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Journal of Nuclear Science and Technology
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Numerical simulations of gas–liquid–particle
three-phase flows using a hybrid method
Liancheng Guo, Koji Morita & Yoshiharu Tobita
To cite this article: Liancheng Guo, Koji Morita & Yoshiharu Tobita (2016) Numerical simulations of gas–liquid–particle three-phase flows using a hybrid method, Journal of Nuclear Science and Technology, 53:2, 271-280, DOI: 10.1080/00223131.2015.1043156
To link to this article: https://doi.org/10.1080/00223131.2015.1043156
Published online: 21 May 2015.
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Journal of Nuclear Science and Technology, 2016
Vol. 53, No. 2, 271–280, http://dx.doi.org/10.1080/00223131.2015.1043156
ARTICLE
Numerical simulations of gas–liquid–particle three-phase flows using a hybrid method
Liancheng Guoa∗, Koji Moritaaand Yoshiharu TobitabaDepartment of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Motooka, Nishi-ku, Fukuoka 819-0395,
Japan;bAdvanced Nuclear System R&D Directorate, Japan Atomic Energy Agency, 4002 Narita, O-arai, Ibaraki 311-1393, Japan
(Received 10 November 2014; accepted final version for publication 1 April 2015)
For the analysis of debris behavior in core disruptive accidents of liquid metal fast reactors, a hybrid com-putational tool was developed using the discrete element method (DEM) for calculation of solid particle dynamics and a multi-fluid model of a reactor safety analysis code, SIMMER-III, to reasonably simulate transient behavior of three-phase flows of gas–liquid–particle mixtures. A coupling numerical algorithm was developed to combine the DEM and fluid-dynamic calculations, which are based on an explicit and a semi-implicit method, respectively. The developed method was validated based on experiments of water– particle dam break and fluidized bed in systems of gas–liquid–particle flows. Reasonable agreements be-tween the simulation results and experimental data demonstrate the validity of the present method for complicated three-phase flows with large amounts of solid particles.
Keywords: numerical simulation; reactor safety; computational fluid dynamics; gas–liquid–particle flows; SIMMER-III; discrete element method
1. Introduction
Three-phase flows of gas–liquid–particle mixtures are complicated multiphase problems, especially for numerical simulations of mixture flows with a large number of solid particles. They are considered to be common phenomena in various industry areas, such as nuclear, chemical, petrochemical, refining, phar-maceutical, biotechnology, food and environmental industries.
When core disruptive accidents (CDAs) occur in sodium-cooled fast reactors (SFRs), core debris may set-tle on the core support structure and form bed mounds. Heat convection and vaporization of coolant sodium will then level this debris bed. This phenomenon can be named as “self-leveling behavior” and is crucial to the relocation of molten core and heat removal capability of the debris bed, which greatly affects the subsequent accident process. To reasonably simulate such transient behavior, as well as thermal–hydraulic phenomena oc-curring during a CDA, a comprehensive computational tool is needed. SIMMER-III [1] is a successful com-puter code that was developed as an advanced tool for CDA analysis in SFRs. It is a two-dimensional (2D), multi-velocity field, multiphase, multicomponent Eulerian fluid dynamics code coupled with a fuel pin model and a space- and energy-dependent neutron
ki-∗Corresponding author. Email: [email protected]; [email protected]
netics model. In recent decades, the SIMMER code has been successfully applied to simulations of key thermal– hydraulic phenomena involved in CDAs, as well as to assessments of reactor safety. However, in simulations of multiphase flows with a large fraction of solid parti-cles, such as the self-leveling behavior described above, the fluid-dynamics models of the code do not consider strong interactions among solid particles and the dis-crete particle characteristics.
