The Pennsylvania State University
The Graduate School
College of Earth and Mineral Sciences
NUMERICAL MODELING OF NATURAL GAS TWO-PHASE FLOW SPLIT
AT BRANCHING T-JUNCTIONS
WITH CLOSED-LOOP NETWORK APPLICATIONS
A Dissertation in
Petroleum and Natural Gas Engineering
by
Doruk Alp
c© 2009 Doruk Alp
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
December 2009
The dissertation of Doruk Alp was reviewed and approved* by the following:
Luis F. Ayala
Assistant Professor of Petroleum and Natural Gas Engineering
Dissertation Adviser
Chair of Committee
Turgay Ertekin
Professor of Petroleum and Natural Gas Engineering
George E. Trimble Chair in Earth and Mineral Sciences
Undergraduate Program Officer of Petroleum and Natural Gas Engineering
Robert Watson
Associate Professor Emeritus of Petroleum and Natural Gas Engineering
Mirna Urquidi-Macdonald
Professor of Engineering Science and Mechanics
John Mahaffy
Associate Professor Emeritus of Nuclear Engineering
R. Larry Grayson
Professor of Energy and Mineral Engineering
George H., Jr., and Anne B. Deike Chair in Mining Engineering
Graduate Program Officer of Energy and Mineral Engineering
*Signatures are on file in the Graduate School.
Abstract
Branching T-junctions are essential components of small and large scale piping systems found in
natural gas and oil pipeline networks, as well as various industrial applications. Two-phase flow
through branching tees (T-junctions) result in pronounced hydraulic losses and uneven phase sepa-
ration following the split of flow stream; ultimately causing profound effects on system performance
and quality of delivered fluids. Current state-of-the-art modeling software available to oil and gas
industry do not address uneven phase separation (route preference) issue and typically do not
provide a distinct T-junction component for modeling purposes. A finite-volume (FVM) based
two-fluid model has been developed for one-dimensional, steady-state analysis of two-phase flow split
and uneven phase separation at branching T-junctions of natural gas networks using steady-state
Euler equations and outlet pressure specifications. Based on a comprehensive review of available
branching T-junction and phase separation models in the literature, and classification of modeling
efforts included in this text, Double Stream Model (DSM) of Hart et al. (1991), essentially a
mechanistic phase separation sub-model for low liquid loading conditions (holdup < 0.06), is applied
at the junction control volume to capture uneven phase separation. In order to have a consistent
model, phasic momentum equations are replaced with gas phase Bernoulli equations at the junction
cell and required gas phase loss coefficients (K-factors) are calculated using Gardel correlations.
Generalized Newton-Raphson method is applied for simultaneous solution of governing equations,
for all the control volumes in the computational grid. Along with the staggered solution of governing
momentum equations at CV edges, this allows accounting for the impact of outlet (downstream)
pressures on flow split as well as closed-loop network applicability. Analysis is focused on regular,
horizontal T-junctions with 90o branching angle. Results from air-water and hydrocarbon mixture
studies are in agreement with literature; model captures pressure rise in the main line following
the flow split (the Bernoulli effect) and uneven phase separation with changing liquid flow rates
and outlet pressures. An important observation is that while pressure in the main line right after
the split (entrance pressure of the run) is always higher than the entrance pressure of the branch
due to Bernoulli effect, this is not necessarily the case for run and branch outlet pressures. When
branch outlet pressure is specified higher than the run outlet pressure, less fluid is diverted into the
branch. However, the turning of flow causes branch entrance pressure to be always smaller than run
entrance pressure. Slight difference in overall hydrocarbon mixture composition in run and branch
arms, after flow split at the tee, suggests that for small holdup values compositional change can be
ignored for practical purposes.
iii
Contents
List of Tables vii
List of Figures viii
Nomenclature x
Acknowledgements xii
Chapters
1 Introduction 1
2 Flow In Conduits 10
2.1 Single-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Single-Phase Steady-State Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Two-Phase Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Homogeneous Equilibrium Model . . . . . . . . . . . . . . . . . . . . . . . . . 20
Separated Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Drift-Flux Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 Mechanistic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Marching Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Flow Split at T-Junctions 26
3.1 Two-Phase Flow Split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Phenomenological and Mechanistic Junction Models . . . . . . . . . . . . . . . . . . 31
3.2.1 Pressure Change Sub-Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Phase Separation Sub-Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.3 The Double Stream Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.4 Correction Factor and Loss Coefficient Correlations . . . . . . . . . . . . . . 42
3.2.5 Film-stop Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
iv
4 Model Conceptualization 46
4.1 Conservative Property: FVM vs FDM . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Single-Phase Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Two-Fluid Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Closure Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Primary Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 Advective Property: Donor Cell Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.7 Staggered Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.8 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.9 Order of Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.9.1 Higher Order Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.10 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.11 Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.11.1 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.11.2 Phase Appearance-Disappearance . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.11.3 Flow Pattern Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.11.4 Alignment of Residuals and Primary Unknowns . . . . . . . . . . . . . . . . . 80
4.12 Two-Fluid Junction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.13 Fluid Composition After Split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Results and Discussion 91
5.1 Single-Phase Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Two-Phase Flow and Secondary Phase Appearance . . . . . . . . . . . . . . . . . . . 97
5.4 Flow Split and Phase Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.1 Hydrocarbon Mixture Uneven Phase Separation . . . . . . . . . . . . . . . . 108
5.5 Open Network Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.6 Single-Phase Closed-Loop Network Application . . . . . . . . . . . . . . . . . . . . . 113
6 Concluding Remarks 116
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
References 120
v
Appendices
A Single-Phase Flow in Pipe Networks 127
A.1 Kirchhoff Analysis for Flow in Networks . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.2 Steady-State Analysis of Pipeline Systems . . . . . . . . . . . . . . . . . . . . . . . . 130
A.2.1 Open Network Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.2.2 Closed Network Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Cross Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Renouard Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Stoner Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.3 Transient Analysis of Pipeline Networks . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.3.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.3.2 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.4 Description of the Network Model for Numerical Solution . . . . . . . . . . . . . . . 135
B Double Channel T-Junction Model 137
B.1 Double Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.2 The Buffer Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.3 Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.4 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B.4.1 Mass Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B.4.2 Momentum Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B.4.3 Parallel Split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
B.4.4 Reduction of Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C Fluid Properties 151
C.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.2 Density calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.3 Fugacity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.4 Enthalpy calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
D Hydrocarbon Data 156
vi
List of Tables
2.1 Conservation principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Correction factor and loss coefficient correlations for a 90o branching tee . . . . . . . 42
3.2 Models compared by Walters et al. (1998) . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Flow pattern dependent parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Primary unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1 Single-phase compressible flow study data . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Two-Phase flow study data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Air-water flow split data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Hydrocarbon mixture flow split data . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5 Hydrocarbon mixture flow change in overall composition after flow split at the
T-junction; pavg = 12636.81 kPa, λG = 0.55 and λL = 1 . . . . . . . . . . . . . . . . 110
5.6 Air-water flow open network data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.7 Single-phase network data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.8 Comparison of pressure results of of the single-phase network . . . . . . . . . . . . . 115
D.1 Hydrocarbon mixture composition and critical properties . . . . . . . . . . . . . . . 156
D.2 Hydrocarbon mixture Peng-Robinson EOS specific parameters . . . . . . . . . . . . 156
D.3 Hydrocarbon mixture binary interacting coefficients . . . . . . . . . . . . . . . . . . 156
D.4 Hydrocarbon mixture Passut and Danner (1972) coefficients . . . . . . . . . . . . . . 157
vii
List of Figures
1.1 Branching T-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Typical phase envelope for natural gases . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Route preference phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Sample 3D computational grids for T-junctions (from Adechy and Issa, 2004; Lahey,
1990) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Typical flow patterns observed in near horizontal gas condensates pipelines . . . . . 18
2.2 Sample flow regime map for horizontal pipes (Shoham, 2006) . . . . . . . . . . . . . 19
3.1 Three-arm junctions: tee and wye . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 T-Junction classification based on flow directions . . . . . . . . . . . . . . . . . . . . 28
3.3 Pressure profiles after single-phase flow split at a branching T-junction . . . . . . . . 28
3.4 Recirculation zones of a branching tee . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Dividing streamlines for two-phase flow in a branching tee . . . . . . . . . . . . . . . 37
3.6 Zones of influence for two-phase flow in a branching tee . . . . . . . . . . . . . . . . 38
3.7 Comparison of K-factors by various correlations . . . . . . . . . . . . . . . . . . . . . 43
4.1 Finite control volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Control volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Force Balance in an Elementary Pipe CV . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Energy Balance in an Elementary Pipe CV . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Control Volume for Stratified Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . 59
4.6 Control Volume for Stratified Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . 60
4.7 Control Volume for Stratified Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . 63
4.8 Coincidental Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.9 Staggered Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.10 Control Volumes for a T-Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 Single-phase compressible flow study pressure profiles . . . . . . . . . . . . . . . . . 92
5.2 Single-phase compressible flow study numerical error profile . . . . . . . . . . . . . . 93
5.3 Two-phase RK and NR pressure profiles . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 Two-phase RK and NR temperature profiles . . . . . . . . . . . . . . . . . . . . . . . 95
5.5 Two-phase RK and NR gas velocity profiles . . . . . . . . . . . . . . . . . . . . . . . 95
5.6 Two-phase RK and NR liquid velocity profiles . . . . . . . . . . . . . . . . . . . . . . 96
5.7 Two-phase RK and NR holdup profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 96
viii
5.8 Phase appearance p profiles - 20 CVs . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.9 Phase appearance T profiles - 20 CVs . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.10 Phase appearance αG profiles - 20 CVs . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.11 Phase appearance αG vs vk profiles - 20 CVs, HTC ∼= 5.6783W K−1m−2 . . . . . . 99
5.12 Phase appearance αG vs vk profiles - 20 CVs, HTC ∼= 1.0W K−1m−2 . . . . . . . . 100
5.13 Phase appearance density and velocity profiles - 20 CVs, HTC ∼= 5.6783W K−1m−2 100
5.14 Phase appearance density and velocity profiles - 20 CVs, HTC ∼= 1.0W K−1m−2 . . 101
5.15 Hydrocarbon mixture p− T profiles - 20 and 40 CVs, HTC ∼= 5.6783 [W K−1m−2] . 102
5.16 Phase appearance αG vs cum. mass transfer profiles - 20 and 40 CVs, HTC ∼=5.6783W K−1m−2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.17 Phase appearance velocity profiles - 20 and 40 CVs, HTC ∼= 5.6783W K−1m−2 . . . 103
5.18 Phase appearance αG and mass transfer per CV profiles - 20 and 40 CVs, HTC∼= 5.6783W K−1m−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.19 Air-water flow pressure profile at T-junction for mL = 0.0057 kg/s . . . . . . . . . . 105
5.20 Air-water flow αG profile at T-junction for mL = 0.0057 kg/s and prun = 102.125 kPa105
5.21 Air-water flow (∆p)tee vs λL plots for different water flow rates . . . . . . . . . . . . 106
5.22 Air-water flow (∆p)out vs λL plots for different water flow rates . . . . . . . . . . . . 107
5.23 Air-water flow λG vs λL plots for different water flow rates . . . . . . . . . . . . . . 107
5.24 Air-water flow (∆p)out vs λG plots for different water flow rates . . . . . . . . . . . . 108
5.25 Air-water complete phase separation curves . . . . . . . . . . . . . . . . . . . . . . . 109
5.26 Hydrocarbon mixture flow λG vs λL plots for different average flow pressures . . . . 110
5.27 Air-water open network pressure profile . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.28 Air-water open network αG profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.29 Single-phase closed-loop network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.30 FVM representation of the single-phase closed-loop network . . . . . . . . . . . . . . 115
A.1 Connection Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2 An Open Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.3 Closed Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
B.1 The double channel model control volumes . . . . . . . . . . . . . . . . . . . . . . . 139
ix
Nomenclature
Roman letters:
A Cross-sectional area [m2]
cp Heat capacity [J kg−1K−1]
cU Heat transfer coefficient [J s−1m−2K−1]
D Pipe diameter [m]
e Intrinsic energy [J kg−1]
e∗ Internal energy [J kg−1]
f Darcy-Weisbach friction factor [dimensionless]
g Gravitational acceleration [ms−2]
gc Gravitational constant [dimensionless for SI]
h Enthalpy [J kg−1]
J Drift Flux [ms−1]
k Momentum correction factor [dimensionless]
K Mechanical energy loss coefficient [dimensionless]
p Pressure [Pa]
Re Reynolds number [dimensionless]
q Volumetric flow rate [m3 s−1]
Q Pipe heat input [J m−2]
si Interfacial area, flow pattern dependent si = ωi∆x [m2]
swL Pipe area wetted by liquid, flow pattern dependent swL = ωL∆x [m2]
T Temperature [K]
T∞ Surrounding (ambient) temperature [K]
u Superficial velocity [ms−1]
v Velocity [ms−1]
V Volume [m3]
z Elevation [m]
Z Gas compressibility factor [dimensionless]
Greek letters:
αG Void fraction [dimensionless]
αL Holdup [dimensionless]
β T-junction branching angle [rad]
δk Distance of phase k from pipe wall [m]
x
ε Pipe roughness [m]
η Joule-Thomson coefficient [K Pa−1]
γG Gas gravity (or gas specific gravity) [dimensionless]
ΓG Rate of mass transfer from liquid-to-gas phase, per unit volume of CV, negative for
condensation and positive for evaporation [kgm−3 s−1]
ΓL Rate of mass transfer from gas-to-liquid phase, per unit volume of CV, positive for
condensation and negative for evaporation [kgm−3 s−1]
λ Branch to inlet mass intake ratio [dimensionless]
Λ Phasic volume fraction [dimensionless]
ωi Phasic interface wetted perimeter [m]
ωL Liquid wetted perimeter [m]
φk Phasic flow multiplier [dimensionless]
φ1 Heat transfer factor [m−1]
φ2 Potential and kinetic energy factor [Km−1]
Φ Two-phase loss multiplier [dimensionless]
ρ Density [kgm−3]
τi Interfacial stress [N m−2]
τw Pipe wall shear [N m−2]
θ Pipe inclination [rad]
σ Radial angle for zone of influence on pipe cross-section [rad]
Subscript:
G Gas phase
L Liquid phase
irr Irreversible
rev Reversible
xi
Acknowledgements
I would like to begin by expressing my deepest gratitude to my advisor Dr.Luis Ayala for bringing
me to Penn State and for his financial support. Needless to say, the four years I have been at Penn
State has been a totally life changing experience which has opened such a path for me that I could
not have thought of otherwise. For this great opportunity, many things I have learned from him and
through the course of this research that have fundamentally changed my understanding of numerical
modeling; for his guidance, understanding and patience with me; I am forever in his debt.
I am truly grateful to Dr.John Mahaffy from whom I have learned a great deal on the subject;
through two courses I took from him and our discussions, as well as additional references he provided
me with during early stages of the study. To a large extend, the approach taken in the study is
based on my learnings from him.
I would like to thank the committee members Dr.Turgay Ertekin, Dr.Robert Watson and Dr.Mirna
Macdonald for their time, interest in the study, valuable comments and input.
I would like to extend my special thanks to Dr.Turgay Ertekin for his understanding, support and
guidance through these years.
I would also like to thank EME staff assistants, and Penn State staff in general, for making my stay
smooth and free of obstacles.
Finally, I would like to express my deep gratitude to my parents Zerrin and Mustafa Alp for their
understanding, continuous and unconditional support overseas through these four years. Their
emotional support and financial assurance has always put my mind at ease and allowed me to focus
my full attention on my studies. As always, I am in their debt.
xii
1
Chapter 1
Introduction
Gaseous hydrocarbon mixtures are brought from wellheads to surface facilities via ‘gathering’ pipe
systems. After necessary treatment, natural gas is delivered to industrial or residential customers
several hundred kilometers away via ‘transmission’ pipelines. Main transmission lines are connected
to a ‘distribution’ grid at the gates of cities or industrial zones. Hence, from wellbore to the end
consumer, natural gas is delivered via complex and integrated network of pipes.
Two-phase flow of hydrocarbon mixtures in a network of pipes could be either a deliberate choice of
the operator or a result of inevitable flow conditions; such as ‘retrograde condensation’ of heavier
hydrocarbons in the case of natural gas lines. Two-phase flow is typically expected in the wellbore
and through out the surface gathering system when oil and gas, water and gas, or oil and water are
initially present and produced simultaneously from the reservoir. On the other hand, depending
on reservoir type, production stage and wellbore conditions liquid hydrocarbons (or water) may
condense in the pipeline as the second phase (or water may condense as the third phase, i.e. water
in emulsion with liquid hydrocarbons); even if inlet stream is single phase.
For on-shore operations, once separation and treatment is completed in centralized facilities close to
well sites, it is preferred to transport oil and gas in separate lines, as single phase fluids. In off-shore
operations, space limitations might impose a need for simultaneous transportation of oil and gas to
on-shore facilities located a considerable distance from the well site.
Branching T-junctions are essential components of small and large scale piping systems found in nat-
ural gas and oil pipeline networks, as well as various industrial applications. Two-phase flow through
branching tees (T-junctions) result in pronounced hydraulic losses and uneven phase separation
following the split of flow stream; ultimately causing profound effects on system performance and
quality of delivered fluids. Furthermore, for the practical purposes of system design and analysis,
almost all other types of junctions joining three or more pipes, as well as specialized sink/source
terms such as wellheads or supply/demand nodes, could then be represented with appropriate
arrangement of consecutive three-arm junctions; in particular tees. Consequently, quantifying
two-phase flow associated hydraulic losses and phase separation through T-junctions is a matter of
serious concern for the design and analysis of various pipe systems.
Industrial applications where two-phase flow split is typically observed include:
• Two-phase flow of refrigerants through distribution headers of multi-pass evaporators or
2
Figure 1.1: Branching T-junction
multi-system air-conditioners (Tae and Cho, 2006).
• Process plants where two-phase streams of chemicals, steam or air-water mixtures are delivered
to various locations in the plant via networks of pipes (Fouda and Rhodes, 1974).
• Conventional (fossil-fueled) power plants where steam is the ‘prime mover’.
• Nuclear reactor coolant loops of various design; i.e. pressurized (PWR), light (LWR) or boiling
(BWR) water reactors (Lahey, 1986) and associated small break discharge scenarios (Smoglie
et al., 1987). Two-phase flow is encountered within reactor coolant systems when water boils
due to large pressure drops; typically caused by accidental bursts of pipes in the primary
coolant loop, known as the loss of coolant accident (LOCA, Todreas and Kazimi, 1990).
• Circulation of geothermal fluids for heating purposes or transportation to geothermal power
plants via pipeline networks (Shoham, 2006).
• Steam injection networks for enhanced oil recovery (EOR) applications (Berger et al., 1997;
Jones and Williams, 1993).
• Condensate (water or liquid hydrocarbon) formation in natural gas gathering and transporta-
tion pipelines (Oranje, 1973).
• Functional phase separators and slug catchers for off-shore platforms and sea floor wellheads
(Margaris, 2007; Azzopardi and Rea, 2000).
It is important to note that through out this text, term ‘phase separation’ is distinctively used
to signify individual distribution of each incoming phase to the outgoing arms of a tee (Fig. 1.1),
while the term ‘phase distribution’ denotes the tendency of gas phase (void fraction) to occupy
certain parts of the pipe cross-section (e.g. due to gravity) –pronounced only in multidimensional
analysis because such details are lost with averaging in one-dimensional analysis (Lahey, 1990; cf.
Chapters 2 and 3). Finally, the term ‘phase split’ stands for the formation of the secondary phase
as a result of phase behavior response to changing pressure and temperature conditions.
In many industrial applications involving two-phase flow, formation of a lighter secondary phase (i.e.
gas) is expected, following a pressure drop in the flow stream that is initially all liquid. For instance,
3
gas phase evolves following pressure drops in oil pipelines. Hence, typically high liquid loading
conditions; i.e. predominantly liquid phase flow, is observed. On the other hand, condensation
of water in steam injection or distribution systems and formation of hydrocarbon liquids (the
condensate) in natural gas lines are low liquid loading conditions (predominantly gas phase flow)
where formation of an heavier secondary phase is observed (the liquid hydrocarbon).
In natural gas pipelines, according to gas composition, and particularly if heavier components are
abundant in the stream (i.e. wet gas), the phase envelope of the hydrocarbon mixture can partially
or entirely enclose the pipe operational region. If operational region is completely within the phase
envelope then two-phase flow would begin right at the very inlet of the system. If operational region
is partially enclosed by the phase envelope then two-phase flow is encountered in the sections of
pipe system where flowing pressure falls inside the phase envelope (Fig. 1.2).
Figure 1.2: Typical phase envelope for natural gases
Therefore, even though single phase conditions are prescribed at the inlet, and thus transportation
commences in entirely gaseous phase, (undersaturated) natural gases tend to drop hydrocarbon
liquids once pressure goes below hydrocarbon dew point conditions in the pipe. Thus, multi-phase
flow can prevail in natural gas transmission lines as well as gathering systems due to retrograde
condensation.
Simultaneous flow of natural gas and condensate is not efficient because presence of a secondary
phase (the condensate) decreases the deliverability of primary phase (gas), due to increased flow
(hydraulic) losses associated with decreased flow area available for the primary phase. Besides;
density, volume and calorific value of both phases vary along the pipeline as flow conditions change
because hydrocarbon phases in contact will continuously alter composition as a result of mass
transfer between the phases. Therefore, mass transfer and fluid re-distribution (i.e. phasic flow path
preference) in the pipeline network have significant influence on fluid and flow properties as well as
overall system performance.
There are two parts to the analysis of two-phase flow problem associated with condensate formation
in natural gas pipelines:
• Prediction and modeling of two-phase flow in straight sections of the network.
4
• Calculation of flow (hydraulic) losses and phase separation at T-junctions.
Within the context of pipe systems, one-dimensional inviscid conservation equations, namely the
Euler equations provide sufficient information for two-phase flow analysis, particularly for straight
pipe sections. Although a full set of multidimensional Navier-Stokes equations can always be
employed to get more detailed information on the flow (i.e. velocity profile, turbulence), such level
of detail is almost never required and would be computationally too demanding for the purposes of
studying large and complex pipeline networks.
In oil and gas industry, hydraulic losses through pipes are typically computed using ‘marching
algorithms’ that propagate the solution from pipe inlet to outlet. Governing one-dimensional Euler
equations are typically simplified and arranged in the form of either a pressure gradient ODE
(Shoham, 2006 – also see Sec. 2.1) or non-conservative set of ODEs (Ayala and Adewumi, 2003)
solved simultaneously at each marching step. However, marching algorithms are not suitable for
modeling closed (looped) networks for which flow related information downstream of the conduit, as
well as flow information from other pipes, should better be incorporated by means of simultaneous
solutions (cf. Sec. 2.2.4 and introduction of Chapter 3). Furthermore, particularly non-conservative
ODE based methods may suffer severe conservation problems with increased marching step size
(Ayala and Alp, 2008).
On the other hand, hydraulic losses at junctions are typically considered negligible in the grand
scheme of large and complicated oil and gas networks where junction volumes are insignificant
next to very long pipelines. Therefore, junctions are simply treated as pressure nodes of indefinite
volume, over which only mass conservation is satisfied and same ‘nodal’ pressure is prescribed as
the inlet or outlet pressure for all the connecting pipes of the junction.
In essence, a pipe network is treated like an electrical circuit where boundary conditions and pipe
‘resistance’ to flow (geometry and surface properties of the pipe: diameter, length, elevation change
and roughness) practically have the sole control on the flow split at junctions and, ultimately, on
the solution (cf. Sec. A.1). For example, at a diverging T-junction, flow split is merely dominated
by specified outlet pressures and flow resistance of the diverging arms while effects of split angle
and tee geometry are ignored. Hence, for example, if both outgoing arms have the same resistance
then flow is assumed to split equally provided that outlet pressures are equivalent too, regardless
of junction geometry and angle of split. This is apparent because, if exit pressures and outgoing
pipe resistances are equal and since the junction node (the ‘knot’) pressure (thus inlet pressures) of
the pipes is same, then same pressure drop over identical pipes can only be attained if flow rates
through the pipes are same too.
In fact, flow split at a T-junction is driven by competing (1) inertial forces (momentum) –in the
axial (inlet) direction, and (2) centripetal forces –in the branch direction, acting on the fluid stream.
The centripetal force is governed by junction geometry and pressure drop that draws the fluid to
the branch. Therefore, a slight difference in flow rates are in order even for the case of identical
outgoing arms and equal outlet pressures, that is unless split is parallel (i.e. no split angle). Again,
this difference in flow rates is practically insignificant for the purposes of single-phase network
5
analysis and typically tee-to-outlet pressure drops1 outweigh other factors, ultimately dominating
single-phase flow split at the tee.
For the case of two-phase flow, however, phases tend to split unevenly with changing proportions
at different flow conditions; inducing significant difficulty for the prediction of hydraulic losses.
Because of (1) the difference in inertia (density and viscosity), (2) associated difference in phasic
wall frictions and due to (3) contributing factors such as interfacial drag; phases are accelerated
differently under same pressure gradient. These differences also govern the inherent ability of a
phase to accomplish a change in flow direction. Thus, phase mass fractions going into branch can
differ significantly as disproportionate phase separation is driven by relative ease of phases to change
direction (Ballyk and Shoukri, 1990; Hwang et al., 1989; Shoham et al., 1987).
Because axial momentum of the lighter phase (i.e. gas) is lower, it responds readily to the pressure
drop in branch direction. While lighter phase is inclined to flow into the branch preferentially;
heavier phase (i.e. liquid) tends not to split at all and flow past straight to the run due to its
relatively high momentum. Consequently, higher fraction of the lighter phase goes into the branch.
However, with a slight change in the conditions (i.e. slight increase in the flow rate of the lighter
phase) heavier phase could entirely flow into the branch thus exhibiting the ‘flip-flop effect’ and
ensuing in the ‘route preference’ phenomenon (Fig. 1.3).
Figure 1.3: Route preference phenomenon
This sensitivity to flow conditions requires that impacts of overall tee geometry, wall surface
properties, inlet (or tee) flow mode and flow pattern on the flow split be properly accounted for in
order to accurately predict (1) disproportionate phase separation after the split and (2) inlet-to-run
and inlet-to-branch pressure changes. Here, it is important to underline that, while outlet pressures
still dominate overall flow split at a tee, afore mentioned factors significantly impact disproportionate
phase separation at the tee. Consequently, different phase separations due to a slight change in
1The pressure drop along the straight section, along the pipe that is connecting the ‘knot’ (tee node) to the outletnode
6
(any one of these) contributing factors can cause different inlet-to-run and inlet-to-branch pressure
drops at the tee, however outlet pressures remain the same.
The flip-flop effect attributed to the route preference of condensate in natural gas pipeline systems
is first reported by Oranje (1973) who studied uneven phase separation problem in natural gas lines.
Oranje (1973) concluded that the change in the direction of lighter phase (gas) entering the branch
induced a pressure drop inside the branch thus creating a suction force driving the secondary phase
(liquid) into the branch.
Later, Hong (1978) observed that as inlet gas flow rate increases (for a fixed liquid viscosity and
flow rate), the centripetal force acting on a unit mass of the liquid increases (perhaps associated
with a higher pressure drop). Consequently more liquid is drawn into the branch. While, for a
fixed gas flow rate, an increase in liquid rate raises the inertial force of the liquid without much
change in the centripetal force. Thus, less liquid tends to turn into branch. Also, an increase in the
liquid viscosity slows down the liquid, lowering axial momentum (inertial force) of the liquid phase;
thus, generally ensues in an increased branch liquid intake. Furthermore, Azzopardi and Whalley
(1982) reported that gas entering the branch drags the liquid film with it in the case of annular flow.
Hence, pressure drop and interfacial drag constitute the centripetal force acting on the fluid stream.
In other words, when branch gas intake is small, centripetal force acting on the liquid phase is also
small compared to the inertial force (the momentum in the inlet direction) of the liquid stream.
Therefore, liquid mostly flows straight through the T-junction directly into the run. When mass
fraction of the branch gas intake exceeds a threshold value (critical gas intake), liquid tends to enter
the branch preferentially.
It is well established with reference to the Bernoulli equation that, when single phase (also two-phase)
split takes place at a T-junction, flow continuing straight in the axial direction to the run experiences
a pressure rise (pressure spike) due to flow area expansion. On the other hand, a pressure drop
is typically experienced by the flow turning to the branch because the magnitude of associated
‘irreversible’ losses are usually greater than the Bernoulli type pressure rise due to flow expansion in
the branch.
Presence of recirculation zones (secondary flows) both in the run and branch, as well as the two-
dimensional nature of flow right after the split (at the T-junction), by default, demand that full
set of Navier-Stokes equations should be employed for multidimensional (i.e. 2D or 3D) analysis
and rigorous definition (and solution) of flow through T-junctions (Fig. 1.4). In fact, following the
incredible progress of computational power and modeling capabilities with Navier-Stokes equations,
two-fluid based multidimensional CFD is likely to become the de facto state-of-the-art for the
modeling of flow through T-junctions. Besides in-house two-dimensional Ellison et al., 1997;
Hatziavramidis et al., 1997 and three-dimensional Adechy and Issa, 2004; Issa and Oliveira, 1994,
flow split and phase separation problem has been studied with general purpose commercial CFD
software as well (Al-Wazzan, 2000; Lahey, 1990; Kalkach-Navarro et al., 1990). Typically, satisfactory
agreement with experimental observations is reported, at least in terms of capturing ‘broad features’
of the split. However, it has also been pointed out that certain details were not captured accurately
7
due to fact that particular phenomena is not represented inherently by the governing equation set
(Adechy and Issa, 2004).
Figure 1.4: Sample 3D computational grids for T-junctions (from Adechy and Issa, 2004; Lahey,1990)
Nevertheless, despite the enormous computational power of the day, modeling all the junctions of a
large, complex pipe network using 3D CFD is a computationally demanding task, and probably
not going to be a feasible approach in the near future for the purpose of modeling large networks.
Moreover, such degree of detail is seldom, if not at all, necessary for the purpose of modeling
network flow. Therefore, proper representation of two-phase flow split mechanism at junctions using
one-dimensional analysis is required.
An important aspect of modeling hydrocarbon condensate route preference in natural gas networks is
the inherent challenge of accounting for the compositional change of the stream. Overall composition
of delivered fluids could differ significantly from original composition after consecutive uneven phase
separations along the way to the delivery station.
Martinez and Adewumi (1997) developed a steady-state open network model by incorporating a
marching algorithm based two-phase flow model with Bernoulli equation based Double Stream
Model (DSM) of Hart et al. (1991) for T-junctions. Changes in stream composition after flow
split at junctions are accounted for using a so called ‘separator approach’ (cf. Sec. 4.13). DSM is
essentially a phase separation sub-model (see Secs. 3.2.1 and 3.2.3) for T-junctions and computes
outgoing phasic flow rates based on inlet conditions of the tee. In that regard, although a separate
model is employed at the tee, ‘forward’ workflow aspect of DSM is similar to marching algorithms
and works effectively in the analysis of open networks. In their study, Martinez and Adewumi (1997)
specified inlet pressures and all inlet/outlet gas flow rates as boundary conditions. Moreover, gas
phase split at each junction is either provided by the user or computed from already specified flow
rate data. The overall network solution has two parts; first, marching algorithm propagates the
solution from pipe inlet to a tee. Then, DSM determines the liquid flow rates to be prescribed as
inlet conditions for outgoing arms of the junction (in addition to specified gas rates). Afterwards,
again the marching algorithm advances the solution along the outgoing arms until a new tee or
a delivery station is reached. Consequently, approach is limited to the analysis of open networks.
Furthermore, in its original form, DSM is not completely adequate in addressing the route preference
phenomenon since flow split, in fact, depends on downstream conditions (pressures) as well.
8
Ottens et al. (2001) developed a finite-difference based transient T-junction model by combining
transient equations that relate pipe liquid level to phasic superficial velocities (derived from phasic
mass and momentum equations) with an advanced version of DSM.
In a recent study, Singh (2009) developed a finite-volume based one-dimensional two-fluid model
(see Chapter 4) for steady-state analysis of isothermal air-water split at T-junctions. Benefiting from
the FVM staggered grid, Euler equations are easily extended over the T-junction control volume
(Spore et al., 2001). Nevertheless, study of Singh (2009) had two inherently contentious aspects: (1)
Over specification of boundary conditions – void fractions are specified at the outlets, in addition to
the pressures. This is attributed to a possible misconception regarding the calculation of two-phase
pressure gradient term in the two-fluid model momentum equations. Please see pertinent discussions
regarding the use of void fraction, or holdup for that matter, when writing phasic pressure gradient
terms for the derivation of phasic momentum equations both in PDE and algebraic forms (cf.
Sec. 4.3.2). (2) Direct substitution of Gardel2 mechanical energy loss coefficients (intended for use
with Bernoulli equations) as momentum correction factors in the phasic momentum equations (cf.
Secs. 3.2.3 and 4.12). Utilized loss coefficients inherently account for the branching angle effect
(which is not accounted for by the non-directional Bernoulli equations otherwise) while momentum
equations already accounted for the directional momentum change with an additional term involving
the branching angle.
In this study, one-dimensional steady-state analysis of two-phase flow split at branching T-junctions
is formulated using the two-fluid model based integrated pipe approach. Governing Euler equations
are discretized over a staggered grid using the inherently conservative Finite Volume Method
(FVM, cf. Chapter 4). The staggered grid arrangement allows seamless extension of Euler equations
over the T-junction control volume and adjoining pipe flow equation set with the Double Stream
Model of Hart et al. (1991), essentially a phase separation sub-model, applied at the junction control
volume. In order to have a consistent model, phasic momentum equations of the tee CV are replaced
with gas phase Bernoulli equations at junction cell volume and required loss coefficients (K-factors)
are calculated using Gardel correlations (cf. Secs. 3.2.3). Focus is kept on the analysis of flow split at
horizontal, regular, 90o angle branching tee arrangements with single-phase, mist and stratified flow
patterns separately. Phases are assumed to be in thermodynamic equilibrium; implying mechanical,
thermal and chemical equilibrium between the phases. As a requirement of steady-state analysis
overall mixture composition remains constant along straight sections of the pipe system; i.e. along
each pipe and each arm of a T-junction. Interaction between the phases is represented by mass and
momentum transfer terms and interfacial friction terms. Due to thermal equilibrium assumption
an overall energy equation is utilized. Peng-Robinson EoS based thermodynamic model is used
for phase behavior predictions (Appendix C). Using a generalized Newton-Raphson technique,
numerical solution is obtained for all discrete points (control volumes) on all three arms of the
junction simultaneously, hence the name ‘integrated pipe’ approach. This approach proves to be
essential for closed-loop network analysis as it circumvents the need for marching through the flow
domain in one direction and readily accounts for changes in downstream conditions, particularly the
2Gardel, A. (1957). Les pertes de charge dans les ecoulements au travers de branchementes en te. BulletinTechnique de la Suisse Romande, 9,122 and 10, 143. (Ottens et al., 2001)
9
outlet pressures in this case.
In Chapter 2, solutions of single-phase steady-state flow in conduits and two-phase flow models are
introduced. In Chapter 3, flow split mechanism and uneven phase separation is discussed along
with a survey of available methods with emphasize on phenomenological and mechanistic junction
models. Appendix A presents conventional single-phase network analysis in oil and gas industry.
In Chapter 4, one-dimensional two-fluid model equations are derived based on FVM. Staggered
grid, boundary conditions, T-junction momentum equations and double stream model integration
is discussed along with the handling of phase appearance-disappearance. Appendix B has an
alternative derivation for a dividing streamline based ‘double channel’ T-junction model for FVM.
Chapter 5 includes results from numerical model comparison with single-phase steady-state analytical
solution, single-phase to two-phase transition handling in a single pipe and two-phase flow split
results for air-water and hydrocarbon two-phase flows through T-junctions.
10
Chapter 2
Flow In Conduits
Analysis of flow in conduits is fundamental to various engineering applications of different scales.
