A.C.Bourdillon

Context Numerical results Conclusion
Coupling of a population balance model with
droplet deposition, entrainment and icing
Arnaud Bourdillon
School of Energy, Environment & Agrifood
Oil & Gas Engineering centre Cranﬁeld University
November 3, 2015

Outline
1 Context of the project
General field of study
Specific field of study
Specific problems of the area
Limitations of the current methods
Objectives of this PhD
2 Numerical results
Droplet diameter predictions
Single-fluid solidification solver
Multi-fluid solidification solver
3 Conclusion and future work
Coupling of a population balance model with droplet icing - A.C.Bourdillon November 3, 2015 1 / 29

Context Numerical results Conclusion General area Specific area Problem Limitations Objectives
General field of study
Oil industry
Pumping of oil through pipelines
Substances pumped
Oil, water, gas, sand...
Induce complex phenomena
Mixing, deposition, separation...
Industry concerns
Life time, efficiency...
Measurable by knowledge of
Velocities, temperatures,
pressures...
Obtained by
Experiments or numerical
simulations (CFD) Alaska pipeline [1]

Specific field of study
Two-phase flows
Water and oil
Interaction between phases
Water droplets transported by oil
Motion of droplets
Drag, lift, virtual mass...
Distribution of droplets
Population balance modelling
Size and shape of droplets
Break-up and coalescence
Necessary to
Accurately describe the flow

Speciﬁc problems of the area
Pipes in extreme conditions
Cold weather, deep water...
Regions of interest
Internal regions
Consequences
Erosion, corrosion...
Solidiﬁcation (icing)...
Can induce
Blocking, breakage of the pipe...
Pipe in extreme condition [2]

Size, shape and distribution of droplets
Validated models in the literature
Droplets distribution: Population balance modelling
Monte-Carlo [18], method of classes [8], method of moments [9],
quadrature method of moments [13], extended quadrature
method of moments [3]....
Droplets size and shape changes
Break-up and coalescence models [10], [11]

Size, shape and distribution of droplets
Droplets distribution: Population balance modelling
Monte-Carlo [18], method of classes [8], method of moments [9],
quadrature method of moments [13], extended quadrature
method of moments [3]....
Droplets size and shape changes
Break-up and coalescence models [10], [11]
Solidiﬁcation of a phase
Phase solidiﬁcation: Icing models
Apparent heat capacity [7], source-based [17], enthalpy-porosity
methods [16] ...

What can be done ?
Droplet size, shape and deposition
prediction
Solidiﬁcation rate prediction

What can be done ?
prediction
What cannot be done ?
Inﬂuence of droplet size, shape
and deposition to the
solidiﬁcation rate

What can be done ?
prediction
What cannot be done ?
Influence of droplet size, shape
and deposition to the
solidification rate
What could it bring ?
More accurate solidification rate
More accurate blocking length
Better flow prediction

Objectives of this PhD
Solidification solver
Develop a 3D, turbulent, transient and multi-fluid solver in
OpenFoam
Population balance solver
Develop a population balance model, along with break-up and
coalescence models in OpenFoam
Coupling population balance and solidification
Develop a coupling between the two previous models in
OpenFoam
Possibly extend further the model
To handle hydrates formation and corrosion-erosion.

Context Numerical results Conclusion Droplet diameter predictions Single-ﬂuid Multi-Fluid
Eulerian model for droplet diameter: Validation case
Phases α ρ µ u
water 0.062 1166 0.0016 0.158
Kerosene 0.938 797 0.0018 2.398
Test case from literature
Numerical simulation: [12]
Experiments: [4].
Initial condition
Water drops injected at the inlet
(ddrop = 1mm)
Objective
Evaluate mean diameter of
droplets, transported by
kerosene, at the outlet
Mathematical models tested
Population balance modelling
(MOM), break-up and
coalescence models

Interphase force effect
FD
, Ftd
FD
, Ftd
, Fvm
FD
, Ftd
, Fℓ
FD
, Ftd
, Fℓ
, Fvm
Verticalposition[m]
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
Sauter mean diameter [μ m]
300 400 500 600 700 800
Outlet top region
Lift force enhances break-up
Outlet bottom region
4 forces slightly increase
maximum droplet diameter

Turbulence modelling of the dispersed phase
2-layer k-ε model
Efﬁcient for intermediate
meshes (y+ 4)
Turbulence response model
Predicts velocity of droplets,
based on velocity of oil
Ct =
µd
µc
Discussion
Issa under-predict k rate
⇒ larger droplets (closer to
experimental data)
Issa and Oliveira
Realizable 2-layer k-ɛ
Verticalposition[m]−0.03
−0.02
−0.01
0
0.01
0.02
0.03
300 400 500 600 700 800 900 1000