To reasonably simulate such complicated multiphase flows, it is necessary to consider the interactions simul-taneously between fluid (gas and liquid) phases, between fluid phases and particles, as well as those between par-ticles themselves in an appropriate manner. Even now, it is still difficult to exactly calculate strong interactions between particles and between particles and the wall in numerical simulations of multiphase flows. In macro-scale computational methods, which are usually built on a fully Eulerian framework, particles in a fluid can be treated as a kind of fluid. Therefore, in the major-ity of related literature, the continuum assumption is applied to the particle phase. For example, the two-fluid model (TFM) [2] regards the particle–fluid mix-ture as the blending of two fluids, and this can predict the time-averaged and instantaneous porosities of the mixture reasonably [3]. To date, TFM has successfully
C
described the hydrodynamics of fluidized bed with liquid–solid flows and gas–solid flows [4]. Later on, some researchers have extended it to a three-phase system [5]. Although the constitutive relations used in this model are usually based on empirical equations, their lack in generality would limit the model applicability. Under the Eulerian framework, the computational domain is composed of multiple mesh cells, and physical variables in each cell are assumed uniform for phases includ-ing particles. When the volume fraction of the particle phase becomes very large in a cell, strong interactions between particles as well as the discrete particle charac-teristics cannot be represented by the continuum model. Other numerical methods treat particles discretely at a mesoscopic level. Among them, application of the dis-crete element method (DEM), originally introduced by Cundall and Strack [6], is investigated actively. Using an explicit force model of DEM, multi-body collisions can be calculated directly with good precision. In ad-dition, DEM can also provide local transient informa-tion about particles, such as their trajectories and veloc-ities. Various studies have applied DEM to the particle simulations [7].
Consideration of liquid–particle and gas–particle interactions is another difficulty in numerical meth-ods for three-phase flows of gas–liquid–particle mix-tures. In Eulerian–Eulerian approaches such as TFM, the solution depends significantly on a proper descrip-tion of interfacial force and solid stress acting on particles [8]. On the other hand, it is more straight-forward to couple DEM with a computational fluid dynamics (CFD) method in a multi-scale modeling scheme, based on an Eulerian–Lagrangian framework. In this coupling method, the gas and liquid phases are modeled as continua and can be solved based on the Eulerian description. With this in mind, Tsuji et al. [9] introduced a DEM–CFD coupling algorithm first into the numerical simulation of a fluidized bed system. Afterwards, this kind of coupling algorithm has been widely applied in the fluid–solid two-phase flows.
To solve the problem described above, in the present study, a hybrid computational method is developed by combining DEM with the multi-fluid model of SIMMER-III code. This approach will be reasonable to simulate the particle behavior, as well as the thermal– hydraulic phenomena, in CDAs. In the coupling al-gorithm, the governing equations of the fluid phases are solved by a semi-implicit time factorization scheme, whereas particle movements are calculated by DEM. The fluid and solid–particle phases are then explicitly coupled through drag force terms in their governing equations. The developed method is applied to analy-ses of two experiments in systems of gas–liquid–particle flows. One is a water–particle dam break, and the other is a three-phase fluidized bed. The simulation results are validated by comparing with available experimental data.
2. Mathematical treatment
2.1. Governing equations
When the heat transfer between multiple phases can be neglected, the governing equations of the fluid phases are the conservation equations of mass and momentum, in terms of the local mean variables over a computa-tional cell. These can be expressed in the following ab-breviated form: ∂αfρf ∂t + ∇ · (αfρf · vf)= 0 (1) ∂αfρf vf ∂t + ∇ · (αfρfvfvf)= −αf∇ p + αfρfg + Sd+ ∇ · (αfτˆf) (2)
where the subscript f denotes the fluid phase contain-ing gas and liquid; t is time; αf, ρf and vf are the
void fraction, gas density and velocity, respectively; ˆτf
is the viscous stress tensor expressed by Newton’s law of viscosity; g is the gravitational acceleration and Sd is
the momentum exchange term between the particle and fluid phases.
The particle phase is treated as discrete, and the mo-tion of particle i is described by Newton’s law of momo-tion as follows: mi d2r i dt2 = Fc,i+ Ff,i+ Fg,i (3) vi = dri dt (4) Ii d2θ i dt2 = jAi j (5) ωi = dθi dt (6)
where mi, ri, vi, Ii, θi and ωi are the mass,
po-sition, translation velocity, moment of inertia, angular displacement and velocity of particle i, respectively; Fc,i
is the contact force between particle i and its neighbor-ing particle or wall; Ff,idenotes the total drag force for
particle i; Fg,i is the gravitational force of particle i and
Ai j is the torque between particles i and j.