Evidently, quantitative description of flow and predicting response to perturbations (i.e. changes
in flow conditions) are essential for engineering designs. This requires determining the values of
governing flow parameters and fluid properties; such as flow rate or velocity, pressure, temperature
and fluid density. Study of flow in conduits is based on the principles of conservation of mass
(continuity), momentum, and energy. Several methods are available for the description of fluid flow
in pipes, with varying levels of detail, complexity and accuracy. Depending on the complexity of
the problem and level of detail desired for the analysis; an entire suite of conservation equations, or
a subset of them, is written along with appropriate simplifying assumptions in order to describe
the flow. It is then possible to quantify parameters governing the flow by solving the conservation
equations. Three-dimensional flow field is governed by Navier-Stokes equations; a comprehensive set
of coupled, nonlinear partial differential equations (PDEs) expressing conservation principles over
the flow domain; the continuum of conduit, and account for viscous and local volume (fluid particle)
forces. For the practical purposes of flow analysis in pipeline networks, where flow property changes
in the radial direction are not as important as property changes along the axial direction, one-
dimensional statements of mass, momentum, and energy conservation provide sufficient detail and
information on the flow. Typically, conservation statements can be generalized for one-dimensional
analysis through the following conservation differential form:
∂
∂t
[Conserved
property p.u.v.
]+
∂
∂x
[Flux of conserved
property p.u.v.
]=
External forcing function p.u.v.
(source/sink)(2.1)
Where; p.u.v. stands for ‘per unit volume’ and ‘flux of conserved property’ is defined as the amount
crossing the unit area, per unit time [(unit of conserved property) L−2 t−1].
Table 2.1 illustrates the relationship among the three conservation principles; where, intrinsic
energy of fluid is sum of its internal, kinetic, and potential energy. It is the energy associated
with fluid at each point of the pipe and accounts for fluid’s internal energy stored at molecular
level (e∗) and additional energy associated due to its velocity (kinetic energy) and location in the
gravitational field (potential energy). Hence, intrinsic energy per unit mass of the working fluid is:
e = e∗ +v2
2+ g∆zel (2.2)
11
Table 2.1: Conservation principles
Mass Momentum Energy
Conserved quantity m P = mv E = me
Conserved quantityp.u.v.
ρ v e
Flux ρv (ρv)v (ρe)v
Source/Sink –Sum of Forces
(Pres. + Fric. + Grav.)Heat input
2.1 Single-Phase Flow
For practical purposes, single-phase flow in wellbores and pipelines is considered unidirectional and
advective where primary concern is the change along the axial direction of flow. Three-dimensional
flow is typically approximated through a one-dimensional field by averaging characteristic flow
properties over the conduit cross-section. Averaging ensues in the omission of viscous and local
volume forces. Hence, Navier-Stokes equations reduce to one-dimensional Euler equations of inviscid
flow. In fact, viscous forces are most significant within the thin layer (the ‘boundary layer’) near
solid surfaces. Nevertheless, thickness of this layer is much smaller compared to the characteristic
length of the solid surface (i.e. pipe length); and volume forces are negligible next to global forces
driving the flow. Consequently, Euler equations of inviscid flow do not account for any dissipative or
diffusive transport (due to viscosity, mass diffusion, and thermal conductivity) along the direction of
flow and transport is assumed to be fully advective. However, effects of fluid viscosity and wall shear,
and wall heat transfer are accounted for through use of appropriate sink/source terms; momentum
and energy forcing functions.
One-dimensional Euler equations of inviscid flow are:
Mass Continuity:dρ
dt+
d
dx(ρv) = 0 (2.3)
Momentum Balance:d
dt(ρv) +
d
dx(ρvv) = −dp
dx− τw
πD
A− ρg sin θ (2.4)
Energy Balance:d
dt(ρe) +
d
dx(ρev + pv) = Q (2.5)
12
Energy balance is usually written in terms of enthalpy flux for an open system:
d
dt(ρe) +
d
dx
(ρ
[h+
v2
2+ g∆zel
]v
)= Q (2.6)
Where;
h = e∗ +p
ρ(2.7)
Please note that no shaft work is considered for pipe flow. Also, no boundary work is present when
a control volume approach is adopted.
Euler equations could be solved analytically only in the limiting case where a perturbation in
the flow is assumed to occur instantaneously everywhere along the conduit (i.e. the rigid column
flow; Larock et al., 2000) or for steady-state conditions. For other flow conditions, Euler equations
must be solved simultaneously, which requires implementation of appropriate numerical schemes.
Numerical solutions provide approximate values of the flow parameters at discrete points in the
flow domain, which is generally sufficient for most practical purposes.
2.1.1 Single-Phase Steady-State Flow
It is customary to introduce simplifications in the conservation equations based on steady-state
assumptions and expected fluid phase behavior in the system. For steady-state conditions, the
governing equations simplify to:
Mass Continuity:d
dx(ρv) = 0 (2.8)
Momentum Balance:d
dx(ρv v) = −dp
dx− τw
πD
A− ρ g sin θ (2.9)
Energy Balance:d
dx
[ρ v
(e+
p
ρ
)]= cU
πD
A(T∞ − T ) (2.10)
Starting from steady-state momentum and mass balances, one obtains:
d
dx(ρv v) = ρ v
dv
dx+ v
0︷ ︸︸ ︷d
dx(ρ v) = −dp
dx− τw
πD
A− ρg sin θ (2.11)
From where following steady-state pressure gradient (pressure drop) equation is obtained:(dp
dx
)total
=
(dp
dx
)fric
+
(dp
dx
)elev
+
(dp
dx
)acc
(2.12)
13
Where:(dp
dx
)fric
= −τw πDA Pressure loss due to friction(dp
dx
)elev
= −ρ g sin θ Pressure loss due to elevation change (gravity)(dp
dx
)acc
= −ρ v dvdx Pressure loss due to acceleration (or kinetic energy change)
Equation 2.12 is customarily used to predict pressure profiles in pipelines when conditions of flow
are assumed to be isothermal. For non-isothermal flows, the energy balance is implemented. For
steady-state conditions, the energy balance yields:
d
dx
[ρ v
(e+
p
ρ
)]= ρ v
d
dx
(e+
p
ρ
)+
(e+
p
ρ
) 0︷ ︸︸ ︷d
dx(ρ v) = cU
πD
A(T∞ − T ) (2.13)
or
ρ vd
dx
(h+
v2
2+ g∆zel +
p
ρ
)= cU
πD
A(T∞ − T ) (2.14)
From where the steady-state enthalpy-gradient equation is formulated as:(dh
dx
)total
=
(dh
dx
)acc
+
(dh
dx
)elev
+
(dQ
dx
)(2.15)
with:(dh
dx
)acc
= −v dvdx Energy loss due to acceleration
(dh
dx
)elev
= −g sin θ Energy loss due to elevation change (gravity)
(dQ
dx
)= cU
πD(T∞−T )m Energy loss to the environment
For the case of oil (liquid) flow, density of oil is typically assumed constant (ρo ≈ constant), which
is largely true for liquids when no significant temperature change is expected. In this case, mass
conservation imposes v ≈ constant for flows through pipes of constant cross-sectional area or
v1A1 = v2A2 when pipe cross sectional area changes.
Since both density and velocity remain constant throughout an oil pipe of constant cross-sectional
area, there are no pressure losses due to kinetic energy changes and Eqn. 2.12 becomes:
dp
dx= −τw
πD
A− ρ g sin θ (2.16)
14
which, when integrated between two points separated by a distance L, becomes:
(pout − pin) = −πf8
q2LρLD5
L− ρLg∆z (2.17)
Or, re-arranged as:
qL =
√8
π
√1
f
√pin − pout + ρLg∆z
ρLLD
52 (2.18)
Where:
f = 4τw2
ρv2Darcy-Weisbach friction factor and f = 4fFanning
Equations 2.17 and 2.18 are typical pressure-loss and flow rate equations used for liquid-system
calculations. When the definition of friction factor for laminar flow conditions (f = 64/Re) is
substituted, the Poiseuille’s equation for laminar liquid flow through a pipe of uniform (circular)
cross-section is obtained.
For the case of natural gas (compressible) flow, fluid density is a strong function of pressure and
temperature. However, it is still customary to neglect contribution of kinetic energy change in the
pressure gradient equation for gas flow because kinetic energy change contribution is typically very
small compared to the friction and elevation terms; hence, one ends up with the Eqn. 2.16 again.
When pressure-gradient equation for gas flow is integrated between two points separated by a
distance L, the density dependency with pressure is accounted for by means of the real gas law
equation. For isothermal flow conditions, this integration yields a stronger dependency of flow rate
on pressure; i.e. squared pressures (Kumar, 1987):
qG = C
(Tscpsc
)√1
f
√p2in − p2
out es
γGTavgZavgLeD
52 (2.19)
Where:
s =2MWair γG ∆z
1000TavgZavgR
g
gcPipe inclination dependent coefficient, s = 0 for horizontal pipe
[dimensionless]
Le = Les − 1
sThe ‘equivalent’ length definition based on s [m]
C =
√1000 gcπ2R
16MWairA unit system dependent constant [m2s−2K−1]
e The base of natural logarithm [≈ 2.718282]
γG Gas gravity (or gas specific gravity) [dimensionless]
15
On the basis of Eqn. 2.19, several gas flow equations have been proposed throughout the years.
Steady-state empirical gas flow equations such as Spitzglass, Weymouth, Panhandle-A, Panhandle-B,
and the IGT/AGA differ in the functional form simply because each equation utilizes a different
expression for the calculation of friction factor in Eqn. 2.19.
The analytical expressions derived above (Eqn. 2.18 and 2.19) allow calculation of pipe pressure
drops for isothermal, single-phase flow systems. However, flow of liquid and gases may not be
well represented by an isothermal model, especially when fluid inlet temperature greatly differs
from ambient temperature, which creates non-adiabatic conditions throughout the length of the
pipe. Even during adiabatic conditions, natural gas pipelines tend to cool with distance (the‘Joule-
Thomson cooling’ effect), while oil lines tend to heat – because of the very distinct Joule-Thomson
coefficient values of liquids and gases. Therefore, for non-isothermal, steady-state, single-phase flow,
the enthalpy-gradient equation is implemented:(dh
dx
)total
=
(dh
dx
)acc
+
(dh
dx
)elev
+
(dQ
dx
)(2.20)
which, integrated between two points in the pipeline separated a distance x, yields the following
explicit expression for the calculation of temperature change as a function of pressure drop:
T (x) = T∞ + (T0 − T∞)e−φ1x + (1− e−φ1x)
(η
φ1
dp
dx− φ2
φ1
)(2.21)
Where:
T0 Fluid temperature at the pipe inlet [K]
T∞ Surrounding (ambient) temperature [K]
φ1 =cUcp
πD
mHeat transfer factor [m−1]
φ2 =v
cp
dv
dx+g
cp
dzeldx
Potential and kinetic energy factor [Km−1]
Equation 2.21 gives the explicit dependency of temperature on (1) pressure drop (dp/dx), (2) heat
transfer from the environment (φ1), and (3) potential and kinetic changes (φ2). When last two
factors two are neglected, Eqn. 2.21 collapses to the following equation (Coulter, 1979):
T (x) = T∞ + (T0 − T∞)e−φ1x + (1− e−φ1x)η
φ1
dp
dx(2.22)
When one further assumes that thermodynamic changes on fluid pressure do not affect the tempera-
ture of the fluid (i.e. Joule-Thomson coefficient is close to zero); Eqn. 2.22 can be further simplified
to:
ln
(T (x)− T∞T0 − T∞
)= −φ1x (2.23)
Equation 2.23 neglects the changes in fluid enthalpy due to pressure, which can be a reasonable
assumption for liquids. For the derivation of Eqn. 2.23, fluid’s enthalpy changes are implicitly
16
evaluated through the simplified expression dh = cpdT instead of the more rigorous dh = cpdT −ηcpdp.
For simultaneous prediction of pressure and temperature changes along a pipeline, one may resort
to using an analytical flow equation for dp/dx calculations and the analytical energy equation for
dT/dx calculations, working concurrently. However, such approach decouples the combined effect
that pressure and temperature changes can have on fluid density. This might be acceptable for oil
or liquid flows but not for natural gas flows.
2.2 Two-Phase Flow
Two-phase flow occurs during production and transportation of oil and gas, where formation of
the condensed liquid (hydrocarbon or water) is determined by the overall mixture composition and
local P-T couple, according to thermodynamic phase behavior.
In order to adequately describe two-phase flow, new variables need to be defined in addition to the
fundamental single-phase flow parameters pressure, velocity and temperature:
Holdup is the liquid phase volume fraction within a volume element.
αL =VL
VL + VG=ALA
(2.24)
Void fraction, on the other hand, is the gas phase volume fraction in a given volume element.
αG =VG
VL + VG=AGA6= qGqL + qG
(2.25)
In that regard, these two parameters are the analogs of liquid and gas phase saturations in reservoir
engineering.
αL + αG = 1 (2.26)
Phasic superficial velocity is defined as the ratio of phasic volumetric flow rate to the whole
conduit cross-sectional area.
uk =qkA
(2.27)
Phasic velocity (intrinsic) is the ratio of phasic volumetric flow rate to the cross-sectional area
available for the phase in conduit.
vk =ukαk
(2.28)
Mixture velocity is the total volumetric flow rate of both phases per unit area.
vmix =qG + qLA
(2.29)
17
Slip velocity is the velocity of gas phase relative to liquid phase.
vslip = vG − vL (2.30)
Drift velocity, however, is the velocity of each phase relative to mixture velocity.
(vk)drift = vk − vmix (2.31)
Drift flux is the per unit area flow rate of each phase, relative to mixture velocity.
Jk = αk(vk − vmix) (2.32)
Phasic volume fraction is distinguished from holdup or void fraction as the ratio of phasic
superficial velocity to the mixture (sum of phasic) superficial velocities.
Λk =ukumix
=uk
uG + uL=
qGqL + qG
(2.33)
Please note that holdup (or void fraction) is equal to phasic volume fraction (αk = Λk) only for
no-slip condition (i.e. vG = vL).
Quality is the ratio of gas mass flow rate to the total mass flow rate.
x =mG
mG + mL=mG
m(2.34)
2.2.1 Flow Patterns
Gas-liquid two-phase flows are typically categorized according to the governing ‘flow pattern’;
distinct geometric arrangements of phasic volumes, the structure and continuity of the phasic
interface in the conduit. Various flow patterns (Fig. 2.1) fall under three major flow types (Shoham,
2006):
1. Dispersed flows: Bubbly (gas bubbles) or mist (liquid droplets) flow patterns, where particle
size secondary phase is ‘dispersed’ in the continuous phase.
2. Transitional flows: Slug, Churn turbulent or dispersed annular flow patterns.
3. Separated flows: Stratified, Annular flow patterns, where phases are assumed to flow in their
separate channels in the conduit.
Flow pattern is a function of phasic flow rates, fluid properties (i.e. density, viscosity and surface
tension), and pipe geometry (i.e. diameter, inclination angle). Thus, different flow patterns appear
at different inclination angles of the pipe. Fortunately, it is possible to establish a family of flow
regimes applicable at different inclination angles with little or no changes at all. However, since
pipeline networks are practically composed of near horizontal pipes and liquid phase (oil) volume
18
fraction is typically well below 50% (low liquid loading condition) in gas condensate lines, a limited
class of flow regimes is adequate for the purpose of this study. Figure 2.1 shows the flow patterns
near horizontal gas condensate lines commonly feature; (1) stratified smooth, (2) stratified wavy,
(3) annular mist and (4) mist (spray) flow.
Figure 2.1: Typical flow patterns observed in near horizontal gas condensates pipelines
Stratified flow occurs when both phases are flowing at slow rates and separated by gravity. The
interface between the phases becomes wavy as gas flow rate increases and establishes stable waves;
i.e. the stratified wavy pattern.
Annular flow appears at high gas flow rates such that gas phase presses down the liquid flowing
at the bottom of the pipe to spread as a thin film over the entire pipe wall. Liquid phase is usually
thicker at the bottom of the pipe and gas phase flows at the center of pipe; gas core, surrounded by
this liquid film. Some liquid may entrain gas phase as droplets, establishing the annular mist flow
pattern.
Mist flow occurs at very high gas flow rates with turbulence, and liquid phase completely entrains
the continuous gas phase. High gas flow rate prevents liquid droplets from precipitating at the
bottom of the pipe to establish the stratified pattern.
A major issue of two-phase flow modeling is the occurrence of diverse flow patterns throughout a long
conduit, and associated characteristic effects on the flow. Accurate prediction of flow patterns in
the conduit is essential because several flow parameters; such as frictional pressure drop, interfacial
area and associated drag-slip, heat and mass transfers are significantly affected by the governing
flow pattern.
Flow patterns are typically identified according to visual observations and flow regime maps are
prepared based on mass flow rates, void fractions and velocities as well as dimensionless variables
and correction factors, in order to make the maps more general. In fact, term ‘flow pattern’ denotes
the visual distinction among various flow topographies while ‘flow regime’ indicates how the flow
19
patterns affect the system. Then, different flow patterns does not necessarily require a change
in the adopted two-phase flow model (or utilized closure relation ships for that matter) unless a
flow regime change is present (Kleinstreuer, 2003); hence the name ‘flow regime’ map. However,
throughout this text both terms are used interchangeably.
Flow regime maps used in the petroleum industry are detailed and comprehensive for a variety of
flow conditions with different mapping criteria. These maps, such as the one presented in Fig. 2.2,
are typically a function of the superficial velocities of each phase and the inclination angle of the
pipe.
Figure 2.2: Sample flow regime map for horizontal pipes (Shoham, 2006)
Downside to the flow regime maps is that they are usually valid for specific flow conditions (i.e.
certain range of inclination angle or flow rates) and it might be necessary to switch maps with
varying flow conditions in a long pipe. Also, transition zones (or interpolation regions) may not
always be present on the map; hence, non-physical, abrupt changes in the flow regime could be
admissible according to the regime map.
2.2.2 Two-Phase Flow Models
In the early designs of gas pipelines, additional pressure drop due to presence of a second phase
was accounted for by introducing an efficiency factor in Eqn. 2.19; the steady-state, analytical flow
equations such as Weymouth, Panhandle etc.
Later on, efforts were focused on extending single-phase pressure gradient equation (Eqn. 2.12, cf.
Sec. 2.1.1) for two-phase flows using empirical correlations. Such methods include the homogeneous
equilibrium model (HEM), the separated flow model (SFM) and the drift-flux model;
all of which, through simplifying assumptions, treat the two-phase flow problem using methods
developed for single-phase flow analysis and do not leave much room for extensive analysis of
20
different flow patterns. Consequently, such approaches have limited applicability and an analytic
description of the process is required for modeling purposes.
Following the progress made in the nuclear industry, efforts to define two-phase flow using funda-
mental conservation laws gained momentum; leading to the development of the two-fluid model
which considers the phases separately using two sets of conservation equations and allows for the
analysis of transients as well (Ishii and Hibiki, 2003; Todreas and Kazimi, 1990).
In fact, HEM, SFM and Drift-Flux models form the group of ‘flow mixture models’ rather suited for
well-mixed dispersed flows (pseudo single-phase, or pseudo two-phase flows for that matter). On the
other hand, the two-fluid model, while has the best overall applicability, is especially required for
the analysis of separated flows. The two-fluid model constitutes the bulk of ‘mechanistic’ modeling
effort which stands for the ultimate, flow regime dependent modeling approach (cf. Sec. 2.2.3).
Homogeneous Equilibrium Model
Homogeneous equilibrium model (HEM, or the homogeneous no-slip model) treats two-phase flow
as if both phases are combined into a homogeneous, pseudo single-phase with gas-liquid mixture
properties; i.e. mixture density and viscosity, based on the assumption that phases are well-mixed
and in thermodynamic equilibrium. Hence, flow is modeled using a modified single-phase pressure
gradient equation (Crowe, 2006; Shoham, 2006).
− dp
dx= −
(τwπD
A
)mix
− ρmix g sin θ + ρmixvmixdvmixdx
(2.35)
The model assumes no-slip condition (vG = vL = vmix) so that phases move at the same velocity.
At no-slip condition αk = Λk. Fluid properties of the pseudo single-phase are volume fraction
dependent average of both phases. Thus, gas-liquid mixture (homogeneous) density is defined as:
ρmix = ΛGρG + ΛLρL = αGρG + αLρL (2.36)
And, gas-liquid mixture (homogeneous) viscosity is:
µmix = ΛGµG + ΛLµL (2.37)
Then, pressure gradient equation becomes:
− dp
dx= −
(τwπD
A
)G
−(τwπD
A
)L
− (αGρG + αLρL) g sin θ +d
dx
[ρG
u2G
αG+ ρL
u2L
αL
](2.38)
Similar modifications are applied for the derivation of mixture energy equation (Shoham, 2006).
21
Separated Flow Model
Separated flow model (or the Lockhart and Martinelli Model) observes two-phase flow problem as
simultaneous flow of each phase in their separate channels (cross-sectional area occupied by each
phase) within the conduit, based on the hydraulic diameter concept. Model is typically limited to
the calculation of frictional pressure drops in horizontal pipes. Two-phase pressure drop is calculated
by multiplying phasic superficial pressure gradient with a two-phase flow multiplier. (Shoham, 2006;
Crowe, 2006).
(dp
dx
)fric
= φ2L
(dp
dx
)S L
= φ2G
(dp
dx
)S G
(2.39)
and (dp
dx
)fric
=
(dp
dx
)k
= φ2k
(dp
dx
)Sk
(2.40)
Where:(dp
dx
)S k
Phasic superficial pressure gradient(dp
dx
)k
Pressure drop induced by phase k
φ2G = 1 + CX +X2 Gas phase multiplier [dimensionless]
φ2L = 1 +
C
X+
1
X2Liquid phase multiplier [dimensionless]
X =φGφL
The Lockhart and Martinelli parameter [dimensionless]
Correlations for X are derived from empirical data. Constant C takes on different values according
to flow types of both fluid, i.e. C = 12 for laminar liquid flow and turbulent gas flow; and holdup is
calculated by an appropriate correlation; i.e. αL = 1− (1 +X0.8)−0.378 (Chisholm, 1967). φk is a
function of phasic hydraulic diameter (flow area available for phase k).
Drift-Flux Model
Although two-phase stream is treated as an homogeneous mixture, based on local flow conditions,
the drift-flux model assumes either a constant or correlation dependent slippage between phases.
Therefore, flow is a function of void fraction and velocity difference between the phases, which gives
the model a better handle on flow pattern dependent modeling.
Pauchon et al. (1994) studied the two-phase flow capabilities of the drift-flux-based code TACITE,
developed for the analysis of transients in multiphase pipeline networks. The code employs separate
continuity equations for gas and liquid phases while mixture momentum and mixture energy equation
are utilized. To account for the missing momentum information, steady-state closure relationships
are used based on flow pattern. Flow regime transitions are based on the continuity of variables so
that solution for different flow regimes are identical through the transition region.
22
Two-Fluid Model
A rigorous approach to the two-phase flow problem is the two-fluid model where a separate set
of conservation equations the Eulerian frame are employed for individual phases; allowing each
phase its own velocity (and pressure or temperature) field based on the ‘interpenetrating continua’
assumption. Mass and heat transfer between phases is represented with appropriate terms in the
governing equations of both phases, without the presence of such interfacial exchange terms, phases
are essentially independent. These interaction terms, in fact, determine the degree of coupling
between phases. Strongly coupled phases tend to be in mechanical and thermal equilibrium (Ishii
and Hibiki, 2003). However, this does not necessarily imply a chemical equilibrium between the
phases. Therefore, utilization of a kinetic model is required in order to account for the kinetics of
mass transfer between the phases, depending on the problem; fluid properties and flow conditions.
Most important aspect of the two-fluid model is that it can account for transient (dynamic) and
non-equilibrium (mechanical and/or thermal, in addition to chemical) conditions by describing phasic
pressure and temperature fields. Furthermore, model is applicable to particle scale three-dimensional
and two-dimensional analysis using Navier-Stokes equations as well as larger scale one-dimensional
pipe flow analysis using Euler equations.
Although better suited for separated flows, the two-fluid approach inherently considers the dispersed
(particle) phase a ‘continuum’ as well, which may create conceptual difficulties for certain flow
conditions. Regardless of the analysis scale, some sort of averaging method is required for the
derivation of governing equations; i.e. volumetric averaging. The averaging process, while bridging
the molecular scale with the continuum model scale, generates new unknowns. In order to close
the equation set supplementary information is required; i.e. suitable constitutive equations that
describe the dynamics of individual phases and their interactions. (Kleinstreuer, 2003). Success of
the model greatly depends on required constitutive equations (closure relations) which incorporate
flow pattern associated effects.
The two-fluid model can be extended to account for multiple phases (hence the multi-fluid model) or
can account for entrained liquid droplets in the gas phase (or gas bubbles in the liquid phase) using
additional field equations; i.e. a third mass and/or momentum balance equation. In the nuclear
industry, typically a 8 equation version of the two-fluid model is utilized for the analysis of coolant
loops; accounting for 3 mass, 3 momentum and 2 energy conservation equations. Additional mass
and momentum equations describe the liquid droplets traveling (entrained) within the gas phase
(Frepoli et al., 2003; Spore et al., 2001). The most comprehensive commercial code available in the
petroleum industry; two-fluid model based OLGA, uses a similar approach and employs a 7 equation
version of the two-fluid model with a single (mixture) energy equation for analysis of various pipe
systems such as wellbores, oil and gas transmission lines, and networks (Bendiksen et al., 1989).
Derivation of governing equations for the one-dimensional, five equation two-fluid model (single
pressure and temperature; assuming mechanical and thermal equilibrium) used in this study are
presented in Sec. 4.3.
23
2.2.3 Mechanistic Modeling
‘Mechanistic’ modeling effort signifies using different (one-dimensional) flow models along a pipe,
based on predicted flow pattern for a particular section of the pipe. General approach is first to
determine the flow pattern for a section and then utilize the best suited model for accurate pressure
drop calculation.
The two-fluid model by itself could be considered a ‘mechanistic model’ through use of proper closure
relations that account for different flow patterns. However, homogeneous or drift-flux models might
be better suited for dispersed flows under certain conditions; or, a specialized slug flow model might
be necessary in order to accurately calculate pressure drop through a particular section of the pipe,
for instance. This is because, success of the seemingly general-purpose ‘mechanistic’ two-fluid model
depends on the availability of proper closure equations. Then, in a broader sense, a ‘mechanistic’
approach may as well require using different flow models (i.e. two-fluid, drift-flux etc.) together.
Overall aim of mechanistic modeling is to provide proper tools better describing the physics,
the hydrodynamics of individual flow patterns, focusing on the accuracy of pressure drops and
void fraction (or hold up) predictions. A comprehensive review of the current state-of-the-art of
mechanistic modeling in the petroleum and natural gas industry has been recently presented in the
text by Shoham (2006) for two-phase flow in oil and gas pipelines.
The fact that different two-phase flow models are to be used side by side requires that adopted
solution method allows appropriate transition between different models as well as prediction of flow
patterns along the pipe. The ‘marching algorithm’ approach, discussed next in Sec. 2.2.4, is suited
for this purpose and used frequently for steady-state, mechanistic analysis of two-phase flow in oil
and gas pipelines and wellbores.
2.2.4 Marching Algorithms
Marching algorithms solve for pressure drops over short increments of pipe, propagating the solution
step-by-step, hence ‘marching’ from inlet all the way to the outlet (Shoham, 2006).
The need for marching algorithms is three-fold:
1. Allows for the implementation of a mechanistic modeling approach that utilize separate flow
models; i.e. first, the flow pattern is determined for next pipe increment and then appropriate
model is used for pressure drop calculation
2. Because solution is focused only on a single increment, computational requirements (i.e.
storage) are significantly low (compared to methods involving simultaneous solution of several
points in a pipe). This becomes especially important considering that transportation pipelines
can be quite long.
3. Transitions in flow patterns are more precisely traced as solution marches forward with small
steps, albeit, at the expense of longer computational times.
24
Ayala and Adewumi (2003) presented a detailed account of how marching algorithm calculations
performed for two-fluid model. Governing PDEs (Euler equations) of one-dimensional, steady-state
two-fluid model are typically arranged into ordinary differential equations (ODEs, non-conservative
form) by expanding the derivatives. ODEs are discretized using finite difference method (FDM) and
numerically integrated over short increments of pipe, in a sequential way, using the Runge-Kutta
technique.
As opposed to typical FDM based approaches where solution is obtained for all the discrete points
simultaneously, Runge-Kutta (RK), ODE solver numerical procedures are suited for space marching
algorithms. In a RK numerical computation, governing equations are rearranged in the explicit
dU/dx ODE form:dU
dx+dF
dx= C (2.41)
Where, U is the vector of primary unknowns (that numerical solution is obtained for). For single-
phase gas flow calculation, for example, U = [p, vG, T ] and then dU/dx becomes:
d
dx
p
vG
T
=
vG
(∂ρG∂p
)T
ρG vG
(∂ρG∂T
)p
1 ρGvG 0
−ρGvG ηG (cp)G ρGv2G ρGvG (cp)G
−1 0
−FfG − FgGQG
(2.42)
In Eqn. 2.42, equations are properly arranged in order to eliminate zeros in the diagonal of
coefficient matrix. Primary unknowns for five-equation formulation of the two-fluid model are
typically U = [p, vG, vL, αk, T ].
The calculation begins at the inlet of the pipe, where U = U0 and RK algorithm calculates the
outlet conditions for the end of first pipe segment based on the following 5th order estimate:
U(x+ ∆x) = U0 +6∑i=1
aFi FFi (2.43)
Where, quantities FFi are computed as functions of vector dU/dx:
FFi = ∆x · dUdx
∣∣∣∣U0 +
i−1∑j=1
dFijFFj
1 ≤ i ≤ 6 (2.44)
Where, aFi and dFij are constant coefficients of the RK method.
Once outlet conditions for a given block are calculated, those values are assigned as the inlet
conditions for the next block and the algorithm keeps on going until end of the pipe is reached. This
constitutes the so called ‘marching algorithm’ via the RK method. This approach can suffer mass
and energy conservation problems since conservative property of the equations has been altered
with the non-conservative ODE arrangement (Ayala and Alp, 2008).
25
Marching algorithms (also known as the pressure traverse computing algorithms) form the basis of
most steady-state two-phase flow software and are presented by textbooks as the standard techniques
used in steady-state two-phase flow analysis for oil and gas pipelines (Shoham, 2006).
While applied successfully for the analysis of typical two-phase, steady-state flow in single pipes
and perhaps open networks (i.e. without loops; Martinez and Adewumi, 1997), space marching
algorithms are not suitable for two-phase single pipe transients, modeling counter-current flow
(re-flood) analysis where both downstream and upstream flow information in the conduit should be
accounted for via simultaneous solution. Moreover, two-phase, steady-state closed network (i.e with
loops) analysis, as well as open network analysis with marching algorithms can be a challenging
task as flow split and uneven phase separation is greatly influenced by pressures downstream of a
T-junction.
26
Chapter 3
Flow Split at T-Junctions
Two-phase flow in a single oil and gas pipeline or wellbore has been thoroughly studied over the
years; a comprehensive text is made available by Shoham (2006), but the problem of multiphase
flow in networks of multiple pipes and loops is still considered largely unresolved due to challenge of
uneven phase splitting at pipe junctions.
Single-phase, compressible or incompressible flow in networks is typically modeled using one-
dimensional, steady-state, analytical expressions which are closed form equations (hydraulic models)
derived from fundamental principles of mass and energy conservation applied over the entire pipeline
length (Sec. 2.1.1). With an analogy to electric circuits, these flow equations are coupled with
Kirchhoff laws to obtain steady-state solutions. Kirchhoff laws stand for conservation of mass at
nodes (junctions of pipes) and conservation of energy around loops. A more detailed review of
single-phase flow in pipe systems is given in Appendix A.
In such network models based on single-phase analysis (i.e. Mucharam and Adewumi, 1991),
two-phase flow effects are loosely accounted for by inducing higher pressure drops in the two-phase
lines via definition of loss coefficients (or pipe ‘efficiency’ factors) or simplified two-phase flow models
(i.e. Beggs and Brill, 1973). Nevertheless, such methods are known to be inadequate for accurate
assessment of overall system performance; besides neglecting secondary phase ‘route preference’ due
to uneven phase separation at junctions .
Use of marching algorithms, within the system wide iteration scheme, for pressure drop calculations
between the nodes (junctions) of a network could be another option. However, this would probably
require ‘marching’ each pipe over again at every system wide iteration step towards satisfying
Kirchoff laws, several times before system converges to a solution. Regardless, this approach
would again fail to acknowledge establishment of preferential secondary phase paths and associated
compositional (or quality) changes of the fluid stream unless the issue of uneven phase separation
at junctions is properly addressed. More elaborate discussion on this matter (why uneven phase
split problem still not accounted for) follows next in this chapter.
A viable approach is the simultaneous solution of two-fluid model based governing equations for
all the discrete points, or control volumes to be more precise, over the computational grid of pipe
network, with special attention to junction control volumes. A good example to such modeling
efforts is the thermal-hydraulic code TRAC which provides a distinct T-junction component used
for the analysis of nuclear reactor coolant loops (Spore et al., 2001).
27
Treatment at junctions is vital to the overall performance of network model since uneven phase
separation and associated ‘route preference’ is governed by split mechanisms at junctions. For the
practical purposes of system design and analysis, almost all other types of junctions joining three or
more pipes, as well as specialized sink/source terms such as wellheads or supply/demand nodes,
could then be represented with appropriate arrangement of consecutive three-arm junctions; in
particular tees.
There are two types of three-arm junctions (Fig. 3.1):
1. T-junctions (tee), where a second line is connected to the body of a straight main line; i.e.
two of the arms are parallel (make 180o angle) with each other,
2. Y-junctions (wye), where no straight main line is present but all the arms make an angle less
than 180o with each other.
tee wye
Figure 3.1: Three-arm junctions: tee and wye
Also, based on the directions of flow through its arms, there are two kinds of three-arm junctions:
1. Dividing (also diverging or separating)
2. Combining (also merging or joining)
In particular, a dividing T-junction is called a ‘branching tee’ when flow enters the main line
through the ‘inlet’, exits through both the ‘run’ (other end of the main line) and the ‘branch’
(side arm). If fluid is injected into the side arm as well then it is called ‘combining tee’. On the
other hand, if two opposing streams coming from both ends of the main line are combined to exit
through the side arm then it is a ‘counter–combining tee’. Finally, if flow enters the junction
only through the side arm and exits through both ends of the main line it is an ‘impacting tee’
(or counter–dividing tee, Fig. 3.2).
Branching T-junctions are essential components of various piping systems of different scales. Two-
phase flow through branching tees result in pronounced hydraulic losses and uneven separation of
phases with the split of flow stream; ultimately causing profound effects on system performance and
quality of delivered fluids. Consequently, quantifying two-phase flow associated hydraulic losses and
phase separation through T-junctions is a matter of serious concern for the design and analysis of
various pipe systems.
A counter-intuitive result of the flow split at branching tees is a sharp pressure increase in the
run direction, right after the split (Hart et al., 1991; Lahey, 1990; Ballyk and Shoukri, 1990).
Figure 3.3 shows typical pressure profile at a branching T-junction following the single-phase
28
Figure 3.2: T-Junction classification based on flow directions
flow split. Presence of such a ‘pressure spike’ is easier to observe based on Bernoulli equation
analysis; according to which an increase in the flow area (A1 < A2 for ρv1A1 = ρv2A2) ensues in
the deceleration of the flow (v1 > v2) causing an increase in the flowing pressure:
1
2ρv2
1 + ρgz1 + p1 =1
2ρv2
2 + ρgz2 + p2 (3.1)
Figure 3.3: Pressure profiles after single-phase flow split at a branching T-junction
An earlier approach that is worth mentioning is Meier and Gido (1978)’s Method of Characteristics
(MOC) based approach for single-phase flow split at an impacting tee (thus, can be extended
to branching tees easily). Mass and momentum conservation equations are first converted to
characteristic equations. Then, model is built around the idea that pressures at all three ends of a
central tee volume of ‘undefined extent’ (i.e. no real volume) are connected via a pressure drop
equation to a single (nodal) pressure that prevails at the center of the tee. Meier and Gido (1978)
model required definition of 3 separate K-factors (loss coefficients) to account for pressure drops
between the central tee node and each of the nearest nodes of all 3 arms. The K-factors of Bernoulli
29
type mechanical energy equation, are derived from experimental data.
Ki =p− pi12ρivivi
i = 1, 2, 3 (3.2)
Where, i denotes arms of the tee.
When modeling two-phase flow in a network of pipes, it is essential to accurately model the flow at
T-junctions as they can effect fluid distribution (i.e. route preference phenomenon Oranje, 1973)
and associated downstream pressure losses in the network significantly. Typical focus of T-junction
modeling has to do with the associated problem of uneven phase separation at the junction. Liquid
preferential routes determine bottlenecks and impact quality and quantity of gas received at delivery
stations.
