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47th International Conference on Environmental Systems ICES-2017-315 16-20 July 2017, Charleston, South Carolina
Numerical Method to Simulate the Performance of
Microgravity Membrane Gas-Liquid Separator
Zhang, W. W.1
Army Aviation Academy, Beijing, China, 101116
Ke, P.2 and Xu, C. L.
3
School of Transportation Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing,
China, 100083
Microgravity membrane gas-liquid separation technology is one of key technologies of
environmental control and life support system, and effective simulation of gas-liquid two-
phase flow is significant for carrying out researches on the miniaturization and lightweight
of the technology or the separator. However, due to the lack of experimental data and
theoretical understanding, it is difficult to give inlet boundary in real gas-liquid two-phase
flow under microgravity. How to obtain gas-liquid interfaces with different geometrical
scales from average two-phase flow parameters is a special multi-scale problem of gas-liquid
interface. Moreover, a membrane thickness with μm order makes it difficult to establish real
geometric model in the simulation, which belongs to a geometric multi-scale problem.
Therefore, a numerical method combining an Eulerian two-fluid model with an interface
probability approximation method (TFM-IPAM) and membrane boundary model is
proposed. 2D simulations for impermeable and permeable straight pipe are carried out, and
the results are validated by microgravity gas-liquid flow patterns obtained by microgravity
experiments in references. There is a good agreement for prediction results by simulations
and experiments. It indicates that the TFM-IPAM can capture and filter the interfaces with
different scales, which can enhance the adaptability of the calculation method and the
computability of the problem; while membrane boundary model can realize the selectivity
and permeability of the membrane. The TFM-IPAM combined with membrane boundary
model is a simple but effective numerical method for simulating the performance of
microgravity membrane gas-liquid separator, and can also be applied for this kind of
simulation technology problems and engineering applications.
Nomenclature
Ab = interfacial area density of bubbles
Ad = interfacial area density of droplets
Afs = interfacial area density of free surface
Ai = membrane area per unit length
a = diameter of tube or length of rectangular section
C0 = distribution parameter of gas phase
C2 = inertia drag coefficient
CD,b = drag coefficient of bubbles
CD,d = drag coefficient of droplets
CD,fs = interface friction coefficient
Dp = average diameter of particles
d = width or diameter
db = Sauter average diameter of bubbles
dd = Sauter average diameter of droplets
1 Engineer, [email protected].
2 Associate Professor, Department of Aircraft Airworthiness, [email protected].
3 Graduate student, Department of Aircraft Airworthiness, [email protected].
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Fd,b = drag force of bubbles
Fd,d = drag force of droplets
Ffs = interface friction
FS,k = surface tension force
g = acceleration of gravity
1/K = viscous resistance coefficient
L = length
Lt = wall thickness
Mk = interface momentum transfers
n = normal vector
p = pressure
pb,i = local pressure at the outer wall of the membrane
pw,i = local pressure at the inner wall of the membrane
Qi = local instantaneous membrane flow
Qv = volume flux
Qva = time-average volume flux
Sk = momentum source
U = velocity difference
u = velocity
us = superfacial velocity
usg = gas phase superfacial velocity
usl = liquid phase superfacial velocity
Weg = gas phase Weber number
Wel = liquid phase Weber number
Wesg = gas phase superfacial Weber number
Wesl = liquid phase superfacial Weber number
xc = critical parameter
α = phase fraction
α0 = critical porosity
αc = critical phase fraction
αg = gas phase fraction
αl = liquid phase fraction
αs = phase fraction with the probability approximation
β = wall contact angle
ε = porosity of porous media
μ = hydrodynamic viscosity
= density
g = gas phase density
l = liquid phase density
ρm = mixed density
σ = surface tension coefficient
τ = stress
τw,g = shear stress of gas phase
τw,l = shear stress of liquid phase
ΔL = unit length
Δpi = local pressure difference
Δti = time step
Δx = grid scale
I. Introduction
ICROGRAVITY gas-liquid separation technology is one of the key technologies for gas and liquid recycling
in the environmental control and life support system (ECLSS). Compared with a dynamic gas-liquid separator,
a membrane static gas-liquid separator has its own unique characteristics, including high separation efficiency, low
power consumption, and no moving parts. The United States, Russia, and the European Space Agency have
developed the corresponding products, which have been successfully applied in the ECLSS on the International
M
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Space Station.1 Under the background of the development of a new generation of the ECLSS, the demand for the
miniaturization and the lightweight of the technologies or products is increasingly strong. Nevertheless, under the
gravity condition, it is difficult to obtain real flow information in the gas-liquid separator under microgravity by the
ground experiment. Furthermore reduced gravity airplane and drop tower experiments can just supply only a few
seconds of microgravity environment, which usually cannot meet the need of a long time performance experiment.