2.2. Fluid dynamics algorithm
The overall fluid-dynamics solution algorithm of the SIMMER-III code is based on a time factorization time-splitting approach. This is a four-step algorithm devel-oped for the advanced fluid dynamics model [10]. In step 1 of this algorithm, intra-cell transfers are solved with-out considering the convection terms. In step 2, the end-of-time-step variables are explicitly estimated to initial-ize the pressure iteration. In step 3, the pressure iteration is conducted to obtain consistent velocity and pressure using a multivariate Newton–Raphson method. The
Journal of Nuclear Science and Technology, Volume 53, No. 2, February 2016 273 iterative calculations in this step are strictly controlled
to reduce the residuals of selected sensitive variables to zero. Finally, in step 4, consistent mass and momen-tum convections are computed based on a semi-implicit method.
2.3. Methodology of DEM
Under the assumption of DEM, particles in 2D sys-tems are assumed circular. The contact forces between the particles, as well as between particles and the wall, are calculated by applying a viscoelastic contact model [11]. For a particle i, the contact force Fc,iis divided into
the normal and tangential components
Fc,i = jFc,i j = jFc,i j,nor+ jFc,i j,tan (7)
where the subscript j denotes the neighboring particles of particle i.
The normal component of the contact force Fc,i j,nor
between particles i and j is described by Hooke’s contact law as follows: Fc,i j,nor= 43 EiEj Ei 1− ν2 i + Ej 1− ν2 j Ri jhi jni j − γnormi jvi j,nor (8)
where mi j =mmii+mmjj and Ri j = RRji+RRjj present the
re-duced mass and radius, respectively, of particles i and
j; E is the Young’s modulus; ν is Poisson’s ratio; hi j =
Ri+ Rj− |ri j| is the overlap length; ri j is the vector of
a relative position; ni j is the unit vector normal to the
contact surface with particle j directed towards particle
i; vi j,noris the normal component of the relative contact velocity between particles i and j, and γnoris the viscous damping coefficient in the normal direction. It should be noted that the present DEM model can flexibly simulate particles with different sizes as the reduced radius is de-fined directly by the radii of two interacted particles in Equation (8). However, the numerical tests in this study are limited to the particles with same sizes.
The tangential force Fc,i j,tanis qualified by
separat-ing the static and the dynamic friction forces as:
Fc,i j,tan= ti jminFc,i j,static,Fc,i j,dynamic (9)
where ti j is the unit tangential vector.
The static friction force identifies friction behavior prior to gross sliding and can be calculated as the sum of shear and viscous damping components
Fc,i j,static =163 · GiGj Ri jhi j Gi(2− νi)+ Gj 2− νj · δi j − γtanmi jvi j,tan (10) where δi j = ui j,tan(t) dt (11) is the integrated tangential displacement vector between particles i and j, G is the shear modulus, vi j,tan is the
relative tangential velocity between particles i and j, and
γtan is the viscous damping coefficient in the tangential direction.
The dynamic friction force describes the friction af-ter gross sliding, and is expressed by Coulomb’s law of friction,
Fc,i j,dynamic= −ς · ti jFc,i j,nor (12)
where ς is the friction coefficient.
The torque of particle i can be calculated by
Ai j = N
j=1, j=i
l× Fc,i j (13)
where l is a vector specifying the position of the con-tact point with respect to the center of particle i, and N denotes the total number of neighboring particles.
2.4. Coupling algorithm
After intra-cell transfers have been calculated (step 1), the governing equations for particles are solved to predict their positions and velocities. For the drag force for fluid phases, it assumes uniform over a computational cell, that is,
Ff,i = β
1− αf
(vf − ¯vp)· Vp (14)
where Vp indicates the volume of one particle and ¯vp
represents the particle velocity averaged over a cell. The interface momentum transfer coefficient β is calculated by the Gidaspow drag model [5] as
β = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 150(1− αf) 2 αf μf d2 p +7 4(1− αf) ρf dp |vf− ¯vp| if αf ≤ 0.8 3 4Cd (1− αf)αf dp ρf|vf− ¯vp|α−2.65f ifαf > 0.8 (15) where μf, dp and Cd are the viscosity coefficient of
fluid, particle diameter and drag coefficient, respectively. Here, Cdis given by Cd = 24 Rep 1+ 0.15Re0.687 p if Rep≤ 1000 0.44 if Rep > 1000 (16)
where Repis the particle Reynolds number. In the
Figure 1. Schematic description of the multi-time-step algo-rithm in the SIMMER–DEM calculation.
particle and fluid phases is defined consistently with Ff,i as
Sd = −β(vf − ¯vp) (17)
Particle acceleration can be calculated according to these forces. Since the time-step sizes in DEM are usu-ally smaller than those in SIMMER for fluid-dynamics calculations, a multi-time-step scheme is used. The cou-pling algorithm is described schematically inFigure 1. The velocity v∗i and temporary position ri∗of particle i driven by the drag forces Ff,i are updated as
v∗i = vni + 1 mi
Ff,i tSIM (18)
r∗i = rni + v∗i tSIM (19) where tSIM is the time-step size used in SIMMER fluid-dynamics calculations. These values are treated as the initial DEM values to calculate the contact force
Ff,i.