3.1 Two-Phase Flow Split
Outlet (downstream run and branch) pressures of a branching tee have major influence on phase
separation, as well as flow split. Consequently, there exists a correspondence, an interdependence
between phase separation and inlet-to-run and inlet-to-branch pressure changes, based on inlet flow
pattern and tee geometry.
Factors affecting two-phase flow split at a T-junction can be summarized as:
• run and branch outlet pressures
• inlet-to-run, inlet-to-branch pressure changes
• inlet stream phasic kinetic energy and/or inertia (momentum in the axial direction)
• inlet flow pattern
• inlet flow mode (turbulent, laminar or transition)
• overall junction geometry (arm inclinations and diameters, split angle, joint corner geometry)
• wall surface properties
• interfacial drag
Ideally, a T-junction model should be consistent and complete in capability; predicting inlet-
to-run and inlet-to-branch pressure changes as well as uneven phase separation at a tee with
reasonable accuracy and consistency. Therefore, there are two components, called ‘sub-models’,
to a T-junction model, that determine overall success and prediction capabilities:
1. Pressure change sub-model
2. Phase separation sub-model
Sub-models are usually tested by substituting experimental data. For instance, experimental phase
distribution data is substituted in the governing equations of pressure change component to match
pressure change information that is typically obtained through extrapolation of observed pressure
profiles for developed flow along the run and branch, thus establishing the pressure change component.
Similarly, pressure change data (often extrapolated from developed flow profiles) is substituted
30
in the governing equations for phase separation component in order to match experimental phase
separation data. However, extrapolation of pressure profiles in order to obtain pressure change data
and separately substituting experimental data in sub-models has the potential danger of leading
to a weak coupling between two components of the model, rendering seemingly separate ‘pressure
change’ and ‘phase separation’ models. Consequently, individual success of sub-models does not
necessarily guarantee consistency for the overall model.
One-dimensional modeling efforts for the analysis of two-phase flow split at T-junctions fall under
three categories (Azzopardi, 1999; Peng and Shoukri, 1997; Lahey, 1986):
1. Empirical correlations; i.e. phase separation sub-model of Seeger et al. (1986).
2. Phenomenological and Mechanistic1 models; i.e. Penmatcha et al. (1996); Ballyk and Shoukri
(1990). Following the discussion of Lahey (1986), approaches such as fluid mechanics based
(hence ‘mechanistic’) models of Penmatcha et al. (1996); Saba and Lahey (1984); Fouda and
Rhodes (1974), are considered under this category.
3. Two-fluid models (control volume based); i.e. TRAC family of thermal-hydraulic codes
(Spore et al., 2001; Steinke, 1996). Although Peng and Shoukri (1997) classifies only two and
three-dimensional models as two-fluid models (i.e. Adechy and Issa, 2004; Ellison et al., 1997),
TRAC extends one-dimensional2 two-fluid model approach over the tee control volume.
Empirical correlations represent the two-phase flow split in a much simplified way that usually
leads to analytical solutions. For instance, Seeger et al. (1986) developed correlations for phase
separation at T-junctions with different branch inclinations. Unfortunately, empirical correlations
are usually valid only in the range of conditions from which they are derived. Besides, accuracy
of predictions typically depends on the amount and quality of experimental data utilized in the
derivation of correlations.
Phenomenological–mechanistic models, on the other hand, represent complicated physics of
flow separation using conservation principles (mass, momentum and/or mechanical energy) and force
balances (i.e. inertial vs. centripetal force balance on fluid particles based on streamline geometry)
that are simplified with assumptions according to physical understanding of the split mechanism. The
simplifications generally lead to models with analytical solutions. Particularly, phenomenological
methods are usually junction type (i.e. diverging, merging or impacting), geometry, and flow
pattern dependent. Also, closure relations or certain parameters such as energy loss coefficients or
momentum correction factors are expected to involve empirical correlations of such dependence.
However, applicability of phenomenological–mechanistic models can be usually extended to wider
range of flow conditions, extrapolated well outside the range on which the model is based.
An intriguing approach is the conformal mapping technique applied to two-phase flow split by
Hatziavramidis et al. (1997). Originally suited for two-phase flows that can be approximated as
potential flow (i.e. irrotational flow of incompressible and inviscid fluids), the model also employs
1Mechanistic T-junction models not to be confused with mechanistic two-phase flow modeling approach2Even for one-dimensional analysis, flow split at a branching tee requires at least two dimensions to be accounted
for; in the directions of (1) run and (2) branch, hence the term ‘one-half’ dimensional (1.5D) model.
31
the concept of the ‘free streamline’ (dividing streamline), defining the boundary of flow split at the
tee (Fig. 3.5).
One-dimensional Two-fluid models are based on the solution of one-dimensional mass and
momentum conservation equations for both phases, in both directions. In essence, two-fluid junction
model is the extension of two-fluid flow model over the junction control volume. As discussed earlier
in Sec. 2.2.2, two-fluid flow model is a rigorous approach for modeling two-phase flow in conduits,
giving rise to the simultaneous solution of coupled conservation equations (mass and momentum
balances) for phases. In fact, what separates two-fluid junction models from mechanistic junction
models is the use of phasic momentum equations in both run and branch directions (hence an
additional, 6th momentum equation) by the two-fluid models (cf. Sec. 4.12).
Success of two-fluid based junction models depends on the description of ‘irreversible’ flow loss
terms; in particular, availability of correlations for appropriate momentum correction k-factors.
Derivation of one-dimensional two-fluid model junction equations are discussed in more detail in
Sec. 4.12
A ‘complete phase separation’ is achieved at ‘critical branch flow split ratio’, when all
of the incoming gas phase is completely extracted through the branch. When complete phase
separation occurs, the T-junction indeed becomes a functional two-phase separator:
λcritical =m3
m1=x1
x3(3.3)
or
m3x3 = m1x1 (3.4)
3.2 Phenomenological and Mechanistic Junction Models
For one-dimensional analysis of steady-state, isothermal two-phase flow through a branching tee,
and assuming that junction geometry (arm inclination and diameters, branching angle, geometry of
joint corners etc.) and wall surface properties (pipe roughness) are known, there are 9 parameters
of primary interest that adequately define (and quantify) flow through the junction, hence the split:
• phasic mass flow rates through each arm (= 2× 3 = 6)
• inlet, run and branch pressures (= 3)
A 10th parameter of particular interest would have been a representative, nodal pressure for the tee.
However, modeling (at least) a two-dimensional pressure distribution with a single pressure node is
a significant challenge (cf. Sec. 4.12 and Appendix B).
For a well posed problem, that is for a unique solution to exist without over specifying the problem,
4 out of 9 parameters can be specified. Then, 5 linearly independent equations are required to
account for remaining 5 unknown parameters.
32
Conventionally, inlet phasic mass flow rates (2) and either run or branch phasic mass flow rates (2),
thus a total of 4 parameters are specified for the analytic solution. In Sec. 4.12 where two-fluid model
based flow split is discussed using a staggered grid arrangement for numerical solution, inlet phasic
mass flow rates and both run and branch outlet pressures are specified as boundary conditions.
Typically, four equations are employed by default when constructing phenomenological–mechanistic
models:
1. mixture mass balance for the tee
2. gas phase mass balance for the tee
3. inlet-to-run mixture momentum balance (or mechanical energy balance)
4. inlet-to-branch mixture mechanical energy balance (or momentum balance)
Mixture (or overall) momentum or mechanical energy balance equations, simplified into inlet-to-run
and inlet-to-branch pressure change equations, indeed constitute the pressure change component of
the model.
A 5th relation, required for completing the equation set, constitutes the phase separation component
of the model determining the degree of separation. In fact, the essential difference between several
T-junction models and what renders them phenomenological or mechanistic is the choice of this 5th
relation and associated constitutive (closure) relations for the calculation of certain parameters (i.e.
two-phase loss multiplier in pressure change equations of homogeneous equilibrium based models
and etc.).
By nature, phenomenological models are expected to be flow pattern dependent. On the other
hand, mechanistic junction models are thought to be flow pattern independent. However, employed
closure relations are frequently flow pattern dependent. Typically, flow pattern specific models
are expected to perform better than independent models. Lahey (1986) provides a list of most
significant branching T-junction studies of the time, categorizing the studies according to analyzed
tee geometry and developed model; i.e. pressure change and/or phase separation model.
3.2.1 Pressure Change Sub-Models
Inlet-to-run and inlet-to-branch pressure changes dominating the flow split and phase separation at
a tee can be calculated using:
1. mechanical energy balances in both directions
2. momentum balances in both directions
3. mechanical energy balance for branch and momentum balance for run direction.
It is well established that (i.e. by reference to Bernoulli equation) when single or two-phase flow
split takes place at a T-junction, flow continuing in the inlet direction through the run experiences
a pressure rise due to deceleration associated with flow area expansion, assuming that main pipe
diameter does not change. On the other hand, typically a pressure drop is experienced by the flow
turning into the branch.
33
Due to limitations of one-dimensional analysis, not all the factors contributing to the pressure change
at a tee can be accounted for explicitly and rigorously in the governing equations. Instead, effects
of factors, such as associated mechanical losses due to change in flow direction or due to secondary
flows, are typically combined into a single parameter (loss coefficient K for mechanical energy
equation and correction factor k for momentum equation) and represented with an additional
term in the pertinent governing equation. In that regard, use of these parameters is similar to the
representation of viscous forces of flow (wall drag) with a friction factor and a friction force for
one-dimensional analysis. Consequently, pressure changes of interest; (1) inlet-to-run pressure rise
and (2) inlet-to-branch pressure change, constitute two parts:
1. Reversible component (∆p)rev: Pressure rise due to Bernoulli effect associated with flow
expansion (or pressure drop in the case of contraction) This is the pressure change naturally
captured by the governing equations without loss coefficients or correction factors.
2. Irreversible component (∆p)irr: Pressure change due to cumulative effects of (a) wall friction,
(b) interfacial drag, (c) recirculation zones (secondary flows, Fig. 3.4) and (d) turning of
flow into the branch. The irreversible component of the total pressure change is completely
accounted for by the loss coefficients in the case of mechanical energy equations. On the other
hand, momentum correction factors predominantly account for the effect of the recirculation
zones and partially account for the effects of turning of flow (momentum transfer), wall and
interfacial frictions; since pertinent loss terms already appear in the governing momentum
equations.
Figure 3.4: Recirculation zones of a branching tee
Overall pressure change depends on relative magnitudes of reversible and irreversible components.
Typically, reversible component along the run direction, the Bernoulli type pressure rise due to
flow expansion, is greater in magnitude compared to pressure drop due to irreversible losses. For
the branch direction, magnitude of the irreversible component, pressure drop associated with flow
turning into branch and other losses, is higher than the pressure rise associated with Bernoulli effect.
34
Governing mechanical energy and momentum equations are usually simplified into pressure change
equations. While pressure equation for inlet-to-run stream is typically based on momentum
conservation, pressure equation for inlet-to-branch stream is based on mechanical energy balance.
Even for single phase flow through a T-junction, it is easier to analyze total pressure change from
inlet-to-branch using mechanical energy conservation because the need to explicitly account for the
directionality of the flow is automatically ruled out. Then, total pressure change can be easily split
into reversible and irreversible components.
∆p13 = (∆p13)rev + (∆p13)irr (3.5)
(∆p13)rev =1
2ρ(v2
3 − v21) (3.6)
(∆p13)irr =1
2K13ρv
21 (3.7)
Where; K13 is the empirical loss coefficient [dimensionless]
On the other hand, for the case of inlet-to-run, Ballyk et al. (1988) base the choice of using a
momentum balance in the run (axial) direction (rather than a mechanical energy balance) to
the better performance of momentum balance predictions in matching experimental data. Then,
inlet-to-run pressure change equation using momentum balance, with momentum correction factor
(k) accounting for an indeterminate axial momentum carried away with the branching flow, is given
as:
∆p12 = k12ρ(v22 − v2
1) (3.8)
However, Buell et al. (1993) reports that breaking the pressure change for inlet-to-run direction
into reversible and irreversible components had the difficulty of generating negative K-factors (loss
coefficients) for certain conditions. Hence, inlet-to-run pressure change can also be expressed, using
mechanical energy balance (with loss coefficient K), without breaking the pressure change into
irreversible and reversible components:
∆p12 =1
2K12ρ(v2
2 − v21) (3.9)
Typically, pressure equations for phenomenological–mechanistic models are extensions of single-
phase mechanical energy or momentum balances. With an analogy to homogeneous equilibrium
model (HEM) of two-phase flow, two-phase flow split through the T-junction is also treated as a
homogeneous mixture. However, two-phase loss multipliers are utilized to account for additional
pressure drop effects (Buell et al., 1993).
Pressure equations for homogeneous flow model can be written in the general form, separately for
(1) inlet-to-run:
∆p12 = k12(ρ2v22 − ρ1v
21) (3.10)
35
and (2) inlet-to-branch:
∆p13 =1
2(ρ∗3v
23 − ρ∗3v2
1) +K131
2
(ρ1v1)2
ρLΦ (3.11)
Where:
ρi Mixture density for arm i = 1, 2, 3 (inlet, run, branch) [kgm−3]
ρ∗3 The ‘homogeneous’ mixture density in branch [kgm−3]
ρL Liquid density (assumed constant through out the tee) [kgm−3]
Φ Two-phase loss multiplier [dimensionless]
Typically, what distinguishes different models are calculation of certain parameters; i.e. mixture
densities, homogeneous mixture density and the loss multiplier (Buell et al., 1993). Follows next is
a brief, chronological review of prominent work on pressure change models.
Fouda and Rhodes (1974) developed a pressure drop model for air–water annular flow in branching
tees, based on separated flow model (SFM) approach. Inlet-to-run pressure change (axial
pressure recovery) is computed by combining phasic momentum equations:
p2 − p1 = ∆p12 = (∆p12)G + (∆p12)L (3.12)
p2 − p1 = ∆p12 = −ksf12 ∆
[m2
AxvL +
m2
A(1− x)vG
](3.13)
p2 − p1 = ∆p12 = −ksf12
(m2
A
)2
∆
[x2
αρG+
(1− x)2
(1− α)ρL
](3.14)
Where, ksf is the separated flow model momentum correction factor [dimensionless].
Inlet-to-branch pressure drop (radial pressure drop) can be computed in a similar fashion but this
time combining phasic mechanical energy balance equations instead of momentum equations because
flow is changing direction (hence Ksf is used):
p3 − p1 = ∆p12 = −Ksf13
2
(m3
A
)2
∆
[x2
αρG+
(1− x)2
(1− α)ρL
](3.15)
Where, Ksf is the separated flow model loss coefficient [dimensionless].
Saba and Lahey (1984) use the homogeneous fluid approach (HEM) with inlet-to-run and inlet-
to-branch mixture mechanical energy balances and splits the pressure change into reversible and
irreversible components. A simplified, ‘homogeneous’ version of the pressure change equation is
then obtained as:
p2 − p1 = ∆p12 = −k12 ∆(ρmixv2mix) (3.16)
Reimann and Seeger (1986) model accounts for a reversible pressure change from inlet-to-throat (a
vena contracta) in the run direction and an additional pressure drop from this throat to a position
36
downstream along the run. Beyond that, the difference between works of Saba and Lahey (1984)
and Reimann and Seeger (1986) is the calculation of two-phase loss multiplier Φ.
Ballyk et al. (1988) use a model similar to Saba and Lahey (1984); homogeneous mixture momentum
equation to determine inlet-to-run pressure drop and homogeneous mixture mechanical energy
balance to determine inlet-to-branch pressure drop for annular flow.
Later, Ballyk and Shoukri (1990) used Fouda and Rhodes (1974) type separated flow momentum
balance for inlet-to-run pressure change and homogeneous flow (mixture) mechanical energy equation
for inlet-to-branch pressure change.
Peng et al. (1993) extend Ballyk et al. (1988) model to annular flow in T-junctions for downwardly
inclined branch arms and develop a pressure drop model based on momentum and mechanical
energy balances.
3.2.2 Phase Separation Sub-Models
Mechanistic junction models employ an additional momentum or mechanical energy equation
as the 5th equation to balance unknowns and this 5th equation determines the phase separation
(Saba and Lahey, 1984; Fouda and Rhodes, 1974).
Fouda and Rhodes (1974) suggest that inlet-to-branch pressure change, in addition to inlet-to-branch
mechanical energy balance (Sec. 3.2.1), could also be estimated using the orifice equation (based on
mechanical energy balance) recognizing that fluid discharge from main line to branch forms a ‘vena
contracta’ (Munson et al., 2009; Bird et al., 2002). If two-phase flow is treated as an homogeneous
mixture then an equation similar to mechanical energy balance (Eqn. 3.15) is obtained. However, a
separated fluid approach (SFM) requires that orifice equation is written individually for phases and
since pressure change should be same for both, model provides an auxiliary relation, a 5th equation
to account for phase separation:
(m3)G = m3 x = CGαA[2ρG(p3 − p1)]12 (3.17)
(m3)L = m3 (1− x) = CL(1− α)A[2ρL(p3 − p1)]12 (3.18)
Where, Ck is the orifice discharge coefficient [dimensionless]
Saba and Lahey (1984) suggest a fluid mechanics based 5th equation to account for phase separation.
However, instead of a control volume based field equation (i.e. momentum or mechanical energy
equation) as in two-fluid models, inlet-to-branch gas phase momentum equation is integrated along
a ‘mean’ (approximate/average) streamline to yield an auxiliary relation for inlet-to-branch pressure
change from which phase separation is determined. The length Ls of the mean streamline is
calculated based on empirical correlations. Gas phase momentum equation is chosen over liquid
phase because gas phase was observed to determine the preferential separation as the ability of gas
to make the turn into branch is the dominant factor.
37
Phenomenological models are typically developed for annular flow pattern in the inlet and
geometrically define proportions of gas and liquid flows extracted from main line into the branch.
There are two fundamental concepts in model development:
1. zones of influence; sections of main line cross-sectional area from which liquid and gas are
diverted into the branch (Fig. 3.6).
2. dividing streamlines that separate incoming (inlet) flow into inlet-to-run and inlet-to-branch
streams for each phase (Fig. 3.5).
Inlet-to-branch streamlines of both phases follow curved paths in order to accomplish a turn into
the branch and typically, phasic streamlines are expected to cross if both phases are to make turn
in the branch direction.
Figure 3.5: Dividing streamlines for two-phase flow in a branching tee
Dividing stream line approach requires geometric analysis of inlet flow pattern to determine ‘zones
of influence’ that affect the outcome of force balance between gas and liquid phases over the
streamline(s) curving into the branch; i.e. competing inertial and centripetal forces. Distance of
streamlines from inlet pipe wall determine zones of influence.
Identifying zones of influence for phases is a semi-empirical step. ‘Zones’, geometric portions of
inlet gas and liquid flows, from which phases extracted to enter the branch are determined by
superimposing cross-sectional phase distribution of the inlet on dividing streamlines. A zone of
influence exists for each phase and dividing streamlines determine both the zone boundary and
phasic (Gas–liquid) interface in the main line. The amount of gas phase extracted to the branch
comes from the area bounded by gas phase dividing streamline and the liquid film boundary.
The concept of ‘zones of influence’ was extended to two-phase flow by Azzopardi and Whalley (1982)
aiming to describe liquid film and gas extraction from the main line for annular flow at a vertical
tee. Based on geometrical considerations only, the ‘zone of influence’ is defined by the angle σ
38
(a) Single-zone model (b) Two-zone model (c) Advanced two-zone model
Figure 3.6: Zones of influence for two-phase flow in a branching tee
covering the sector of material extraction, the liquid film arc of inlet cross-section (Fig. 3.6). The
model constitutes an alternative 5th equation derived from geometrical relations provided by ‘zone
of influence’ concept.
Azzopardi and Whalley (1982) model assumes liquid and gas are extracted from the same zone of
influence. This suggests coincidental dividing streamlines for liquid and gas phases. However, due
to the significant difference in axial momentum fluxes of two phases, the gas and liquid are expected
to have different dividing streamlines.
Recognizing short comings of the single zone of influence used by Azzopardi and Whalley (1982)
and purely geometric approach lead to its definition, Shoham et al. (1987) defined separate zones of
influence for phases, for the analysis of stratified and annular flows. Consequently, phasic zones of
influence are delineated by separate gas and liquid dividing streamlines. Position of liquid streamline
(i.e. distance from pipe wall) is determined by a simplified force balance for competing centripetal
and inertial forces acting on the liquid film, thus establishing the 5th equation for phase separation
component.
Hwang et al. (1988) combined Saba and Lahey (1984) model with zone of influence concept, again
defining a separate zone of influence per phase. Phase separation (gas flow into the branch) is
governed by force balance equations written per phase and derived from a ‘modified version’ of
Euler’s equation of motion (momentum equation for a particle traveling along a streamline). In this
case, an empirical correlation is utilized to determine phasic ‘mean’ streamlines as opposed to the
force balance employed by Shoham et al. (1987). While not lumped into a single equation, whole
phase separation sub-model itself substitutes for the ‘5th equation’ needed for closure.
Both Shoham et al. (1987) and Hwang et al. (1988) models assume that radial (horizontal) distance
of dividing streamlines from pipe wall, defined by (δL, δG) remain constant and independent of
vertical position on the plane of cross-section, in accordance with the original intent of vertical
39
annular flow analysis. However, the major difference between the models is how the positions of
streamlines is determined.
In an attempt to improve the performance of dividing streamlines approach of Hwang et al. (1988)
for annular flow in horizontal, equal-diameter T-junctions, Ballyk and Shoukri (1990) computes
the position of dividing streamlines at varied elevations of the inlet cross-section, introducing the
concept of ‘development length’.
Hart et al. (1991) with Double Stream Model (DSM), and then Ottens et al. (1994)3with Advanced
Double Stream Model (ADSM) further developed the concept of dividing streamlines for separated
flow patterns (i.e. stratified flow) using a purely fluid mechanics based approach. Both models
derive a governing equation for phase separation based on phasic Bernoulli (mechanical energy)
equations and associated loss coefficients. More detailed discussion on Double Stream Model follows
in Sec. 3.2.3.
Peng and Shoukri (1997) further developed and generalized the model of Ballyk and Shoukri (1990)
to account for the effect of gravity, hence branch inclination, on phase separation.
Penmatcha et al. (1996) combined the Hwang et al. (1988) and Shoham et al. (1987) models for
stratified wavy conditions. Separate momentum equations are written for phasic streamlines and
flow areas required for the equations are determined by inlet flow geometry. Marti and Shoham
(1997) extended this model for reduced T-junctions with different branch arm inclinations.
3.2.3 The Double Stream Model
Hart et al. (1991) developed Bernoulli equation based Double Stream Model (DSM) to predict flow
split at T-junctions based on the dividing (separating) streamline concept which assumes phases
flow along their individual channels or streamlines. DSM is essentially a phase separation sub-model
for separated flows. Although Bernoulli equation comes with the inherent simplifying assumption of
constant phasic density all throughout the T-junction volume, it is a reasonable assumption for an
adequately small tee volume.
Bernoulli equations are written for inlet-to-run and inlet-to-branch directions, per phase:
1
2(ρk)1(vk)
21 + (ρk)1gz1 + p1 =
1
2(ρk)2(vk)
22 + (ρk)2gz2 + p2 +
irreversible loss term︷ ︸︸ ︷1
2(ρk)1(vk)
21Kk12 (3.19)
1
2(ρk)1(vk)
21 + (ρk)1gz1 + p1 =
1
2(ρk)3(vk)
23 + (ρk)3gz3 + p3 +
irreversible loss term︷ ︸︸ ︷1
2(ρk)1(vk)
21Kk13 (3.20)
Where:
3Author wishes to acknowledge Dr.Marcel Ottens (Department of Biotechnology, Delft University of Technology,Netherlands) for providing the article, which has not been possible to obtain otherwise
40
(ρk)i = ρk Phase density, constant throughout the tee volume [kgm−3]
Kk Energy loss coefficient for phase k (defined in Tab. 3.1) [dimensionless]
In this arrangement, outgoing phasic velocities (vk)2, (vk)3 are simply controlled by outlet pressures
and irreversible loss terms. Besides the velocities, calculation of mass flow rates going through each
arm, hence actual phase separation, requires void fraction information:
(mk)1 = (ρk)1(vk)1(αk)1A1
(mk)2 = (ρk)2(vk)2(αk)2A2
(mk)3 = (ρk)3(vk)3(αk)3A3
(3.21)
At this point, it is recognized that leaving void fractions (or holdups) as unknowns would require
additional equations to be included in the solution, as is the case with other phenomenological–
mechanistic junction models. This fact gives rise to another simplification via the assumption
(αk)1∼= (αk)2
∼= (αk)3, based on the previous assumption of adequately small tee volume. With the
assumption of constant void fraction through out the tee, DSM establishes that not only outgoing
velocities but also the actual flow split is, in fact, controlled by outlet pressures and irreversible loss
terms.
Four (4) Bernoulli equations (one per phase, per tee arm direction) are re-arranged into a closed
form ‘phase separation equation’ relating gas phase branch mass intake to the liquid phase branch
mass intake:
(a4 − 1)λ2L + 2λL − κ
[(a4 − 1)λ2
G + 2(λG − λ0)]
+1
βL(FrL)13− 2λ0 (3.22)
Where:
a =D2
D3Main line (inlet) to branch diameter ratio [dimensionless]
FrL13 =(vL)2
1
g(D1 −D3)Modified Froude number for liquid phase [dimensionless]
βk Constant depending on inlet phasic flow mode [dimensionless]
βk = 1.00 if flow is turbulent
βk = 1.54 if flow is laminar
κ =ρGv
2G
ρLv2L
Ratio of phasic inlet kinetic energies [dimensionless]
λk =(mk)3
(mk)1Phasic branch mass intake fraction [dimensionless]
λ0 =1
2(1 + (KG)12 − (KG)13) Junction gas phase energy dissipation factor [dimensionless]
Equation 3.22 is further simplified by taking phasic loss coefficients equal (KGi = KLi), based on
41
the assumption that phases are flowing along their separate streamlines (or channels). Nevertheless,
with this assumption DSM became limited to very small holdup values (αL ≤ 0.06) for accurate
phase separation predictions.
The most important point is, because it is based on mechanical energy equations, model had no
true way of recognizing the effect of split angle unless accounted for by other means; in this case by
the irreversible loss terms (and associated loss coefficients). Although Bernoulli equation is applied
on a streamline that curves into the branch, inherently the equation has no means of observing the
directional change in flow. This is to be expected since Bernoulli equation is indeed a non-vectorial
mechanical energy equation derived from a directional momentum balance with certain simplifying
assumptions Please see Munson et al. (2009) for the details of Bernoulli equation derivation from
momentum balance.
To make it more clear, without irreversible loss terms, Bernoulli equation can not account for a
directional change in flow (at least in its usual form and perception) and therefore DSM can not
account for the effect of split angle when irreversible losses are ignored. In a way, Steinke (1996)
also points out to this misconception lead by inadvertent approximation of split angle effects via
irreversible loss terms. However, Steinke (1996) discussion, or Spore et al. (2001) for that matter,
is quite interesting that Bernoulli equation is arranged in a form to resemble a non-conservative
momentum balance accounting for flow direction.
K-factors (loss coefficients) typically depend solely on surface properties (roughness) and geometry
of the conduit. The geometry element is expected to inherently introduce the effect of split angle.
Nevertheless, there is no certainty that a K-factor obtained by matching experimental data for
particular flow conditions, for a particular flow pattern and tee structure, would yield satisfactory
results for significantly different flow conditions and flow pattern if actual physics of the flow is not
properly acknowledged.
Ottens et al. (1994) relieved DSM from the small holdup constraint by deriving appropriate
correlations for liquid phase K-factors (KL) and establishing the Advanced Double Stream Model
(ADSM).
The advantage of DSM (or ADSM thereof), is, while it is certainly necessary to account for the
details of mechanisms controlling the split (i.e. flow pattern geometry), macroscopic balances should
be satisfied at all times and thus a satisfactory model employing simple energy or momentum
balances is preferable to flow pattern dependent methods. Nevertheless, when flow pattern geometry
and fluid distribution in the main line is not accounted for, DSM and similar mechanistic models
are likely to fail if branch diameter is smaller than main line diameter and/or side port orientation
is not horizontal (position of the opening for branch on the main line surface, not the inclination of
branch arm). Especially, if model has no way of recognizing liquid level or elevation of the highest
point the liquid film within the main line reaches it could still predict liquid phase going into the
branch while it is not physically possible. One-dimensional two-fluids model are also subject to
same problem unless a means to account for flow geometry is incorporated in the model.
42
3.2.4 Correction Factor and Loss Coefficient Correlations
For single-phase flow through T-junctions hydraulic loss coefficients (K-factors) and/or momentum
correction factors (k-factors) are dependent on overall junction geometry and branch mass intake
fraction:
λ =m3
m1
Single-phase T-junction parameters (k-factors and K-coefficients) are typically correlated with
experimental data based on polynomial fits:
k =∑
anλn−1
or
K =∑
bnλn−1
Hence, these correlations are specific to experimental conditions and do not account for T-junction
geometry or wall surface properties explicitly. Polynomial coefficients an and/or bn differ between
correlations depending on flow conditions, junction geometry and surface properties for which the
experimental data is obtained. Table 3.1 lists single-phase correlations from three different studies
(Hwang et al., 1988; Ballyk et al., 1988; Buell et al., 1993) for horizontal branching tees with 90o
angle. Buell et al. (1993) and Ballyk et al. (1988) use momentum equations for inlet-to-run direction.
Table 3.1: Correction factor and loss coefficient correlations for a 90o branching tee
inlet-to-run:
Hwang (1988) K12 = 0.3419− 1.223
(m2
m1
)+ 0.8638
(m2
m1
)2
Ballyk (1988) k12 = 0.704− 0.320λ− 0.028λ2
Buell (1993) k12 = 0.57− 0.102λ+ 0.107λ2
inlet-to-branch:
Hwang (1988) K13 = 1− 0.8285λ+ 0.6924λ2
Ballyk (1988) K13 = 1.081− 0.914λ+ 1.050λ2
Buell (1993) K13 = 1− 0.982λ+ 1.843λ2 − 0.717λ3
Gardel (1957), however, derived K-factor correlations for single-phase flow that accounts for changes
43
in the branching angle as well:
K12 = 0.03(1− λ)2 + 0.35λ2 − 0.2λ(1− λ) (3.23)
K13 = 0.95(1− λ)2 + λ2
[1.3 cot
(1
1θ
)1− 0.9
√r]
+ 0.8λ(1− λ) cot
(1
2θ
)(3.24)
Comparison of single-phase flow K-factor correlations of several authors is given in Fig. 3.7.
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 0.2 0.4 0.6 0.8 1
K13
λG
Gardel (1957)
Ballyk (1988)
Buell (1993)
Figure 3.7: Comparison of K-factors by various correlations
Besides empirical correlations (e.g. Gardel), authors such as Oka and Ito (2005); Bassett et al. (2001)
derive single-phase loss coefficient correlations based on momentum balances written over CVs at the
tee. Pressure change predictions of mechanical energy equations are related to momentum equations
neglecting momentum correction factors and thus analytic expressions for hydraulic (energy) loss
coefficients (K-factors) can be obtained.
However, all the aforementioned correlations are for single-phase flow conditions and for two-phase
conditions these parameters are observed to be dependent on flow pattern and flow mode (i.e.
turbulent or stratified) as well. For instance, El-Shaboury et al. (2007) express phasic loss coefficients
for an impacting tee as functions of Reynolds number.
Similarly, Ottens et al. (1994) derived correlations for liquid phase loss coefficients to complement
Gardel’s single-phase (gas) correlations (for use with ADSM) based on liquid phase Reynolds number
and gas phase K-factors:
(KL)12 =
(11.48 +
1063
1− (ReL)2− 12.67κ
)(KG)12 (3.25)
44
(KL)13 =
(1.247− 69.26
1− (ReL)3− 0.198κ
)(KG)13 (3.26)
Where, κ is the gas phase to liquid phase ratio of inlet kinetic energies and ReL is the liquid phase
Reynolds number.
3.2.5 Film-stop Phenomenon
In annular flow, a substantial part of liquid is flowing as a thin liquid film spread along the conduit
wall. It is, therefore, expected that the liquid film in the main tube segment that is intercepted by
the branch cross-sectional area will be the first component to be extracted into the branch.
During annular flow, as the liquid film approaches the side port of a branching tee, a sudden increase
in the liquid amount going to branch is observed corresponding to a small increase in gas off-take
(Roberts et al., 1995). This ‘step’ increase in liquid amount is attributed to liquid film slowing down
as inlet-to-run pressure spike along the main line increases and ultimately coming to a complete
halt, hence the name ‘film-stop’, at some threshold pressure value, lending itself to easy withdraw
through the side port.
Roberts et al. (1995) extended Azzopardi and Whalley (1982) model to account for the film-stop
effect in annular and semi-annular flows. DSM of Hart et al. (1991) is also applicable to such flow
conditions but does not account for the film-stop effect thus fails to match experimental data beyond
a certain range.
Increasing the branch flow, which corresponds to decreasing the branch downstream pressure, causes
more gas to be extracted through the branch, raising the branch quality. Increasing the branch
flow is associated with a pressure rise through the junction run due to flow expansion. This adverse
pressure rise yields an additional force which tends to drive the flow through the branch. Because
the gas (lighter) phase in the inlet arm has lower axial momentum than the liquid (heavier) phase,
it tends to respond more readily to the decreasing pressure through the branch and the increasing
pressure through the run. Accordingly, increasing the branch flow tends to result in an increase in
the branch flow quality.
For annular flow condition at horizontal tees, inclination of the branch has a significant influence on
the separation of phases because of the non-uniform distribution (on the inlet cross-section) of the
liquid film, attributed to gravity.
3.2.6 Summary
Lahey (1986) concluded that none of the existing models was successfully applicable to a whole
range of flow conditions (different inlet flow patterns and flowing pressures) without first generating
experimental data for loss coefficients or correction factors and recommended a composite model as
an interim solution.
Because Saba and Lahey (1984) model predicted phase separation for high branch mass off-take
45
ratios (0.5 ≤ m3m1≤ 1.0) correctly, Lahey (1986) suggested using Azzopardi and Whalley (1982) model
for the range (0.0 ≤ m3m1≤ 0.05) and a the empirical correlation of Zetzmann4 for the remaining,
uncovered range (0.05 ≤ m3m1≤ 0.5).
Buell et al. (1993) compared experimental results with different phase separation and pressure
change sub-models:
1. Fouda and Rhodes (1974), Separated Flow Model (SFM)
2. Saba and Lahey (1984), Homogeneous Flow Model (HFM)
3. Reimann and Seeger (1986)
4. Hwang et al. (1988)
5. Ballyk et al. (1988)
Recognizing that performance of models varied over branch mass intake fraction, Buell et al. (1993)
also acknowledged the difficulty for a single pressure change model to satisfactorily predict pressure
changes for all flow conditions at a tee, and concluded that ultimately a suite of flow pattern specific
models may be required.
Walters et al. (1998)5 compared performances of 4 pressure change and 4 phase distribution
sub-models for different inlet/branch diameter ratios (reduced tees) (Tab. 3.2).
Table 3.2: Models compared by Walters et al. (1998)
Pressure change: Phase distribution:
Fouda and Rhodes (1974) Shoham et al. (1987)
Saba and Lahey (1984) Azzopardi (1988)
Reimann and Seeger (1986) Hwang et al. (1988)
Hwang et al. (1988) Hart et al. (1991)
Walters et al. (1998) concluded that SFM of Fouda and Rhodes (1974) gave the best overall pressure
drop predictions while DSM of Hart et al. (1991) gave good results for stratified, wavy and annular
flow regimes albeit limited to small holdup values.
4Zetzmann, K. (1982). Phasenseparation und Druckfall in zweiphasen Durchstromten vertikalen Rohrabzweigungen,Doctorate Thesis, University of Hannover, FRG. (Lahey, 1986)
5Azzopardi, B.J., (1988). An additional mechanism in the flow split of high quality gas–liquid flows at a T-junction.UKAEA Report AERE-R 13058. (Walters et al., 1998)
46
Chapter 4
Model Conceptualization
4.1 Conservative Property: FVM vs FDM
Numerical solution to Euler equations can be achieved through mathematical approximation of
governing PDEs using algebraic equations where differential operators are replaced with difference
operators. This leads to the formulation of algebraic Finite Difference Equations (FDEs), also
known as discrete or discretization equations since they require information from discrete points in
the flow domain. Hence, continuous information available in the exact solution of PDEs is replaced
with discrete, approximate information of the FDEs. Approximate values of flow parameters are
then computed for discrete points (nodes) in the domain, which is the basis of the Finite Difference
Method (FDM).
