implementation of a one-group interfacial · pdf fileknowledge of the two-phase flow...

2011 International Nuclear Atlantic Conference - INAC 2011 Belo Horizonte,MG, Brazil, October 24-28, 2011 ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN ISBN: 978-85-99141-04-5

*Corresponding Author

IMPLEMENTATION OF A ONE-GROUP INTERFACIAL AREA TRANSPORT EQUATION IN A CFD CODE FOR THE SIMULATION

OF UPWARD ADIABATIC BUBBLY FLOW

F. Pellacani1, S. Chiva2,* , C. Peña2 and R. Macián1

1 NTech Lehrstuhl für Nukleartechnik

Technische Universität München Boltzmannstr. 15

85748 Garching, Germany [email protected], [email protected]

2 Departamento de Ingeniería Mecánica y Construcción

Universitat Jaume I Campus del Riu Sec

12080 Castellón de la Plana, Spain [email protected]

ABSTRACT

In this paper upward, isothermal and turbulent bubbly flow in tubes is numerically modeled by using ANSYS CFX 12.1 with the aim of creating a basis for the reliable simulation of the flow along a vertical channel in a nuclear reactor as long term goal. Two approaches based on the mono-dispersed model and on the One-Group Interfacial Area Transport Equation (IATE) model are used in order to maintain the computational effort as low as possible. This work represents the necessary step to implement a Two-Group Interfacial Area Transport Equation that will be able to dynamically represent the changes in interfacial structure in the transition region from bubbly to slug flow. The drag coefficient is calculated using the Grace model and the interfacial non-drag forces are also included. The Antal model is used for the calculation of the wall lubrication force coefficient. The lift force coefficient is obtained from the Tomiyama model. The turbulent dispersion force is taken into account and is modeled using the FAD (Favre Averaged Drag) approach, while the turbulence transfer is simulated with the Sato´s model. The liquid velocity is in the range between 0.5 and 2 m/s and the average void fraction varies between 5 and 15%.The source and sink terms for break-up and coalescence needed for the calculation of the implemented Interfacial Area Density are those proposed by Yao and Morel. The model has been checked using experimental results by Mendez. Radial profile distributions of void fraction, interfacial area density and bubble mean diameter are shown at the axial position equivalent to z/D=56. The results obtained by the simulations have a good agreement with the experimental data but show also the need of a better study of the coalescence and breakup phenomena to develop more accurate interaction models.

1. INTRODUCTION Two-phase flow occurs in a wide range of industrial application. It is important among many other systems in water-cooled nuclear reactors. Here the presence of bubbles influences the density of the moderator and so the reactivity response of the system. In these situations knowledge of the two-phase flow conditions is paramount for determining the reaction kinetics. Bubbly flow, in a nuclear reactor, is generated in subcooled boiling flow condition. It occurs when the local clad wall temperature during the heating of the subcooled coolant is above the saturation temperature and sufficiently high for bubbles nucleation to occur. Considering boiling we should not forget that mass, momentum, and energy transfer (single and two-phase) involving a solid wall, liquid, and vapor are tightly coupled. In the present contribution, bubbly flow under isothermal condition assumption has been considered. This has been done to limit the complexity of the problem and to test the implementation strategy

INAC 2011, Belo Horizonte, MG, Brazil.