In the DEM calculation, particle velocities and po-sitions are updated using an explicit scheme,
vki+1 = vki + 1 mi Fg,i + Fc,i tDEM (20) rki+1 = rki + vki+1 tDEM (21) ωk+1 i = ω k i + 1 Ii jAi j tDEM (22) θk+1 i = θki + ωki+1 tDEM (23) where k and tDEMare the step number and the time-step size of the DEM calculation, respectively. Finally,
after (k+ 1) tDEMreaches tSIM, the particle positions and velocities are updated as
rni+1= rki+1 (24)
vni+1= vik+1 (25) As the particle phase is treated as one of the liq-uid components under the continuum assumption in the original multi-fluid model, the particle phase is tempo-rally defined as a static incompressible component in steps 2–4. This treatment excludes the particle phase’s influence on the convection and pressure calculations in the present algorithm. At the end of step 4, the volume fraction of the particle phase is updated to the end-of-time-step values in every cell according to the DEM re-sult as αn+1 p = cell,iVpn,i+1 Vcell (26) where Vcell is the volume of a computational cell in SIMMER-III.
2.5. Control of time-step size
Time-step size is a very important element for accu-racy and stability of calculations. In addition, it should be considered in terms of the implied computing cost. In the SIMMER-III code [1], time-step size is mainly opti-mized based on the Courant condition and the number of pressure iterations in step 3.
For the DEM calculation, two main conditions limit the time-step size [6]. First, the interaction between a particle and the surrounding fluid cannot influence the particle’s immediate neighboring particles during a time step. Second, in each time step, only one collision pro-cess can occur between two neighboring particles. Usu-ally, the time-step size given by the second constraint is much smaller.
2.6. Parallelization algorithm
When a nuclear accident occurs in a real reactor, the formed debris bed usually contains millions of particles. To directly simulate all particles’ movements in such a situation, the calculation time might be years. There-fore, some high-speed parallelization algorithms, such as message passing interface (MPI) or graphics processing unit (GPU), will be introduced to solve the calculation time problem in the future work.
3. Numerical simulation
As a validation procedure, the developed method was applied to two typical gas–liquid–particle mixture
Journal of Nuclear Science and Technology, Volume 53, No. 2, February 2016 275
Figure 2. Schematic view of the experimental apparatus of the dam-break experiment.
flows, which are a dam break with solid particles [12] and a three-phase fluidized bed [13]. The simulation re-sults were compared with corresponding experimental data available in the literature.
3.1. Dam break experiment
3.1.1. Experimental conditions
The schematic view of the experimental apparatus is illustrated inFigure 2. In the experimental runs, a rect-angular water tank made from transparent acrylic resin was used as the main test section. Its width, height and depth were 0.26, 0.26 and 0.1 m, respectively. A dam board was hung 0.064 m away from the left tank wall, which held the mixture of water and solid particles in a stagnation state at the beginning of the experiment. When experiments began, the dam board was pulled out from the water tank with a vertical velocity of about 3 m/s by an external force system. The mixture started to move due to gravity. Since the contact time between the motive board and the mixture was very short, the influence of the contact was neglected in the simulation. After the board was pulled out, a high-speed camera was used for recording the movement of the solid particle– water mixture. In the selected experimental runs, the wa-ter height was commonly set to 0.128 m, with a different particle bed height each time. The solid particles used in the experiment were made of plastic, whose physical properties are listed inTable 1.
Table 1. Physical properties of solid particles used in the dam-break simulations.
Particle type Plastic balls
Density (kg/m3) 1010
Diameter (m) 0.006
Elasticity modulus (Pa) 1.1 × 109
Poisson’s ratio 0.39
Friction coefficient 0.3
Figure 3. The computational domain of the dam-break sim-ulation (color bar: volume fraction of water).