With some physical insight, it is realized that numerical solution could also be achieved by initially
dividing the physical domain of flow into imaginary, discrete (non-overlapping) computation sub-
domains, namely Control Volumes (CVs), that are fixed boundary, open systems over which the
continuum approach and the conservation principles prevail. The algebraic governing equations are
then obtained by applying conservation principles over these CVs. The fluid and flow properties at
the center of each CV typically represent the volumetric average of these values for that CV. This is
the basis of Finite Volume Method (FVM). In fact, writing algebraic equations for a CV can be
seen as a preliminary step in deriving the governing PDEs of flow and thus the algebraic equations
should collapse to the PDEs when a CV of infinitesimal size is considered.
Finite Volume Method equations are also described as integrated forms of the PDEs. However, and
because some type of finite difference (FD) approximation might be required for the definition of
flux values at each CV face, some authors describe the FV approach as an FD method applied to the
integrated form of the flow equations; i.e. Integral (or Integrated) Finite Difference Method (IFDM).
This more general approach to obtain the FVM discretized equations are especially necessary
when complex; irregular geometries are adopted as CVs. Then, governing PDEs are integrated in
a piece wise continuous manner over the CVs and the integral is then evaluated with necessary
approximations.
Governing algebraic equations of the FVM express the conservation principles over finite CVs while
governing PDEs articulate conservation principles over infinitesimal CVs. Assuming infinite machine
precision, FVM retains the ‘conservative property’ at the algebraic equation level as well as at the
47
PDE level. However, because they are derived using mathematically defined differential operators;
algebraic equations of the FDM may not retain the conservative property built in the original PDE
formulation when applied to the finite computational grid. Since FVM algebraic equations possess
the conservative property, conservation of mass, momentum and energy is always satisfied (ignoring
limited computer precision) over any group of CVs and thus over the entire domain regardless of
the size or number of CVs, and not only at the never-attained limit of infinitesimal CV size. This
constitutes the fundamental difference between FDM and FVM. The extensive discussion by Roache
(1998) on the conservative property is highly recommended for the interested party.
In practice, however, what actually matters is to retain the conservative property of the original
PDEs in the final form of the governing algebraic (discretized) equations, which is to retain flux
terms intact and by doing so, ensure that ‘flux matching’ concept prevails at the faces of CVs.
4.2 Single-Phase Flow Equations
All governing equations for fluids in motion are based on the principles of mass conservation,
Newton’s second law, and the first law of thermodynamics applied to a control mass system
(CM). These fundamental physical principles are outlined below. It is important to recognize that
conservation statements are straightforwardly defined for systems of fixed mass (i.e. control mass
systems – Lagrangian view), but they can be readily extended to system of fixed volume, variable
mass (i.e. control volumes – Eulerian view) using the Reynolds Transport Theorem (Munson et al.,
2009; Kleinstreuer, 2003; Todreas and Kazimi, 1990). This discussion is given below.
For a system containing a fixed amount of matter (i.e. the control mass system CM), the three
statements of conservation can be written as:
∂MCM
∂t= 0
i.e. by definition of a control mass system CM, total amount of mass
inside the system is constant (conserved) and does not change with
time. M =∫ρdV in general, and M = ρV for systems of uniform
density.
d ~PCMdt
= ~FCMi.e. the sum of all external forces acting on a CM equals its rate
of change of linear momentum. Linear momentum is defined as
~P =∫ρ~vdV , while ~P = M~v for systems of uniform density.
dECMdt
=(Q− W
)CM
i.e. the rate of change of total energy in the CM equals the amount of
heat energy entering the CM minus the work done by the CM.
48
In this expression;
E =
∫ρedV in general
E = me for systems of uniform density
e = e∗ +v2
2+ g∆z intrinsic energy content per unit mass
e∗ system internal energy
v system velocity
∆z system elevation with respect to a datum z0
Q rate of heat transfer
(positive when added to the system)
W rate of work
(positive when done by the system)
These equations, as stated, apply to systems that maintain fixed amount of mass. In fluid dynamics,
we look at systems of fixed volume stationary in space or ‘control volumes’, rather than systems of
fixed amount of mass. Evidently, in the control volume representation, the amount of fluid contained
inside the CV changes as fluid flows through the CV. The conservation statement that provides
the link between the fixed mass system description and its equivalent control volume description is
called the Reynolds transport theorem, which states that:
dNCM
dt=
rate of change of N
in the CM
=
rate of change of N
in a CV
+
rate of change of flux of N
out of the CV
(4.1)
The Reynolds transport theorem is the statement that relates a CV representation to a CM
representation. dNCMdt refers to the rate of change of any extensive property N within the fixed
mass system representation or CM. The statement above assumes that the CM and CV coincide
at a given time and then tracks how the initial fixed amount of mass leaves the CV at a later
time. The Reynolds transport theorem is a fundamental theorem of fluid dynamics and it is used
in formulating the governing equations of fluid flow for a CV representation or Eulerian frame of
reference. It is important to note that in the theorem above, N represents any extensive property
of the system, such as mass (M), linear momentum (~P ), total energy (E) or entropy (S).
4.2.1 Conservation of Mass
Let us consider the extensive property total mass; M = (ρA∆x), within this finite CV shown in
Fig. 4.1. In a finite volume representation, node point values of properties at x are assumed to be
averages over the entire control volume. The Reynolds transport theorem applied to this CV states:
dMCM
dt=
rate of change of
mass in a CV
+
rate of flux of
mass crossing
the CS
(4.2)
49
Figure 4.1: Finite control volume
Where, CS control surface or surface that defines the CV
Conservation of mass for this CM imposesdMCM
dt= 0, thus the statement above can be re-written
for a discrete period of time ∆t as:
0 =
change of mass
in the CV
during ∆t
+
net flow of mass
leaving the CS
during ∆t
(4.3)
Where:change of mass
in a CV
during∆t
= (ρA∆x)t+∆tx − (ρA∆x)tx
advective mass flux
entering the CS
during∆t
= (ρ~vA)t′
x−∆x2
∆t
advective mass flux
leaving the CS
during∆t
= (ρ~vA)t′
x+ ∆x2
∆t
In above expressions, t′ represents any suitable time reference within the time interval t′ = t and
t′ = t+ ∆t. This selection can make the resulting algebraic numerical scheme to be fully implicit
(for the case t′ = t+ ∆t), fully explicit (for the case t′ = t) or a hybrid of these two. Substituting
these expressions into Eqn. 4.3, one obtains:
[(ρA∆x)t+∆t
x − (ρA∆x)tx]
+[(ρ~vA)t
′
x+ ∆x2
∆t− (ρ~vA)t′
x−∆x2
∆t]
= 0 (4.4)
50
or equivalently, by dividing Eqn. 4.4 by ∆x and ∆t:
(ρA)t+∆tx − (ρA)tx
∆t+
(ρ~vA)t′
x+ ∆x2
− (ρ~vA)t′
x−∆x2
∆x= 0 (4.5)
Equation 4.5 represents the 1D Finite Volume representation of conservation of mass for advective
flow implemented in this study. Please note that if one takes the limit ∆t,∆x→ 0 then the PDE
form of conservation of mass equation is readily obtained.
∂(ρA)
∂t+∂(ρ~vA)
∂x= 0 (4.6)
which, for a system of constant cross-sectional area, becomes:
∂ρ
∂t+
∂
∂x(ρv) = 0 (4.7)
An alternate way of obtaining the FVM equation is to start with the corresponding PDE represen-
tation (Eqn. 4.6) and carry out its integration over time (t to t+ ∆t) and discrete space (x− ∆x2 to
x+ ∆x2 ) to obtain:
x+ ∆x2∫
x−∆x2
t+∆t∫t
∂(ρA)
∂tdt
dx+
t+∆t∫t
x+ ∆x
2∫x−∆x
2
∂
∂x(ρ~vA)dx
dt = 0 (4.8)
Because the sequence of integration is irrelevant, the most convenient sequence that groups the
relevant differentials is elected. Thus;
x+ ∆x2∫
x−∆x2
[(ρA)t+∆t − (ρA)t
]dx+
t+∆t∫t
[(ρ~vA)x+ ∆x
2− (ρ~vA)x−∆x
2
]dt = 0 (4.9)
In a discretized domain, node point values are assumed to be averages over the domain x− ∆x2 <
x < x+ ∆x2 , and the first integral is easily evaluated. For the second integral a decision must be
made regarding whether fluxes (ρ~v) will be considered constant and equal to their values at time
level t (explicit scheme), or t + ∆t (implicit scheme) or some combination (Crank-Nicholson or
Semi-Implicit). By using a generic time reference t′, one obtains:
[(ρA)t+∆t
x − (ρA)tx] x+ ∆x
2∫x−∆x
2
dx+[(ρ~vA)t
′
x+ ∆x2
− (ρ~vA)t′
x−∆x2
] t+∆t∫t
dt = 0 (4.10)
51
or
(ρA)t+∆tx − (ρA)tx
∆t+
(ρ~vA)t+∆tx+ ∆x
2
− (ρ~vA)t+∆tx−∆x
2
∆x= 0 (4.11)
Eqn. 4.11 is identical to Eqn. 4.5. In either expression the time dependency of the flux terms has
not been explicitly defined. This decision would render the resulting numerical scheme implicit,
explicit or Crank-Nicholson or semi-implicit.
4.2.2 Conservation of Momentum
Figure 4.2: Control volume
Let us now consider the case of conservation of linear momentum. For the CV depicted in Fig. 4.2,
one can define:
Extensive property = Linear momentum = ~P = M~v = (ρA∆x)~v
Intensive property = Momentum per unit mass = ~v
Flux of
Extensive property = mass flux × intensive property = (ρ~v)~v
by advection
For the case of the momentum balance equation, Reynolds transport theorem states:
d ~PCMdt
=
time rate increase
of momentum
in CV
+
net rate of flux of
momentum out
the CS
(4.12)
where d ~PCMdt = ~FCM , as defined by the second law of Newton applied to the CM. The application of
the Reynolds transport theorem assumes that the CM and CV coincide at the same initial time,
which allows one to write ~FCMs = ~FCV . The conservation statement defined by Eqn. 4.12 can thus
52
be written for a discrete period of time ∆t as:
~FCV ∆t =
increase of momentum
in CV
during ∆t
+
net flow of momentum
crossing the CS
during ∆t
(4.13)
Where:change of momentum
in a CV
during ∆t
= (ρA∆x~v)t+∆tx − (ρA∆x~v)tx
momentum flux
entering the CV surface at LHS
during ∆t
= (ρ~v~vA)t′
x−∆x2
∆t
momentum flux
leaving the CV surface at RHS
during ∆t
= (ρ~v~vA)t′
x+ ∆x2
∆t
Thus Eqn. 4.14 is obtained;
~FCV ∆t =[(ρ~vA)t+∆t
x ∆x− (ρ~vA)tx∆x]
+[(ρ~v~vA)t
′
x+ ∆x2
∆t− (ρ~v~vA)t′
x−∆x2
∆t]
(4.14)
Equation 4.14 states that the sum of all forces (surface and body forces) acting on the finite CV over
the finite period of time ∆t equals the accumulation of linear momentum in the CV plus the net
flux of momentum though the CV surface that takes place over during the time period ∆t. Forces
(surface and body forces) acting on the CV of fluid of interest are shown in Fig. 4.3. Body Forces
( ~FB) act on the entire system (i.e. gravitational forces) while Surface Forces ( ~Fs) are transmitted
through an area or surface. The net force acting on the CV is hence given by:
Figure 4.3: Force Balance in an Elementary Pipe CV
53
~FCV = ~FS + ~FB (4.15)
Surface forces can be classified as:
1. Tangential forces: Those tangential to flow surface of CV (i.e. friction or shear force).
2. Normal forces: Those perpendicular to the flow surface of CV (i.e. pressure). Pressure force
is isotropic (i.e. it has the same magnitude in all directions) and it is always considered to be
applied in compression (not tension).
From the free body diagram in Fig. 4.3, one can readily derive:
~FCV =
Net force
in the direction
of flow
= −[(pA)x+ ∆x
2− (pA)x−∆x
2
]− [τws]− [Mg sin θ] (4.16)
Where:
θ Angle of inclination [rad]
s CV surface in contact with pipe or total wetted area [m2]
τw Shear stress [Nm−2]
Shear stress (τw) in 1D flow is typically evaluated as a function of the dimensionless friction factor
coefficient. By definition, friction factor is a measure of the ratio of shear stress to inertial force per
unit area. Darcy-Weisbach friction factor is defined as:
f = 4τw
12ρv
2(4.17)
Substituting the force balance in Eqn. 4.16 into the conservation statement in Eqn. 4.14, one can
obtain:
[(ρ~vA)t+∆t
x ∆x− (ρ~vA)tx∆x]
+[(ρ~v~vA)t
′
x+ ∆x2
∆t− (ρ~v~vA)t′
x−∆x2
∆t]
= (pA)t′
x−∆x2
− (pA)t′
x+ ∆x2
+ [−τws− ρA∆xg sin θ]t′
x ∆t (4.18)
Equation 4.18 can now be re-written per unit volume and time as shown below:
(ρ~vA)t+∆tx − (ρ~vA)tx
∆t+
(ρ~v~vA)t′
x+ ∆x2
− (ρ~v~vA)t′
x−∆x2
∆x
= −(pA)t
′
x+ ∆x2
− (pA)t′
x−∆x2
∆x+ [−τwLw − ρgA sin θ]t
′
x (4.19)
Equation 4.19 represents the 1D linear momentum conservation equation for a finite volume element.
54
In this expression, Lw is wetted perimeter (Lw = πD). When one takes the limit as ∆t,∆x→ 0,
the PDE form of conservation of momentum is obtained:
∂(ρ~vA)
∂t+∂(ρ~v~vA)
∂x= −∂(pA)
∂x− τwLw − ρgA sin θ (4.20)
One can also derive the FVM expression in Eqn. 4.19 by direct integration of the PDE in Eqn. 4.20,
as shown in the previous section for the case of mass conservation.
4.2.3 Conservation of Energy
Figure 4.4: Energy Balance in an Elementary Pipe CV
For the case of energy conservation in a CV (Fig. 4.4), one defines:
Extensive property = Energy = E = Me = (ρA∆x)e
Intensive property = Energy per unit mass = e = e∗ + v2
2 + g∆zel
Flux of
extensive property = mass flux × intensive property = (ρ~v)e
by advection
Where:
∆zel System elevation with respect to a datum z0 Since x represents the distance
from center of CV to the entrance of pipe: ∆zel = x sin θ
θ Pipe inclination
For the case of energy conservation over the control volume in Fig. 4.4, the Reynolds transport
theorem states:
d ~ECMdt
=
time rate of change of
energy in CV
+
net rate of
energy leaving
through the CS
(4.21)
55
Where;∂Esys
∂t =(Q− W
)sys
, as required by the first law of thermodynamics and(Q− W
)sys
=(Q− W
)CV
because CM and CV are assumed to coincide at the initial time.
For a finite period of time ∆t Eqn. 4.21 becomes:
(Q− W
)CM
∆t =
change of energy
in CV
during ∆t
+
net flow of energy
out of CS
during ∆t
(4.22)
Where:change of energy
in CV
during ∆t
= (ρA∆xe)t+∆tx − (ρA∆xe)tx
energy flux
entering the CV surface at LHS
during ∆t
= (ρ~veA)t′
x−∆x2
∆t
energy flux
leaving the CV surface at RHS
during ∆t
= (ρ~ve)t′
x+ ∆x2
A∆t
change in energy
in CV due to heat and work
during ∆t
=(Q− W
)CV
∆t
Thus;(Q− W
)CV
∆t = +[(ρeA)t+∆t
x ∆x− (ρeA)tx∆x]
+[(ρ~veA)t
′
x+ ∆x2
∆t− (ρ~veA)t′
x−∆x2
∆t]
(4.23)
Heat transfer with the surroundings (Q) in pipe flow is typically calculated as a function of an
overall heat transfer coefficient (cU ). The overall heat transfer coefficient (cU ) relates the total heat
transfer (Q) to the surface area available for heat transfer (s) and temperature difference (∆T ), as
shown below:
Q = cUs∆T (4.24)
Where:
s = πD∆x
∆T = (T∞ − T )
Q is positive when heat enters the CV (T∞ > T ) and negative if heat is leaving the CV (T∞ < T ). cU
is a required design parameter in the calculation of energy balances of comprehensive pipeline models.
It characterizes the heat interaction between the flowing fluid and the environment (i.e. materials
surrounding the pipeline). In non-insulated pipes, pipe walls are considered infinitely conductive
56
because they do not pose any significant heat flow resistance due to the typically large conductivity
of pipe materials and usually small wall thicknesses. In such situations, cU basically equals the
effective convective coefficient between external side of the pipe and the environment. Values of cU
ultimately depend on the environmental conditions around the pipe and for example, whether the
pipe is buried or not. For a buried pipe, typical values of cU are 0.1 to 2 BTU/ft2 − hr−o F , while
it could vary between 2 and 20 BTU/ft2 − hr −o F when the pipeline is directly exposed to the
atmosphere.
For buried pipes, cU ends up being more or less equal to the conductivity of the ground or soil. This
statement assumes that soil temperature is prescribed at a distance L = 1ft from the pipe external
diameter, since cU = k/L for approximately constant heat flow areas. Thermal conductivity of
earth materials is found around k = 1.4BTU/ft2 − hr −o F , with values changing with respect
to soil type, soil density, moisture content, salt concentration, among others. While soil thermal
science literature provide several correlations for conductivity as a function of these properties,
reasonable values of cU can be determined from known values used for old lines placed in similar
soil conditions. Most authors, however, would assume the average value of 1 BTU/ft2 − hr −o Ffor buried pipes for preliminary calculations. When a pipeline is directly exposed to the atmosphere,
cU becomes practically equal to the convection heat transfer coefficient between the pipe external
surface and the air surrounding it. A pipe can be exposed to either natural (free) convection or
forced convection in windy environments. For reliable estimations of cU for pipes under external
forced convection, one can use correlations of ‘convective coefficient’ for flow across cylinders found
in standard heat transfer books. Typical values of heat transfer coefficients for forced gas convection
can be around 2 to 50 BTU/ft2 − hr −o F . When pipeline goes through a water body (i.e. river),
this value can be greater than 20 because heat transfer coefficients for forced convections in liquids
can be found within the range of 2 to 2000 BTU/ft2 − hr −o F .
In the absence of rotating shafts, work can be done on a CV by forces or agents that can influence
fluid expansion/compression or cause the deformation of the CV surface. The pressure force, for
instance, causes the fluid to enter and leave the control volume and does work, which is typically
called flow energy or flow work in the thermodynamic analysis of open systems. Flow work
is done by the inlet pressure to push the fluid into the control volume, and work is done by the
control volume to push the fluid out of the system. The presence of terms accounting for flow
work in an energy balance gives rise to enthalpy terms, when combined with internal energy values.
The friction force, on the other hand, can not do work because it is a stationary force unable to
cause deformation of the CV boundaries or any fluid expansion/compression. The friction force
is a stationary force applied at the boundary, and stationary forces do not do work. Wall shear
or frictional effects are often associated with heat generation inside the CV, but internal heat
generation due to friction is largely neglected in this type of formulations.
In a flow system, the rate at which work is done by a force is equal to the product of that force
times the velocity component in the direction of the force:
W = ~F · |~v| (4.25)
57
Where, velocities are evaluated at the fluid point where the force is being applied. Therefore, the
work done by pressure force can be described as:
WCV = −[(pA~v)t
′
x+ ∆x2
− (pA~v)t′
x−∆x2
](4.26)
The convention positive work (+W ) for forces in the direction of flow (work done by the fluid) and
negative work (−W ) for forces acting against the flow (work done on the fluid) is maintained. It is
important to point out that, other than pressure, none of the forces present in the free body diagram
of the CV (Fig. 4.3) can be considered to do work on the fluid. They do not cause deformation
of the CV surface or fluid expansion/compression. Such is the case for both friction and gravity
forces. The contribution of gravity (or the gravity force) to the energy balance is, however, fully
captured by the definition of total intrinsic energy (e), where the displacement of the fluid through
the gravitational field comes associated with changes in fluid’s potential energy.
By introducing the definitions of heat transfer and work into the energy conservation statement of
Eqn. 4.23, one obtains:
[(ρeA)t+∆t
x ∆x− (ρeA)tx∆x]
+[(ρ~veA)t
′
x+ ∆x2
∆t− (ρ~veA)t′
x−∆x2
∆t]
− [cULw∆x∆T∆t]x −[(pA~v)t
′
x+ ∆x2
− (pA~v)t′
x−∆x2
]∆t = 0 (4.27)
Or equivalently, dividing the equation Eqn. 4.27 by ∆x and ∆t;
[(ρeA)t+∆t
x − (ρeA)tx]
∆t+
[(ρ~veA)t
′
x+ ∆x2
− (ρ~veA)t′
x−∆x2
]∆x
+
[(pA~v)t
′
x+ ∆x2
− (pA~v)t′
x−∆x2
]∆x
− [cULw∆T ]x = 0 (4.28)
Or, in terms of enthalpy (h) flux:
[(ρeA)t+∆t
x − (ρeA)tx]
∆t+
[(ρ~vhA)t
′
x+ ∆x2
− (ρ~vhA)t′
x−∆x2
]∆x
− [cULw∆T ]x = 0 (4.29)
Equation 4.28 represents the 1D statement of conservation of energy for a finite control volume
implemented in this study. Again, this expression collapses to the PDE form as shown in Eqn. 4.30
after taking the limits ∆t,∆x→ 0:
∂(ρeA)
∂t+∂(ρ~veA)
∂x+∂(pA~v)
∂x− [cULw∆T ] = 0 (4.30)
58
Or;∂(ρeA)
∂t+∂(ρ~vhA)
∂x− [cULw∆T ] = 0 (4.31)
Where; Lw = πD
4.3 Two-Fluid Model Equations
In this study, a one-dimensional, single-pressure two-fluid model is developed with the assumption
of thermodynamic equilibrium; i.e. mechanical, thermal and chemical equilibrium. Mechanical
equilibrium suggests that phasic pressures are same at any point over the cross-section of the
conduit. Similarly, local thermal equilibrium means that phases are at the same temperature over
the cross-section; hence, it is sufficient to use a single energy equation ensuing in the 5 equation
single-pressure two-fluid model.
The two-fluid model treats each phase in a multiphase flow environment as a separate fluid with
its own set of governing equations. In the most detailed case each phase is considered to have its
own velocity, temperature and pressure. In this study, however, the time lag associated with the
attainment of thermal and mechanical equilibrium between the phases is considered insignificant
in comparison to the characteristic time it would take for pressure and temperature conditions to
change during flow along the axis of pipe. Therefore, pressure and temperature non-equilibrium is
neglected and pressure and temperature are assumed to be same for both phases.
It is important to note that some sort of an averaging process (e.g. volume) is almost always
involved in the derivation of governing two-fluid model PDEs (and FDEs thereof), regardless of
the flow dimensions involved; i.e. for 3D and 2D models with Navier-Stokes equations as well.
The need for an averaging process essentially stems from the fact that the two phases are indeed
interpenetrating and there are moving boundaries between the phases to account for; at macroscopic
as well as microscopic (local) scale. A ‘local instant’ formulation is possible without resorting to
an averaging and results in the ‘direct numerical simulation’ approach. However, besides being
computationally too demanding, for most practical applications only ‘average’ flow properties are
of interest. Governing equations in this text are derived using the ‘macroscopic balance’ approach
because it is easier to visualize, has a physical basis and thus simpler to comprehend. Nevertheless,
it can mask the averaging process if one is not careful. Hence, it should be remembered at all times
that flow properties (e.g. ρ, v, α) represented in the equations of this text are average properties
both at microscopic (local) and macroscopic (control volume) scale. Extensive detail on various
averaging processes can be found in the texts by Prosperetti and Tryggvason (2007); Kolev (2005);
Ishii and Hibiki (2003); Todreas and Kazimi (1990).
4.3.1 Conservation of Mass
One might write separate conservation statements for each of the flowing phases found in the CV in
Fig. 4.5. In the case of mass conservation for the liquid, one considers:
59
Figure 4.5: Control Volume for Stratified Two-Phase Flow
Extensive property = Liquid mass = ρLVL = ρL(αLA∆x)
Superficial liquid mass flux =Liquid mass flow
Total area=ρL ~vLAL
A= ρL ~vLαL
Intensive property = liquid mass per unit mass = 1 (one)
The statement of conservation of liquid mass, based on Reynolds Transport theorem is written as:
dMCM
dt= 0 =
increase of liquid mass
in CV
over period ∆t
+
net efflux of liquid
out of CS
during period∆t
(4.32)
Where:increase of liquid mass
in CV
over period ∆t
=[(ρLαLA)t+∆t
x ∆x− (ρLαLA)tx∆x]
net efflux of liquid
out of CS
during period ∆t
=
−(ρL ~vLαLA)t′
x+ ∆x2
∆t
mass leaving CS via advection
+(ρL ~vLαLA)t′
x−∆x2
∆t
mass entering CS via advection
+(ΓLA∆x)t′x∆t
mass crossing CS due to mass transfer
Here; t′ is any suitable time, either t′ = t or t′ = t+ ∆t, which renders the numerical scheme either
explicit or implicit. Therefore, the 1D Finite volume statement of mass conservation for the liquid
phase can be written as:
(ρLαLA)t+∆tx − (ρLαLA)tx
∆t+
(ρL ~vLαLA)t′
x+ ∆x2
− (ρL ~vLαLA)t′
x−∆x2
∆x= ΓLA = −ΓGA (4.33)
60
Equivalently for the gas phase we can write:
(ρGαGA)t+∆tx − (ρGαGA)tx
∆t+
(ρG ~vGαGA)t′
x+ ∆x2
− (ρG ~vGαGA)t′
x−∆x2
∆x= ΓGA = −ΓLA (4.34)
The PDE form of the continuity equation for the two-fluid model can be obtained as one takes the
limit ∆t,∆x→ 0 in these expressions. For a generic phase k we have:
∂
∂t(ρkαkA) +
∂
∂x(ρk ~vkαkA) = ΓkA (4.35)
Where; αG + αL = 1 and ΓG + ΓL = 0.
4.3.2 Conservation of Momentum
Figure 4.6: Control Volume for Stratified Two-Phase Flow
In the case of momentum conservation for the liquid phase, one identifies:
Extensive property = Linear momentum = ρLVL ~vL = ρL(αLA∆x) ~vL
Intensive property = Momentum per unit mass = ~vL
Flux of = Liquid mass flux × intensive property = (ρL ~vLαL) ~vL
extensive property
The momentum conservation statement for the CV in Fig. 4.6 is thus, for the liquid phase:
~FCV ∆t =
increase of momentum
in CV
during ∆t
+
net efflux of momentum
leaving the CS
during ∆t
(4.36)
Where:
61increase of momentum
in CV
during ∆t
= (ρL ~vLαLA∆x)t+∆tx − (ρL ~vLαLA∆x)tx
net momentum efflux
out of CS
during ∆t
= −
momentum leaving CS
via advection
+
momentum entering CS
via advection
+
momentum gained by CV
due to mass transfer
Momentum leaving CS via advection −(ρL ~vL ~vLαLA)t′
x+ ∆x2
∆t
Momentum entering CS via advection +(ρL ~vL ~vLαLA)t′
x−∆x2
∆t
Momentum gained by CV
as a result of mass transfer+(ΓLA∆x~vc)
t′x∆t
Here; ~vc = ~vL if mass transfer is from gas-to-liquid (ΓL > 0) and ~vc = ~vG if mass transfer is from
liquid-to-gas (ΓL < 0) as discussed by Shoham (2006); Todreas and Kazimi (1990).
The net force acting on the liquid CV is given by:
~FCV ∆t =[(pαLA)t
′
x−∆x2
− (pαLA)t′
x+ ∆x2
]+ [τisi − τwLswL − ρLαLA∆xg sin θ]t
′
x ∆t (4.37)
Although Todreas and Kazimi (1990) leaves the derivation at this stage, it is important to note that
as a consequence of the aforementioned averaging process, which is required for the derivation of
two-phase flow equations (see introduction of Sec. 4.3), holdup (αL, or void fraction αG for that
matter) is assumed to be constant all throughout the control volume.
(αL)t′x = (αL)t
′
x−∆x2
= (αL)t′
x+ ∆x2
Hence; pressure gradient accelerating the liquid volume is defined as the fraction of overall pressure
gradient available for the liquid phase (across the CV) and it is customary to write the holdup
outside the parenthesis to avoid later misconceptions. This matter is further elaborated after
staggered grid is introduced in Sec. 4.7 and then following the discussion of boundary conditions in
Sec. 4.8.
~FCV ∆t = (αL)t′x
[(pA)x−∆x
2− (pA)t
′
x+ ∆x2
]+ [τisi − τwLswL − ρLαLA∆xg sin θ]t
′
x ∆t (4.38)
62
Then, 1D Finite Volume equation for liquid phase momentum balance is:
(ρL ~vLαLA)t+∆tx − (ρL ~vLαLA)tx
∆t+
(ρL ~vL ~vLαLA)t′
x+ ∆x2
− (ρL ~vL ~vLαLA)t′
x−∆x2
∆x
= (ΓLA~vc)t′x − (αL)t
′x
(pA)t′
x+ ∆x2
− (pA)t′
x−∆x2
∆x+ [τiLi − τwLLwL − ρLαLAg sin θ]t
′x (4.39)
The PDE counterpart of this equation, for a generic phase k, obtained at the limit ∆t,∆x→ 0 as:
∂
∂t(ρk ~vkαkA) +
∂
∂x(ρk ~vk ~vkαkA) + αk
∂
∂x(pA) = δiτiLi − τwkLwk − ρkαkAg sin θ + ΓkA~vc (4.40)
Where; δi = 1 for k = L and δi = −1 for k = G since τi is positive for the usual case ~vG > ~vL and
τi is negative for the case ~vG < ~vL, by the interfacial friction factor definition given in Eqn. 4.42.
Hence, interfacial force is ‘adding’ to the momentum of liquid phase by accelerating it, i.e. by
dragging it with the faster gas phase velocity.
Phase wall shear stresses are calculated using single phase friction factors calculated for the
corresponding phase velocity:
fk =
[τw
12ρ|~v|~v
]k
(4.41)
Interfacial shear stress is calculated as a function of interfacial friction factors, which are defined
using an inertia term that employs the slip velocity between the phases and the density of the
continuous phase (assumed gas in this study):
fi = 4τi
12ρG| ~vG − ~vL|( ~vG − ~vL)
(4.42)
It is important to note again that a consequence of one-dimensional modeling is the loss of certain
details regarding the flow; i.e. information pertinent to the ‘eddies’ in the case of turbulent flow.
Nevertheless, associated hydraulic losses along the axial direction of the conduit are captured
through the use forcing functions (friction forces) with one-dimensional Euler equations. Friction
force terms are calculated using friction factors (Eqns. 4.50 and 4.51) based on Reynolds number of
the flow which is an indicator of the flow mode; i.e. laminar, transition or turbulent (cf. Sec. 4.4).
4.3.3 Conservation of Energy
Based on the thermal equilibrium assumption, a single, overall (mixture) energy equation is adequate
for the two-fluid model approach.
63
Figure 4.7: Control Volume for Stratified Two-Phase Flow
Extensive property = Energy = Ek = Mkek = (ρkAk∆x)ek
Intensive property = Energy per unit mass = ek = e∗k +v2k2 + g∆zel
Flux of
extensive property = mass flux × intensive property = (ρk ~vkαk)ek
by advection
Where;
∆zel System elevation with respect to a datum z0 Since x represents the distance
from center of CV to the entrance of pipe: ∆zel = x sin θ
θ Pipe inclination
The conservation statement for the liquid CV becomes:
(Q− W
)CV
∆t =
increase of Total energy
in CV
during ∆t
+
net flux of energy
leaving CS
during ∆t
(4.43)
Where:increase of total energy
in CV
during ∆t
= [ρGαGA∆teG + ρLαLA∆teL]t+∆tx − [ρGαGA∆teG + ρLαLA∆teL]tx
net flux of energy
leaving CS
during ∆t
=+ (ρk ~vkαkAek)
t′
x+ ∆x2
∆t
energy leaving CS via advection
+ (ρk ~vkαkAek)t′
x−∆x2
∆t
energy entering CS via advection
Q = [cUsw∆t]t′
x
W = (pAG ~vG + pAL ~vL)x−∆x2− (pAG ~vG + pAL ~vL)x+ ∆x
2
64
Please note that work done against the liquid CV incorporates the pressure work contribution and
that of the interfacial friction force. However, since overall energy equation incorporates work done
on both phases, interfacial friction force terms cancel each other. Based on these considerations,
Eqn. 4.43 becomes:
[(ρGαGeG + ρLαLeL)A]t+∆tx − [(ρGαGeG + ρLαLeL)A]tx
∆t
+[(ρG ~vGαGeG + ρL ~vLαLeL)A]t
′
x+ ∆x2
− [(ρG ~vGαGeG + ρL ~vLαLeL)A]t′
x−∆x2
∆x
= −(pAG ~vG)t
′
x+ ∆x2
− (pAG ~vG)t′
x−∆x2
∆x−
(pAL ~vL)t′
x+ ∆x2
− (pAL ~vL)t′
x−∆x2
∆x+ [cULw∆t]t
′
x (4.44)
Or, in terms of enthalpy flux:
[(ρGαGeG + ρLαLeL)A]t+∆tx − [(ρGαGeG + ρLαLeL)A]tx
∆t
+[(ρG ~vGαGhG + ρL ~vLαLhL)A]t
′
x+ ∆x2
− [(ρG ~vGαGhG + ρL ~vLαLhL)A]t′
x−∆x2
∆x
+ [cULw∆t]t′
x (4.45)
As discussed previously, a two-fluid model for steady one-dimensional flow can be composed of
up to six conservation equations. Such sort of models would include two equations accounting for
continuity, two equations accounting for momentum balance and two equations for energy balance
in the liquid and gas phases. However, use a combined (mixture) energy equation for the system
(by lumping the liquid and gas phase balances together) leads to the five equation version of the
two-fluid model. The use of combined energy equation is predicated upon the assumption of zero
interfacial temperature gradient (i.e. thermal equilibrium) and it could also be obtained by summing
up individual, phasic energy conservation equations. The PDE version of the combined energy
statement can be obtained as one takes the limits ∆t,∆x→ 0, as shown:
∂
∂t[(ρLαLeL + ρGαGeG)A] +
∂
∂t[(ρL ~vLαLhL + ρG ~vGαGhG)A] = cULw∆t (4.46)
It is important to note that heat transfer coefficient cU should indeed become a function of flow
pattern, gas and liquid phase conductivities since liquid and gas phases typically have different heat
conductivities. However, in this study cU is assumed constant for practical purposes based on the
assumption of low liquid loading conditions.
4.4 Closure Relationships
The use of a one-dimensional two-fluid model requires describing the interaction between phases
themselves and the pipe wall, which depends on the nature of the flow pattern describing the
gas-liquid distribution over the pipe cross-section. One of the important phase interactions that
65
needs to be properly described is the interfacial mass transfer term. Interfacial mass transfer between
the phases can be calculated in different ways which depend on the nature of flowing fluids and
version of the two-fluid model. Two immiscible phases, for example, would not exchange mass and
this calculation is omitted. TRAC, for example, calculates mass transfer between the phases as a
function of heat input and latent heat required to vaporize the liquid phase (Spore et al., 2001).
For the case of natural gas condensation in a pipeline, mass transfer from gas phase to liquid phase
within two neighboring cells is computed based on thermodynamic equilibrium gas mass fractions of
neighboring cells (Mucharam et al., 1990):
ΓL =[(ρGvGαG + ρLvLαL)A](fmgi+1 − fmgi)
Adx(4.47)
Where; fmgi is the (thermodynamic) equilibrium gas mass fraction for the flow stream (hence the
composition) crossing the cell face j and evaluated at p− T conditions of upstream cell i; while,
fmgi+1 is the equilibrium gas mass fraction for the same flow stream evaluated at p− T conditions
of downstream cell i+ 1. Equilibrium gas mass fractions are computed via an EOS based on p− Tconditions of pertinent cell.