Orbital aircrafts and microgravity rockets can achieve the long-term microgravity environment, but the cost of
experiments is very high. Hence the limitations of the experiment make computational fluid dynamics (CFD)
simulation an important research method of the product design, which can help to fully understand the internal flow
pattern of gas-liquid separator under microgravity and evaluate its working performance. Microgravity membrane
gas-liquid separator depends on the selective permeability of the membrane to achieve gas-liquid separation, and its
performance is related to the membrane-liquid contact area and the local pressure difference on both sides of the
membrane. Therefore obtaining real inlet flow pattern and establishing appropriate membrane boundary model are
significant for the performance simulation by CFD.
However, due to the lack of experimental data and theoretical understanding, it is difficult to give the real inlet
boundary of the gas-liquid two-phase flow under microgravity for the simulation. How to calculate the gas-liquid
interfaces with geometric scales from the average two-phase flow parameters is a special multi-scale problem of
gas-liquid interface. As is discussed in Ref. 2, deriving from the limit of the grid scale and the lack of the model
boundary respectively, the numerical models of gas-liquid two-phase flow, such as volume of fluid (VOF) and two-
fluid model (TFM), cannot be applied to solve the computability problem of multi-scale gas-liquid interface.
Fortunately several coupled and embedded models based on the TFM have been developed. A coupled model that
combines the Eulerian two/multi-fluid model with the interface-capturing method is an effective approach to
investigate the computability of multi-scale problem. Although there have been some successful applications of
coupled models,3-9
the coupling of two mathematical models with different numbers of equations in the same
computational domain remains a very complex problem.
Besides that, an embedded model has also been widely adopted in the research of computability of multi-scale
problem and is much more convenient from the mathematical point of view. The key concept of an embedded model
is that the TFM is applied throughout the entire computational domain, and when large-scale interfaces are
encountered, an additional interface-sharpening algorithm is implemented to sharpen and identify the geometric
positions of the interfaces (or called geometric boundaries), and then reasonable interface transfers of mass,
momentum, and energy (or called physical boundaries) are included at the geometric boundaries. The first type of
the embedded models to obtain sharp large-scale interfaces in the TFM is to use numerical methods. Minato et al.
proposed an extended TFM in which the downstream difference scheme of the VOF method is used.10
And then,
Štrubelj and Tiselj proposed an artificial compression equation (essentially the mass-source-correction method)
based on the conservative level set method for interface sharpening after the application of a high-resolution scheme
to further reduce the numerical diffusion of the interface.11
The interface obtained by these models is still smeared
though several cells, and is regarded as the concept of the zone, which make it hard to implement accurate local
physical boundaries. The second type of the embedded models to distinguish between small- and large-scale
interfaces in the TFM is by using a physical model. Several popular models, such as the algebraic interfacial area
density (AIAD) model,12
the interfacial area concentration (AIC) model,13
and the SIMMER model,14
were
developed by implementing momentum exchange dependent on different interfacial area densities in accordance
with the flow morphology. Although local physical boundaries are included by the physical model, large-scale
interfaces still cannot be captured and sharpened due to no controlling of the numerical diffusion. The third type of
the embedded models considering numerical methods and physical models simultaneously is introduced. By using
the AIAD, Hänsch et al. developed a generalized two-phase flow (GENTOP) approach and described a clustering
method to decrease the numerical diffusion, in which the cluster force is a function of the phase-fraction gradient
and is implanted into the momentum equations as the source (essentially the momentum-source-correction
method).15
Similarly, Coste put forward the Large Interface Model (LIM), in which the interface is sharpened and
detected using the refined gradient method (essentially the mass-source-correction method), followed by the
application of the appropriate closure laws for interface transfers.16
Recently Mimouni et al. provided a multi-field
approach with interface tracking, called the Large Bubble Model (LBM), in which interface sharpening is achieved
by solving the interface sharpening equation used by Štrubelj, and interface identification is achieved by the phase
fraction, similar to the AIAD.17
The above three models has been initially verified and applied to simulate gas-liquid
two-phase flow in the chemical and nuclear industry. However, these mass/momentum-source-correction methods
for interface sharpening do not carry any real physical meaning, and may lead to additional uncertainty for the TFM.