of the One-Group Interfacial Area Transport Equation (IATE) in Ansys CFX. The IATE was proposed , among others, by Kocamustafaogullari and Ishii [1]. The interfacial area concentration ( IAC ) is a key parameter in modeling the interfacial transfer terms in the two-fluid model due to mechanical and thermal non-equilibrium between the two phases. The IAC is defined as the area of an interface per unit mixture volume. In conventional approaches, IAC models have been developed for full developed flow, steady state, and established as a flow map. Then, it is well known, that formulation introduce instabilities and discontinuities in the prediction of the IAC in the transition between regimes. In order to resolve these problems, an IATE has been developed. This new approach consider the IAC as a transportable magnitude, defined by the IATE, the IAC is solved in any point, including the transition between regimen, without the need of any preestablished shape. In the recent years, several researchers have studied the source term of the IATE, where the dynamic of the bubbles are taken on account, and breakup and coalescence play an important role. On the other hand, the development of new measurement techniques led to better descriptions of the physical phenomena and of the forces acting on the phases. The improvement of the physical models helped the setting up of interfacial forces model that able to reproduce the distribution of the phases in a given system describing the interfacial forces such has the drag force [2], the lift force [3] and the wall lubrication force [4,5]. These improvements lead to the possibility of obtaining results with a high detail level. The influence of interfacial forces drag and non-drag forces models on the simulation of adiabatic bubbly vertical upward flow experiment proposed by Hibiki et al. [6] have been already presented in a previous work [7]. In that contribution ANSYS CFX 12 has been used for solving the two-fluid model and the relevant closure relations using the so called “monodisperse” approach. In such a simulation approach a known mean bubble diameter for the gas phase is defined based on experimental data at beginning of calculation and the other flow parameters are calculated depending on it. Several authors in the past years dealt with the implementation of a one-group or two-group interfacial area equation in CFD codes. Yao and Morel [8] in 2004 presented a revisited model for coalescence and breakup and implemented them in the CFD code NEPTUNE. In order to test the accuracy of the model they reproduced cases of the DEDALE experiment and compared the results of their model with those of Wu et al. [9] Hibiki and Ishii [6] and Ishii and Kim [10]. Sari et al. [11] in 2009 implemented both bubble number density equation and the interfacial area density equation in Fluent and tested the models of Wu et al. [9], Hibiki and Ishii [12] and Yao and Morel [8] on the Serizawa [13], Hibiki and Ishii [14] and other experiments. In 2010, Wang [15], in her PhD thesis, implemented both one and two-groups interfacial area density equations in Fluent testing the existing models with original coefficients and also proposing some modification for a better fitting of the experimental data.

2. MATEMATICAL MODELS

2.1. Two-Fluid Model

2.1.1. Mass Conservation

The work presented in this paper are based on the two-fluid model Eulerian–Eulerian approach. The liquid phase is considered as the continuous phase and the gas phase is


considered as dispersed. The flow is “isothermal” and no interfacial mass transfer takes place. The continuity equation of the two-phases presented by Ishii [16] and Drew and Lahey [17] can be written as

0

iii

ii Ut

(1)

2.1.2 Momentum Conservation

The momentum equation for the two-phase mixture can be expressed as it follows:

i

T

iiiiiiiiiiiiii FUUgPUU

t

U

(2)

The term Fi in eq. (2) represents the total interfacial force acting on the phases. Closure laws are needed to calculate the momentum transfer of the total interfacial force.

2.2 Modelling of the Interfacial Forces

Four interfacial forces have been considered during the analysis. The drag force FD, has been modeled using the Grace model [2]. The non-drag forces considered are the lift force FL, the wall lubrication force FWL, and the turbulent dispersion force FTD. The virtual mass force was neglected since tests conducted by Frank et al. [5] showed that its influence is of minor importance in comparison with the amplitude of the other drag and non-drag forces.

TDWLLDi FFFFF (3)

In the next, each of the correlations needs empirical closure relations for the various force coefficients.

2.2.1 Drag Force

The drag force accounts for the drag of one phase on the other and the coefficient that has been used is that of Grace et al. [2].

(4)

2.2.2 Lift Force

Due to velocity gradient, bubbles rising in liquid are subjected to a lateral lift force. This is modeled according to the Tomiyama [3] formulation


. . . (5) For the evaluation of the lift coefficient CL the value is calculated according to Tomiyama [5]. The evaluation of the lift coefficient is based on the definition of the modified Eötvös number,

. . (6)

where dh is the maximum horizontal bubble dimension that is calculated using this empirical expression:

√1 0.163 ∙ . (7)

The lift coefficient proposed by Tomiyama has this form

min0.2888 ∙ tanh 0.121 ∙ , 44 10

0.27 10 (8)

Where , the Eötvös number function, is defined as

0.00105 0.0159 0.0204 0.474 (9)

The behavior of the Tomiyama lift coefficient is a function of the bubble diameter db. A change of sign occurs, for air-water at atmospheric conditions, when the bubble reaches the critical diameter of ca. 5.8 mm. Bubbles with a diameter smaller than this value will be pushed toward the wall. Bubble with a diameter bigger than 5.8 mm will be moved toward the pipe centerline. At higher pressures the critical diameter becomes smaller. Since the bubble size of the experiments considered for calculation is in general lower than 4 mm, for numerical stability a constant value for the lift coefficient of 0.228 has been used.

2.2.3 Wall Lubrication Force

Due to surface tension, a lateral force appears to prevent bubbles attaching on the solid wall. The wall lubrication force has been modeled as it follows:

F F α . ρ . . C . U U . n . n . n (10)

The wall lubrication coefficient in the Antal formulation [4] has the following expression:

, 0 (11)

The values used for and are -0.0064 and 0.016 as proposed by Krepper and Prasser [18] The two lubrication force coefficients were chosen in order to obtain these two effects: achieve a higher absolute value of the wall lubrication force and also extend its action not only at the near wall region. The implementation of the wall lubrication force is necessary for


the adiabatic two-phase flows, as it reproduces the void fraction peak near the wall [19]. Krepper [20] reports that its use at high-pressure wall boiling conditions may be questionable, but it is of primary importance when considering isothermal upward bubbly flow at atmospheric pressure and room temperature.