It is important to recognize that this mass transfer model assumes thermodynamic equilibrium and
ignores kinetic aspect of the mass transfer process by considering conclusion of mass transfer within
the time it takes for flow stream to traverse the CV. While this may not be the best solution for
studies involving small sized CVs where significant or sharp changes in pressure and temperature
between neighboring cells is expected, it becomes a reasonable assumption for studies with large
CVs where the time it takes for flow to enter and leave the CV is long enough for mass transfer to
be complete or where change in pressure and temperature between neighboring CVs is insignificant.
Momentum transfer accompanying the mass transfer is defined by the momentum transfer term
present in Eqn. 4.39:
Momentum source term for phase k = ΓkA~vc (4.48)
Where; ~vc = ~vL if mass transfer is from gas-to-liquid (ΓL > 0) and ~vc = ~vG if mass transfer is from
liquid-to-gas (ΓL < 0) as discussed by Shoham (2006); Todreas and Kazimi (1990).
Another important parameter, which accounts for the momentum exchange between the phases, is
the interfacial force (or interfacial stress) as defined in Sec. 4.3.2. Relative motion of the phases
is usually described as the difference between the gas phase velocity and that of the interface.
Interface velocity is typically approximated as being equal to the liquid phase velocity and surface
area between the phases is defined as the ratio of the surface area of contact between the phases
and the total volume of the fluid. Calculations of such interfacial parameters as friction factors,
surface areas as well as contact surfaces between each phase and the pipe wall and associated wall
frictions are carried out based on the flow regime.
These flow regime dependent closure relationships simply involve calculations based on flow geometry
and can be found in detail in several text books (i.e. Shoham, 2006) or in the text by Ayala (2001),
66
hence will not be repeated here extensively. However, calculation of pertinent parameters for two of
the utilized flow patterns is shown for the sake of completeness.
Wall shear force calculation:
Fwk = Awkfwk4
ρk|vk|vk2
(4.49)
For laminar flows wall friction factor (Darcy-Weisbach) is calculated as:
f =64
Rek(4.50)
For turbulent flows wall friction factor (Darcy-Weisbach) is calculated using Chen (1979) formula:
1√f
= −2 log10
[ε
3.7065Dk− 5.0452
Reklog10
(1
2.8257
(ε
Dk
)1.1098
+5.8506
Re0.8981k
)](4.51)
Where:
ε Pipe roughness [m]
Dk Phasic hydraulic diameter [m]
Rek Phasic Reynolds number [dimensionless]
Phasic hydraulic diameter is defined as:
Dk = 4Volume of phase k
Area wetted by phase k(4.52)
Interfacial friction force calculation:
Fi = Aifi4
ρG|vG − vL|(vG − vL)
2(4.53)
Flow pattern dependent phasic hydraulic diameter, wetted area and interfacial area calculations are
given in Tab. 4.1 for mist and stratified flow patterns, where rdrop is mean droplet radius.
Table 4.1: Flow pattern dependent parameters
Flow pattern: DG = DL = AwG = AwL = Ai =
Stratified Flow
(παG
π − θ + sin θ
)D
(παLθ
)D
(π − θπ
)4
D
(θ
π
)4
D
(sin θ
π
)4
D
Mist Flow D D αG4
DαL
4
D
3(1− αG)
rdrop
67
4.5 Primary Unknowns
Single-phase flow in a conduit can be described by three physical parameters, namely; pressure
(p), velocity (v) and temperature (T ), where temperature is no longer an unknown in the case
of isothermal flow. Governing algebraic equations are solved for these three primary parameters,
hereafter referred to as primary unknowns or simply the ‘primaries’. The choice of primary unknowns
is based on the fact that they are all fundamental and directly measurable properties of the flow
and fluid. Density and fluid energetic content (i.e. internal energy or enthalpy) are derived from
p− T information by applying a suitable Equation of State (EOS). In this work, the Peng-Robinson
EOS is implemented for fluid property predictions (Peng and Robinson, 1976). Detailed information
about the implementation of Equations of State and associated fluid property prediction is found in
Appendix C.
For the case of two-phase flow, two additional primary unknowns are added to the system, liquid
phase velocity (vL) and void fraction (α). A list of primary unknowns and related secondary
parameters are given in Tab. 4.2 and
Table 4.2: Primary unknowns
Primary unknown Secondary parameter
Pressure (p) Phasic density (ρk, through EOS)
Gas phase velocity (vG) —
Liquid phase velocity (vL) —
Void fraction (αG) —
Temperature (T ) Phasic density (ρk, through EOS)
4.6 Advective Property: Donor Cell Scheme
Looking at the algebraic equations developed for FVM, it is recognized that values of advected
properties, particularly mass (or density), momentum (or momentum per unit mass – velocity)
and energy (or energy per unit mass – specific energy), are required both at the center (as the CV
average) and at the edges of CVs. Unfortunately, not all of these values can be left as unknowns in
the system of governing equations. Otherwise, number of equations and number of unknowns are
not balanced. Therefore, in order to balance number of unknowns and the equations, typically edge
values are written in terms of cell center values through various approximation schemes. Here, it
should be noted again that cell center values of parameters in a CV also represent the volumetric
average of these values for that CV.
Focusing on the continuity equation and assuming that all required velocity values are available (i.e.
velocity field is known a priori and not a concern); approximation of density values at cell edges is
needed. A linear interpolation between the cell center densities of two neighbor CVs carries the
68
potential danger of yielding unrealistic results, especially for transient analysis, as evidenced by the
donor cell exercises provided by Patankar (1980).
One remedy to this problem is the donor cell scheme, also known as the upwind method or upstream
weighting. Donor cell is, in essence, based on the foundation that fluid passing through a point
carries the information from its upstream but has little or no information about the downstream.
Therefore, upstream weighted averaging (100% upstream in the context of this text) is favored
over linear averaging (which corresponds to central difference scheme) in the case of advective flow
modeling, specifically for the case of transient analysis.
4.7 Staggered Grid
When deciding for the nodes where primary unknowns will be evaluated in FVM equations, following
the donor cell discussion, pressure and temperature are solved for at the cell centers since density is
calculated based on p− T information in the case of compressible flow. The intuitive approach is
to place all primary unknowns (p, v and T ) at the same node, cell center, as a convenient choice.
This approach is typically known as the ‘co-located’ or coincidental grid, where all unknowns are
calculated at the center of the same grid block and typically is the default setting for FDM based
models. A typical coincidental grid is shown in Fig. 4.8. In this arrangement, mass, energy, and
momentum CVs coincide and there is a single computational grid for the placement of unknown
variables.
Figure 4.8: Coincidental Grid
Placing pressures and velocities at the same node has the potential danger of producing wavy
(checkerboard) patterns for pressure and velocity values, particularly in the case of incompressible
flow. This potential danger associated with the use of coincidental grid is discussed in detail by
Patankar (1980). In this type of grids, calculation of the pressure forces driving the flow necessarily
employ an average of pressure values of neighboring cells centers, causing the actual CV pressure
value to be leapfrogged in the calculation. Checkerboard pressures can not be effectively dissipated
in these co-located numerical grids, since pressure information does not appear explicitly in the
governing equations but rather appears in the form of a momentum source (pressure drop) or
through fluid densities. In the case of incompressible flow, continuity equation can neither help to
deduce the pressure information (since density is not a function of pressure in this case) nor be
used to determine the velocity field (both pressure and velocity are unknowns). The difficulty of
wavy pressure patterns is likely to be eliminated in the case of compressible flow where density is
an explicit function of pressure. In those cases, pressure values would be constrained by density
values (or vice a versa) through the use of an appropriate EOS and the physical connection between
velocities and pressures are then established by the FVM equations.
69
As presented by Patankar (1980), the most traditional approach to avoid the checkerboard pressure
problem is the implementation of staggered grids in the computational domain. A staggered grid
implies the definition of non co-located, staggered location for the primary unknowns. Typically,
cell pressures are evaluated at the cell centers but fluid velocities are defined at the cell edges.
This implies the definition of two, displaced numerical grids, as shown in Fig. 4.9, one used for the
implementation of mass and energy FV equations (p-cells) and the other for the implementation of
the momentum FV equations (v-cells). The basis of this approach is that neither the nodes where
primary unknowns are calculated nor the CVs for the conservation equations have to coincide in
order to obtain a numerical solution.
Figure 4.9: Staggered Grid
Besides preventing checkerboard type pressure fields, the use of a staggered grid arrangement brings
additional advantages. The apparent, direct advantage is that velocities are now evaluated between
pressure couples that are actually driving the flow and thus a solid physical link is established.
In fact, paying closer attention to the FVM equations and their derivation, it is observed that
the natural location for velocities is the cell edges. This is because edge velocities are frequently
needed in the governing equations for the calculation of fluxes at cell edges. The staggered approach
will also facilitate the formulation of conservation equations for T-junctions, which is discussed in
Sec. 4.12.
The staggered grid approach is also applied in both OLGA (Bendiksen et al., 1989) and TRAC
(Spore et al., 2001). While details of the application of this computational grid were not available
for OLGA, the staggered grid approach utilized in this study is slightly different than the form
applied in TRAC. TRAC employs the non-conservative form of the momentum equations meaning
that momentum equations are written specifically at the edges of cells. The non-conservative form
of the momentum equations are derived from the conservative FVM equations and one advantage
of using the non-conservative form is that it eliminates the need for extensive use of cell center
velocities, which have to be approximated using edge values, in momentum equations. Nevertheless,
in this study, the conservative form of the momentum equations is employed in order to assure and
reinforce rigorous conservation.
It is important to point out that Rhie and Chow (1983) have presented a pressure treatment that
allows the formulation of coincidental grids that do not suffer from the checkerboard pressure problem.
In their coincidental grid application, all primary unknowns are evaluated at cell centers and CVs
of the conservation equations coincide (Fig. 4.8). Their method requires the implementation of an
70
additional pressure equation that decouples implicit pressure-velocity information of the governing
equations and allows the formulation of an explicit link for pressure information. Co-located methods
are particularly useful with complex, irregular geometries.
A low accuracy, simple alternative to staggered grid is using forward differences for momentum flux
terms and backward differences for pressure gradient term (Wesselling, 2001). However, in this
study staggered grid is applied in order to extend governing Euler equations over the interfaces of
tee control volume.
As a consequence of the staggered grid arrangement, momentum equations are now written for a
displaced momentum CV overlapping two neighboring mass CVs. This requires that the average
holdup (or void fraction) for the momentum CV (multiplying the pressure gradient term as discussed
when two-phase momentum PDE is derived in Sec. 4.3.2) described as a function of neighboring
mass CV holdups. In this study, a volumetric average of neighboring mass CV holdups (reduces to
a linear average when diameter is same for both mass CVs) is employed (Frepoli et al., 2003).
4.8 Boundary Conditions
A potential inconvenience of the staggered grid approach is the handling of boundary conditions
(BC). Because of the displaced nature of the staggered grid for mass and momentum equations
(see Fig. 4.9), it can be seen that first momentum cell is not directly connected to pipe inlet
information while center of last momentum CV exactly coincides with pipe outlet. This last cell,
moreover, has half of its volume defined outside the actual pipe physical domain. As a consequence
of this staggered arrangement, none of the momentum equations written for the system explicitly
incorporate information defined at the pipe inlet (i.e. inlet pressure). Consequently pressure is
specified at pipe outlet while velocity (or mass flow rate) and temperature are specified at pipe
inlet. Then, the flow problem is stated as:
• Flow rate qin is injected at temperature Tin at point ‘inlet’ to be delivered at pressure pout, at
point ‘outlet’.
In fact, this arrangement of boundary conditions is quite useful for the analysis of branching
T-junction problem as outlet pressures have significant influence on the flow split.
Nevertheless, it should be noted that for flow specification in the reverse direction, pressure becomes
specified at the inlet while flow rate is now being specified at the outlet. Furthermore, by adding an
additional momentum CV at the beginning of the grid, thus increasing number of momentum CVs
one more than mass CVs, it could be possible to link inlet pressure specification to the pipe. Of
course, this would require removing the outlet pressure specification.
Prosperetti and Tryggvason (2007) discuss that flow at the pipe outlet should essentially mimic an
adequately long flow domain where outflow has no effect on the solution; i.e. as if there is no real
outlet but outlet location is a point within an infinitely long conduit where fully developed flow
is taking place and ‘boundary layer’ effects are insignificant. This is typically accommodated by
assuming that flow profile is relatively smooth around the outlet and flow continues beyond the
71
outlet without being disturbed (i.e. no jumps in the pressure and velocity profiles). Such a smooth
outlet condition is typically imposed on the velocity profile since pressure is already specified as
BC. Prosperetti and Tryggvason (2007) lists four general outflow boundary conditions that allow a
smooth exit of the flow; (1) convective, (2) parabolic, (3) zero gradient and (4) zero second gradient
(zero curvature).
In this study, phasic velocities at the edge of the last momentum CV, temperature and void fraction
values for the imaginary boundary mass CV (i.e. ghost or phantom cell) are all calculated based on
the zero second gradient condition (zero curvature, i.e. free flow). In other words, these properties
are directly calculated as a linear extrapolation of the values of previous two CVs.
Inlet flow rate specification is converted to inlet phasic velocities based on void fraction specification
and phasic density calculation at inlet p− T conditions. For isothermal studies, inlet pressure does
not require updating. However, for thermal studies inlet pressure is also updated using the zero
curvature method. Otherwise, profiles of flow parameters display jumps, disturbances over the first
couple of cells; associated with the sudden expansion of gas phase into the pipe and Joule-Thompson
cooling.
On the other hand, there are three options for inlet void fraction specification; (1) fixed (2) calculated
internally based on no-slip condition, which also sets inlet phasic velocities equal (3) updated based
on zero curvature. While first and second options cause disturbances in flow profiles near the
inlet, third option adheres completely with the smooth, fully developed inlet flow condition, hence
preferred for steady-state analysis.
Developed model also has the option of specifying outlet void fractions as boundary conditions,
solely for checking purposes. However, an outlet void fraction (or holdup) specification is quite
a counter-intuitive attempt since; typically, there is no true way of controlling or knowing it in
advance, unlike pressure or flow rate. This is more clear when a branching tee is considered; as
phase separation at the junction is not known in advance, outlet void fraction specification(s) would
perhaps impose a certain phase separation ratio on the solution. Besides, specification of void
fraction at the pipe outlet has been observed to cause no significant effect on the profiles of flow
parameters for single pipe study, at least according to the two-fluid model formulation employed in
this thesis.
Singh (2009) specifies outlet void fractions (both of them for the case of a branching tee). However,
through the course of this study, it has been observed that the only time void fraction specification
effects flow profiles of a single pipe is when (and if) the void fraction term is included within the
pressure gradient term (Eqn. 4.39). That is to say, two different void fractions (i.e. void fractions of
neighboring mass CVs) are utilized when calculating the pressure force accelerating the flow stream
within the momentum CV. In fact, only then it is possible to match single pipe, two-phase results
from Singh (2009) with outlet void fraction specifications. This brings the discussion back to the
average holdup issue discussed in Sec. 4.3.2.
72
4.9 Order of Accuracy
From FDM point of view, numerical solutions are approximate solutions of the governing PDEs
where differential operators are replaced with finite difference operators. A finite difference operator
is in essence a truncated Taylor series and, by the formal definition, order of accuracy is the order
of highest order term in this truncated Taylor series. Hence, order of accuracy is a measure of
the accuracy of the numerical solution (i.e. how many digits after the decimal point matches the
analytical solution). How to determine theoretical order of accuracy using Taylor series analysis is
available in great detail in the text by Roache (1998) therefore it will not be repeated here.
From practical stand point, the order of accuracy indicates how much the numerical solution can
improve when one increases (e.g. doubles) the number of steps or CVs (or decreases the step or
CV size) of the solution. Evidently, numerical solution gets closer to the analytical solution with
increasing number of steps (i.e. finer grid). Practical (actual) order of accuracy of a numerical
method can be estimated through Richardson extrapolation (Roache, 1998). It is particularly
important to check the order of accuracy with this technique since practical order of accuracy
is usually lower than the theoretical value determined by Taylor series analysis. Furthermore,
enhancements done in the numerical method with best of intentions may end up lowering the
practical order of accuracy.
4.9.1 Higher Order Approximations
The accuracy of a numerical solution improves with increasing number of CVs, at the expense
of computational cost and time. Typically, there is an optimum number of cells beyond which
improvements are no longer significant
‘First-order accurate’ implies that numerical error associated with the discretization of PDE(s)
decreases linearly with with step size; may it be time or space. For instance, with a discretization
method ‘second-order accurate’ in space, discretization error is expected to decrease quadratically
as CV (step) size is halved.
One important objective in numerical simulation is to decrease the number of optimum cells
necessary to get a reliable solution, and this can be accomplished by implementing higher-order
approximations. Higher-order approximations include more information from the neighboring CVs
(or nodes) when calculating the solution at a particular cell (or node). This can be observed from the
Taylor series perspective of replacing differential operators. However, a more physical way of looking
at higher order approximations is realizing that these are indeed polynomial approximations of the
changes in variables through each cell. On the other hand, a relatively lower order approximation
(i.e. first-order) is merely a linear approximation of these changes. One good example of higher
resolution (higher-order) schemes is the QUICK (Quadratic Upstream Interpolation for Convective
Kinematics) scheme by Leonard (1979) where edge densities needed for the governing FVM equations
are evaluated as:
ρi+1/2 =3
8ρi+1 +
3
4ρi −
1
8ρi−1 (4.54)
73
ρi−1/2 =3
8ρi +
3
4ρi−1 −
1
8ρi−2 (4.55)
Utilization of higher resolution schemes ensues in estimating the change in a variable through a
CV with polynomials. This frequently introduces the wiggles (cf. Sec. 4.10) around shock fronts
and even stability problems. For instance, the study of Jelinek (2005) shows that stability issues
surface for the case of two-phase flow in nuclear systems suggesting that higher-order schemes are
not always suitable. Therefore, in this study, ‘theoretically’ second-order accurate upwind scheme
(the donor cell) is utilized. The donor cell scheme is known to be stable and robust albeit at the
expense of significant numerical diffusion (Frepoli et al., 2003; Spore et al., 2001), which is not a
major concern in the current context of steady-state two-phase flow split at T-junctions.
4.10 Transient Analysis
Transient analysis is concerned with the propagation of a perturbation, an induced change in
flow conditions, traveling through the conduit as a shock front, or as a slower change in the flow
conditions; i.e. a change in the injected fluid density or composition. Sudden changes in the flow
(i.e. change in boundary conditions such as the injection rate or temperature) cause shock fronts
(discontinuities); while slow, gradual changes cause smoother waves (slow transients) in the profiles
of flow parameters. Both types of perturbations are transmitted within the conduit, analogous to
waves. For instance, the information of the change in pressure will travel at the speed of sound waves
in a fluid while a change in the inlet temperature, retaining the wave nature of the perturbation,
travels at the speed of flow when conduction is ignored. Evidently, capturing the propagation of
perturbations which move at different speeds is particularly important when modeling compressible,
thermal flow in long pipes.
The wave equation conveys the information that perturbation is taking place in a continuous medium
and a part of the fluid moves as a whole (i.e. a particle in the flow). In the case of Euler equations,
the information of continuous medium is expressed by the equation of continuity (derived from
conservation of mass) while the information that a part of the fluid moves as whole is supplied
by the equation of motion (derived from Newton’s second law, conservation of linear momentum
Lindsay, 1931).
The conservative form of Euler equations tend to smooth such jumps in the numerical solutions
obtained for flow profiles, despite the inviscid nature of advective Euler equations. This difficulty is
known as numerical diffusion or numerical viscosity1 and typically due to averaging of fluxes or flow
parameters to obtain second order accurate approximations. In fact, from Taylor series point of
view, the second order terms that end up being included in the solution are the terms that resemble
viscosity/diffusivity terms.
Utilization of higher resolution schemes (i.e. QUICK, cf. Sec. 4.9.1) allows higher order approxima-
tions and decrease the effects of numerical diffusion, thus capture shock fronts better. However,
such schemes cause non-physical wiggles (oscillations, overshoots and undershoots) in the shock
1Not to be confused with artificial viscosity
74
front profiles of parameters. These wiggles are not due to stability issues but they are merely
mathematical results of the polynomials utilized in the numerical solutions by the higher resolution
schemes (Roache, 1998).
The non-physical wiggles in the solution imply that fluxes of conserved properties are not physical.
Hence, one remedy of preventing these wiggles is to limit computed fluxes to realistic values (values
that can be realized) and make the solution Total Variation Diminishing (TVD – Harten, 1983).
However, typical applications of flux limiters cause practical order of accuracy of higher resolution
schemes to decrease around the shock fronts. Furthermore, it can cause convergence problems.
Another remedy is to introduce an artificial viscosity term in the solution (that will ultimately limit
the flux). However, the adjustment required for such an artificial parameter renders the method
problem specific and prevents it from being a generalized solution (Roache, 1998).
Another method of uniformly retaining higher order accuracy while eliminating the wiggles is
the Essentially Non-Oscillatory (ENO) Schemes that use an adaptive stencil compared to the
fixed stencils of typical higher order schemes. In other words, an ENO scheme decides how many
additional points and in which direction (upstream or downstream) to include in the polynomial
approximation of the variable depending on the position of the shock front, hence prevents possible
oscillations (Harten et al., 1987).
Another option to maintain higher order accuracy is using an Adaptive Mesh Refinement (AMR)
algorithm. The AMR algorithm checks the coarse grid solution of a domain for certain criteria,
such as order of accuracy based on Richardson analysis. The algorithm, then spots the cells over
which the criterion is not met and establishes a finer grid within those cells and repeats the solution.
Therefore, the regions that require smaller CVs for precision are automatically picked and processed
accordingly.
According to the utilized time integration scheme (i.e. explicit, semi-implicit or implicit), stability
of the numerical solution depends on CFL condition (Courant number) which limits the maximum
allowable time-step duration based on grid spacing, flow velocity and velocity of the fastest traveling
perturbation (speed of sound –the pressure wave):
v∆t
∆x+ c0≤ 1 (4.56)
or
∆t ≤ ∆x+ c0
v(4.57)
Where; c0 represents the speed of sound.
Using an implicit scheme removes, in theory, this limit which can be quite constraining. However,
due to the difficulty of capturing wave propagation of different speeds, i.e. flow (void fraction,
density) and pressure wave accurately, it is typically recommended that a material CFL condition is
applied at all times:v∆t
∆x≤ 1 (4.58)
75
or
∆t ≤ ∆x
v(4.59)
4.11 Numerical Treatment
Volume averaged two-fluid model equations (the governing equations), after proper discretization,
could be solved numerically with any method suitable for non-linear system of equations, such
as Newton-Raphson. Due to computational intensity of this direct approach, different classes of
algorithms were developed (Prosperetti and Tryggvason, 2007):
1. Segregated algorithms: Governing equations are solved sequentially, such as SIMPLE
(Semi-Implicit Method for Pressure-Linked Equations) similar to IMPES (Implicit Pressure
Explicit Saturation) algotihm in reservoir simulation. These algorithms are typically suitable
for relatively slow transients and for flows that are not highly coupled and thus not requiring
a simultaneous solution. Typically, continuity (mass balance) and momentum equations are
arranged to form the pressure Poisson equation. During iterations, first the pressure equation
is solved to obtain pressures at iteration level k+1, then pressures at k+1 are utilized to
calculate the velocity field at k+1. Successive pressure corrections are utilized to adjust
densities and velocities to satisfy required conservation at convergence.
2. Partially coupled algorithm: Sequential solution of governing equations using segregated
algorithms causes accumulation of errors if time and spatial step sizes are adequately large.
Forcing iteration process to limit such accumulations with higher degree of convergence
typically ensues in increased computational times. Instead, simultaneous solution of the
pressure Poisson equation with coupled momentum equations yields velocities and pressures.
An automatic time-step control is needed to prevent volume fraction errors from growing
beyond an acceptable limit.
3. Coupled algorithms: Simultaneous solution of governing non-linear equations. Typically
suited for flows where interaction between phases is very strong or problems with short time
scales (fast transients).
4.11.1 Newton-Raphson Method
The set of coupled, non-linear algebraic (discretized) equations that govern the fluid flow are solved
simultaneously using a simple, generalized Newton-Raphson iterative procedure (coupled algorithm).
A brief description of the Newton-Raphson scheme is outlined next.
Let us assume that x∗1 and x∗2 form the actual solution set for equations f1(x1, x2) = 0 and
f2(x1, x2) = 0 and xk1, xk2 are initial approximations (guesses) to this solution. Newton-Raphson
procedure utilizes a turncated Taylor series expansion (Eqn. 4.60 and Eqn. 4.61) in order to come
up with the necessary increments ∆x1 and ∆x2 that can improve the current guesses closer to the
76
actual solution of the system of equations (Eqn. 4.63).
f1(xk1 + ∆xk1, xk2 + ∆xk2) = f1(xk1, x
k2) + ∆xk1
∂f1
∂x1+ ∆xk2
∂f1
∂x2+
(∆xk1)2
2!
∂2f1
∂x21
+(∆xk2)2
2!
∂2f1
∂x22
+ · · ·
f2(xk1 + ∆xk1, xk2 + ∆xk2) = f2(xk1, x
k2) + ∆xk1
∂f2
∂x1+ ∆xk2
∂f2
∂x2+
(∆xk1)2
2!
∂2f2
∂x21
+(∆xk2)2
2!
∂2f2
∂x22
+ · · ·
(4.60)
∆xk1∂f1
∂x1+ ∆xk2
∂f1
∂x2
∼= −f1(xk1, xk2)
∆xk1∂f2
∂x1+ ∆xk2
∂f2
∂x2
∼= −f2(xk1, xk2)
(4.61)
The system of equations in the form of Eqn. 4.61 can be expressed in a matrix formation and solved
for the increments ∆xk1 and ∆xk2 (Eqn. 4.62). Finally, ∆xk1 and ∆xk2 are utilized to get improved
approximations to the solution.
∂f1
∂x1
∂f1
∂x2
∂f1
∂x3· · · ∂f1
∂xn
∂f2
∂x1
∂f2
∂x2
∂f2
∂x3· · · ∂f2
∂xn
∂f3
∂x1
∂f3
∂x2
∂f3
∂x3· · · ∂f3
∂xn
......
.... . .
...
∂fn∂x1
∂fn∂x2
∂fn∂x3
· · · ∂fn∂xn
k
∆x1
∆x2
∆x3
...
∆xn
k
= −
f1
f2
f3
...
fn
k
(4.62)
x∗1 = xk1 + ∆xk1
x∗2 = xk2 + ∆xk2(4.63)
Similarly, algebraic forms of governing equations are written in the ‘residual’ form (Eqn. 4.64) and
then their derivatives with respect to the primary unknowns are numerically computed (Eqn. 4.65)
to construct the Jacobian (coefficient matrix) for the matrix solution (Eqn. 4.66).
Rmssk = 0
Rmomk = 0
Renek = 0
(4.64)
∂Ri∂p∼=Ri(p+ δp, T, v)−Ri(p, T, v)
δp(4.65)
77
∂Rmss1
∂p1
∂Rmss1
∂v1
∂Rmss1
∂p2· · · ∂Rmss1
∂vn
∂Rmom1
∂p1
∂Rmom1
∂v1
∂Rmom1
∂p2· · · ∂Rmom1
∂vn
∂Rmss2
∂p1
∂Rmss2
∂v1
∂Rmss2
∂p2· · · ∂Rmss2
∂vn
......
.... . .
...
∂Rmom1
∂p1
∂Rmom1
∂v1
∂Rmom1
∂p2· · · ∂Rmom1
∂vn
k
∆p1
∆v1
∆p2
...
∆vn
k
= −
Rmss1
Rmom1
Rmss2
...
Rmomn
k
(4.66)
The perturbation amount required to numerically evaluate derivatives should be a function of (1)
the actual variable value and (2) significant digits that can be recognized as a perturbation by the
residual. Besides, perturbation should be small enough to get a good estimate of the analytical
derivative (Frepoli et al., 2003).
δxn = max[σ, (σ × xn)] (4.67)
Where; typically σ = 10−6 but different multiplier (σ) values could be used for different primary
unknowns based on the order of magnitude of the pertinent residual and the primary unknown.
More elaborate discussion on the subject can be found in the text by Dennis and Schnabel (1996).
A fixed multiplier value of σ = 10−8 has proved to be adequate for the purposes of this study.
Convergence is achieved when maximum absolute values of both residuals and the improvements
are below certain limits:
1. |R|max < εresidual = 1e−8
2. |∆x|max < εimprovement = 1e−8
4.11.2 Phase Appearance-Disappearance
A typical problem encountered when conducting multiphase flow studies is the handling of phase
appearance-disappearance throughout the system.
In the case of two-fluid models, a phase can ‘disappear’ for two reasons:
1. Phase can be left behind, i.e. ‘hold-up’, as a result of the hydrodynamic requirements; i.e.
due to low interfacial drag, high wall friction and inertia.
This is a major concern for transient analysis since phases can not be completely ‘hold-up’ at
steady-state conditions due to conservation of mass. This is easier to observe when a system
is considered without mass transfer (i.e. air-water); where all the injected phases has to reach
to the outlet of the conduit for steady-state to be achieved.
2. Phase can disappear as a result of the mass transfer, which is a concern for both steady-state
78
and transient analysis.
Furthermore, because void fraction is a primary unknown and none of the governing equations
mathematically bound its value within the physical limits of [0, 1], void fraction can assume values
beyond the physical limits. This is simply a consequence of the N-R process when iterations swing
around the actual solution; typically when actual solution is located very close to the physical
limits of [0, 1]. Nevertheless, true convergence can only be achieved when conservation is physically
sound. That is to say, if void fraction is persistently going above the upper limit of one (1) or going
below the lower limit zero (0), then this means that hydrodynamic conditions require single-phase
conditions in that particular CV.
When void fraction is equal to zero (or near zero; αG ∼= 0, i.e. practically all liquid) or one (or near
one; αG ∼= 1, i.e. practically all gas); mass and momentum equations of the disappearing phase can
no longer provide flow information, and current equation set becomes ‘singular’ as balance equations
for the disappearing phase no longer respond to perturbations. Consequently, numerical derivatives
of the missing phase residuals can not be calculated. At this point, solution of the jacobian is
difficult if not impossible as the matrix itself also becomes singular; i.e. diagonal contains zero (or
near zero elements when void fraction is very close to the limits), causing difficulties with the matrix
inversion process.
One of the options for preventing governing equation set from becoming singular is to prevent the
disappearing phase from ‘truly’ disappearing due to hydrodynamics; by artificially increasing the
magnitude of momentum forces (i.e. drag force) as the void fraction approaches the physical limits.
In that regard, this option resembles the presence of capillary forces in reservoir engineering that
prevent phases from being completely drained from a reservoir block during the simulation; that is,
unless phases disappear due to mass transfer. Similar approach can also be taken to limit the mass
transfer in order to leave behind a trace amount of the disappearing phase and setting secondary
phase fluid properties equal to those of the primary phase.
In this study, phase appearance-disappearance problem is handled by switching balance equations of
the disappearing phase with ‘dummy’ equations as described by Frepoli et al. (2003). For instance,
upon disappearance of the liquid phase, liquid mass balance equation (which no longer provides any
useful information) is replaced by a dummy void fraction equation:
RmssL = 1− αG = 0 (4.68)
This dummy equation serves two purposes:
1. Fixes the value of αG = 1, which has become a ‘bogus’ parameter in the case of single-phase
flow; hence, preventing it from assuming unphysical values as a result of the iteration process;
which, otherwise would cause conservation problems for the gas phase mass and momentum
balances.
2. Ensures that one (1) is registered in the diagonal of the jacobian as a result of the numerical
79
evaluation of the derivative of dummy residual with respect to void fraction. Therefore,
prevents jacobian from becoming a singular matrix.
Fixing the value of αG = 1 renders the liquid phase momentum equation invalid (0 = 0). Since there
is no actual use for liquid phase velocity for single-phase gas flow conditions, a similar treatment; i.e.
a simple dummy equation with the sole purpose of registering 1 at the jacobian diagonal, could be
applied. However, Frepoli et al. (2003) suggest a better treatment for the liquid phase momentum
equation; by replacing void fraction in the original liquid momentum equation with a constant upper
limit value (i.e. αmax = 0.999999) above which flow is practically single-phase from numerical point
of view. Then, liquid velocity is calculated using this new, ‘dummy’ liquid momentum equation for
a droplet velocity that would have prevailed should liquid phase re-appeared as a dispersed (mist)
flow pattern. Frepoli et al. (2003) concludes that this approach prevents the need for ‘accelerating’
(or decelerating) the liquid droplets from an entirely irrelevant ‘bogus’ liquid velocity to the actual
droplet velocity when liquid phase re-appears. Therefore, the method provides the Newton-Raphson
iteration with a reasonable initial guess. Frepoli et al. (2003) method is adopted in this study
in order to provide a good estimate of the liquid velocity profile for the case of single-phase to
two-phase transition in steady-state analysis.
Because this study is focused on steady-state analysis, primary concern is to address phase
appearance-disappearance due to mass transfer. Phasic mass transfer term, defined by Eqn. 4.47
in Sec. 4.4, is calculated based on thermodynamic equilibrium assumption, ignoring kinetics of
the transfer mechanism and assuming that the time it takes flow stream to traverse the CV is
enough to establish thermodynamic equilibrium, and complete the mass transfer. Consequently,
steady-state analysis requires secondary phase to be present (or absent) in the CV, regardless of the
hydrodynamic conditions, once thermodynamic conditions predict its appearance (or disappearance).
For instance, between two neighboring CVs, if p− T couple falls outside of the two-phase region
(Fig. 1.2) in the downstream cell, calculated mass transfer term should remove all liquid phase and
add it to the gas phase. This is simply a requirement of (and complies with) both the steady-state
assumption and the thermodynamic equilibrium assumption. In this case, dummy void fraction
equation (Eqn. 4.68) serves a third purpose; it drives the value of αG to 1 (along with the mass
transfer term) upon predicting the disappearance of the liquid phase based on the thermodynamics.
Equation switching is applied as follows:
1. At the beginning of an iteration, control volumes are marked (flagged) as single-phase or
two-phase based on cell p− T conditions. This may ensue in neighboring single-phase and
two-phase cells, which poses no problem since discretized governing equations are able to
handle such discontinuities.
2. Iteration (construction of the jacobian) is completed based on phase flag assigned at the
beginning of the iteration; i.e. equation set employed for a CV corresponds to its phase flag.
No equation switching is allowed when numerically evaluating the residual derivative; i.e.
perturbed thermodynamic conditions may indicate a phase change but equation set is not
switched.
80
3. Following the update of primary variables, cells are re-evaluated for the phase flag and the
equation switch now takes place if new thermodynamic conditions require and calculated gas
mass fraction (fmg) for the CV is beyond the practical limits [10−6, 0.999999].
4. Strictly for steady-state purposes, a cell is forced to use the single-phase or two-phase equation
set solely based on its flag which is assigned according the new thermodynamic conditions.
It is important to note that, aforementioned equation switching algorithm is valid only for steady-
state conditions where void fraction could only become 1 (or liquid holdup become 0, for that matter)
due to thermodynamic mass transfer between the phases. For steady-state conditions, second phase
(holdup or void fraction thereof) simply cannot disappear completely due to conservation of mass,
unless one of the phases is completely removed by the mass transfer.
Equation switching process becomes more detailed and complicated in the case of transient analysis
where cells are flagged based on an intermediate hydrodynamic void fraction prediction before the
iteration commences (Frepoli et al., 2003; Spore et al., 2001).
4.11.3 Flow Pattern Transition
Everything else being the same, i.e. p, vG, α, vL, T ; magnitudes of calculated wall (fwk) and
interfacial friction forces (fi) can differ significantly for different flow patterns. Such differences in
the magnitudes could be large enough to cause discontinuities in the momentum residuals should
there be a change in the flow pattern. These jumps can eventually cause problems when evaluating
derivatives of momentum residuals if a flow pattern change is predicted after the perturbation.
Unless pattern transition is smoothened, problems can surface, ultimately delaying or preventing
N-R iteration from converging.
Pattern transition could be handled in a manner similar to single-phase to two-phase transition.
That is to say, no flow pattern transition is permitted during an iteration step but only after
the iteration is completed following the successful construction of jacobian. However, this would
probably slow down the convergence rate. Another approach could be to smoothen friction force
calculations using some sort of an averaging technique. For instance, Spore et al. (2001) uses
weighted averages of the friction forces calculated individually for the transition flow patterns:
(Fw)equation = (1− σ)(Fw)mist + σ(Fw)stratified (4.69)
Where; σ is a weighing factor.