Furthermore the interface identification by the phase fraction (such as the GENTOP approach and the LBM) is too
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simple to provide the accurate position of the interface; while it by the phase-fraction gradient (such as the LIM)
involves multiple contrast and screening of the data, and the calculation process is more complex.
Therefore Eulerian two-fluid model with an interface probability approximation method (TFM-IPAM) is
proposed here. It uses a high precision scheme named compressive interface capturing scheme for arbitrary meshes
(CICSAM)18
to control interface numeral diffusion for interface sharpening, and uses a probability approximation
method to identify the interfaces and make uncertainty interfaces caused by unavoidable numerical diffusion into
certainty interfaces with the probabilistic sense. It aims to provide a simple and effective engineering simulation
technology in complex gas-liquid two-phase flow systems of manned space and other areas.
In addition, the membrane thickness with the order of μm makes it difficult to establish a true geometric model
when dealing with the boundary of the membrane. It is necessary to describe the membrane properties by means of
mathematical models as a physical boundary condition for the gas-liquid two-phase flow in the separator. Vieira et
al. used the momentum source method to achieve the two-phase penetration process of the ceramic membrane in the
simulation of oil-water separation,19
but did not reflect the selectivity of the membrane. Sun et al. used the mass
source method to describe the single-phase permeation process of the membrane in the study of membrane gas-
liquid separation simulation,20
but did not consider the effect of membrane penetration on the local pressure and
velocity, and the case of the two-phase flow. Here a momentum source method is developed for simulating the two-
phase penetration and selectivity characteristics of the membrane simultaneously in order to avoid the multi-scale
geometric problem.
This paper is organized as follows. First, the TFM-IPAM and the membrane boundary model mentioned are
detailed. 2D simulation cases for the validation and analysis of inlet flow pattern and membrane boundary model are
presented in Section Ⅲ. Finally, conclusions are drawn and future work is given.
II. Model and Method
A. Geometric Model
The structure of membrane gas-liquid separator includes gas-liquid mixture inlet, gas outlet, separation
membrane, porous plate and separation channel,21
shown in Figure 1. Here the separation membrane is a selective
and permeable membrane with liquid passing and no gas passing, which is a core component of membrane gas-
liquid separator. The porous plate is a support member of the membrane, which increases mechanical strength.
Gas-Liquid
Mixture Inlet
Gas
Outlet
Separation MembranePorous Plate
L
d
Separation
Channel
Figure 1. The structure of membrane gas-liquid separator.
The macroscopic process of gas-liquid separation can be described as bellow: gas-liquid mixture is pumped into
the separation channel by the action of external transport pressure. After passing through the porous plate and
contacting the membrane, liquid phase is adsorbed, and passed through the porous of the membrane and into the
cavity by pressure difference on both sides of the membrane. Gas phase is discharged from the gas outlet. While the
microscopic mechanism of gas-liquid separation can be interpreted by the preferential adsorption-capillary flow
model. Due to the hydrophilic property of the membrane material and the design of the pores, liquid phase is
preferentially adsorbed, and then forms thin liquid film in the pores and on the surface of the membrane, which
blocks the penetration of gas phase.
Here for the sake of generality, a straight pipe is used for the validation and analysis cases. The length L and the
diameter d of the channel are 200 mm and 10 mm respectively. In order to weaken the impact of inlet and outlet, the
extended length L is added for both of them. In addition, the back pressure of the membrane is set to -50 kPa; the
back pressure of gas outlet is set to 0 kPa; surface tension coefficient σ equals to 0.07 N / m; wall contact angle β
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equals to 60 °; taking into account limited computer resources and previous testing results, grid scale Δx is set to 0.5
mm, and it is enough to describe the flow phenomenon.