2.2.4 Turbulent Dispersion Force

A turbulent dispersion force has been considered to take into account the turbulence assisted bubble dispersion. The turbulent dispersion force model that has been used is the Favre averaged Drag force (FAD) [21]. This force is modeled as:

. . ,

, (12)

where Ccd is the momentum transfer coefficient for the interphase drag force. The model depends on the details of the drag correlation used. Sct is the turbulent Schmidt number for continuous phase, it is taken to be 0.9. CTD is a multiplier, and a constant of 1.0 has been used.

2.2.5 Bubble Induced Turbulence

The Bubble induced turbulence has been taken into account according to Sato’s [22] model.

2.3 Modelling of the Interfacial Area Concentration

To obtain a reliable calculation of the momentum transfer of the total interfacial force an accurate calculation of the interfacial area is prerequisite. The interfacial transfer terms contained in equation 2 are proportional to a geometric parameter, the interfacial area concentration ai defined as the total interfacial area per unit two-phase flow mixture volume. In this contribution a dynamic approach for the evaluation of the interfacial area density is used: it is called IATE approach. This was mainly developed by Ishii [16] to capture the changes in the interfacial structure.

2.3.1 Interfacial Area Transport Equation (IATE)

The IATE represents the evolution, transport, in the interfacial area between the phases, one considered continuous (liquid), the other considered dispersed (gas). The possible changes are due to several interaction mechanisms. Wu el al. [9] individuated five mechanisms responsible for them. They are: (1) coalescence due to random collisions driven by turbulence, (2) coalescence due to wake entrainment, (3) breakup due to the impact of turbulent eddies, (4) shearing-off of small bubbles from larger cap bubbles, (5) breakup of large cap bubbles due to interfacial instabilities. If we consider bubbly flow cases with a relative low void fraction level (up to around 15 to 20%) were no cap bubbles are present and the liquid velocity is moderate, the interactions between bubbles can be simplified and only the coalescence due to random collisions and breakup due to the impact of turbulent eddies should be significant for calculation. The IATE is derived from the Boltzmann transport


equation and it is described in for an isothermal flow in steady state conditions, the IATE can be reduced to ( i.e. Wu et. al. [9] ):

ii

jj

ii va

va

3

2 (13)

where the φj are the terms that take into account the interaction mechanisms cited above. The second term on the RHS of equation 13 represents the variation in bubble volume and takes into account expansion bubble are undergoing due to pressure changes along the considered domain. The breakup and coalescence terms considered are those proposed by Yao and Morel [8]. They are coalescence due to random collisions driven by turbulence (RC) and breakup due to the impact of turbulent eddies (TI). Their expressions and coefficients are summarized in Table 1.

Table 1: Breakup and coalescence source terms from Yao and Morel (xx)

Phenomena Expressions Coefficients

Coalescence (RC)

cr

C

cr

C

sm

C

i

RC We

WeK

We

WeK

DK

a 3

23/1

max

3/13/1

max

3/11

3/12

1

2

exp1

12

smsmlDD

We3/2

2

86.2

1

CK

922.12

CK

017.13

CK

52.0max

24.1cr

We

Breakup (TI)

cr

crB

smB

iTI We

We

We

WeK

DK

aexp

11

1112

2

3/11

3/1

1

2

6.1

1

BK

42.02

BK

The Sauter mean diameter is defined as the diameter of a sphere that has the same volume/surface area ratio as a particle of interest:

ism a

D6

(14)

It is worth to say that the local average bubble diameter used for calculation is the Sauter mean diameter calculated using equation 14. Then, it is not a constant value fixed at beginning of calculation, like in monodispersed approximations, ant it is dependent on the flow conditions through the modification applied by the sources, sinks and convective transport.