3.1.2. Simulation conditions
Three experimental cases with different heights of the particle bed, 0.08, 0.112 and 0.128 m, were simu-lated in the present study. The corresponding numbers of solid particles consisting of the particle bed for the three cases were 130, 180 and 210, respectively. The rectangu-lar tank with the water–particle mixture bed was mod-eled by a 2D computational system. The initial compu-tational domain in the case of the particle bed height of 0.08 m is shown inFigure 3. The initial positions of the particles were distributed regularly in the bed.
It should be noted that the depth or layers in the deep direction of the particle bed in the simulations under the present 2D assumption (one layer) were not consistent with those in the experiments (about 17 lay-ers). As observed by Li et al. [14], for small-scale three-dimensional (3D) rectangular particle beds with a spher-ical shape, a 2D flow can develop when the bed thickness is smaller than 20 times of a particle diameter. There-fore, the present difference in the bed depth might bring a negligibly small influence on the dam-break behavior. Regarding the mesh configuration, 17 mesh cells with a width of 0.016 m were used both in the horizontal and in the vertical direction. The maximum and min-imum time-step size for the SIMMER fluid dynamics calculation were 10−4 and 10−6 s, respectively. For the time-step used in DEM, a constant value of 10−6 s was selected based on the computational experience.
3.1.3. Simulation results
As indicated by Liu [12], it was difficult to directly compare the exact vertical height of the dam between images recorded in the experiments and snapshots ob-tained by the simulation. However, it seems to be enough to discuss the general mixture movement. Figure 4
presents the snapshots of simulation results compared with the corresponding experimental images at 0.1, 0.3
Figure 4. Transient comparison of the mixture flow.
and 1.0 s in the case of the initial bed height of 0.112 m. It can be seen that the present hybrid simulation roughly predicts the evolving movement of the mixture flow, including the collapse and collision behaviors of the mix-ture as well as the water sloshing motion. As depicted in
Table 2, the moments when the front head of the mixture arrives at the right tank wall are compared between ex-perimental and simulation cases and reasonable agree-ments are obtained. Moreover, the simulation reason-ably reproduces the movement tendency of the mixture observed in the experiments that the front head of the mixture takes more time to arrive at the right wall with the increase in the initial particle bed height.
As defined schematically inFigure 5, the distance be-tween the left tank wall and the front head of the mix-ture, XD, was used as a parameter to compare the flow movement.Figures 6–8show the variations of XD ver-sus time in three cases. It can be seen that the simula-tion results are in quantitative agreement with those ob-tained in the experiments. These results demonstrate the
Table 2. Comparison of the time for arriving at the right tank wall.
Case 1 2 3
The height of particle bed (m) 0.08 0.112 0.128
Experiments (s) 0.21 0.22 0.25
Simulations (s) 0.21 0.23 0.25
Figure 5. Distance between the left wall and the flow front edge, XD.
Figure 6. Comparison of XD with the initial height of the particle bed (0.08 m).
Figure 7. Comparison of XD with the initial height of the particle bed (0.112 m).
Journal of Nuclear Science and Technology, Volume 53, No. 2, February 2016 277
Figure 8. Comparison of XD with the initial height of the particle bed (0.128 m).
Figure 9. Sensitivity comparisons of XD with different mesh sizes.
fundamental validity of the present method for the sim-ulation of gas–liquid–particle mixture flows.
3.1.4. Effect on mesh size
Since spatial resolution is usually one of the impor-tant factors to greatly influence the computational ac-curacy, a sensitivity analysis was performed with three different mesh sizes to verify its impact. In the simula-tion cases, the diameter and overall height of the parti-cles were set up to 0.001 and 0.012 m, respectively. The widths of the cells were 0.016, 0.012 and 0.008 m, respec-tively. The corresponding numbers of cells were 306, 484 and 1224. Other parameters were same as described in
Section 3.1.2. The values of XD with different mesh sizes in the simulation cases are compared inFigure 9. It can be seen from the results that minimal variation of XD is obtained by changing the mesh size, which means the effect on mesh size is very small for the present study.
Table 3. Physical properties of solid particles used in the flu-idized bed simulations.
Particle type Glass beads
Density (kg/m3) 2475
Diameter (m) 0.003
Elasticity modulus (Pa) 1.0 × 1010
Poisson’s ratio 0.25
Friction coefficient 0.3
3.2. The fluidized bed experiment
3.2.1. Experimental conditions
In the fluidized bed experiment [13], three-phase flu-idization was observed in a cylindrical Plexiglas column of diameter 0.1 m and height 1.5 m. The air and water were charged concurrently and uniformly from the col-umn bottom. The solid particles used in the experiment were glass beads, whose physical parameters are listed in
Table 3. The movements of the particles were measured by a particle tracking system.