B. TFM-IPAM
A typical gas-liquid two-phase flow in the separator is represented by liquid phase and gas phase with
continuous and dispersed morphologies. Considering no phase change and heat transfer, the whole frame of the
TFM-IPAM includes basic transport equations, an interface probability approximation method (IPAM), interface
momentum transfers and turbulence closure relations, described in Figure 2. The IPAM is used to divide entire
computational domain into three parts: interface layer and dispersed flow zones (bubbly flow zone and droplet flow
zone). In the interface layer, the IPAM treats large-scale interfaces (them between continuous gas and liquid phase)
as explicit geometric boundaries basing on the grid scale and includes interface friction at the positions of interfaces,
while in the dispersed flow zones, the IPAM deals small-scale interfaces (them between continuous gas/liquid phase
and dispersed liquid/gas phase) with implicit physical scale basing on the averaged field and supplements traditional
interphase force models. Moreover both interface momentum transfers and turbulence closure relations are
necessary conditions for solving basic transport equations. Herein, the Sauter mean diameter of dispersed liquid/gas
phase is chosen as an implicit physical scale, and is usually defined through experimental measurements, theoretical
calculations, or reasonable assumptions. For the distribution of implicit physical scales, it can be calculated by
introducing a population balance model or an interfacial area density transport equation, and will be realized in the
TFM-IPAM in future. Multi-scale
interfaces
Small-scale
interfaces
Large-scale
interfaces
Transport
equations
Geometric boundary Physical boundary
Traditional
interphase force
Interface friction
Implicit physical scale
based on the averaged
field
Explicit geometric scale
based on the grid scale
TFM
Turbulence
closure
relations
IPAMInterface momentum
transfers Figure 2. The whole frame of the TFM-IPAM.
a. Transport equations
The TFM based on the averaged field is described. The fields used in the TFM have been subjected to a variety of
averaging operations, including time, space, and ensemble averaging. Each phase has a corresponding set of
governing equations, including continuity equation, momentum equation and energy equation, which are coupled by
phase-interaction terms in the form of source terms in their respective equations. For an incompressible and
isothermal two-phase flow, governing equations of the TFM can be defined by Ishii and Hibiki as follows:22
0k k k k kt
u
(1)
k k k k k k k k k k k k kpt
u u u g M
(2)
where index k denotes gas phase (g) or liquid phase (l). αk denotes volume fraction of the corresponding phase. ρk, uk
and p are density, velocity and pressure at a given point respectively. t is time. τk is stress. g is acceleration of
gravity. The first term on the right side of Eq. (2) is pressure gradient term. The second term includes viscous stress
and Reynolds turbulence stress. In this paper, the latter is closed by k-ω discrete turbulence model, and the Troshko-
Hassan model23
is used to consider turbulence interactions. The third item denotes gravity, and it is ignored under
microgravity. The last term Mk represents interface momentum transfers. In the porous zone the above equations are
multiplied by the porosity ε.
b. IPAM
The IPAM includes two parts: the strategy of controlling interface numerical diffusion for interface sharpening
and the algorithm of interface probability approximation for interface identification.
As is discussed in Ref. 18, the geometric reconstruction algorithm can completely eliminate numerical diffusion
of the interface, but it has following problems: it is usually based on quadrilateral and hexahedral mesh cells, and it
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is difficult to apply to other grid element types; The reconstruction of internal fluid distribution in the grid cell
becomes very complex as the dimension increases. This is also one of the reasons why such methods are
computationally intensive and computationally prone to collapse in engineering applications. In addition, the
mass/momentum-source-correction methods have changed original phase fraction field or velocity field. Therefore,
in order to improve adaptability of the method in complex structure and not introduce additional uncertainty, a high
precision compression difference scheme CICSAM of the VOF method is applied to deal with the diffusion of large-
scale interfaces in the TFM-IPAM.
However, the CICSAM can effectively control but cannot completely eliminate the numerical diffusion, and the
interface across several grids still has uncertainty. Therefore, the algorithm of interface probability approximation is
proposed to capture large-scale interface, which described by a single-layer grid. And, the interface layer and the
discrete flow region are calibrated. It provides accurate position information for local interface momentum transfers.
First, the algorithm is based on an equivalence relation, that is, in the physical sense, a phase fraction is a volume
content of a phase in the cell, which is equivalent to a probability that a phase appears in the probability sense. For
example, when a phase fraction in a grid cell is greater than critical phase fraction (0.99 in this paper), the
probability that a phase appears in the cell is considered to be 1, based on the probability approximation, and we
determines that the phase occupies this cell. Secondly, it is based on an analogy relationship, that is, the difference
between the isolines/isosurfaces of the phase fraction can be regarded as a distance function in the sense of phase
fraction. Therefore, a mathematical description of the probabilistic approximation of phase fraction and a sharpening
adjustment of the interface can be realized by means of the Heaviside function in the Level-set method,24
which can
help to simplify the process of approximation and adjustment.
When |1-α|<|1-αc| (α>αc), phase fraction with the probability approximation αs equals to 1; When |1-α|>|αc| (α<1-
αc), αs equals to 0; When |1-αc|≤|1-α|≤|αc| (1-αc≤α≤αc), αs is set to a range form 0-1 by the normalization process.