3. IMPLEMENTATION OF ONE-GROUP IATE IN ANSYS CFX The IATE is solved by using the User defined additional variable under the form of transport equation offered by Ansys CFX. In fact the software allows defining a transport equation for


any scalar quantity by a field velocity. The general form of the transport equation for the interfacial area density ai in multiphase flow calculations is shown in equation 15.

ii

iaai

gt

gtagii

i TSaSc

Dvat

a

,

,

(15)

If the additional transport equation is associated to the gas phase:

ai is the conserved quantity interfacial area density per unit volume of the actual phase α is the void fraction D id the kinematic diffusivity for the scalar to be transported and can be set freely by

the user Sc is the Turbulent Schmidt number (per default 0.9) μ is the Turbulent eddy viscosity S is the external volumetric source term T is the total source due to inter-phase transfer across interfaces with other phases and

is not considered in the adiabatic case The first term on the LHS of equation 15 is the term taking into account the time changes of the interfacial area density, the second is the convective term and the third is the diffusive term. 3.1 Differences between the Theoretical Model and the Ansys CFX Additional Transport Equation Equation 15 differs from equation 13 because of the presence of the diffusive term and of the void fraction inside the derivative terms.In a previous work also Prabhudharwadkar et al. [23] pointed out these differences and suggested a possible strategy to overcome the constraints of the software since the form of equation 15 cannot be modified by the user. 3.1.1 The Elimination of the Diffusive Term

The strategy to reduce or eliminate the influence of the diffusive term is to reduce the influence of the diffusive term without eliminating it, as suggested by Prabhudharwadkar et al. [23] and the Customer Support of Ansys, by setting a low value for the kinematic diffusivity D (1e-15) and a sufficient high value for the Schmidt Number (100 or above). Turbulent Schmidt number is defined like it follows

iag

gtgt D

Sc ,,

(16)

3.1.2 The Calculation of Derivatives in the Source Term In order to calculate gradients and divergences present in the final form of the IATE equation implemented, two different set of subroutines have been written. The first set is composed by only “User CELs” and make use of the internal command of the software.“User CEL” functions allow to create functions in addition to the predefined CEL functions already


defined in the code. A “User CEL” function passes an argument list to a subroutine, and then uses the returned values from the subroutine to set values for the quantity of interest [24]. The second set of subroutines calculates the gradient making use of a self implemented discretization tool. It is setup by both “User CELs” and “Junction Boxes”.The different processes realized by this second set of subroutines are the mapping of the mesh points and the needed operations for the calculation of the gradient based on the value of the variable in 3 or 5 neighbor points. The whole process works for serial and parallel run. It has been developed to have a direct control on the numerical method used for gradients and divergences calculation. 3.1.3 The Transformed Source Term The presence of the void fraction multiplied by the transported variable in equation 15 leads to the necessary implementation of a supplementary term on the RHS of equation 17. This term takes in account the presence of supplementary mixed derivative terms in equation 15 with respect of equation 13. The final form of the source term to be implemented in equation 15 to be equivalent to equation 13 is shown hereunder:

iiii

jj

a vt

avt

aS

i

3

2 (17)

The first term of the RHS is implemented as “Source” it means that it is applied to the bulk of the fluid and then multiplied for the relative volume fraction automatically by the code. The second term is defined as “Fluid Source” and is applied directly to the related fluid.

4. VALIDATION OF THE IMPLEMENTED MODEL The interfacial forces and the IATE models have been compared to the experiment of Mendez 2008 [25]. 4.1 Mendez 2008 The experimental work was performed using a thermo-hydraulic loop placed at Polytechnic University of Valencia ( Spain ). The loop is schematically illustrated in Figure 1. It consists of a vertical tube test section, length of 3340 mm, with a 52 mm inner diameter constant section, an upper plenum and a lower plenum where air and water are mixed. The water was circulated by two centrifugal pumps. The air was supply by a compressor, and it was introduced to the test section through a porous sinter element with an average pore size of 10 mm installed below the mix chamber. The flow conditions covered most of a bubbly flow region, including finely dispersed bubbly flow and bubbly-to-slug transition flow regions. The set-up of the Mendez [25] experiment is illustrated schematically in Fig. 1.

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In fig. 5, 6, 7, the results for the interfacial area density, the void fraction and the Sauter mean diameter are shown. Case A, “monodisperse”, is the default case in ANSYS-CFX where the gas phase is considered as dispersed bubbles with a unique constant spherical diameter. For the case B. the IATE has been used without taking into account coalescence and breakup phenomena. Case C and D have been simulated taking into account both bubble interaction mechanisms multiplying the original Yao & Morel expressions by a constant factor, case C with coalescence and breakup mechanism expression multiply for constant factors, RC= 0.5 and TI = 2.0 respectively, and case D with factors, RC = 0.2 and TI = 5.0