4.11.4 Alignment of Residuals and Primary Unknowns
The proper order, proper arrangement of residuals (rows) and primary unknowns (columns) when
constructing the jacobian matrix is an important aspect of the problem in order not to produce
zero (or near zero) elements in the jacobian diagonal. Along those lines, considering single-phase
incompressible flow, it is apparent that pressure should be aligned with primary phase momentum
81
equation while velocity should be aligned with the mass balance since mass balance would not
recognize pressure perturbations in the case of incompressible flow.
Extending the approach for two-phase flow conditions, keeping primary phase arrangement same as
before, secondary phase velocity is aligned with the secondary phase momentum equation while
void fraction is aligned with the secondary phase mass balance since, in the absence of secondary
phase, its mass balance is replaced with a dummy void fraction equation. Here, it is important to
recognize that which phase being primary or secondary is not the main concern in the sense that
momentum equation of the disappearing phase will be replaced with a momentum equation using a
constant, ‘limit’ void fraction (i.e. α = 0.999999 or α = 10−6); hence, it will still be responsive to
pressure and velocity perturbations.
An important observation to note is; instead of primary phase mass balance, initially a combined
mass balance (arithmetic summation of phasic mass balances) was utilized along with the secondary
phase mass balance. Idea was to simplify the alignment of residuals and primary unknowns if
primary phase were to disappear instead of the secondary phase, since combined mass balance is
always (i.e. single-phase gas, single-phase liquid or two-phase conditions) a valid equation. For
single pipe studies this approach successfully generated the same results as were with individual
phasic mass balances. However, later it has been observed to create mass balance problems at the
tee CV. While no additional analysis was performed to pinpoint the exact reason of this problem,
this approach is simply left out as an option.
4.12 Two-Fluid Junction Model
The FVM based formulation, along with the staggered grid arrangement, allows extension of
governing 1D Euler equations over the T-junction volume regardless of the junction geometry (i.e.
irregular shape). This would allow a seamless algorithm for the network analysis. Therefore, as
shown in Fig. 4.10, tee structure is divided into control volumes.
Figure 4.10: Control Volumes for a T-Junction
82
An important point to underline is that tee cell has 3 interfaces; 2 with neighboring cells of the
main line (inlet and run arms) and one with the side line (branch). Conservation equations for this
CV are written regarding all three interfaces. Flow leaving the tee CV through its arms (or the
split itself for that matter) is to be determined by momentum balances that regard directionality of
the flow.
Change in momentum along each axial flow direction at the tee should be associated with the
presence of forces in these directions. If a coincidental mesh had been used (i.e. velocities at cell
centers along with pressures), then defining effective pressure forces along any of the two primary
directions of flow could have been complicated. This is because, it would then have been necessary
to superpose projection of pressure force in one direction on the other direction which is specifically
a problem for the case of a merging tee structure. Moreover, it would have been necessary to place
a second velocity within the tee CV (perhaps at the cell center as well) in order to account for the
momentum flux in the branch direction. Here, the convenience of the staggered grid is obvious, by
evaluating the velocities at the faces of the tee CV all these difficulties are worked out.
Nevertheless, there are three aspects of the 1D staggered formulation that complicates what, perhaps
otherwise, would have been a straight-forward application:
1. Accurate conservation of momentum in the tee CV.
Among the 3 principle conserved quantities (i.e. mass, momentum and energy), momentum is
the only one with vector attribute. Since it is a vector quantity, like force, momentum must
be conserved in all 3 primary directions of the flow. However, 1D momentum equations within
the Euler set of equations prevails only in one direction. This is not a problem for the case of
a pipe because only concern is the flow in the ‘single’ axial direction of the pipe. On the other
hand, in the case of a tee, tee CV to be more specific, there are two axial directions to account
for. Therefore, momentum conservation has to be accounted for both along the run (x) and
along the branch (x′) axial directions in order to account properly for momentum conservation
at the tee CV. Furthermore, writing the second momentum equation is in fact a necessity in
order to balance the number of equations of the system to the number of unknowns.
2. Proper representation and approximation of 3D pressure and velocity fields (distributions)
within the tee CV, in 1D.
Flow ‘loss’ terms included in 1D momentum equations are expected to make up for the essential
details lost by approximating 3D pressure and velocity distributions in 1D, such as irregular
geometry, flow area changes (diameters of the lines are not necessarily equal), secondary flows
(recirculation zones) and change in flow direction due to branching of flow.
3. Proper definition of momentum fluxes.
Due to staggered nature of momentum CVs, three momentum cells will be enclosing three faces
(edges) of the tee CV (mass CV). While faces of inlet and run momentum cells are parallel
to each other branch momentum cell’s influx face will be making an angle with the faces of
83
inlet and run momentum cells. Besides, incoming momentum flux is to be split among the
influx faces of run and branch momentum cells. Therefore, proper definition of (1) momentum
flux across all three of the ‘momentum faces’ and (2) branch momentum flux contribution for
conservation of momentum along the run direction are essential. Contribution from this third
interface could be observed as a sink/source term as well.
For practical purposes, and with the assumption of a reasonably small tee CV, density and other fluid
properties can perhaps be safely approximated as constants through out the tee cell. Assumption of
a small enough CV could provide the basis for the assumption of a same void fraction for all outgoing
streams. But this is also a consequence of the finite volume concept; that is, phasic mass fluxes
leaving the tee CV typically have the same fluid properties (i.e. density) and same void fraction
since flow is originating from the same ‘pool’. However, if void fractions of outgoing streams are
assumed to be same then, necessary adjustments for outgoing flow rates could only be established
through phasic velocities as previously pointed out with the DSM discussion in Sec. 3.2.3.
From FVM modeling perspective, a tee is merely a 3 pipe system. Both branch and run can be
visualized as separate pipes with outlet pressure specifications. Then, solution for an outgoing arm
solely depends on the flow going through it and the outlet pressure specification. This is because,
in the staggered grid arrangement inlet pressure and void fraction specifications have no significant
influence on the solution; particularly for isothermal, steady-state analysis. However, what really
matters is the phasic flow rates going through arms, where flow rates are supposed to be determined
by void fraction and phasic velocities of the tee CV.
Therefore, once flow split (and phase separation) is determined for the tee CV, that is phasic flow
rates for outgoing arms are known, pressure profiles of outgoing arms is solely based on specified
outlet pressures. Nevertheless, momentum equations at outgoing tee interfaces should link the
pressures obtained for the first cells of outgoing arms (next to tee CV) to the tee CV pressure which
becomes the outlet pressure specification for the inlet arm when considered a separate, third pipe.
Hence, an accurate solution can be obtained in the main line (inlet arm) as well.
If this pressure link is not correct; that is, it might be possible to obtain an incorrect tee cell pressure
with a seemingly correct flow split unless proper momentum equations are utilized, connecting all 3
arms of the tee, solution in the inlet arm would be incorrect as well.
Governing mass and momentum equations for the tee cell and associated momentum CVs are
obtained as follows for the tee structure given in Fig. 4.10:
Mass equation for the T-cell:
Vp3(ρn+1p3 − ρnp3)
∆t= [ρp2vv2Av2 − ρp3vv3Av3 − ρp3vv6Av6]n+1 (4.70)
84
Momentum equation for cell v2:
Vv2
[(ρv2vv2vv2)n+1 − (ρv2vv2vv2)n
]∆t
=
ρp1vv1vv1Av1 − ρp2vv2vv2Av2
+ (p2Ap2 − p3Ap3)
−ρv2Av2∆xv2g sin θ
− (τws)v2
n+1
(4.71)
Momentum equation for cell v3:
Vv3[(ρv3vv3vv3)n+1 − (ρv3vv3vv3)n]
∆t=
(ρp2vv2Av2 − ρp3vv6Av6) vv2 − ρp3vv3vv3Av3
+ (p3Ap3 − p4Ap4)
−ρv3Av3∆xv3g sin θ
− (τws)v3
+irr. loss
n+1
(4.72)
Where; the term [−ρp3vv6Av6], representing mass loss to the branch, is used for the calculation of
actual momentum flux going straight into the run.
Momentum equation for cell v6:
Vv6[(ρv6vv6vv6)n+1 − (ρv6vv6vv6)n]
∆t=
(ρp3vv6A
∗v6) vv2 cos(180− β)− ρp3vv6vv6Av6
+(p3A
∗p3 − p6Ap6
)−ρp3Av6∆xv6g sin θ
− (τws)v6
+irr. loss
n+1
(4.73)
Where; A∗p3 is the cross-sectional area of the mass flux within the T-cell, along the branch direction.
There could be different approaches on the derivation of momentum equations; for instance if
momentum influx to CV v3 is calculated based on the mass flux at edge v3, then there would be no
need to account for branch contribution. Nevertheless, correct form of the momentum equations
should work effectively without the loss terms. One way of testing momentum equations, without
the loss terms, is to see if they predict a pressure spike (pressure rise) for the first cell of run and if
pressure drops would be equal for both outgoing arms when there is no split angle; i.e. branch is
parallel.
Using momentum equations at the tee CV, instead of Bernoulli equations, allows accounting for
(to a degree) branching angle effects without requiring any loss terms. Therefore, inlet-to-branch
pressure drop predicted by momentum equations (without loss terms) should represent the reversible
part of pressure drop.
Calculation of momentum loss terms is, in fact, another challenge since, to the best of author’s
knowledge, and unlike Bernoulli equation energy loss terms, there is no general, explicit description
for these terms. Besides, calculation of these loss terms should be dependent on the derivation of
85
momentum equations and calculation of momentum fluxes at the faces of tee momentum cells. Based
on mechanistic junction models and Bernoulli equation loss terms (Sec. 3.2.1) typically momentum
loss terms are expected to be in the form:
loss12 = k12 ρtee v21
loss13 = k13 ρtee v21
(4.74)
The k-factors utilized with this definition of momentum loss terms are perhaps slightly different than
the k-factors used with the inlet-to-run pressure drop equation (Eqn. 3.8) in Sec. 3.2.1. Moreover,
because inlet-to-branch k-factor correlations are not available in the literature, they should be
obtained before this approach is pursued any further.
Steinke (1996) discusses that momentum correction k-factors can be written in terms of energy
loss coefficient K-factors by arranging non-conservative form of the momentum equations (used in
TRAC; Spore et al., 2001) in the form of Bernoulli equations. For single-phase problems, appropriate
momentum k-factors can be obtained this way. However, extending the approach for two-phase
conditions may not be straight-forward:
• Two-phase K-factors (energy loss coefficients) are not widely available in the literature. It is
typically assumed that single-phase K-factors should suffice in capturing two-phase pressure
drops; i.e. via separated flow model or two-phase loss multipliers used with homogeneous
mixture approach (cf. Sec. 3.2.1).
• Transformation of Euler type momentum equations into Bernoulli type mechanical energy
equations requires certain assumptions; (1) Steady flow, (2) incompressible flow, (3) flow along
a streamline or (4) irrotational flow (cf. Munson et al., 2009 pages 279 - 284).
Theory of an alternative FVM formulation, one that accounts for different void fractions for outgoing
streams, is presented in Appendix B. The theory is an extension of Steinke (1996) discussion based
on the need for using two separate pressures within the tee CV in order to better represent 3D
pressure distribution in the tee CV for combining tees. Steinke’s approach is adopted for branching
tees with the ‘buffer zone’ concept that introduces certain simplifications in the governing model
equations.
Another approach is to use Bernoulli equations (mechanical energy balance) at the tee CV, based
on phasic streamline approach, instead of momentum equations. Then, phasic Bernoulli equation
for inlet-to-run stream ([p1 − p2]):
1
2ρkv
2k1 + ρkgzk1 + p1 =
1
2ρkv
2k2 + ρkgzk2 + p2 +
1
2Kk12ρkv
2k1 (4.75)
And, phasic Bernoulli equation for inlet-to-branch stream ([p1 − p3]):
1
2ρkv
2k1 + ρkgzk1 + p1 =
1
2ρkv
2k3 + ρkgzk3 + p3 +
1
2Kk13ρkv
2k1 (4.76)
86
Where;
vk1 =qk1
αk1A1(4.77)
vk2 =qk2
αk2A2(4.78)
vk3 =qk3
αk3A3(4.79)
From aforementioned equations, it seems as if Bernoulli equations account for exit (outgoing) void
fractions at a tee, not directly but through velocities. In practice, velocities and void fractions are
distinct primary unknowns of the FVM numerical solution. After an iteration, updated values of
the primaries are directly substituted in the governing equations. However, it is recognized that
these Bernoulli equations are not really dependent on void fractions if values of velocities are to be
substituted directly (or vice a versa). Therefore, these equations (at least in their current form)
do not carry information on void fractions of outgoing stream. Consequently it is assumed that
outgoing void fractions are same as the tee cell itself.
Phasic Bernoulli equations could replace phasic momentum equations derived previously (Eqn. 4.71
to Eqn. 4.73) and then could be solved along with mass balances for the tee CV; in order to account
for exit velocities of the tee cell. This is, indeed, the basis of Double Stream Model of Hart et al.
(1991) introduced in Sec. 3.2.3. However, observations regarding this transformation have shown that
the assumption KG = KL, which led to the derivation of final, closed form DSM equation (Eqn. 3.22),
renders simultaneous solution of phasic Bernoulli equations and mass balances inconsistent. Because
DSM is essentially a phase separation sub-model, its implementation for the tee CV; connecting
DSM pressures and velocities to the FV momentum grid and prediction of pressure drop across
the faces of tee has to be worked out. Consequently, DSM is incorporated in the FVM solution
by solving only gas phase Bernoulli equations (for both run and branch directions). Gas phase
Bernoulli equations link run and branch inlet pressures to the tee cell pressure which becomes the
outlet pressure specification for inlet arm. Liquid phase momentum equations are replaced with
dummy equations that set outgoing liquid velocities to the velocities determined by closed form
DSM equation (Eqn. 3.22) which is solved separately after completing each iteration in order to
determine phase separation. Solving the DSM equation separately, in fact, slows the convergence
rate (i.e. increases number of iterations); yet, has been observed to be stable. Ultimately, two-phase
flow separation problem is addressed by incorporating an already verified, mechanistic junction
model.
It is important to repeat once more that due to limitations of one-dimensional analysis, not all the
factors contributing to the pressure change at a tee can be accounted for explicitly and rigorously in
the governing equations. Instead, for instance, mechanical losses due to change in flow direction
or effect of secondary flows (recirculation zones, Fig. 3.4), are typically combined into a single
parameter (loss coefficient K for mechanical energy equation and correction factor k for
momentum equation) and represented with an additional irreversible loss term in the pertinent
governing equation. In that regard, use of these parameters is similar to the representation of viscous
forces of flow (i.e. wall drag) with a friction factor and associated friction force in one-dimensional
87
analysis.
4.13 Fluid Composition After Split
For the practical purpose of steady-state analysis, stream composition is assumed to be constant
along each pipe element and along each arm of a tee during an iteration. In other words, all
the CVs enclosed by a straight section of the network and the tee cells themselves have particular
compositions assigned at the beginning of an iteration and these compositions remain constant
during all calculations until the iteration is completed. Compositions are updated before new
iteration commences; according to phase split and/or flow merge at tee cells.
For a particular pipe, composition is assigned to all of it’s CVs based on it’s inlet stream, may it
be a boundary condition prescribed in the input file (if pipe is connected to a boundary node) or
outlet composition of an upstream network element (in this case another pipe or a tee) connected
to the inlet of this pipe.
For a T-junction; first, overall fluid composition arriving at the tee CV is calculated according to
incoming streams. Once the overall composition at the tee cell is determined, compositions for
outgoing arms are obtained based on the ‘separator approach’; i.e. using outgoing phasic velocities
(Martinez and Adewumi, 1997).
Following the completion of each iteration and after updating primary unknowns (and correcting
boundary conditions as necessary), pipe, tee arm and tee CV (tee cell) compositions are updated.
Then, secondary parameters; i.e. phase mass fraction (fmg), density, viscosity etc. (properties based
on composition) are computed according to newly assigned composition. The ‘update’ procedure
for the compositions is described next.
First, total composition arriving at the tee cell is computed by adding up number of moles of each
component flowing into the cell with each incoming stream. Total number of moles flowing-in
through each arm’s interface can be easily calculated by dividing total mass flow (combination of
both phases) at that face by the mixture molecular weight of the upstream cell (due to donor cell
scheme).
Total mass entering the tee cell is a summation of mass entering through each incoming arm:
min =
N (=3)∑j=1
mj (4.80)
Where, N is total number of incoming arms and for the particular case of branching tee, N = 1.
Total mass entering through arm j is given by:
mj = [mG + mL]j (4.81)
88
And, mass carried in by each phase is:
[mG = ρG vG αA]j
[mL = ρL vL (1− α)A]j(4.82)
Mass entering through each arm could also be expressed as a function of number of moles:
mj = nj
[∑ciMWi
]j[
mG = nG∑
xiMWi
]j[
mL = nL∑
yiMWi
]j
(4.83)
Mixture molecular weight for arm j:[MWmix =
∑ciMWi
]j
(4.84)
Gas phase molecular weight for arm j:[MWG =
∑xiMWi
]j
(4.85)
Liquid phase molecular weight for arm j:[MWL =
∑yiMWi
]j
(4.86)
Therefore, number of moles entering the tee cell through arm j can be written as:
nj =mj∑ciMWi[
nG =mG∑xiMWi
]j[
nL =mL∑yiMWi
]j
(4.87)
Or, substituting definition of mj from Eqns. 4.81 and 4.82:[n =
(ρg vg αg + ρg vg (1− αg))A∑ciMWi
]j[
nG =ρg vg αg A∑xiMWi
]j[
nL =ρl vl (1− αg)A∑
yiMWi
]j
(4.88)
89
Number of moles entering through arm j could also be given as:
nj = [nG + nL]j (4.89)
Total number of moles entering the tee cell:
nin =N∑j=1
nj (4.90)
Where, N is the number of arms with incoming flow to the tee, for the particular case of branching
tee, N = 1.
Number of moles of component i coming into tee cell through arm j:
[ni = n ci]j (4.91)
Total moles of component i entering tee cell:
(ni)in =
N∑j=1
[ni]j =
N∑j=1
[nci]j (4.92)
Now that total moles of all the components arriving at the tee cell are known, overall composition
at the tee cell is known and it is ‘flashed2’ at tee cell p− T conditions to determine outgoing phase
composition and fluid properties.
Overall compositions of outgoing arms are then determined according to the amount of each phase
that goes through an arm, hence depends on the phasic velocities and void fractions (please see
‘double channel’ development in Appendix B for further discussion on the theory of separate void
fractions for each outgoing arm) associated with each outgoing face of the tee cell since density and
composition of each phase is assumed to be constant through out the tee CV.
Mass of gas phase leaving the tee cell through arm j:
[mG = ρG vG αGA]j (4.93)
Mass of liquid phase leaving tee CV through arm j:
[mL = ρL vL (1− αG)A]j (4.94)
2Equation of state based determination of second phase formation and associated computation of chemicalcomponent distribution among gas and liquid phases is called ‘flash’ calculations.
90
Total mass leaving the tee cell through arm j:
[m = (ρG vG αG + ρL vL (1− αG))A]j (4.95)
Number of moles leaving the tee cell through arm j within gas phase:
[nG]j =[ρGvGαGA]j
[∑xiMWi]tee
(4.96)
Number of moles leaving tee cell through arm j within liquid phase:
[nL]j =[ρL vL (1− αG)A]j
[∑yiMWi]tee
(4.97)
Number of moles of component i leaving the tee cell through arm j within gas phase:
[(nG)i]j = [nG]j [xi]tee =[ρG vG αA]j
[∑xiMWi]tee
[xi]tee (4.98)
Number of moles of component i leaving the tee cell through arm j within liquid phase:
[(nL)i]j = [nL]j [yi]tee =[ρL vL (1− α)A]j
[∑yiMWi]tee
[yi]tee (4.99)
Total number of moles of component i leaving the tee cell through arm j:
[ni = (nG)i + (nL)i]j (4.100)
Total number of moles leaving the tee cell through arm j:
[n = nG + nL]j (4.101)
Now that number of moles of all the components going through arm j is known, composition for
arm j can be easily obtained as: [ci =
nin
]j
(4.102)
91
Chapter 5
Results and Discussion
5.1 Single-Phase Compressible Flow
Pressure profiles generated by the numerical model for an isothermal, single-phase compressible
study are compared with the analytical solution provided by Tian-Adewumi design equation:
dA2MW
f m2Z RT(p2
2 − p21)− d
fln
(p2
2
p21
)+ L = 0 (5.1)
Tian-Adewumi design equation differs from the steady-state gas flow equation (Eqn. 2.19) introduced
in Sec. 2.1.1 in the sense that:
1. Tian-Adewumi equation (Eqn. 5.1) expresses the flow in terms of mass flow rate while
steady-state gas flow equation (Eqn. 2.19) expresses the flow in terms of volumetric flow rate.
2. Kinetic energy terms ignored during the derivation of steady-state gas flow equation (Eqn. 2.19)
are included in the derivation of Tian-Adewumi equation (Eqn. 5.1).
Tian-Adewumi equation is derived from the very same Euler mass and momentum equations without
any simplifications (Tian and Adewumi, 1992). Thus, model and the analytical results should be in
very good agreement. Specifics of the study; pipe and fluid properties and boundary conditions
are given in Tab. 5.1. Pipe properties are taken from Tables 6.4 and 6.5 of Ayala (2001). However,
injected fluid is chosen as air with a constant gas compressibility (z-factor) of Z = 0.9 both for the
Tian-Adewumi equation and for the numerical model in order to have a fair comparison. Otherwise,
the design equation requires externally computed z-factors to be provided over each increment.
Comparison of pressure profiles given in Fig. 5.1 is in good agreement for the over 300 km long
pipeline. Numerical solution seems to deviate from the analytical solution towards the inlet of the
pipe where inlet pressure prediction by Tian-Adewumi equation is ∼= 15103 kPa and numerical
model predicts an inlet pressure of ∼= 15008 kPa with 20 CVs and ∼= 15054 kPa with 40 CVs. Error
in pressure prediction with 20 CVs is ∼ 0.628% at the outlet (where difference is the most significant)
and decreases to ∼ 0.321% when 40 CVs is employed by the model. The fact that error is at most
halved by doubling the number of cells suggests practical order of accuracy of the numerical model
is indeed lower than the theoretical value; as expected. Figure 5.2 shows the percentage error change
across the pipe length suggesting that numerical error increases with pipe length. An important
92
Table 5.1: Single-phase compressible flow study data
Pipe length 321.869 km
Pipe diameter 0.70485 m
Pipe roughness 0.24384× 10−3 m
Gas flow rate 170.550 kg/s
Gas MW (Air) 29 g/mole
Gas compressibility 0.9
Flow temperature 388.705 K
Ambient temperature 299.816 K
Heat transfer coefficient 5.6783 W/m2/K
Outlet pressure 9643.697 kPa
observation is that based on the shape of the curves (Figure 5.2) error does not grow beyond a
certain value however increasing towards the pipe inlet.
15,05415,103
15,008
9,000
10,000
11,000
12,000
13,000
14,000
15,000
16,000
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Pre
ssu
re [
kPa]
x [km]
Tian-Adewumi
20 Cell
40 Cell
Figure 5.1: Single-phase compressible flow study pressure profiles
93
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Pre
ssu
re E
rro
r [%
]
x [km]
20 Cell
40 Cell
Figure 5.2: Single-phase compressible flow study numerical error profile
5.2 Two-Phase Flow
For two-phase analysis, the gas condensate fluid described by Ayala and Adewumi (2003) is
implemented in this study. The hydrocarbon mixture is composed of three ‘lumped’ components
where molar composition, component properties, binary interaction coefficients for EOS calculations
and parachor parameters for interfacial tension calculations are given in Tabs. D.1, D.2, D.3 and
D.3 respectively, in Appendix D. The mixture is injected at the inlet of the pipeline described in
Tab. 5.2.
Table 5.2: Two-Phase flow study data
Pipe: SI Units US Units
Length 16093.44 m 52,800 ft (10 miles)
Diameter 0.3048 m 12 in
Roughness 3.048e-4 m 0.001 ft
Heat Trans. Coef. 5.6783 J m2 s−1K−1 1 BTU ft−2 hr−1 oF−1
Ambient temperature 26.67 oC (299.8167 K) 80 oF (539.67 R)
Gas injection rate 65.5483 m3s−1 200 MMscfD
Inlet pressure 13.65MPa 1980 psia
Inlet temperature 60.183 oC (333.3 K) 140 oF (600 R)
94
Two-phase flow performance of the FVM based two-fluid model (NR - with Newton-Raphson
solution technique) has been assessed for mist flow in the entire pipe and compared against a
Runge-Kutta based marching two-fluid model (RK). The pipe is discretized anywhere in between 2
CVs (dx = 26,400 ft) to 52,800 CVs (dx = 1 ft) in order to study the robustness of the RK and NR
methods for two-phase flow predictions. The operational conditions in this pipeline are such that
the gas is being injected to the pipeline very close to dew point conditions. Therefore, retrograde
hydrocarbon condensation is triggered upon pressure depletion, and multiphase conditions are found
throughout the entire pipe. The relatively small quantities of liquids formed in this case and the
significant amounts of turbulence in the gas phase impose mist flow (dispersed liquid) conditions
through out the pipe.
Figures 5.3 to 5.7 show the agreement between NR and RK methods when RK method uses very
small step size (i.e. dx = 1 ft) however NR model has significantly large spatial discretization (dx
= 528 ft); and how the RK solution deviates from the actual profile if computational grid is any
coarser (i.e. dx > 2 ft). Further details on the two-phase flow performance assessment of the FVM
model and comparison against RK based marching algorithm can be found in the text by Ayala
and Alp (2008).
1500
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
0 10000 20000 30000 40000 50000 60000
Pre
ssu
re (
psi
)
Distance (ft)
NR (dx = 528 ft)
RK (dx = auto)
RK (dx = 1 ft)
RK (dx = 2.5 ft)
RK (dx = 5 ft)
RK (dx = 10 ft)
Figure 5.3: Two-phase RK and NR pressure profiles
95
575
580
585
590
595
600
0 10000 20000 30000 40000 50000 60000
Tem
pe
ratu
re (
R)
Distance (ft)
NR (dx = 528 ft)
RK (dx = auto)
RK (dx = 1 ft)
RK (dx = 2.5 ft)
RK (dx = 5 ft)
RK (dx = 10 ft)
Figure 5.4: Two-phase RK and NR temperature profiles
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
22.0
22.5
0 10000 20000 30000 40000 50000 60000
Ve
loci
ty (
ft/s
)
Distance (ft)
NR (dx = 528 ft)
RK (dx = auto)
RK (dx = 1 ft)
RK (dx = 2.5 ft)
RK (dx = 5 ft)
RK (dx = 10 ft)
Figure 5.5: Two-phase RK and NR gas velocity profiles
96
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
0 10000 20000 30000 40000 50000 60000
Ve
loci
ty (
ft/s
)
Distance (ft)
NR (dx = 528 ft)
RK (dx = auto)
RK (dx = 1 ft)
RK (dx = 2.5 ft)
RK (dx = 5 ft)
RK (dx = 10 ft)
Figure 5.6: Two-phase RK and NR liquid velocity profiles
0.000
0.005
0.010
0.015
0.020
0.025
0 10000 20000 30000 40000 50000 60000
Ho
ldu
p (
Frac
tio
n)
Distance (ft)
NR (dx = 528 ft)
RK (dx = auto)
RK (dx = 1 ft)
RK (dx = 2.5 ft)
RK (dx = 5 ft)
RK (dx = 10 ft)
Figure 5.7: Two-phase RK and NR holdup profiles
97
5.3 Two-Phase Flow and Secondary Phase Appearance
Two-phase flow capability of the model is demonstrated again for mist flow pattern along with the
handling of phase appearance-disappearance. For this purpose, the same hydrocarbon mixture with
lumped components (Appendix D) is injected at the inlet of the pipeline described in Tab. 5.1.
Figures 5.8 and 5.9 show pressure and temperature profiles obtained under different thermal
conditions (HTC: heat transfer coefficient cU , [W K−1m−2]) and for isothermal conditions. The
reason of increased pressure drop along the pipeline is more clear when void fraction profiles in
Fig. 5.10 is observed; as the amount of secondary phase (liquid hydrocarbon) increases, it induces
additional pressure drop along the line. Pressure and temperature profiles obtained for HTC
= 5.6783 [W K−1m−2] is in good agreement with Figure 6.8 of Ayala (2001).
Void fraction plots in Fig. 5.10 show how two-phase conditions prevail at different distances from
the inlet, based on p− T conditions along the pipeline. As expected, with increased heat loss to the
environment (increasing HTC and greater temperature drop) liquid hydrocarbons released earlier in
the pipe and model captures this phenomenon successfully.
9,000
10,000
11,000
12,000
13,000
14,000
15,000
16,000
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
pre
ssu
re [
kPa]
x [km]
HTC = 5.6783
HTC = 1.0000
HTC = 0.0000
Isothermal
Figure 5.8: Phase appearance p profiles - 20 CVs
Phasic velocity profiles given in Figs. 5.11 and 5.12 indicate that gas velocity and dummy liquid
velocity values are almost identical in the single-phase region. This is to be expected since in the
absence of liquid phase, liquid phase properties such as density and viscosity are simply set equal
to the gas phase properties by the EOS module (otherwise EOS module does not provide any
information on the absent phase).
98
280
300
320
340
360
380
400
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Tem
pe
ratu
re [
K]
x [km]
HTC = 5.6783
HTC = 1.0000
HTC = 0.0000
Isothermal
Figure 5.9: Phase appearance T profiles - 20 CVs
0.9550
0.9600
0.9650
0.9700
0.9750
0.9800
0.9850
0.9900
0.9950
1.0000
1.0050
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Vo
id F
ract
ion
[-]
x [km]
HTC = 5.6783
HTC = 1.0000
HTC = 0.0000
Isothermal
Figure 5.10: Phase appearance αG profiles - 20 CVs
99
The staggered nature of the momentum CVs along with the Frepoli et al. (2003) type dummy
momentum equation causes a jump in liquid velocity (not plotted in the figures) right before the
momentum CV where liquid phase appears the first time (∼ 50 km in Fig. 5.11). This jump is
not physical and only a consequence of the current formulation; i.e. densities substituted in the
staggered momentum equations is a weighted average of neighboring mass CVs. In the case of
dummy liquid momentum equation written for the inlet face of a mass CV where liquid phase
appears the first time; the upstream liquid density is simply set equal to the gas density (by choice)
and downstream density is the actual liquid determined by the EOS. This arrangement, along with
the dummy momentum equation is the source of jump and indicates that current implementation of
the dummy momentum equation, while not essential for steady-state analysis, needs improvement
since its purpose is to accelerate the iteration process by preventing such jumps in the profiles that
could ensue in convergence or stability issues.
Another important thing to notice is the effect of steep temperature drop during the first 100 km of
the pipe; the density increase accompanying the temperature drop (Fig. 5.13) is slowing down the
phases, hence the initial descend in the velocity profiles of Fig. 5.11 is not observed in Fig. 5.12 where
temperature drop is not that sharp due to smaller heat transfer coefficient (cU = 1 [W K−1m−2],
Fig. 5.14).
2.5
2.7
2.9
3.1
3.3
3.5
3.7
0.955
0.960
0.965
0.970
0.975
0.980
0.985
0.990
0.995
1.000
1.005
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Ve
loci
ty [
m/s
]
Vo
id F
ract
ion
[-]
x [km]
Void fraction [-]
Liquid velocity [m/s]
Gas velocity [m/s]
Figure 5.11: Phase appearance αG vs vk profiles - 20 CVs, HTC ∼= 5.6783W K−1m−2
Figures 5.15 and 5.16 compare p−T and void fraction vs cumulative mass transfer profiles obtained
with 20 and 40 CVs. No significant difference is observed in the profiles besides the slight change in
the location of secondary phase appearance; within [40− 50] km with 20 CVs and [45− 55] km with
40 CVs, as a result of finer griding. It is also observed that jump in the liquid velocity profile is still
100
3.1
3.3
3.5
3.7
3.9
4.1
4.3
0.980
0.982
0.984
0.986
0.988
0.990
0.992
0.994
0.996
0.998
1.000
1.002
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Ve
loci
ty [
m/s
]
Vo
id F
ract
ion
[-]
x [km]
Void fraction [-]
Liquid velocity [m/s]
Gas velocity [m/s]
Figure 5.12: Phase appearance αG vs vk profiles - 20 CVs, HTC ∼= 1.0W K−1m−2
2.5
2.7
2.9
3.1
3.3
3.5
3.7
0
50
100
150
200
250
300
350
400
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Ve
loci
ty [
m/s
]
De
nsi
ty [
kg/m
3]
x [km]
Gas density [kg/m3]
Liquid density [kg/m3]
Liquid velocity [m/s]
Gas velocity [m/s]
Figure 5.13: Phase appearance density and velocity profiles - 20 CVs, HTC ∼= 5.6783W K−1m−2
101
3.1
3.3
3.5
3.7
3.9
4.1
4.3
90
140
190
240
290
340
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Ve
loci
ty [
m/s
]
De
nsi
ty [
kg/m
3]
x [km]
Gas density [kg/m3]
Liquid density [kg/m3]
Gas velocity [m/s]
Liquid velocity [m/s]
Figure 5.14: Phase appearance density and velocity profiles - 20 CVs, HTC ∼= 1.0W K−1m−2
of the same magnitude when velocity profiles are compared (Fig. 5.17).
With steep temperature drops (i.e. due to heat loss to the environment) stability problems have
been observed and required providing better initial conditions to the system. In fact, mass transfer,
associated momentum transfer and heat loss terms; all evaluated at current iteration level (k+1),
increase the non-linearity of the problem and cause difficulties in convergence. Tricking the way
around by lagging the calculations of transfer and heat loss terms one iteration behind could help
attain convergence; yet would definitely increase number of iterations.
Another important observation is; using small CVs (fine grid) also ensues in (1) smaller transfer
terms because values of gas mass fractions (fmg) of neighboring cells are closer to each other (cf.
Sec. 4.11.2) and (2) smaller heat loss terms because pipe surface area is lesser. While fluxes entering
and leaving the CVs remain the same, as these terms get smaller in magnitude, it is easier for the
system to digest associated non-linearity, albeit at the expense of increased computational load due
to increased number of CVs. Comparison of mass transfer terms per CV; given in Fig. 5.18, clearly
indicate the difference in the magnitude of calculated mass transfer terms however cumulative mass
transfer is same (Fig. 5.16).
Also, looking at Fig. 5.11 and Fig. 5.16 the effect of hydrodynamic holdup is clearly observed; as
gas-to-liquid mass transfer diminishes towards the last 100 km of the pipeline, void fraction begins
to ascend (or holdup begins to decrease for that matter) indicating that liquid is slowly being left
behind not being able to catch up with the gas flow.
102
250
270
290
310
330
350
370
390
410
9,000
10,000
11,000
12,000
13,000
14,000
15,000
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Tem
pe
ratu
rer
[K]
Pre
ssu
re [
kPa]
x [km]
P 20 cell [kPa]
P 40 cell [kPa]
T 20 cell [K]
T 40 cell [K]
single phase
Figure 5.15: Hydrocarbon mixture p− T profiles - 20 and 40 CVs, HTC ∼= 5.6783 [W K−1m−2]
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
0.955
0.960
0.965
0.970
0.975
0.980
0.985
0.990
0.995
1.000
1.005
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Cu
mu
lati
ve M
ass
Tras
nfe
r [k
g/s
]
Vo
id F
ract
ion
[-]
x [km]
Void frac. 20 cell [-]
Void frac. 40 cell [-]
Mass Trans. 20 cell [kg/s]
Mass Trans. 40 cell [kg/s]
Figure 5.16: Phase appearance αG vs cum. mass transfer profiles - 20 and 40 CVs, HTC ∼=5.6783W K−1m−2]
103
2.5
2.7
2.9
3.1
3.3
3.5
3.7
3.9
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00
Ve
loci
ty [
m/s
]
x [km]
velG 20 cell [m/s]
velL 20 cell [m/s]
velG 40 cell [m/s]
velL 40 cell [m/s]
single phase
Figure 5.17: Phase appearance velocity profiles - 20 and 40 CVs, HTC ∼= 5.6783W K−1m−2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.955
0.960
0.965
0.970
0.975
0.980
0.985
0.990
0.995
1.000
1.005
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Mas
s Tr
ansf
er
pe
r C
V [
kg/s
]
Vo
id F
ract
ion
[-]
x [km]
Void frac. 20 cell [-]
Void frac. 40 cell [-]
Mass Trans. 20 cell [kg/s]
Mass Trans. 40 cell [kg/s]
Figure 5.18: Phase appearance αG and mass transfer per CV profiles - 20 and 40 CVs, HTC∼= 5.6783W K−1m−2
104
5.4 Flow Split and Phase Separation
We begin flow split and phase separation analysis using air-water mixture. Data of the tee structure
(from Hart et al., 1991 and Singh, 2009) is given in Tab. 5.3. Flow is assumed to occur isothermally
and no mass exchange is considered; i.e. no water vapor or air dissolution in the water (mass
transfer = 0). Stratified flow pattern is assumed to prevail in the junction in accordance with DSM.