Thus, first approximation of α and first sharpening adjustment of the interface are achieved. Then this approximation
and adjustment is carried out N times (generally N can take 10), and the cells with α from 0-1 can be eliminated
(Due to numerical diffusion of the interface is not completely symmetrical, there may be very few cells with α from
0-1). It allows phase fraction gradient in the field to be at two extremes, 0 or 1/Δx, and the cells with non-zero phase
fraction gradient are marked as large-scale interface (the interface layer). Here the expression of the Heaviside
function after the transformation of the abscissa is expressed as follows:
s
0
1 1sin
2 2 2
1
E E
c
c c
c
1
1
(3)
Wherein, Φ=α-0.5; Ε=αc-0.5.
c. Interface momentum transfers
Because of different-scale interfaces treated in different ways, corresponding interface momentum transfers are
different. In the interface layer (large-scale interfaces), interface friction is included. The general formula of
interface friction can be expressed by:
2
fs D,fs fs m
1
2C A F U (4)
Wherein, interfacial area density Afs = |▽αl|; mixed density ρm = αlρl + αgρg; gas-liquid velocity difference |U| = |ul -
ug|. Reference to AIAD model,12
interface friction coefficient CD,fs is defined:
l w,l g w,g
D,fs 2
l
2C
U
(5)
Where the shear stress of gas phase and liquid phase is similar to the wall shear force, and is written as:
w, l,gi
i i i
u
n (6)
In dispersed flow zones (small-scale interfaces), traditional interphase forces are implemented. Traditional
interphase forces include drag force and non-drag forces, and the drag force Fd,k is usually considered to be the
major component of phase-interaction term Mk.
In bubbly flow zone, the drag force is:
2
d,b D,b b l
1
8C A F U (7)
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Wherein, Ab equals to 6αg/db, and db is set to 1/2Δx; drag coefficient CD,b is given by the model of Grace et al..
25
In droplet flow zone, the drag force is written as:
2
d,d D,d d g
1
8C A F U (8)
Wherein, Ad equals to 6αl/dd, and dd is set to 1/2Δx; drag coefficient CD, d is given by the model of Ishii and Zuber.
26
Furthermore, surface-tension force FS, k is also included in Mk, and can be calculated using the continuum
surface force (CSF) model proposed by Brackbill et al..27
C. Membrane Boundary Model
Considering the following model hypothesis: (a) the membrane is considered as a porous wall with a constant
thickness (far less than equivalent diameter of the channel), and defined as an external virtual computing domain,
shown in Figure 3; (b) flow distribution in the membrane is ignored; (c) flow resistance of the porous plate is
ignored; (d) thin liquid film forms in the pores and on the surface of the membrane, that is, the virtual computing
domain is filled with liquid phase.
Pb,i
Pw,i
External virtual
computing domain
Internal channel
computing domain
Figure 3. Virtual and channel computing domain.
The permeability of the membrane is realized by adding the source term to the momentum equation of liquid
phase in the virtual computing domain. For homogeneous porous media, the momentum source term is
2( )2
k kk k k k
C
K
S u u
(9)
Wherein, μ is hydrodynamic viscosity; 1/K and C2 is viscous resistance coefficient and inertia drag coefficient of the
porous media respectively, and both of them are determined by the Ergun semi-empirical relation.28
2
2
2 3 3
t p p
1 1.75 1150pu u
L D D
(10)
Where Dp is average diameter of particles; ε is porosity of porous media; Lt is wall thickness, and equals to 1 mm in
this case. Comparing Eq. (9) with Eq. (10), we can obtain
2
2 3
p
11 150
K D
(11)
2 3
p
13.5C
D
(12)
According to experimental values of the membrane: pressure difference Δp is 50 kPa, and mass flux of liquid
phase is 0.033 kg/(m2·s), linear volume flux can be established as follows:
3
v l2.4 10 /Q p
(13)
Substituting Eq. (13) into Eq. (10), and assuming that internal flow in the membrane is laminar, then, the second
term on the right side of Eq. (10) is negligible. Thus porosity parameters (Dp = 0.15 mm and ε = 0.016) are matched.
Further Substituting Dp and ε into Eq. (11), 1/K = 1.49 × 1015
m-2
is obtained.
The selectivity of the membrane is achieved by adding the maximum source term to the momentum equation of
gas phase in the virtual computing domain. Here 1/K is set to 1020
m-2
.
