Saturation in Liquid/Gas Coalescence
Presented by Mark Hurwitz, Pall Corporation
P. Aceves Sanchez, N. Bailey, M. Bruna, P. Cesana, B. Chakrabarti,
J. Chakraborty, C. J. Chapman, L. Cummings, M. Dallaston, J. Dietz, P. Dondl,
C. Finn, I. Griffiths, J. Herterich, C. Holloway, M. Hurwitz, A. Hutchinson,
R. Kgatle, D. Khoromskaia, A. Krupp, V. Lapin, N. Letchford, A. Lewis,
A. Manhart, M. Moore, A. Münch, H. Ockendon, J. Oliver, D. Paccagnan,
A. Patel, B. Piette, C. Please, R. Purvis, T. Rafferty, M. Saxton, D. Stuerzer,
Y. Sun, A. Tamsett, J. Uddin, T. Vo, Y. Wei, M. Zagórowska
ESGI100
University of Oxford
April 11, 2014
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The problem
If the concentration of the droplets on the fibres becomes too large,
we have saturation and the gas flow may re-entrain the droplets.
We aim to model the processes occuring in the coalescer in order
to understand saturation.
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Our approach
1
Macro-scale model of the coalescer
2
Micro-scale considerations for droplets within the filter
3
Constitutive relations for macro-model determined by
micro-scale considerations
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Macro-scale: Canonical one-dimensional problem
The quantities αlg , αlm , αg , ug , ul , p vary with position and time
according to
αlg + αlm + αg = 1,
∂
∂
(ρg αg φ) +
(ρg αg ug φ) = 0,
∂t
∂x
∂
∂
(ρl αlg φ) +
(ρl αlg φug ) = −fd ,
∂t
∂x
∂
∂
(ρl αlm φ) +
(ρl αlm φul ) = fd ,
∂t
∂x
∂
∂p µg
[(ρg αlg + ρg αg )φug ] = −
− φug ,
∂t
∂x kg
∂
∂p
(ρl αlm φul ) = −
− Flm .
∂t
∂x
(1)
(2)
(3)
(4)
(5)
(6)
Here, fd = fd (αlg , αlm , ug ) and Flm = Flm (αlm , ug , ul , λlm ) are
determined by micro-scale considerations.
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Macro-scale: Simple case in 1D
∂
∂ x̃
∂
∂ x̃
∂
∂ x̃
∂ P̃
α̃lg
∂ x̃
!
∂ P̃
∂ x̃
!
∂ P̃
α̃g
∂ x̃
!
4
α̃lm
Concentrations along the filter
1
= Aα̃lg ,
0.8
= −Bα̃lg ,
0.6
αlm
αlg
0.4
= 0,
αlg0 α̃lg + αα̃lm + α̃g = 1,
0.2
0
0
0.2
0.4
0.6
0.8
1
x̃ (Along width of filter)
where
A=
λL2 µg
,
kg φρl (P1 − P0 )
B=
λL2 µlm αlg0
.
3 αφρ (P − P )
klm
g
1
0
We assumed that α̃g ≈ 1 ⇒ P is linear.
å Analytical solutions for αlg and αlm .
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Macro-scale: Simple case in 2D
∂
∂ x̃
"
−
∂
∂ x̃
4
α̃lm
∂ P̃
∂ x̃
α̃lg
!
+ 2
∂ P̃
∂ x̃
∂
∂ ỹ
∂
∂ x̃
α̃g
!
∂
∂ ỹ
!#
+ 2
∂ P̃
∂ ỹ
!
4
α̃lm
∂ P̃
∂ x̃
+ 2
α̃lg
−E
∂
∂ ỹ
∂ P̃
∂ ỹ
!
= Aα̃lg ,
∂
(α̃lm ) = Bα̃lg ,
∂ỹ
!
∂ P̃
α̃g
= 0,
∂ ỹ
Contour plot of αlm
where
µlm Lρlg
.
E= 3
klm (P1 − PO )
ỹ (Along filter height)
1
0.25
0.8
0.2
0.6
0.15
0.4
0.1
0.2
0.05
0
0
0.2
0.4
0.6
0.8
x̃ (Along filter width)
1
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Micro-scale considerations I
Capture rate of droplets:
fd = αlg ρl vg (1 − φ)(df + dd )
(7)
Penetration length λ = fd /vg ≈ 0.2 mm
Stokes flow along fibres
Assume fibres coated by liquid
Stokes flow of liquid along the fibre
R0 : fibre radius, R1 = R0 + h radius of coated fibre
Flow:
2π ∂p (R12 − R02 )(3R12 − R02 ) R14
R1
Q=
−
ln
µ ∂x
16
4
R0
(8)
Experimental data → h ≈ 0.1µm.
Coated fibres can sustain large fluid flow.
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Micro-scale considerations II
Droplet displacement
Balance surface tension variation when moving on fibre
against drag force on droplet:
rd =
(Nf − 1)γdf
3ηug
(9)
Droplets with radii below rd do not move.
Continuum equation to implement this:
αlm = nλ
∂αlm ∂ul αlm
+
= ug αlg (1 − φ)
∂t
∂x
(
0,
ul =
∂p
1
ug + ∂x
3πµrd η ,
(10)
Vdrop < Vdrop minimum
Vdrop > Vdrop minimum
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Markov approach I
One-dimensional lattice model that includes crossings and
open space
Particles move freely in the open space and as a function of
size on the crossings

k

e k is small
Two regimes; r(k) = Crossover behaviour

 −3
k
k is large
Saturation transition dependent on parameters such as
crossover and influx of particles
Do we provide a tool to help design a coalescent? Find
optimal operating conditions?
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Markov approach II
The mathematics; Markov generator
X
Lf (η) =
r(ηx )(f (η 0 ) − f (η)) Paticles on crossings
X
+
u(f (η 0 ) − f (η)) Term due to particles in open space
+ u(f (η + δ1 ) − f (η)) Term due to incoming particles
Defining saturation - When a single particle is of N (the total
number of particles)
Controlling the average particle number d
dt E(N (t)) = u − E(ηL r(ηL ))
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Markov approach: working coalescer
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Markov approach: saturated coalescer
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Results
Multiple macro-scale models to describe the physical
processes going on in the coalescer:
Continuum model
Markov approach
Two possible micro-scale mechanisms for flow of liquid
droplets through the fibre network:
Stick/slip idea
Stick/film-flow idea
Simulations for continuum and Markov models
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