numerical method to simulate the performance of

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].

International Conference on Environmental Systems

2

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


3

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


4

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


5

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


6

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)


7

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)


8

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.


9

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


10

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.


11

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


12

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


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


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


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


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.


13

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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numerical method to simulate the performance of

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