When the flow reaches dynamic equilibrium, time-average value of the membrane flux is used as the evaluation
parameter of separation performance:
va
i i
i i
Q tQ
A t
(14)
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Wherein, Qi is local instantaneous membrane flow; Ai is membrane area per unit length, where Ai=πdΔL for the
circular section pipe; unit length ΔL equals to Δx; Δti is time step.
Local instantaneous membrane flow Qi can be calculated by two methods. The one is basing on the penetration
velocity of liquid phase cross the membrane.
l,i i i iQ Au
(15)
Where, αl,i is local phase fraction of liquid phase in the cell adjacent to the inside of the membrane; ui is local
penetration velocity of liquid phase cross the membrane, and the direction is outwardly perpendicular to the
membrane.
The other is basing on pressure difference on both sides of the membrane.
3
l, v l, l2.4 10 /i i i i i iQ AQ A p
(16)
Where, local pressure difference on both sides of the membrane Δpi = pw,i-pb,i. The results obtained by Eq. (15)
can be regarded as simulation values, while obtained by Eq. (16) can be regarded as engineering values. Therefore
the mutual validation between simulation and engineering values can be used to verify the effectiveness of
membrane boundary model.
D. Numerical Procedures
The phase-coupled SIMPLE algorithm29
is used to solve the transport equations of the TFM. The coupling terms
are treated implicitly and form part of the solution matrix. The pressure–velocity coupling is based on total volume
continuity, and the interfacial coupling terms are fully incorporated into the pressure-correction equation. According
to our preliminary research,30
except for the CICSAM applied for the discretization of the volume-fraction equations,
the effect of numerical scheme for the spatial and temporal discretization of transport equations can be ignored in
the TFM-IPAM. Here, for the spatial discretization, a second-order upwind scheme is chosen for the discretization
of the momentum equations, and the gradients used to discretize the convection and diffusion terms in the transport
equations are evaluated based on the Green-Gauss gradient method. For the temporal discretization, a two-order
implicit scheme is used.
The linear system is solved using the point implicit Gauss–Seidel method combined with the block algebraic
multi-grid method. These solver procedures are all implemented using FLUENT® version 14.5 (ANSYS, Inc.,
Canonsburg, PA, USA). The procedure for the IPAM is implemented by embedding user-defined functions into the
FLUENT solver. The entire solution algorithm for the TFM-IPAM is summarized in Figure 4. After one iteration, if
the residuals do not converge, then the IPAM is implemented. Geometric boundaries and momentum transfers of
interfaces are updated each iteration.
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Initialize all variables
Convergence?
Start next time step
Update boundaries and coupling terms
Reconstruct the volume fluxes
Build and solve the pressure-
correction equation from total
volume continuity
Correct volume fluxes, velocity,
and share pressure
Solve for volume fractions enforcing
realizability conditions and update
properties.
Yes
No Loop
Phase coupled SIMPLE algorithm
Storing phase
volume fraction
Using the
modification of the
Heaviside function
Identifying the
geometric positions of
large-scale interfaces
Implementing local
interface momentum
transfers
Interface probability
approximation method
Solve phase-coupled discretized
momentum equations
Figure 4. The entire solution algorithm for the TFM-IPAM.
III. Validation and Analysis
A. Inlet Flow Pattern
Firstly validation and analysis are carried on inlet flow pattern. The purpose is checking microgravity flow
patterns between simulation results and references, and verifying the effectiveness of the TFM-IPAM in solving the
simulation problem of inlet flow pattern. Therefore impermeable straight channel is chosen as a simple example, and
operating parameters are designed to cover three main microgravity flow patterns. Four kinds of verification cases
for microgravity flow pattern in impermeable straight channel are designed, shown in Table 1. Here, superficial
velocities of gas and liquid phase are calculated by inlet flow, section area and gas phase fraction at the inlet. The
inlet flow is set to 1 L/min for first three cases and 22.15 L/min for the last case. Gas phase fraction αg at the inlet is
set to 0.1, 0.5, 0.9 and 0.96 respectively.