Critical diameter value
Wecrit
= 0.05
Wecrit
= 0.15
Wecrit
= 0.25
Verticalposition[m]
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
200 400 600 800 1000
Critical diameter value
Diameter past which break-up
Formulation
dcrit = 5.6
2σWecrit
ρc
3/5
ε−2/5
Interpretation
Wecrit ⇒ dcrit
dcrit ⇒ Brup rate
Brup rate ⇒ dmean
Discussion
Wecrit linked to coalescence

Validation of the study
Simmons & Azzopardi (2001)
Lo & Rao (2007)
Present simulations
Cummulativevolumefraction[-]
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000

Conclusion the study
Good comparison with experimental data for 0.5 < αd < 1
Larger discrepancies with experimental data for 0 < αd < 0.5

Conclusion of the study
Interphase forces of great importance
Turbulence modelling of great importance
Break-up model probably needs further enhancement

Interphase forces of great importance
Turbulence modelling of great importance
Break-up model probably needs further enhancement
Further studies
Similar standard method of moment have been implemented in
OpenFOAM recently [15]
Solver to be enhanced by the previous ﬁndings

Single-fluid solidification solver: Validation case
Phases λ ρ T[K]
water 0.56 999.8 Tliq = 273.3
Ice 2.26 916.8 Tsol = 273.0
Cylinder D Ti Tcold
82.8 mm 21◦C −18◦C
Experiments: [6]
Present numerical simulations:
OpenFoam
Initial conditions
Cylinder filled with water at 21◦
C.
Walls are cooled down at −18◦
C
Objective
Compute phase change of water
from liquid to solid
Mathematical models tested
Solidification (enthalpy-porosity)

Solidiﬁcation solver: Formulation
Enthalpy based model
∂ρH
∂t
+ u
∂ρH
∂x
+ v
∂ρH
∂y
= λ∆T
H = h + ∆H ⇒ H =
T
Tref
cpdT + α L
Energy equation
∂ρcpT
∂t
+ u
∂ρcpT
∂x
+ v
∂ρcpT
∂y
= λ∆T + St
St = −L
∂ρα
∂t
+ u
∂ρα
∂x
+ v
∂ρα
∂y

Solidiﬁcation solver: Formulation
Momentum equation
∂ρu
∂t
+ u
∂ρu
∂x
+ v
∂ρu
∂y
= −
∂p
∂x
+ µ∆u − Smx
∂ρv
∂t
+ u
∂ρv
∂x
+ v
∂ρv
∂y
= −
∂p
∂y
+ µ∆v − g[ρ(T) − ρ] − Smy
Mushy region
Smx = FmDc
α2
s
α3 + ε
u
Smy = FmDc
α2
s
α3 + ε
v
Fm = 0.5 +
arctan[cs(αs − αscrit
)]
π
Slurry region
µ =



µ αs = 0 ⇒ liquid region
µ 1 −
Fs
A
−2
αs ≤ αcrit and αs = 0 ⇒ slurry region
µ α + µsαs αs > αcrit and αs = 1 ⇒ mushy region
µs αs = 1 ⇒ solid region

Validation of single-ﬂuid solidiﬁcation
a) Ice layer at t = 500 s
b) Temperature at t = 500 s
c) Ice layer at t = 750 s
d) Temperature at t = 750 s

Validation of single-ﬂuid solidiﬁcation
Experimental (Chen & Lee (1998))
IcingFoam
IcingFoamSlurryMushy
Temperature[○
C]
−15
−10
−5
0
5
10
15
20
Time [s]
0 1000 2000 3000 4000 5000
a) Top part
IcingFoam
Temperature[○
C]
−15
−10
−5
0
5
10
15
20
Time [s]
0 1000 2000 3000 4000 5000
b) Centre part
IcingFoam
Temperature[○
C]
−15
−10
−5
0
5
10
15
20
Time [s]
0 1000 2000 3000 4000 5000
c) Side part
IcingFoam
Temperature[○
C]
−15
−10
−5
0
5
10
15
20
Time [s]
0 1000 2000 3000 4000 5000
d) Bottom part

Good comparison with [6] experimental solution

Discussion
Both slurry and mushy solver give similar results
Slight discrepancies with experiments are due to differences in
operating conditions

Discussion
Both slurry and mushy solver give similar results
Slight discrepancies with experiments are due to differences in
operating conditions
Further and current studies
Solver enhanced to a 3D, turbulent and multi-fluid solidification
flow solver

Multi-fluid solver: Validation case
Phases λ ρ T
water 0.6 999.8 Tliq = 273.3
Ice 2.26 916.8 Tsol = 273
cube L = Thot Tcold
38 mm 283 − 278 263
Experiments: [14].
Present simulations with
OpenFOAM multi-fluid
solidification solver.
Objective
Predict ice-layer growth rate.
Compare results with
experiments of [14].