Besides, typically phase separation is equal (i.e. λG = λL) for dispersed flow patterns such as mist
flow since dispersed phase is entrained in the continuous phase. While gas flow rate is kept constant,
simulation is repeated with different liquid flow rates in order to observe the effect of increased
liquid presence on uneven phase separation. For a given water flow rate branch and run outlet
pressures are separately increased (from the initial 101.325 kPa) to observe the effects of outlet
pressure on the solution.
Table 5.3: Air-water flow split data
Arm length L 15 m
Number of CVs nCV 5
Arm diameter D 0.051 m
Pipe roughness ε 0.005 m
Gas flow rate mG 1.973× 10−2 kg/s
Air MW MWair 29 g/mole
Gas compressibility Z 0.9
Water density ρw 998 kg/m3
Flow temperature T 288.71 K
Run outlet pressure prun 101.325 kPa
Branch outlet pressure pbranch 101.325 kPa
Typical pressure and void fraction profiles for the tee structure are given for water flow rate of
0.0057 kg/s in Figs. 5.19 and 5.20 respectively. Model captures pressure rise in the run entrance as
expected (Fig. 5.19). And Fig. 5.20 shows how significantly void fraction can change (in the run)
following an uneven phase separation at the junction when most of the gas phase is diverted into
the branch (for λL = 0.52 and λG = 0.90).
Looking at Fig. 5.21, it is observed that there is always a pressure increase at the run entrance and
the pressure difference between run and branch entrances is always positive; i.e. (prun > pbranch)tee.
More liquid tends to turn into branch direction as this pressure differential increases. Also, with
increasing liquid flow rate (as the liquid momentum increases) it is harder to divert liquid phase
into the branch. This is to be expected since same amount of pressure drop now has to overcome
greater liquid inertia in order to draw liquid phase into the branch.
105
101.2
101.4
101.6
101.8
102.0
102.2
102.4
102.6
102.8
103.0
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0
Pre
ssu
re [
kPa]
x (m)
inlet
run
branch
Figure 5.19: Air-water flow pressure profile at T-junction for mL = 0.0057 kg/s
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.00
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0
Vo
id f
ract
ion
[-]
x (m)
inlet
run
branch
Figure 5.20: Air-water flow αG profile at T-junction for mL = 0.0057 kg/s and prun = 102.125 kPa
106
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070
λ L
(prun- pbranch)tee [kPa]
mL=1.427e-4 kg/s
mL=1.814e-3 kg/s
mL=3.140e-3 kg/s
mL=5.700e-3 kg/s
Figure 5.21: Air-water flow (∆p)tee vs λL plots for different water flow rates
Plotting a similar graph for outlet pressures tells more on the uneven phase separation. Looking at
Fig. 5.22, the effect of liquid flow rate is observed in a different manner; with increasing water flow
rate less liquid tends to flow into branch when both outlet pressures are equal (prun = pbranch or
prun − pbranch = 0). When run outlet pressure is increased (to the right of the y axis in Fig. 5.22)
more of the both phases are diverted into the branch as expected since fluids prefer flowing towards
a lower potential; in this case, into the branch. Similarly, when branch outlet pressure is increased
(to the left of the y axis in Fig. 5.22) lesser amounts of both phases go into the branch. It is easier
to observe the flip-flop effect when water flow rate curves of mL = 1.427e−4 and mL = 1.814e−3
[kg/s] are considered; slight change in the outlet pressure differential causes water turn into branch
entirely. This signifies how sensitive uneven phase separation can be in response to downstream
(outlet) pressures.
When λG vs λL is plotted (Fig. 5.23) curves with characteristics to Fig. 5.22 is observed. Suggesting
that primary (continuous) phase split is not significantly effected. Plotting (∆p)out vs λG (Fig. 5.24)
verifies the fact that for small holdup values primary phase split is simply governed by outlet
pressure differential as is the case with single-phase flow split.
107
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000
λ L
(prun - pbranch)out [kPa]
mL=1.427e-4 kg/s
mL=1.814e-3 kg/s
mL=3.140e-3 kg/s
mL=5.700e-3 kg/s
prun > pbranch
prun < pbranch
Figure 5.22: Air-water flow (∆p)out vs λL plots for different water flow rates
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
λ L
λG
mL=1.427e-4 kg/s
mL=1.814e-3 kg/s
mL=3.140e-3 kg/s
mL=5.700e-3 kg/s
Figure 5.23: Air-water flow λG vs λL plots for different water flow rates
108
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000
λ G
(prun - pbranch)out [kPa]
mL=1.427e-4 kg/s
mL=1.814e-3 kg/s
mL=3.140e-3 kg/s
mL=5.700e-3 kg/s
prun > pbranch
prun < pbranch
Figure 5.24: Air-water flow (∆p)out vs λG plots for different water flow rates
Figure 5.25 is another widely used representation of phase separation; the ‘complete phase separation’
curve. This curve shows at what total mass flow rate fraction all the liquid will be diverted into
the branch. It is important to note that this representation differs slightly from the discussion in
Sec. 3.1 which was aimed for total gas phase separation. Consequently, mathematical description of
the liquid phase separation needs to be arranged as follows:
m1(1− x1) = Inlet liquid mass rate
m3(1− x3) = Branch liquid mass rate(5.2)
5.4.1 Hydrocarbon Mixture Uneven Phase Separation
In this section, the hydrocarbon mixture (Appendix D) used for the analysis of two-phase flow in
Sec. 5.3 is utilized again for uneven phase separation analysis. Table 5.4 describes the tee structure.
In order to ensure that liquid hydrocarbon is present in the flow stream throughout arms of the
whole tee structure, average operating p− T conditions are chosen to fall within the phase envelope
(i.e. p = [12536 − 12696] kPa) and, given the length of arms (150m), flow is assumed to occur
isothermally at T = 299.82 K and without mass transfer.
Based on average operating pressure different amounts of liquid is formed within the inlet arm and
consequently different phase separation curves are obtained as shown in Fig. 5.26. As an outcome of
the retrograde condensation (keeping the operational temperature constant) more liquid is formed
109
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
(1-x3)/(1-x1)
m3/m1
mL=1.427e-4 kg/s
mL=1.814e-3 kg/s
Complete Separation
Figure 5.25: Air-water complete phase separation curves
Table 5.4: Hydrocarbon mixture flow split data
Arm length L 150 m
Number of CVs nCV 5
Arm diameter D 0.70485 m
Pipe roughness ε 0.24384× 10−4 m
Gas flow rate qG 178.29 m3/s
Flow temperature T 299.82 K
110
in the line with decreasing average operating pressure; i.e. inlet void fraction values lie in the range
[0.98− 0.95]. Hence, slopes of phase separation curves decrease with decreasing operational pressure
(more liquid), as observed previously for air-water mixture (Fig. 5.23).
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
λ L
λG
p = 12536.81 kPa
p = 12636.81 kPa
p = 12696.81 kPa
Figure 5.26: Hydrocarbon mixture flow λG vs λL plots for different average flow pressures
Table 5.5 lists overall composition of hydrocarbon streams going through each of the arms of the
tee when all the liquid phase is diverted into the branch. Consequently, mole fractions of heavier
components (i.e. Lumped-3 and Lumped-2) are greater in the branch stream; compared to inlet
and run streams, since heavier components condense within the liquid phase; while, fraction of
the lighter component (i.e. Lumped-1) has decreased in the branch stream. However, the changes
in the compositions are not significant and points to the conclusion that for small holdup values
compositional change can be ignored for the purposes of network analysis.
Table 5.5: Hydrocarbon mixture flow change in overall composition after flow split at the T-junction;pavg = 12636.81 kPa, λG = 0.55 and λL = 1
Mole fraction:
Component: Inlet Run Branch
Lumped-1 0.7658 0.7679 0.7641
Lumped-2 0.2240 0.2226 0.2251
Lumped-3 0.0102 0.0095 0.0108
111
5.5 Open Network Application
The significance of uneven phase separation is further demonstrated for air-water mixture flowing
through a relatively short, open network with two 90o (horizontal) branches joining a main line.
Network data is given in Tab. 5.6.
Table 5.6: Air-water flow open network data
Main line L 60 m
Main line num. of CVs nCV 20
Branch (1 & 2) L 15 m
Branch num. of CVs nCV 5
Arm diameter D 0.051 m
Pipe roughness ε 0.005 m
Inlet gas flow rate mG 1.973× 10−2 kg/s
Air MW MWair 29 g/mole
Gas compressibility Z 0.9
Inlet water flow rate mL 2.23× 10−2 kg/s
Water density ρw 998 kg/m3
Flow temperature T 288.71 K
Run outlet pressure prun 109.39 kPa
Branch outlet pressure (1st) pbranch1 110.925 kPa
Branch outlet pressure (2nd) pbranch2 109.39 kPa
Pressure distribution in the network is given in Fig. 5.27. Important observation to note is the
pressure rise (the spike) in the main line right after each split (around 20m and around 52m). Void
fraction (αG) distribution of the network is given in Fig. 5.28; demonstrating how significantly main
line void fraction can change as a result of uneven phase separation.
112
109.0
109.5
110.0
110.5
111.0
111.5
112.0
112.5
0 10 20 30 40 50 60 70
Pre
ssu
re [
kPa]
x [m]
Inlet
Run
Branch
Branch 2
110.93
110.94
110.95
110.96
110.97
15 17 19 21 23 25
109.50
109.52
109.54
109.56
109.58
109.60
47 49 51 53 55
Figure 5.27: Air-water open network pressure profile
0.945
0.95
0.955
0.96
0.965
0.97
0.975
0.98
0 10 20 30 40 50 60 70
Pre
ssu
re [
kPa]
x [m]
Inlet
Run
Branch
Branch 2
Figure 5.28: Air-water open network αG profile
113
5.6 Single-Phase Closed-Loop Network Application
A simple, single phase, steady-state network presented in Fig. 5.29, is solved using the model
proposed in this study. All required data is shown in Fig. 5.29 (i.e. supply/demand and delivery
pressures) and given in Tab. 5.7 (i.e. fluid and pipe properties). Results are compared to the
traditional gas network solution that uses generalized analytical gas flow equations. It is important
to note that upstream and downstream nodes are assigned randomly to describe the necessary
pipe interconnectivity and to provide an initial guess for flow directions. Actual flow directions
are ultimately determined by the models and both methods expected to be in agreement. Given
the inlet flow rate and exit pressures, the network is first solved using the FVM based model to
predict exit flow rates and then these exit flow rates are fed back in the conventional model (based
on Kirchhoff’s laws and analytic equations) in order to compare pressure distribution predictions
for the network. Pipe properties are kept same for the whole network.
Adopting the FVM model of this network required some effort since the nodes, where both a
supply/demand has been assigned and 3 pipes meet, need to be replaced with two consecutive tee
structures. However, the cells of these structures should be small enough not to significantly change
the network topology. This configuration; consecutive tee junctions, is also expected to be a better
representation of actual field conditions where pipes are typically connected using 90o tees and it
is not practical to connect more than 3 pipes to the same pipe junction. The FVM arrangement
built for the network is presented in Fig. 5.30. The shaded cells (gray and light green) are each
1m long while all the other cells are 2414/3 = 804.67m long. In fact, Fig. 5.30 demonstrates how
similar in essence the FVM representation of a pipeline network to the typical single layer reservoir
representations in simulation studies.
Exit flow rates obtained by the FVM model, as well as the input rate, are supplied to the analytical
model. Pressure specification at only one node (in this case at node 3) is sufficient for the analytical
solution. Pressure distribution obtained by the FVM code is compared to the analytical solution
in Tab. 5.8. Pressures seem to be in good agreement with the analytical results. Up to this point,
the proposed FVM model has been successfully applied to the study of single-phase networks and
multiphase flow conditions in single pipes and tees.
114
Figure 5.29: Single-phase closed-loop network
Table 5.7: Single-phase network data
Parameter SI Units US Units
Pipe length 2414 m 7920 ft
Pipe diameter 0.1016 m 4.00 in
Pipe roughness 1.8288E-04 m 0.0072 in
Gas gravity (γG) 0.5524
z-factor (Z) 0.90
Operating Temperature 23.89 oC 75.00 oF
115
Figure 5.30: FVM representation of the single-phase closed-loop network
Table 5.8: Comparison of pressure results of of the single-phase network
Node Pressure (Analytic) Cell Pressure (FVM) Deviation
# kPa psia # kPa psia %
N1 1974.452 286.37 1 1939.340 281.28 1.7783
N2 1283.459 186.15 37 1279.124 185.52 0.3378
N3 1103.161 160.00 23 1103.204 160.01 -0.0039
N4 1159.078 168.11 7 1162.302 168.58 -0.2782
N5 1116.054 161.87 17 1116.336 161.91 -0.0252
N6 1089.303 157.99 29 1090.359 158.14 -0.0970
N7 1011.047 146.64 11 1014.038 147.07 -0.2958
N8 1010.771 146.60 54 1013.140 146.94 -0.2343
N9 1008.358 146.25 33 1010.805 146.60 -0.2426
116
Chapter 6
Concluding Remarks
Current state-of-the-art modeling software available to oil and gas industry do not address uneven
phase separation (route preference) issue and typically do not provide a distinct T-junction component
for modeling purposes. This study includes a comprehensive review of branching T-junction and
phase separation models available in the open literature and classification of modeling efforts. In that
regard, a FVM based one-dimensional, two-fluid model has been developed for steady-state analysis
of low-liquid loading two-phase flow split at branching T-junctions with the purpose of uneven phase
separation and route preference analysis in gas condensate networks. The model uses steady-state,
one-dimensional Euler equations and outlet pressure specifications. Uneven phase separation at
the junction has been captured through incorporating a fluid mechanics based phase separation
sub-model at the junction; double stream model (DSM) of Hart et al. (1991), where Bernoulli type
mechanical energy equation (gas phase) ultimately replaced phasic momentum equations at the
junction CV and closed form DSM equation is solved separately in order to determine amount of
phase separation.
Developed model captures the expected pressure rise in the run direction following the flow split
at the junction and predicts different amounts of liquid phase separation with increasing liquid
flow rate and different outlet pressure specifications, in agreement with experimental data and
trends available in the literature. With increasing liquid flow rate, lesser amounts of the liquid
phase goes into the branch as a result of higher inertia and axial momentum; while, decreasing
branch outlet pressure (or increasing run outlet pressure for that matter) has been observed to
divert more of the liquid phase into the branch by creating a greater centripetal force on the liquid
phase. Depending on the phase separation, significantly different void fractions can ensue in the
branch or run directions compared to inlet void fraction (Fig. 5.28). Compositional changes in the
outgoing hydrocarbon mixtures following the split have been observed to be insignificant for the
particular case of low-liquid loading conditions (i.e. αL < 0.06, Tab. 5.5)
6.1 Conclusions
Conclusions based on the most prominent challenges of this study and respective observations
follows below:
• No apparent difficulties have been observed related to the assignment of gas phase properties as
liquid properties when liquid phase is absent (for hydrocarbon mixtures). Typically, following
117
a pressure (or temperature) perturbation, when constructing the jacobian matrix, severe
discontinuities are anticipated to surface when EOS module decides formation of the liquid
phase and computes actual liquid properties in order to replace bogus liquid phase properties
that were equivalent of gas phase. Despite the fact that system continues to use the same
equation set associated with the phase flag assigned to a CV at the beginning of an iteration,
significant changes in the phasic properties (i.e. density and viscosity) could cause problems.
• While single-phase to mist flow pattern (two-phase) transition had no problems, convergence
problems have been detected when a direct transition from single-phase to stratified flow
pattern is artificially induced. This is due to larger differences in gas phase and liquid phase
velocities in stratified flow and associated magnitudes of wall and interfacial friction forces.
stability and convergence problems when trying to switch from single-phase to stratified flow
pattern directly.
• The dummy momentum equation, along with the use of gas phase properties for the absent
liquid phase and averaging of neighboring densities to be substituted in the liquid momentum
equation, has been observed to cause an artificial jump in the liquid phase profile.
• Without the irreversible losses, utilizing phasic momentum equations at the T-junction CV has
been observed to predict a fixed phase separation (i.e. λL/λG = constant) for different flow
rates. However, with the inclusion of loss terms (Ottens et al., 1994) convergence problems
has been noted.
• Replacing phasic momentum equations with Bernoulli equations and taking phasic loss
coefficients to be equivalent (the inherent assumption of DSM) or computing liquid phase loss
coefficients based on Ottens et al. (1994) correlations (basis of advanced DSM) have been
found out to be inconsistent. Instead, closed form DSM equation is solved separately and
phasic momentum equations are replaced with gas phase Bernoulli equation.
• The DSM benefits from the fact that macroscopic mechanical energy balances should be
satisfied at all times. Nevertheless, when flow pattern geometry and phasic distribution in the
main line is not accounted for, DSM and similar mechanistic approaches (i.e. one-dimensional
momentum equations) are likely to fail if branch diameter is significantly smaller than main
line diameter and/or side port orientation1 is not horizontal. Especially, if model has no way
of recognizing the highest elevation the liquid film (or the liquid surface) within the main
line, it could still predict liquid phase going into the branch while it is not actually physically
possible; i.e. liquid level lower than the entrance of side port opening.
• The distances from tee center to CV faces have been observed to have an effect on convergence;
as these distances get smaller system required initial conditions closer to the actual solution;
else, either convergence rate has slowed or failed completely.
• Phases are not necessarily at thermodynamic equilibrium initially (i.e. upon entering the
CV); especially when a secondary stream merges with a primary stream at a combining
1Location of the branch entrance on the main line surface, not the inclination of branch arm
118
T-junction. There is a time factor involved for the establishment of chemical equilibrium
among the merging streams within a CV. Hence, chemical equilibrium may not be established
before phasic streams leave the current CV and move to the next one at new p-T conditions.
For sufficiently long CVs however; chemical equilibrium assumption is reasonable based on
volumetric averages utilized in FVM. Furthermore, over a fine grid where change in p − Tis insignificant between neighboring CVs; assumption of thermodynamic equilibrium is not
expected to cause any divergence from the actual physics unless sharp changes are observed in
p− T ; i.e. due to sudden enlargement at a T-junction. This is the only source of anticipated
sharp changes in p− T parameters in steady-state analysis, apart from merging of streams at
a T-junction. Regardless, a kinetic model for mass transfer, accounting for the time it takes
fluid streams to traverse the CV, should establish the correct amount of mass transfer without
the chemical equilibrium assumption.
• Heat loss to the surroundings is calculated based on a simplified model that inherently assumes
both phases to be in thermal equilibrium which could introduce errors in the prediction of
overall heat transfer to the environment when using finer grids following the argument in the
previous paragraph.
6.2 Recommendations
Based on aforementioned conclusions following recommendations are in order for future work:
• Although providing better initial guesses (initial conditions) has almost always alleviated the
convergence problems that occurred during the course of this study, the need for a globally
convergent Newton-Raphson algorithm; one that introduces obtained improvements in fractions
whenever iterations begin to diverge from the locality of actual solution, is apparent in order
to assure convergence when initial conditions are outside the the radius of convergence.
• For steady-state conditions, use of following equation in place of the missing phase momentum
equation should eliminate the jump observed in the absent phase velocity profile, when
single-to-two phase transition occurs:
0 = vG − vL (6.1)
• A ‘front tracking’ algorithm, similar in theory to the ‘level tracking’ algorithm discussed by
Aktas (2003), could be useful in predicting the actual location of phase appearance within a
long CV.
• Impact of (1) near T-junction griding and (2) distance of CV faces from the tee center; on the
pressure rise in the run direction and convergence issues is to be further inspected.
• Inclusion of advanced DSM to relieve the current model from the small holdup constraint
and accounting for the liquid level in the main line would improve the model applicability
significantly.
119
• It could be possible to obtain better correlations for phasic momentum correction factors (or
phasic energy loss coefficients) for two-phase flow at a T-junction through a separate 3D CFD
study.
• Proper modeling of two-phase flow through counter-combining and impacting T-junctions,
in addition to branching T-junctions, is a must for a full scale, closed-loop gas condensate
network analysis.
• Discussion in the last two paragraphs of Sec. 6.1 suggests that a grid sensitivity analysis is
essential for T-junctions along with the sensitivity analyses for changes in mixture composition,
void fraction, outlet pressures and a wider range of gas-liquid flow rates.
120
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127
Appendix A
Single-Phase Flow in Pipe Networks
Pipeline systems (networks) exist in several engineering applications such as water distribution
systems in civil engineering, oil and gas pipelines in petroleum engineering and reactor cooling
systems in nuclear engineering.
Single-phase, compressible or incompressible flow in networks is typically modeled using one-
dimensional, steady-state, analytical expressions which are closed form equations derived from
fundamental principles of conservation of mass and energy applied over the entire pipeline length.
By analogy to electric circuits, these flow equations are coupled with Kirchhoff laws to obtain
steady-state solutions. Kirchhoff laws stand for conservation of mass at nodes (junctions of pipes)
and conservation of energy around loops.
From very simple to very complex pipeline systems, every network is composed of two basic
components (Szilas, 1985):
1. Node: Nodes are junctions where two or more pipes meet and supply/demand is introduced
to the system (i.e. wellheads, delivery stations etc.). Kolev (2005) uses the term ‘knots’ to
distinguish junction nodes from supply/demand nodes.
2. Node Connecting Element (NCEs): All the elements laying between nodes are considered as
NCE, such as legs (straight pipe sections), compressors, chokes and etc.
There are three distinct connection types for NCEs (Fig. A.1):
1. Series connection; outlet of one NCE is connected only to the inlet of another NCE
2. Parallel connection; inlet of one NCE is connected to the inlet of another NCE and their
outlets are connected to each other as well
3. Loop connection; three or more NCEs are connected in series to form a geometrically closed
structure (outlet of the last NCE is connected to the inlet of first NCE in the loop). However,
there is no restriction on the number of connections that the nodes of the loop can have.
The two kinds of pipeline systems can be built with these connection types; loopless systems and
looped systems. Loopless systems are better known as spanning trees or open networks where the
NCEs of the system form no closed loops Fig. A.2.
128
Figure A.1: Connection Types
Figure A.2: An Open Network
Looped systems can be further classified as single loop systems in which there is only one loop
formed by the NCEs and multiple loop systems in which several loops exist (Fig. A.3).
In a network which has a single or no input source, number of loops (around which independent
energy equations can be written) is equal to the number of NCEs minus number of nodes plus one
(Eqn. A.1).
Num. of loops = Num. of NCE−Num. of nodes + 1 (A.1)
Figure A.3: Closed Networks
129
If the network has more than one input source, number of loops is simply equal to the number of
NCEs minus number of nodes (Eqn. A.2).
Num. of loops = Num. of NCE−Num. of nodes (A.2)
A.1 Kirchhoff Analysis for Flow in Networks
Conventionally, before modeling a pipeline system a mathematical hydraulic model for every NCE in
the pipeline system is defined; usually in the form of a pressure drop versus flow rate (throughput)
relationship. Then, by recognizing an analogy between pipeline systems and electric circuits,
Kirchhoff’s laws are applied to the pipeline system.
The hydraulic model for steady-state, compressible, single-phase gas flow under isothermal conditions,
shown in Eqn. A.3, is the closed form equation derived from principle conservation equations as
previously presented in Sec. 2.1.1 (Eqn. 2.19).
qG = C
(Tscpsc
)√1
f
√p2in − p2
out es
γGTavgZavgLeD
52 (A.3)
This equation relates pressure drop across the pipe to the gas flow rate, pipe length, diameter
and roughness. This hydraulic model for single-phase compressible flow through straight pipes is
typically arranged in the form of Eqns. A.4 and A.5 with an analogy to Ohm’s law for conductors
in an electric circuit.
∆(p2) = q2GR (Resistance) ⇔ V (Voltage) = I (Current)R (Resistance) (A.4)
∆(p2) = p2in − p2
out = q2G (
pscTsc
)2 f
C2
γGTavgZavgLeesd5︸ ︷︷ ︸
Resistance R
(A.5)
where: qG is the flow rate through the leg (straight pipe section), ∆(p2) is the difference of squared
inlet and outlet pressures of the pipe and represents the potential for flow, R is the ’resistance’ term
in units of p2/q2G with an analogy to circuitry. The ‘resistance’ term accounts for the cumulative
effect of pipe properties and flow variables such as diameter, friction factor, fluid properties etc.
Kirchhoff’s first law, also known as the ‘node law’ is indeed an application of conservation of
mass at a node where algebraic sum of mass flow coming into and going out of a node is zero
Eqn. A.6. It is also possible to observe the first law as a node/junction continuity equation
nconn∑i
(qG)i = 0 (A.6)
Where; (qG)i is the flow in the i’th NCE connected to the node and nconn is the number of NCEs
connected to the node.
130
Kirchhoff’s second law or the ‘loop law’ is the application of conservation of energy around a
loop where algebraic sum of potential change (i.e. pressure drop) along each NCE in the loop, with
a consistent sign convention, is zero. Hence it is also known as the energy loop equation (Eqn. A.7).
With an analogy to circuitry, this second law could be interpreted as the movement of a particle
around the loop; a particle begins the movement at a certain node and always follows a certain
direction (clockwise or anti-clockwise). One loop is completed when the particle arrives at the
starting node. Then, (potential) energy of the particle should not change since no work has been
done or since particle arrived at the same reference point of same potential.
nLoop∑i
∆(p2)i = 0 (A.7)
Where; ∆(p2)i is the potential change in the i’th NCE of the loop and nLoop is the number of NCEs
in the loop.
A.2 Steady-State Analysis of Pipeline Systems
Flow in pipeline systems is typically assumed to be at steady-state in order to simplify the modeling
task. In this section, the most traditional approaches for steady-state analysis of pipeline network
are presented.
A.2.1 Open Network Analysis
In a ‘loopless’ system with n nodes and m = n− 1 pipes, if parameters to compute NCE resistances
(R) are present as well as necessary initial and boundary conditions, pressure at any node of the
network could be computed in a sequential manner. The necessary initial and boundary conditions
include a proper set of flow rates in to and out of the nodes and any of the terminal pressures (any
one of the network inlet or outlet pressures). Then, Beginning with the NCE connecting the node of
known terminal pressure to the network; hydraulic equations for each NCE are solved successively
along with the application of Kirchhoff’s first law.
A.2.2 Closed Network Analysis
Cross Method
For looped system analysis, one of the early methods is again an iterative procedure, proposed by
Cross for the analysis of a low pressure looped system and later extended by Hain for high pressure
systems (Szilas, 1985).
Let us consider the analysis of a single loop system where flow rate and pressure at the inlet and
input/output rates at the nodes are known but pressures at the nodes and flow rates in the pipes
are to be determined. This analysis requires the application of the Kirchhoff’s second law, in which
a consistent sign convention for flow direction is needed. Conventionally, clockwise direction is
131
positive. Initially; a flow rate is assumed for one of the lines emerging from the inlet node and then
node continuity equation is applied at every successive node till one loop is completed in a certain
direction. Therefore, flow rates in pipes are computed and node pressures can be easily obtained.
If calculated flow rates for the lines differ from the actual rates by an ∆q amount due to initially
assumed flow rate in the first pipe, then Kirchhoff’s second law is applied for the loop and initially
assumed inlet flow rate is corrected by an ∆q amount as expressed in Eqn. A.8:
(qG)actualinlet = (qG)computedinlet + ∆q (A.8)
Kirchhoff’s second law for a loop is arranged as shown in Eqn. A.9 by substituting the analytical
flow equation in the resistance form as given in Eqn. A.7.
nLoop∑i
∆(p2)i =
nLoop∑i
(q2G)iRi (A.9)
Equation A.9 is further modified to accommodate the computing procedure as shown in Eqn. A.10
and can be re-written again for the correction ∆q as presented in Eqn. A.11.
nLoop∑i
(qG)i|qG|iRi =
nLoop∑i
δi(q2G)iRi = 0 (A.10)
nLoop∑i
(qG)i + ∆q |(qG)i + ∆q|Ri =
nLoop∑i
δi (qG)i + ∆q2Ri (A.11)
Where, δi is a sign function to account for adopted sign convention correctly.
Subtracting Eqn. A.10 from Eqn. A.11 yields Eqn. A.12:
nLoop∑i
δi
(q2G)i + 2(qG)i∆q + ∆q2
Ri = 0 (A.12)
For |∆q| << | qG|, ∆q2 ≈ 0 then Eqn. A.12 simplifies to Eqn. A.13:
nLoop∑i
δi
(q2G)i + 2(qG)i∆q
Ri = 0 (A.13)
Finally required correction for the flow rates can be obtained from Eqn. A.14:
∆q =
nLoop∑i
δi(q2G)iRi
−2
nLoop∑i
δi(qG)iRi
(A.14)
132
In a multiple loop system, initially; flow rates are assumed for individual pipes in loops. Correction
amount ∆q is individually computed for loops and pipe flow rates are corrected loop after loop.
Flow rates in pipes shared by two (or more) loops are corrected using the ∆q determined for every
loop containing the pipe. In other words; flow rate in this shared pipe is corrected each time a
correction done in any of the loops it belongs to.
Renouard Method
The Cross method converges rather slowly for a multiple loop system. One reason of the slow
convergence is the individual corrections applied for each shared pipe. Because loops are addressed
independently; an improvement of the flow rate in a shared pipe by one of the loops can be attenuated
by another loop sharing the same pipe. In order to overcome this convergence difficulty Renouard
(Szilas, 1985) offered a variant of the Cross method where interaction of the loops sharing common
elements are taken into account. In this method, flow correction for a pipe is made using the sum
of individual corrections from the loops sharing the same element and this requires simultaneous
solution of the loop equations.
Stoner Method
Stoner introduced a method capable of treating looped systems with different types of NCEs and
the system response to any parameter of these NCEs while Cross method was limited to flow rate
and node pressure analysis. The basis of the analysis is the simultaneous solution of all network
governing equations using a multivariable Newton-Raphson iterative procedure (Szilas, 1985).
A pipeline system of n nodes and m NCEs yields a system of n equations with (2n+m) variables
when node continuity equations (Kirchhoff’s second law) are written at each node. While the
(2n+m) variables are; n node pressures, n supply/demand rate at nodes and either m NCE flow
resistance or m NCE flow rate; this system of n equations can be solved for any set of n variables
provided that other (n+m) variables are known.
As described before; continuity equation at any node can be expressed as shown in Eqn. A.15:
Fj = (qG)s +
nconn∑i
(qG)i = 0 (A.15)
Where:
(qG)s Flow in to or out of node j (supply/demand)
(qG)i Flow in the i‘th NCE connected to node j
nconn Number of pipes connected to node j
Fj Node continuity equation at node j
Flow in any NCE that is connected to a node can be further defined using the mathematical
hydraulic model for the NCE. Hence, node continuity equation itself becomes a function of all the
133
parameters controlling the flows in the NCEs reaching the node and can be presented as shown in
Eqn. A.16.
Fj = Fj(x1, x2, x3, ..., xn) = 0 (A.16)
Similar equations written for every node in the system constitutes a non-linear system of equations
since hydraulic model expressions involve powers and multiplications of unknowns. Nevertheless,
system can be solved with various iterative techniques (such as Newton-Raphson) as long as involved
equations are linearly independent.
A.3 Transient Analysis of Pipeline Networks
Flow in gas pipeline systems is frequently at transient (unsteady-state) and though steady-state
modeling may be adequate for several instances; the aim of transient modeling is the analysis of
planned modifications to the system as well as unplanned situations such as a leak causing a sudden
pressure drop in the system.
Transient analysis of a pipeline system is carried out in two parts; first, transient flow in NCEs
must be modeled then node continuity can be applied. The hydraulic model for straight pipes –
Eqn. 2.19 derived in Sec. 2.1.1 - no longer describes the flow accurately since the simplifications
lead to its derivation are based on the assumption of steady-state conditions. When modeling
transient flow in a single pipe, it is not possible to simplify the governing conservation equations.
Hence, simultaneous solution of the two fundamental equations; mass and momentum conservation
equations, is required. For the case of non-isothermal flow; a third equation, energy conservation
equation, is needed as well.
In typical network transient analysis, Euler equations (mass and momentum conservation) are
applied over the pipes while junctions are still treated solely as nodes at which only the conservation
of mass (Kirchhoff’s first law) is applied. This implies the assumption of steady-state conditions
at the nodes and is reasonable for the practical purposes of studying oil and gas networks where
volumes of junctions are insignificant compared to the very long pipes.
The fundamental equations, whether in conservative form or not, constitute a system of non-
linear partial differential equations (NL-PDE) which may be solved in several different ways.
Most frequently adopted techniques include Finite Difference Method (FDM) and Method of
Characteristics (MOC) and its variations.
The node continuity equations applied at every node will also need to be solved simultaneously. The
fact that each qi in every node continuity equation will be computed using one of the aforementioned
methods makes the computation procedure very laborious. Conventionally, assuming an up front
insignificant loss in accuracy, some simplifications are done on the equations defining the transient
flow in the pipe. These assumptions include isothermal conditions, neglecting elevation changes and
removing some of the terms in momentum equation which are usually smaller than other terms by
an order of magnitude.
134
For the pipes connecting to the same node, simplified form of mass conservation equations are
written at the nodes. These equations are summed together and all the flux terms ( ∂∂x(ρ v)) are
collected on one side. After rearranging, the transient form of node continuity equation, Eqn. A.17,
is arrived:
Vj kdpjdt
= (qG)s,j +
nconn∑i
(qG)i,j (A.17)
Where:
(qG)s,j Flow in to or out of node j (supply/demand)
(qG)i,j Flow in the ith NCE connected to node j
nconn Number of pipes connected to node j
Vj Pseudo node volume which is equal to
the sum of half the volumes of every pipe connected to node j
pj Pressure at node j
k Represents the remaining terms
Simplified and arranged form of momentum equation is substituted in place of flow rates in this new
form of the node continuity equation (Eqn. A.17). A new system of NL-PDEs is obtained following
the same steps at every node.
A.3.1 Finite Difference Method
Governing NL-PDEs of the network are first transformed into algebraic equations using spatial
and temporal discretization schemes. Later on, algebraic equations can be solved for the values
of variables at certain points in the flow domain. FDMs are classified according to temporal
discretization scheme utilized.
In an Explicit Method; NL-PDEs are transformed into algebraic equations where unknown values of
variables at time t+ ∆t depend on the values at time t which are known. Hence algebraic form of
conservation equations can be solved individually in a straight forward manner.
In an Implicit Method; NL-PDEs are transformed into a system of algebraic equations where values
of unknown variables at time t + ∆t depend on the values of variables at neighboring points at
t+ ∆t. Therefore; simultaneous solution of the equations are needed.
Crank - Nicholson Method may be considered as a combination of implicit and explicit methods
where values of unknown variables at a time t + ∆t depend on both the values of variables at
neighboring points at time t+ ∆t and the values of variables at time t. So, simultaneous solution of
the equations is still needed.
A.3.2 Method of Characteristics
In the method of characteristics, by finding paths (characteristic lines or characteristics) in the plane
of natural coordinates (independent variables) of the equation system, the NL-PDEs are reduced to
135
ordinary differential equations (ODE) which can then be solved easier numerically (Larock et al.,
2000).
A.4 Description of the Network Model for Numerical Solution
Natural Gas transmission networks can be represented as a directed graph which provides all
the necessary connection information. Connections between nodes are easily described using a
connection matrix and loops are defined by a loop matrix. Loop matrix can be extracted from the
connection matrix using a connected graph which is a ‘loopless’ graph. This connected graph is
called the matching matrix and in essence contains two individual matrices derived from connection
matrix. One of these matrices (the matrix of tree branches) contains the information of all NCEs
except the ones creating loops in the system. NCEs causing loops in the system are collected in
another matrix (the matrix of chord branches) that is added to the end of tree matrix to form the
matching matrix. Conventionally NCEs form the columns and nodes form the rows in all of these
matrices (Eqn. A.18).