Table 1 Microgravity flow patterns comparison between simulation results and references
Case usl/(m·s-1
) usg/(m·s-1
) usl/usg Wel Weg Wel/Weg Microgravity flow patterns
Simulation references
1 0.19 0.02 9.00 0.46 0.05 9.00 Bubble Bubble31-36
2 0.11 0.11 1.00 0.25 0.25 1.00 Slug Slug31-36
3 0.02 0.19 0.11 0.05 0.46 0.11 Slug Slug31-36
4 0.20 4.50 0.04 0.48 10.76 0.04 Annular Annular37
First, relative real distribution of inlet flow patterns are obtained by the TFM-IPAM. Figure 5 shows the
simulation results of four kinds of microgravity flow patterns. In Figure 5, red represents liquid phase, blue
represents gas phase, and the other colors represent the mixed phase. As superficial velocity of gas phase usg
increases, predicted flow patterns by simulation from case 1 to case 4 are bubble flow, slug flow, slug flow and
annular flow in proper sequence. Meanwhile, the results show that inlet flow pattern is given in the form of uniform
mixing of gas and liquid phase, but gas-liquid two-phase flow automatically forms the flow pattern matching inlet
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flow parameters, under the constraints of geometric boundaries and interface momentum transfers in the TFM-
IPAM. The TFM-IPAM successfully captures typical large-scale interface features in the flow, has the filtering
effect on small-scale interfaces, realizes the hierarchical processing of the interface for multi-scale interface, and
effectively avoids poor adaptability and computability of traditional simulation methods.
Inlet section Middle section Outlet section
Figure 5. Simulation results of microgravity flow patterns (from up to down: bubble, slug, slug and annular). Second, we compare simulation results with the flow pattern obtained by microgravity experiments in references
from Ref. 31 to Ref. 37. Here, the Zuber-Findlay porosity model38
is applied to predict the transition between bubble
flow and slug flow under microgravity. The transition condition is
csl sg
c
1 xu u
x
(17)
c 0 0x C (18)
Where, xc is critical parameter; distribution parameter of gas phase C0 and critical porosity α0 are determined
experimentally or calculated by the experimental empirical relationship. The semi-analytic Weber number model
proposed by Zhao and Hu37
is applied to predict the transition between slug flow and annular flow under
microgravity. The transition condition is
cl g
c
1 xWe We
x
(19)
Where, Wel=usl/u0=(Weslρg/ρl)1/2
is Weber number of liquid phase, u0=(ρga/σ)1/2
is characteristic velocity, a is
diameter of the tube or the length of the rectangular section, σ is surface tension, and Wesl=ρlusl2a/σ is superficial
Weber number of liquid phase; Weg=usg/u0=(Wesg)1/2
is Weber number of gas phase, and Wesg=ρgusg2a/σ is superficial
Weber number of gas phase.
Figure 6 compares simulation results with the results of flow graphs determined by experiments under
microgravity. Because of different experimental working fluids, diameters and tube types by different researchers, it
is difficult to get a unified parameter values for Eq. (17) and Eq. (18). Therefore there is a certain degree of
uncertainty, and the point of case 2 is located in an approximately parallel segment. Howsoever simulation results of
four cases are in good agreement with the prediction results of two flow graphs. It verifies the effectiveness of the
TFM-IPAM in solving the simulation problem of inlet flow pattern.
10-2
10-1
100
10-2
10-1
100
Case 2
Colin33
Bousman35
Zhao36
Bousman35
Colin32
Zhao34
Duckler31
Case 3
Case 1
Slug
usl
, m
/s
usg, m/s
Bubble
100
101
10-1
100
Case 4
Zhao37
Annular
Slug
Weg
We
l
a), the transition between bubble flow and slug flow b), the transition between slug flow and annular flow
Figure 6. Comparison between simulation and flow pattern31-37.
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B. Membrane Boundary Model
Usually the membrane performance defined as the relationship of liquid mass permeability and the pressure
difference on both sides of the membrane can be obtained by component-level single-phase flow ground experiment.
However, under microgravity and two phase flow conditions, the performance of the membrane and membrane gas-
liquid separator cannot be easily achieved by ground experiments. Therefore membrane boundary model applied in
microgravity membrane gas-liquid separator simulation is particularly important. The preliminary effectiveness
validation of membrane boundary model is carried out by comparing simulation values with engineering values.
Here permeable straight channel is chosen as a simple example, and operating parameters are designed to meet
normal flow conditions in the separator,21
not exactly the same as the former in Table 1. The total permeation flow
of liquid phase ∑Qi is chosen as an index for the validation. Verification of cases and results is shown in Table 2.
The results show that the maximum error of ∑Qi is less than 9.0 %, and the average error of ∑Qi is less than 3.0 %.
Here, the error is defined as |simulation value - engineering value| / simulation value × 100 %. Therefore
membrane boundary model is verified quantitatively. More detail analysis are carried out as below.
