patterns as directed graphs

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