Validation of multi-ﬂuid solidiﬁcation
Experimental (Kowaleski & Cybulski)
IcingMultiPhaseFoam (fs = 0.5)
Icelayerheight[-]
0.01
0.1
Time [s]
500 1000 1500 2000 2500 3000
a) Results (Thot = 5◦
C)
Experimental (Kowaleski & Cybulski)
IcingMultiPhaseFoam (fs = 0.5)
Icelayerheight[-]
0.01
0.1
Time [s]
400 600 800 1000 1200
b) Results (Thot = 10◦
C)
c) Temperature iso-contour (t = 1800 s)
d) Velocity and ice-layer (t = 1800 s)

Conclusion
Conclusion of this work
Droplet motions, size and shape changes have been validated
Parameters influencing droplets behaviour have been analysed
Single-fluid solidification solvers have been validated
Multi-fluid solidification solver has been validated

Conclusion
Current and future work
Coupling between PBM and multi-ﬂuid solidiﬁcation solvers

Conclusion
Current and future work
Coupling between PBM and multi-ﬂuid solidiﬁcation solvers
Further possible extension of this work
Application of the solver to hydrates and errosion/corrosion
problems.

Comparison with experiments for ice formation in cavities
a) Experiments [14] (t = 2600 s) b) Present solver (t = 2600 s)
Thank you, any questions ?
Complete work
Bourdillon.A.C, Verdin.P.G and Thompson.C.P. Numerical simulations of water freezing processes in
cavities and cylindrical enclosures. Applied Thermal Engineering, 75, January 2015, 839-855.
Bourdillon.A.C, Verdin.P.G and Thompson.C.P. Numerical simulations of drop size evolution in a horizontal
pipeline. International Journal Of Multiphase Flow, 78, January 2016, 44-58.

References I
Google image: dict.space.4goo.net/dict=pipeline.
Google image:
http : //www.ﬁlemagazine.com/thecollection/
archives/2007/01/pipelineofd re.html.
Computational models for polydispersed particulate and
multiphase systems.
Cambridge, 2013.
Azzopardi.B and Simmons.M.
Drop size distributions in dispersed liquid-liquid pipe ﬂow.
International Journal of Multiphase Flow, 27, 2001.

References II
Bourdillon.A, Verdin.P, and Thompson.C.
Numerical simulations of water freezing processes in
cavities and cylindrical enclosures.
Applied thermal engineering, 75, 2015.
Chen.S and Lee.T.
A study of supercooling phenomenon and freezing
probability of water inside horizontal cylinder.
International Journal of Heat and Mass Transfer, 41, 1998.
Hashemi.H and Sliepcevich.C.
A numerical method for solving two-dimensional problems
of heat conduction with change of phase.
Chemical Engineering Progress, 1967.

References III
Hounslow.M, Ryall.R, and Marshall.V.
A discretized population balance for nucleation, growth and
aggregation.
Alche journal, 1988.
Hulburt.H and Katz.S.
Some problems in particle technology- statistical
mechanical formulation.
Chemical Engineering and Science, 1964.
Liao.Y and Lucas.D.
A literature review of theoretical models for drop and bubble
breakup in turbulent dispersions.
Chemical engineering science, 2009.

References IV
Liao.Y and Lucas.D.
A literature review on mechanisms and models for the
coalescence process of ﬂuid particles.
Chemical engineering science, 2010.
Lo.S and Rao.P.
Modelling of droplet breakup and coalescence in an
oil-water pipeline.
International Conference on Multiphase Flow, 13, 2007.
McGraw.R.
Description of aerosol dynamics by the quadrature method
of moments.
Aerosol science and technology, 27, 1997.

References V
Michalek.T and Kowaleski.A.
Simulations of the water freezing process - numerical
benchmarks.
International Journal of computational fluid dynamics, 2002.
Traczyk.M.
Numerical computations of liquid-liquid dispersions at high
phase fractions.
PhD thesis, Cranfield university, 2014.
Voller.V and Prakash.C.
A fixed grid numerical modelling methodology for
convection-diffusion mushy region phase-change problems.
Heat and mass transfer, 1987.

References VI
Voller.V and Swaminathan.C.
General source based method for solidiﬁcation phase
change.
Numerical Heat Transfer, 1991.
Zhao.H, Kruis.F, and Zheng.C.
A differentially weighted monte carlo method for
two-component coagulation.
Journal of computational physics, 2010.

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