Table 2 ∑Qi comparison between simulation values and engineering values
Case usl /(m·s-1
) usg /(m·s-1
) Flow patterns ∑Qi
Simulation /(mL·s-1
) Engineering /(mL·s-1
) Error /%
1 0.21 0.00 Single phase 0.2100 0.2100 0.0
2 0.19 0.02 Bubble 0.2102 0.2098 0.2
3 0.11 0.11 Slug 0.1630 0.1600 1.8
4 0.02 0.19 Slug 0.0432 0.0394 8.8
Figure 7 gives local phase fraction of liquid phase αl,i in the cell adjacent to the inside of the membrane varying
as the length of separation section. It indicates that according to operating parameters at the inlet and limited
separation section, flow pattern in four cases are single phase flow, bubble flow, slug flow and slug flow
respectively.
0 50 100 150 2000.0
0.2
0.4
0.6
0.8
1.0
Case 4 Case 3 Case 2
αg=0.1 αg=0
Length of separation section, mm
αg=0.9 αg=0.5
αl,
i
Case 1
Figure 7. Local liquid phase fraction varying as the length of separation section.
Figure 8 shows the variation curves of local permeation flow of the membrane. First, the simulation curves have
a high degree of consistent trend with the curves by engineering calculation. Second, comparing Figure 7 and Figure
8, the same varying trend between αl,i and Qi illustrates that local phase fraction of liquid phase αl,i plays a decisive
role in local permeation flow of the membrane. Therefore keep appropriate flow pattern in the channel and increase
liquid phase fraction on the surface of the membrane, can improve the permeability of the membrane and the
efficiency of the separator. Three, the increase in the difference between simulation and engineering values of the
permeation flow can be explained as follow: engineering method only considers the contribution of static pressure
difference on both sides of the membrane; while the contribution of dynamic pressure of liquid phase at the
membrane and the static pressure difference are both included in simulation method. And as gas phase fraction of
the inlet increases, the friction between gas phase and liquid phase increases and makes dynamic pressure of liquid
phase at the membrane larger. Therefore the difference between simulation and engineering values increases. On the
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other hand, it reflects that the application of membrane boundary model in simulation is effective, and simulation
method is more accurate than engineering method. Here a fact can also be recognized that simulation method is
based on the momentum source method, and engineering method is more equivalent to the mass source method.
0 50 100 150 2005.1
5.2
5.3
5.4
5.5
Qi,
10
-4m
L/s
Length of separation section, mm
Engineering
Simulation
Case 1
αg=0
0 50 100 150 2005.1
5.2
5.3
5.4
5.5
Engineering
Simulation
Qi,
10
-4m
L/s
Length of separation section, mm
Case 2
αg=0.1
a), case 1 b), case 2
0 50 100 150 200
0
1
2
3
4
5
6
7
Engineering
Simulation
Qi,
10
-4m
L/s
Length of separation section, mm
Case 3
αg=0.5
0 50 100 150 200
0
1
2
3
4
5
6
7
Engineering
SimulationQ
i, 10
-4m
L/s
Length of separation section, mm
Case 4
αg=0.9
c), case 3 d), case 4
Figure 8. Comparison between simulation and engineering values. Further Figure 9 gives a partial enlarged view of permeation velocity vector of liquid phase. Liquid permeation
velocity vector has a distribution on the interface between external virtual computing domain and internal channel
computing domain. However, this velocity distribution on the interface cannot be obtained by the mass source
method.20
Therefore, compared with the mass source method, the momentum source method is more realistic in
simulating the permeation process of the membrane. In addition, there is no gas phase permeation velocity on the
membrane, which shows that the model can effectively block gas penetration and realize the selectivity of the
membrane for liquid phase.
Figure 9. Permeation velocity vector of liquid phase.
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IV. Conclusion
A numerical method combining the TFM-IPAM and membrane boundary model is proposed for simulating the
performance of microgravity membrane gas-liquid separator. The TFM-IPAM has the function of capturing and
filtering different scale interfaces, which can enhance the adaptability of the calculation method and the
computability of multi-scale problem of gas-liquid interface; while membrane boundary model based on the
momentum source method can realize the selectivity and permeability of the membrane, and describe relatively real
permeation process of the membrane. The model and method can be applied to effectively solve this kind of
simulation technology problem.
It is true that the TFM-IPAM is only a preliminary attempt to find a simple but effective engineering simulation
method for multi-scale problem of gas-liquid interface in microgravity gas-liquid separator. Further, first, more
specific quantitative verification with experiments and methods of others, including the effects of interface
momentum transmission, turbulence effects, and so on, will be carried out. Second, the TFM-IPAM considering the
scale distribution of discrete flows will be developed.
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