A =
a1,1 a1,2 a1,3 . . . a1,m
a2,1. . .
.... . .
an,1 . . . an,m
︸ ︷︷ ︸
NCEs(m)
nodes(n) (A.18)
Any element ai,j of the matrix A will be assigned:
• +1 if edge j emerges from node i
• −1 if edge j ends in node i
• 0 if edge j is not connected to node i
Once the network is defined by the connection and loop matrices, node continuity equation and
energy loop equation can be expressed as matrix operations to be carried out in the computer.
Kappos and Economides (2004) provide a simple, iterative algorithm for the analysis of pipeline
networks. The algorithm neither requires prior knowledge of split node location nor good initial
estimates of the solution. Instead, code successively computes node pressures, pipe flow rates and
flow directions.
Sung et al. (1998) developed a network optimization tool based on a minimum cost spanning tree
approach (in which occurrence of loops are not allowed in the network) for the optimum path and
constrained derivative approach for the optimum diameter. Their solution algorithm for the network
flow includes the conventional Newton-Raphson method.
Ke and Ti (2000) extended the conventional analogy with electrical networks to analyze transients
in an isothermal gas pipeline network. In this approach, other circuitry concepts like capacitance
136
and inductance are accounted for. A Major benefit of this approach is that governing equations are
transformed in to a set of first order ODE instead of dealing with second order (hyperbolic) PDE,
hence computational costs are greatly reduced while reasonably accurate results were obtained.
While it is possible to reduce governing equations to a single, closed form relation (the hydraulic
model) for single-phase steady-state flows it has not been yet shown that a similar approach can be
taken without making any simplifying assumptions which eventually lead to loss of an adequate
mathematical description of the system and the two-phase flow phenomenon. However, without
the closed form hydraulic models for pipes or NCEs, it is not convenient to form an analogy
to the circuitry leaving Kirchhoff technique as an unfeasible option for the network analysis of
two-phase flow. Instead, a proper solution technique needs to accommodate simultaneous solution
of conservation equations for pipes and/or NCEs.
137
Appendix B
Double Channel T-Junction Model
Presence of a pressure spike, a pressure increase in the main line, following the flow split at a
T-junction is recognized through simple analysis of Bernoulli equation. The decrease in mass flow
rate along the main line of the tee (after the split) has an effect similar to flow area expansion along
a single pipe. Therefore, Bernoulli equation analysis indicates a pressure increase in the run. This
is most clear when irreversible losses are neglected:
1
2ρ1v
21 + ρ1gz1 + p1 =
1
2ρ2v
22 + ρ2gz2 + p2 +
1
2ρ1v
21k12 (B.1)
For;
2A1 = A2
ρ1 = ρ2
z1 = z2
12ρ1v
21k12
∼= 0
Then, by conservation of mass:
ρ1v1A1 = ρ2v2A2
ρ1v1A1 = 2ρ1v2A1
(B.2)
v2 =v1
2(B.3)
Solving Bernoulli equation for p2 requires that p2 > p1:
12ρ1v
21 + p1 = 1
2ρ1(v12 )2 + p2
12ρ1v
21 + p1 = 1
8ρ1v21 + p2
p2 = 38ρ1v
21 + p1
p2 > p1
(B.4)
Proper treatment and capture of flow split at a T-junction requires acknowledging that outgoing
arms of the tee do not necessarily share the same single pressure and same single void fraction
although their extended inlets originate from within the same control volume; i.e. tee node or the
knot (the tee cell itself in FVM grid).
The fact that arms of the tee should ‘feel’ different pressures can simply be observed from experimental
data by linear extrapolation of pressure profiles of run and branch arms towards the tee (Ballyk
138
and Shoukri, 1990).
Flow going out through each arm (or the split itself for that matter) is to be determined by
momentum balances that regard directionality of the flow. From FVM point of view, once the split
is determined, pressure profiles of outgoing arms is solely based on specified outlet pressures for the
arms. Nevertheless, utilized set of equations should tie pressures of outgoing arms, as well as the
flow rates, to the main line so that an accurate solution can be obtained in the main line as well.
Depending on the equations that establish the connection between tee cell inlet and tee cell
outlets, both the split and thereof pressure profiles along all arms of the tee would be different.
In other words, without proper accounting for the actual physics, different solutions would be
possible. However, purpose of the momentum equations written specifically for the tee (not
the regular momentum equations written for faces) is to predict correct split amounts so that
both outlet pressure specifications and momentum conservation within the tee cell are satisfied
properly/adequately.
From FVM perspective, the tee component itself is merely a 3 pipe system where flow coming out
of main pipe is split between branch and run. Both branch and run can be visualized as separate
pipes with outlet pressure specifications. Then, solution for an outgoing arm merely depends on the
flow going through it and it’s outlet pressure specification. This is because, in the staggered grid
approach inlet pressure specification has no significant influence on the solution (particularly for
isothermal analysis). However, void fraction (along with velocities) has control over the flows going
through arms.
From this point of view, it seems as if outgoing arms can be considered to feel a single pressure at
the tee and therefore only purpose of momentum equations written for the tee is to predict outgoing
flow rates (or velocities and void fractions for that matter). Perhaps this would have been the case
if any possible effect tee pressure might have on outgoing flow rates was to be ignored.
Nevertheless, even if a single tee cell pressure had no effect on outgoing flow rates, the connection
between pressures in the first cells of outgoing arms and tee cell has to be properly established
simply because, as a separate pipe, solution in the main pipe depends on the tee cell pressure which
becomes it’s outlet pressure specification.
In other words, it might be possible to obtain an incorrect tee cell pressure with a seemingly correct
flow split unless proper momentum equations are utilized to connect all 3 arms of the tee, thus
leading a different solution in the main line. In essence, this means not only the velocities of outgoing
faces but also outgoing pressures have to be properly linked to inlet pressure and velocities through
specialized equations for the tee.
To repeat it once more, even if the effect of a singular tee cell pressure on flow split (outgoing
velocities) could be neglected, the correct solution for the main pipe still depends on the link that
would be established between pressures felt by outgoing arms and the tee cell.
Another important point is that, in the case of two-phase flow, outgoing arms may as well have
139
separate/different void fractions, despite the fact that flow originates from the same pool - tee
CV itself. Unless two-phase pattern in the tee cell is mist (or bubbly) flow, behaving like an
homogeneous fluid, assumption of a single void fraction for both branch and run arms could be
misleading/erroneous Perhaps, it would have been a valid assumption for an homogeneous two-phase
flow model though. Instead, whether void fractions of outgoing streams are same or not should be
determined through application of conservation principles.
The whole point of this discussion is to emphasize that derivation of proper equations for flow split
should begin by recognizing outgoing pressures, void fractions and velocities as unknowns. These
unknowns are to be solved for via specialized equations written for the tee and not by regular
edge/face momentum equations. However, it is shown later in the following sections that equations
derived for tee could collapse to a specialized set of edge/face momentum equations for outgoing
faces/edges of tee cell.
At this stage of the analysis, there are a total of 10 unknowns to account for at a tee (Fig. B.1):
p31, p32, p33, α31, α32, α33, vG3, vL3, vG6, vL6.
B.1 Double Channel Model
For the derivation of proper tee split equations and to generate adequate number of equations
accounting for all unknowns, a variation of Steinke (1996) approach is adopted (shall be referred to
as Double Channel Model, DCM). Based on author’s original derivation for combining T-junctions,
(1) main pipe to run and (2) main pipe to branch are flows considered to occur through separate
channels;by dividing the tee CV into two smaller CVs, similar to the separating (or dividing)
streamlines approach adopted by mechanistic junction models. Doing so, Steinke recognizes that an
additional pressure at the tee (pT ), as well as portion of main line cross-sectional area (A313) should
be accounted for.
Figure B.1: The double channel model control volumes
140
Flow turning into branch can be treated as (contracting) bent pipe flow. Hence, presence of
additional pressure pT is simply a requirement of the bent pipe flow. This can be observed by
considering momentum balance over a bent pipe of uniform cross-sectional area. Ignoring irreversible
losses due to wall friction or bent of flow, momentum conservation in the direction of run (x-direction)
requires that momentum of the incoming fluid be balanced by a pressure increase at about where
flow deflects in the bent. In other words, a pressure force in the opposite direction is required
to slow down the incoming momentum in the run direction, before diverting the flow. Without
acknowledging such a pressure increase in the bent around the deflection point, momentum of
incoming stream in the run direction (x-direction) could only be balanced by irreversible losses. This
would mean imitation of actual physics of flow curvature by blending its effect within irreversible
losses.
Despite generating two additional unknowns (pT and A313), imagining of the flow channels has
other advantages that is worth the effort:
1. Could help facilitate a better understanding of flow mechanics at the tee.
2. Separate momentum equations can be written for each channel, providing additional equations
for the solution, as opposed to writing momentum balances over the whole tee itself (which,
in theory, should yield similar results).
3. There is no need to account for the contribution to/from branch flow when writing momentum
balances over individual flow channels, while it would have been necessary to do so in the case
of writing momentum balances for the whole tee cell or outgoing edges.
B.2 The Buffer Zone
For practical purposes and as a requirement of the momentum balance around the inlet face of the
tee, it is reasonable to assume that tee densities (one per phase, ρG3 and ρL3) and the tee void
fraction α31 prevail in the tee cell way before the split occurs. By creating an imaginary ‘buffer zone’
before flow splits into channels within the tee, it is possible to convert incoming mass flow based on
upstream phase densities and velocities to mass flow based on phase densities and velocities of tee
cell. Further more, it could also be assumed that all the mass transfer (and momentum transfer
thereof) taking place at the tee occurs within this buffer zone. Nevertheless, this convenience comes
at the price of recognizing a new set of velocities (vG31 and vL31) at the entry point of channels.
Extending flow channels to the inlet face of the tee and ignoring such an imaginary buffer zone
might have helped avoid creating new velocities as unknowns. Yet, it might have as well been
difficult to account for tee densities, void fraction and the momentum balance over the inlet face.
Also, assuming that incoming momentum based on upstream properties would be equal to the
momentum at the end of buffer zone (based on tee densities, and velocity) before the split would
not be correct as can be seen from a single phase compressible flow analysis:
141
mass balance over the buffer zone:
ρ2v2A2 = ρ31v31A31 (B.5)
where A31 = A2;
v31 =ρ2v2
ρ31(B.6)
If ρ2 = 2ρ31 then 2v2 = v31
Hence;
(ρ2v2A2)v2 6= (ρ31v31A2)v31 (B.7)
At this stage, there are a total of 14 unknowns with the inclusion of (vG31 and vL31).
B.3 Simplifying Assumptions
In order not to further complicate the problem, a series of simplifying assumptions are introduced
before commencing the derivation of governing equations:
1. Assuming that imaginary flow channels are adequately short (or the tee volume itself is
adequately small), density and void fraction changes as well as mass transfer (and momentum
transfer thereof), are neglected within the channels (or entirely within the tee).
This implies, due to conservation of mass, change of velocity over a channel is caused by
a change in flow area. Constant void fraction assumption along a channel is more of a
convenience, because of which the bent pipe treatment for branch channel and calculation of
pressure forces in both channels are simplified.
Also, ignoring mass transfer in the channels requires that all the mass transfer supposed to
occur in the tee takes place in the buffer zone, unless completely ignored for the sake of
simplicity.
2. Pressure gradient for the run channel is approximated as p31− p32∼= p31− p4, thus eliminating
p32 as an unknown.
3. Pressure gradient for the branch channel beyond the deflection point is approximated as
pT − p33∼= pT − p6, thus eliminating p33 as an unknown.
Approximating pressure gradients with downstream cell pressures provides the basis for extending
momentum equations written for channels to further include the volume of edge momentum cells
of the tee, over which regular edge momentum balances would have been written. This helps
establishing an improved physical link (1) between outgoing velocities at tee faces and the pressure
gradients by placing velocities actually inside the pressure gradient, and (2) between pressure profiles
of outgoing arms and the tee cell pressures.
142
B.4 Governing Equations
At this point, for the sake of completeness, governing equations are developed without any additional
simplifications to the model with separate void fractions accounted for the arms of the tee. Additional
simplifications are to be introduced should there be a need; either to balance number of available
equations or for the purposes of easier implementation.
First step is to acknowledge all 13 unknowns at a tee: p31, pT , α31, α32, α33, vG3, vL3, vG6, vL6,
vG31, vL31, A312 and A313. T31 of the cell is ignored since it is associated with energy balance by
default.
B.4.1 Mass Balances
Linearly independent mass balance equations that can be written for the system are:
1) Gas phase mass balance for branch
mssG23 −mssG6 = 0
ρG3vG31α33A313 − ρG3vG6α33A6 = 0(B.8)
2) Liquid phase mass balance for branch
mssL23 −mssL6 = 0
ρL3vL31(1− α33)A313 − ρL3vL6(1− α33)A6 = 0(B.9)
3) Gas phase mass balance for run
mssG22 −mssG3 = 0
ρG3vG31α32A312 − ρG3vG3α32A3 = 0(B.10)
4) Liquid phase mass balance for run
mssL22 −mssL3 = 0
ρL3vL31(1− α32)A312 − ρL3vL3(1− α32)A3 = 0(B.11)
5) Gas phase mass balance for buffer zone
mssG2 −mssG31 − Γ3 = 0
ρG2vG2α2A2 − ρG3vG31α31A2 − Γ3 = 0(B.12)
6) Liquid phase mass balance for buffer zone
mssL22 −mssL3 + Γ3 = 0
ρL2vL2(1− α2)A2 − ρL3vL31(1− α31)A2 + Γ3 = 0(B.13)
Please recall that in Sec. B.3 flow channels are assumed to be adequately short enough to ignore
143
mass transfer within: Γchannel ∼= 0. Similarly, transfer within the buffer zone could be ignored
(Γ3∼= 0) for convenience
Important auxiliary relations to recognize as independent equations are:
7)
A2 = A31 = A312 +A313 (B.14)
8)
α31A3 = α32A312 + α33A313 (B.15)
Equation B.15 is a requirement of phasic mass balance in the buffer zone, before the split: 8)
ρG3vG31α31A2 = ρG3vG31α32A312 + ρG3vG31α33A313 (B.16)
Or;
8)
ρL3vL31(1− α31A2) = ρL3vL31(1− α32)A312 + ρL3vL31(1− α33)A313 (B.17)
It is very important to recognize that, despite being written for separate phases, Eqns. B.16 and
B.17 do not constitute linearly independent equations. This is quite clear as both equations collapse
to same Eqn. B.15 should mass flux terms be canceled in both equations.
It should be noted that phasic mass balances over the whole tee do not constitute linearly independent
equations because they can be easily obtained by adding mass balances of the same phase from above
list. Thus, they would not provide additional information to the system. However, for convenience
and purposes of easier implementation, any phase mass balance from above list could be replaced
by a mass balance of the the same phase over the whole tee:
7) Gas phase mass balance over whole tee (Eqns. B.12 + B.16):
ρG2vG2α2A2 − ρG3vG3α32A3 − ρG3vG6α33A6 − Γ3 = 0 (B.18)
8) Liquid phase mass balance over whole tee (Eqns. B.13 + B.17):
ρL2vL2(1− α2)A2 − ρL3vL3(1− α32)A3 − ρL3vL6(1− α33)A6 + Γ3 = 0 (B.19)
B.4.2 Momentum Balances
It is possible to write 6 linearly independent momentum equations. Assuming that length of branch
channel along the run driection is short enough to neglect friction and additional irreversible losses
along this section, friction and any additional irreversible losses are only attributed to the section
of channel along the branch direction. Slightly complicated alternatives include: (1) partitioning
of friction and irreverible losses according to length and (2) accounting for separate friction and
irreverible losses for the run and branch sections of the branch channel. Also, gravitational forces
144
are ignored along both channels.
1) Gas phase momentum balance for branch, in run direction
momG33 −momG6 cos θ + (p31 − pT )α∗33A∗313 + (pT − p6)α∗33A
∗6 cos θ
−fwG33 cos θ − fi33 cos θ − 12k13mom
∗G33 cos θ = 0
(B.20)
2) Liquid phase momentum balance for branch, in run direction
momL33 −momL6 cos θ + (p31 − pT )(1− α∗33)A∗313 + (pT − p6)(1− α∗33)A∗6 cos θ
−fwL33 cos θ + fi33 cos θ − 12k13mom
∗L33 cos θ = 0
(B.21)
3) Gas phase momentum balance for branch, perpendicular to run direction
−momG6 sin θ + (pT − p6)α∗33A∗6 sin θ − fwG33 sin θ − fi33 sin θ
−12k13mom
∗G33 sin θ = 0
(B.22)
4) Liquid phase momentum balance for branch, perpendicular to run direction
−momL6 sin θ + (pT − p6)(1− α∗33)A∗6 sin θ − fwL33 sin θ + fi33 sin θ
−12k13mom
∗L33 sin θ = 0
(B.23)
5) Gas phase momentum balance for run
momG32 −momG4 + (p31 − p4)α∗32A∗3 − fwG32 − fi32 − 1
2k12mom∗G32 = 0 (B.24)
6) Liquid phase momentum balance for run
momL32 −momL4 + (p31 − p4)(1− α∗32)A∗3 − fwL32 + fi32 − 12k12mom
∗L32 = 0 (B.25)
Where:
momG4 = (ρG3vG3α32A3)vG3
momG6 = (ρG3vG6α33A6)vG6
momL4 = (ρL3vL3(1− α32)A3)vL3
momL6 = (ρL3vL6(1− α33)A6)vL6
A∗312∼= A312
A∗313∼= A313
A∗3∼= A3
A∗6∼= A6
α∗32 is a weighted (volumetric) average of α32 and α4 and
α∗33 is a weighted (volumetric) average of α33 and α6 as is the case for a typical momentum balance
equation around an edge.
145
Due to staggered grid and donor cell arrangements momG3 = (ρG2vG2α2A2)vG2. Hence, momG3 has
to be partitioned based on flow areas and void fractions in order to obtain momG32 and momG33
per channel. This is recognized by analysis of mass and momentum partition right before the split
(Eqn. B.16):
mssG31 = mssG32 +mssG33
ρG3vG31α31A2 = ρG3vG31α32A312 + ρG3vG31α33A313
ρG3vG31α31A2 = ρG3vG31α31A2α32A312
α31A2+ ρG3vG31α31A2
α33A313
α31A2
mssG31 = mssG31α32A312
α31A2+mssG31
α33A313
α31A2
Where;
mssG32 = mssG31α32A312
α31A2
mssG33 = mssG31α33A313
α31A2
Similarly,
momG3 = momG32 +momG33
momG3 = momG3α32A312
α31A2+momG3
α33A313
α31A2
Then,
momG32 = momG3α32A312
α31A2(B.26)
momG33 = momG3α33A313
α31A2(B.27)
Where momG3 = (ρG2vG2α2A2)vG2.
Instead of writing momentum balances along orthogonal axes (x-y) of run (primary) direction,
it is equivalent to write momentum balances along orthogonal axes (x′-y′) of branch (secondary)
direction. Then, a secondary equation set written for orthogonal axes of branch could replace the
current set. However, it should be remembered that equation sets are not linearly independent from
one other.
Second set of alternative momentum balances:
1) Gas phase momentum balance for branch, in branch direction
momG33 cos θ −momG6 + (p31 − pT )α∗33A∗313 cos θ + (pT − p6)α∗33A
∗6
−fwG33 − fi33 − 12k13mom
∗G33 = 0
(B.28)
146
2) Liquid phase momentum balance for branch, in branch direction
momL33 cos θ −momL6 + (p31 − pT )(1− α∗33)A∗313 cos θ + (pT − p6)(1− α∗33)A∗6−fwL33 + fi33 − 1
2k13mom∗L33 = 0
(B.29)
3) Gas phase momentum balance for branch, perpendicular to branch direction
momG33 sin θ + (p31 − pT )α∗33A∗313 sin θ = 0 (B.30)
4) Liquid phase momentum balance for branch, perpendicular to branch direction
momL33 sin θ + (p31 − pT )(1− α∗33)A∗313 sin θ = 0 (B.31)
Please note that phasic momentum balances for the run channel could be decomposed into x′
and y′ directions (that is unless x′ = x, depending on the split angle) by taking cos and sin of
all the components in the equations. Yet, in essence, the equations simply would not be linearly
independent, hence not repeated in the second set.
Typically, as it is the case with mass balances, writing momentum balances for the whole tee volume
would not bring any new information to the system because same information is already included
within run and branch momentum balances. For instance, the gas phase momentum balance in the
run direction for the whole tee volume is simply equivalent to mathematical summation of (1) gas
phase momentum balance in run direction for the run and (2) gas phase momentum balance in run
direction for the branch.
It is easier to write overall momentum balances instead of phasic balances for the tee, considering
the complications associated with void fractions multiplying pressure gradients (that is if pressure
forces calculations are to be consistent with the rest of the moodel), alternative momentum balances
can be written for the system are:
1) overall momentum balance for whole tee, in run direction:
summation of Eqns. B.20 + B.21 + B.24 + B.25
momG3 +momL3 − (momG4 +momL4)− (momG6 +momL6) cos θ
+(p31 − pT )A∗313 + (p31 − p4)A∗3 + (pT − p6)A∗6 cos θ
−fwG32 − fwG33 cos θ − 12k12mom
∗G32 −
12k13mom
∗G33 cos θ
−fwL32 − fwL33 cos θ − 12k12mom
∗L32 −
12k13mom
∗L33 cos θ = 0
(B.32)
2) overall momentum balance for whole tee, perpendicular to run direction
147
summation of Eqns. B.22 + B.23
−(momG6 +momL6) sin θ + (pT − p6)A∗6 sin θ − fwG33 sin θ − fwL33 sin θ
−12k13mom
∗G32 sin θ − 1
2k13mom∗L32 sin θ = 0
(B.33)
A second set of alternative momentum balances are as follows:
1) overall momentum balance for whole tee, in branch direction
summation of Eqns. B.28 + B.29 + (B.25 + B.25) cos θ
(momG3 +momL3) cos θ − (momG4 +momL4) cos θ − (momG6 +momL6)
+(p31 − pT )A∗313 cos θ + (p31 − p4)A∗3 cos θ + (pT − p6)A∗6−fwG32 cos θ − fwG33 − 1
2k12mom∗G32 cos θ − 1
2k13mom∗G33
−fwL32 cos θ − fwL33 − 12k12mom
∗L32 cos θ − 1
2k13mom∗L33 = 0
(B.34)
2) overall momentum balance for whole tee, perpendicular to branch direction
summation of Eqns. B.30 + B.31 + (B.25 + B.25) sin θ
(momG3 +momL3) sin θ − (momG4 +momL4) sin θ
+(p31 − pT )A∗313 sin θ + (p31 − p4)A∗3 sin θ
−fwG32 sin θ − 12k12mom
∗G32 sin θ
−fwL32 sin θ − 12k12mom
∗L32 sin θ = 0
(B.35)
Where:
momG3 = (ρG2vG2α2A2)vG2
momL3 = (ρL2vL2(1− α2)A2)vL2
It is also important to note that, because momentum is a vectoral quantity, mathematical summation
of equations in both directions are not necessarily equivalent to vectoral summation of the equations.
In other words, for instance, only vectoral summation of momentum balance in run direction and
momentum balance perpendicular to run direction is mathematically equivalent to momentum
balance in branch direction.
For branch channel, it is a convenient alternative to use momentum equations in the branch and
run directions instead of branch (or run) direction and with its associated perpendicular direction
momentum balance.
So far it has been possible to write more than 13 linearly independent equations to balance number
of unknowns.
148
B.4.3 Parallel Split
When branch is parallel to run (split angle is 0o), momentum conservation orthogonal to run (or
momentum conservation orthogonal to branch if second set of momentum balances are considered)
does not provide any useful information because none of the momentums or forces would have
orthogonal components (sin(0o) = 0), thus equation becomes singular. Also, there is no longer an
actual pT to account for; when branch is parallel to run pT = p31 indeed and pT becomes a dummy
variable.
From the perspective of numerical solution, if branch is parallel to the run, while constructing the
jacobian (coefficient) matrix which hosts numerical derivatives of the residuals (governing equations),
these perpendicular momentum equations for branch are likely to cause jacobian to be singular as
well .
In order to prevent this singularity and retain dummy equations in the system when split angle is
0o, orthogonal momentum balance of a phase could be substituted with a mathematical summation
of momentum balances of the same phase in both directions.
However, when branch is parallel to run, the fact that pT = p31 may pose other requirements for the
dummy equation replacing singular equations. As pT becomes a dummy variable when split angle is
0o, it is necessary to fix its values to p31. This is typically established via a dummy equation that
explicitly sets variable to the desired value and, for the purposes of jacobian matrix solution, that
does not generate a zero when its derivative is taken with respect to the variable:
pT − p31 = 0 (B.36)
B.4.4 Reduction of Unknowns
For isothermal analysis of regular, straight pipe; there are 2 equations solved per CV (phasic mass
balances) and 2 equations solved per face (phasic momentum balances). Therefore, as long as
governing equations permit, it is convenient to express several of the 13 unknowns with explicit
equations involving remaining unknowns of the tee, which shall become the primary unknowns.
This is to reduce the number of equations solved for a tee and match the general scheme of 2 mass
equations per CV and 2 momentum equations per downstream face.
First, using Eqn. B.14:
A312 = A2 −A313 (B.37)
Then, vG31 can be defined separately using flow channel gas mass balances (Eqns. B.8 and B.10):
vG31 =vG3A3
(A2 −A313)
vG31 =vG6A6
A313
149
Then A313 is obtained by setting two expressions of vG31 equal:
A313 = A2vG6A6
vG3A3 + vG6A6(B.38)
Then;
A312 = A2 −A2vG6A6
vG3A3 + vG6A6= A2
vG3A3
vG3A3 + vG6A6(B.39)
And final expressions for vG31 is:
vG31 =vG3A3 + vG6A6
A2(B.40)
Because of constant void fraction (α) assumption, phasic mass balances for the flow channels alone
do not provide information to express α32 and α33 explicitly. Therefore, it is necessary to utilize
momentum balances to determine α32 and α33.
Nevertheless, liquid mass balance equations (Eqns. B.9 and B.11) can be employed to obtain either
vL3 or vL6 explicitly with new definitions of vL31 and A313 at hand.
A better option to express vL31 is to use Eqn. B.9 and the new definition of A313 (Eqn. B.38):
vL31 =vL6
vG6
vG3A3 + vG6A6
A2= vL6
vG31
vG6(B.41)
Using Eqn. B.11 and the definition of A312:
vL31 =vL3
vG3
vG3A3 + vG6A6
A2= vL3
vG31
vG3(B.42)
It is possible to get vL6 by setting both expressions of vL31 equal:
vL6 = vL3vG6
vG3(B.43)
Also, from Eqn. B.15:
α31 =α32A312 + α33A313
A2(B.44)
or
α31 = α32 − (α32 − α33)vG6A6
vG3A3 + vG6A6(B.45)
At this point, 6 unknowns has been defined in terms of other unknowns using 5 (out of 7) mass
balances and 1 auxiliary relation (Eqn. B.14).
Using Eqn. B.33 pT can be defined as:
pT = p6 +momG6 +momL6 + fwG33 + fwL33 + 1
2k13mom∗G32 + 1
2k13mom∗L32
A∗6(B.46)
150
Using Eqn. B.33 p31 can be defined as:
pT = p6 +momG6 +momL6 + fwG33 + fwL33 + 1
2k13mom∗G32 + 1
2k13mom∗L32
A∗6(B.47)
151
Appendix C
Fluid Properties
Nomenclature
Roman letters:
a Attraction parameter [Pam6mol−2orPam6mol−2]
b Co-volume [m3mol−1]
c Component mole fraction [dimensionless]
kij Binary interacting coefficient [dimensionless]
mi A function of accentric factor ω [dimensionless]
p Pressure [kgm−1 s−2]
A A function of (aα) [dimensionless]
B A function of co-volume (b) [dimensionless]
MWβ Molecular weight of phase β [gmol−1]
R Gas constant [8.31447 J K−1mol−1]
T Temperature [K]
V Molar volume [m3mol−1]
V TCβ Volume translation coefficient of phase β [gm3 kg−1mol−1]
Z Compressibility factor [dimensionless]
Greek letters:
α Dimensionless parameter [dimensionless]
ε Pipe roughness [m]
ω Accentric factor [dimensionless]
φβi Fugacity coefficient for component i in phase β [dimensionless]
ρβ Density of phase β [kgm−3]
Ωai Component and EOS specific constant [dimensionless]
Ωbi Component and EOS specific constant [dimensionless]
Subscript:
c Critical property
152
i Individual component
m Mixture
r Reduced property
β Phase β
C.1 Equation of State
An equation of state (EOS) relating the pressure, volume and temperature (PVT) properties of a
mixture is required for the purpose of numerical modeling of the flow. Because we are only concerned
with hydrocarbon (HC) components, the Peng-Robinson EOS is utilized for the computation of
needed state functions in the model.
p =RT
V − bm− (aα)mV 2 + 2bmV − b2m
(C.1)
Attraction parameter (a) is defined as shown:
ai = ΩaiR2T 2
ci
Pci(C.2)
Dimensionless parameter alpha (α) is defined as:
αi = [1 +mi(1− T 0.5ri )]2 (C.3)
Function miis defined as:
mi =
0.374640 + 1.542260ωi − 0.269920ω2
i ωi ≤ 0.49
0.379642 + 1.485030ωi − 0.164423ω2i + 0.016666ω3
i ωi > 0.49(C.4)
Hence;
(aα)i = ΩaiR2T 2
ci
pci[1 +mi(1− T 0.5
ri )]2 (C.5)
For a mixture, (aα)m is computed through the application of quadratic mixing rule:
(aα)m =
n∑i
n∑j
cicj(1− kij)((aα)i(aα)j)0.5 (C.6)
Co-volume parameter (b) is defined as:
bi = ΩbiRTcipci
(C.7)
153
For a mixture, bm is computed through the application of quadratic mixing rule:
bm =n∑1
cibi (C.8)
Law of corresponding states enables the definition of real gas equation with the introduction of Z
factor (compressibility factor):
pV = ZRT (C.9)
The EOS in Eqn. C.1 can be re-arranged in the cubic form by utilizing z-factor (Z) concept of real
gas equation and then solved for the unknown z-factor.
Z3 − (1−B)Z2 + (A− 2B − 3B2)Z − (AB −B2 −B3) = 0 (C.10)
where: A is given by:
A = (aα)p
(RT )2(C.11)
and B is defined as:
B = bp
RT(C.12)
C.2 Density calculation
The phase densities are calculated using an arranged form of the real gas law:
ρβ =pMWβ
ZβRT(C.13)
The density values computed using the z-factors from PR-EOS needs to be corrected utilizing the
volume translation (volume shift) concept:
ρcorrectedβ =ρβMWβ
MWβ − V TCβρβ(C.14)
VTC of a phase is calculated as shown:
V TCm =
n∑1
ciV TCi (C.15)
C.3 Fugacity calculation
Fugacity, a measure of chemical potential, is utilized as an indication of chemical equilibrium. Phases
are in chemical equilibrium when fugacity of each component is same in both phases. Fugacity of
154
component i in phase β is calculated as:
fβi = φβicβip (C.16)
Fugacity coefficient is calculated as a function of parameters of the cubic EOS:
lnφβi =BβiBβ
(Zβ − 1)− ln(Zβ −Bβ)
+Aβ
2√
2Bβ
BβiBβ− 2
Aβ
N∑j=1
cβj(1− kij)√AβiAβj
ln
(Zβ + (1 +
√2)Bβ
Zβ − (1−√
2)Bβ
)(C.17)
C.4 Enthalpy calculation
Mixture enthalpy is calculated by adding ideal enthalpy to the enthalpy of departure:
hm = hidealm + hdeparturem (C.18)
Departure enthalpy is obtained by:
hdeparturem =1000
MWm
[RT (Z − 1) +
T d(aα)mdT − (aα)m
2√
2bmln
(Z + (
√2 + 1)Bm
Z − (√
2− 1)Bm
)](C.19)
The derivative termd(aα)mdT of Eqn. C.19 is obtained as shown:
d(aα)mdT
=−R2√T
n∑i
n∑j
cicj(1− kij)
[mi
√ΩaiTciPci
(aα)0.5j +mj
√ΩajTcjPcj
(aα)0.5i
](C.20)
Where:
hdeparturem [Jkg−1]d(aα)mdT
[Pa2m6mol−2K−1]
Component ideal enthalpy is evaluated by using the Passut and Danner (1972) correlation:
hideali = cf(APD +BPDT + CPDT
2 +DPDT3 + EPDT
4 + EPDT5)
(C.21)
Where; APD, BPD, CPD, DPD, EPD, FPD are coefficients provided by Passut and Danner (1972)
and:
hideali Component ideal enthalpy [Jkg−1]
cf Conversion factor [2.326000000768900103Jkg−1 −BTUlbm−1]
T Temperature [R]
155
And, finally, mixture ideal enthalpy is obtained as:
hidealm =n∑1
ciMWi
MWmhideali (C.22)
156
Appendix D
Hydrocarbon Data
Lumped hydrocarbon mixture data is converted to SI units from Ayala (2001). Please refer to
Appendix C for nomenclature of this chapter.
Table D.1: Hydrocarbon mixture composition and critical properties
Molar MW pc Tc Vc
Component fraction g/mole kPa K m3/kg
Lumped-1 0.7658 16.20 4588.25414 189.89 6.1286E-03
Lumped-2 0.2240 39.37 4548.67823 350.44 4.6571E-03
Lumped-3 0.0102 130.00 1585.79418 722.22 4.1721E-03
Table D.2: Hydrocarbon mixture Peng-Robinson EOS specific parameters
Component ω Ωa Ωb Parachor
Lumped-1 1.075E-2 4.5724E-1 7.7796E-2 7.6520E+1
Lumped-2 1.342E-1 4.5724E-1 7.7796E-2 1.3424E+2
Lumped-3 1.550E-1 4.5724E-1 7.7796E-2 3.6276E+2
Table D.3: Hydrocarbon mixture binary interacting coefficients
Lumped-1 Lumped-1 Lumped-1
Lumped-1 0.0000 0.0280 0.0280
Lumped-2 0.0280 0.0000 0.0280
Lumped-3 0.0280 0.0280 0.0000
157
Table D.4: Hydrocarbon mixture Passut and Danner (1972) coefficients
Component A B C D E F
Lumped-1 -5.5083 5.6072E-1 2.7943E-4 4.1205E-7 -1.5057E-10 1.9329E-14
Lumped-2 2.6694 2.1301E-1 1.6983E-4 2.3631E-7 -1.2312E-10 1.8461E-14
Lumped-3 30.263 -2.1654E-1 4.5852E-4 -9.7970E-8 1.0465E-11 -3.1320E-16
Vita
Doruk Alp graduated from Yukselis Koleji highschool (Ankara) in 1997. Based on his score in the
nationwide university exam, taken by over 600 thousand students that year, he was placed in the
Petroleum and Natural Gas Engineering program at Middle East Technical University (Ankara).
During undergraduate years, Doruk had internships with Turkish Petroleum Corporation (1999)
and NV Turkse Perenco (2000), briefly held treasurer position of SPE METU student chapter
and worked on ‘Gas Pipeline Deliverability Analysis’ for his graduation project with Dr.Mahmut
Parlaktuna. Doruk received the BS degree with honors in Petroleum and Natural Gas Engineering
in June 2001.
Following his graduation, Mr.Alp was admitted to the MS program in Petroleum and Natural Gas
Engineering at METU. During the course of his study, Doruk was TA for more than 5 different
course (with associated lab and recitation hours) including; fluid mechanics, well logging, natural gas
engineering, rock & reservoir properties, reservoir engineering and engineering graphics. He worked
on numerical modeling of ‘Gas Production from Hydrate Reservoirs’ with Dr.Mahmut Parlaktuna
(METU) and Dr.George Moridis (Lawrence Berkley National Lab. - LBNL) as his advisors and
received the MS degree in Petroleum and Natural Gas Engineering in July 2005.
Mr.Alp was admitted to the PhD program in Petroleum and Natural Gas Engineering at Penn
State in August 2005. He became a graduate assistant to Dr.Luis F. Ayala for the production
process engineering lab and class. Doruk had a reservoir engineering internship with Chevron ETC
(Houston) in 2008.
At the time of writing, Doruk has accepted a reservoir engineering position with Chevron ETC,
pending his graduation.