Figure 6: Radial distribution of Void Fraction

Looking at the results of fig. 5, at low void fractions, where the bubble interactions rate is low, solution B leads to results in line with C, D and A. Increasing the void fraction, solution B shows its inadequacy always over predicting the experimental data and the results of solution A. Taking into account breakup and coalescence (case C and D) it is possible to lower the results of case B. Case C shows a too high coalescence rate and almost always under predict the experimental data and case A took as reference. The best results have been obtained by configuration D where the RC term has been reduced by a factor of 5. Both case C and D show a low level of breakup in the wall near region even if the original model of Yao and Morel has been multiplied up to a factor of 5. This is reflected in a slightly under

0

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F01G01 F01G02 F01G03

F02G01 F02G02 F02G03

F03G01 F03G02 F03G03


predicted IAC in the wall near region and in a resulting peak of the Sauter mean diameter (fig. 7) near the wall. This trend reproduces quite well that of the experimental data but a higher breakup rate could lower further the results for the Sauter Mean Diameter in that region. Regarding the void fraction prediction (see fig. 6) both monodisperse and IATE approaches lead to very similar results and the differences are concentrated near the wall. Monodisperse and One-group IATE show their limits near the transition region (cases F03G02 and F03G03 figures 5 and 6) where they over predict the void fraction in the core region. In these cases, whre transition from bubbly to slug flow is taking place, the actual models considered for calculation are not able to reproduce in an effective way the dynamics of the flow.

Figure 7: Radial distribution of Sauter Mean Diameter

5. CONCLUSIONS

The One-group interfacial area transport equation has been implemented successfully in the general CFD code Ansys-CFX. The implementation strategy has been presented and discussed. To take into account the bubble interaction mechanisms the model of Yao and

0

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Sa

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r M

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0.004

0.006

0.008

0.01

0 0.005 0.01 0.015 0.02 0.025

Sa

ute

r M

ea

n D

iam

ete

r [m

]

Radius [m]

0

0.002

0.004

0.006

0.008

0.01

0 0.005 0.01 0.015 0.02 0.025

Sa

ute

r M

ea

n D

iam

ete

r [m

]

Radius [m]

0

0.002

0.004

0.006

0.008

0.01

0 0.005 0.01 0.015 0.02 0.025

Sa

ute

r M

ea

n D

iam

ete

r [m

]

Radius [m]

0

0.002

0.004

0.006

0.008

0.01

0 0.005 0.01 0.015 0.02 0.025

Sa

ute

r M

ea

n D

iam

ete

r [m

]

Radius [m]

0

0.002

0.004

0.006

0.008

0.01

0 0.005 0.01 0.015 0.02 0.025

Sa

ute

r M

ea

n D

iam

ete

r [m

]

Radius [m]

0

0.002

0.004

0.006

0.008

0.01

0 0.005 0.01 0.015 0.02 0.025

Sa

ute

r M

ea

n D

iam

ete

r [m

]

Radius [m]

0

0.002

0.004

0.006

0.008

0.01

0 0.005 0.01 0.015 0.02 0.025

Sa

ute

r M

ea

n D

iam

ete

r [m

]

Radius [m]

F01G01 F01G02 F01G03

F02G01 F02G02 F02G03

F03G01 F03G02 F03G03


Morel [8] has been implemented and two different sets of correction factors have been analyzed. The accuracy of the results has been tested against a wide range of experimental data with liquid velocities from 0.5 to 2 m/s and gas volume fraction from 5 to 15%. In general the IATE approach with coalescence and breakup models from Yao and Morel leads to results in line with the monodisperse calculation. One of the main advantage of this modeling strategy is that the interfacial area is not assumed from spherical bubbles, since a transport process is used to determinate its value, then more precise calculations can be done. The dimension of dispersed phase is not fixed at the beginning of calculation but is a result of the process using the interfacial area density and the void fraction obtained. It has been possible to obtain results for the Sauter mean diameter and Interfacial area density in relative good agreement with the experimental data. Coalescence and breakup mechanism have an important role in the calculation, and as it has been showed, interfacial area and Sauter mean diameter are very sensitive to the expressions and models used. A better analysis of the coalescence and breakup formulations and empiric coefficients is needed to better fit the experimental data object of this contribution and to extend the range of applicability of the model to other conditions. Regarding the bubbly to slug transition region, neither the monodisperse nor the IATE approach is able to reproduce effectively the dynamics of the flow in the core region. They show their limit and more accurate models are needed. They are for example a polydisperse (i.e MUSIG) or Two-group Interfacial Area Transport Equation approach..

ACKNOWLEDGMENTS Part of this research was supported by the “Plan Nacional de I + D + I”, Project ENE2010-21368-C02-02/01.

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implementation of a one-group interfacial · pdf fileknowledge of the two-phase flow...

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