3.2.2. Simulation conditions
In the present simulation, a Cartesian 2D system was used to model the cylindrical column, as shown in
Figure 10, where the color bar represents the volume fraction of water. As verified by Panneerselvam et al. [15], the complex three-phase fluidized bed phenomenon in 3D column can be reasonably reflected in light of that in a 2D system. They studied the mixture flow with dif-ferent gas and liquid superficial velocities, and found that same regimes developed in both 2D and 3D sim-ulations. Referring to the similarity between the present study and [15], the 2D assumption in the simulation can be accepted to compare with the corresponding experi-mental results. At the beginning of the calculation, only glass particles were distributed regularly so as to form the bed with an initial height of 0.35 m in the column, which was filled with air. For the SIMMER mesh, 11 mesh cells with a width of 0.009 m and 125 mesh cells with a width of 0.012 m were used in the horizontal and vertical direction, respectively. Referring to the injected flow rates of air and water in the experiment, the su-perficial water velocity was set constantly to 0.006 m/s, whereas the superficial air velocity was 0.012 m/s at the bottom boundary cells. The time-step sizes used in the SIMMER and DEM calculations were same as those used in the simulations presented inSection 3.1.2.
3.2.3. Simulation results
As pointed out by Chen et al. [16], the dynamic solid movement in three-phase fluidized beds showed a tran-sition or spiral flow for the present gas and liquid super-ficial velocities’ region. The typical mean velocity vec-tors of solid particles in the simulation at various time intervals are shown inFigure 11. It can be seen from the
Figure 10. The computational domain of the fluidized bed (color bar: volume fraction of water).
results that the spiral flow of particles were reproduced in the simulation.
The mean particle velocities in the vertical and hori-zontal directions were compared quantitatively with the experimental data [13]. Referring to the experimental setting,Figures 12and13depict these variables at a bed height of 0.2 m, where r indicates the radial distance from the particle position to the column center, and R is the column diameter. As shown inFigure 12, the sim-ulation results show reasonable agreements on velocity magnitude and variation tendency, where the flow re-versal occurs at r/R≈0.70. InFigure 13, the simulation results and experimental data both exhibit the approxi-mate zero velocity. It can be seen from these figures that the behaviors of the particle bed, which are represented by the mean particle velocities, are also reproduced rea-sonably well.
Figure 11. Snapshots of positions and velocity vectors of the particle bed (color bar: volume fraction of water).
Figure 12. Comparison of particle mean velocity in the ver-tical direction.
Figure 13. Comparison of particle mean velocity in the hor-izontal direction.
Journal of Nuclear Science and Technology, Volume 53, No. 2, February 2016 279
4. Concluding remarks
An Eulerian–Lagrangian coupling method, based on the multi-fluid model of the SIMMER-III code and DEM, was developed to reasonably simulate tran-sient behavior of the particle phase, as well as thermal– hydraulic phenomena occurring during a CDA. As a validation procedure, the method is applied to the sim-ulations of gas–liquid–particle three-phase flows. Two typical systems including a dam-break experiment and a three-phase fluidized bed are selected as benchmark cases. In the simulations of the dam-break experiment, the results reasonably predict the evolving movement of the mixture flow, including the collapse behaviors of the mixture as well as the water sloshing motion. Quanti-tative agreements are obtained on comparing the dis-tance between the front head of the mixture and the left tank wall. In the simulations of the fluidized bed, the flow pattern and mean velocities of the particle bed are reasonably reproduced. Through reasonable agreements between the simulation results and the corresponding experimental data available in the literature, the validity of the present method is demonstrated.
The hybrid method will be finally applied in the anal-ysis of a CDA, which should consider the phase change and heat transfer of fuel and coolant components as well as the reactor scale calculation. Therefore, the heat and mass transfer calculation will be introduced straight-forwardly with the related models in the SIMMER-III code in the succeeding study. In addition, supercomput-ing algorithms, such as MPI or GPU, will be introduced to solve the calculation time problem caused by large amounts of particles.
Acknowledgments
The present study is the result of the “Development of Evaluation Methodology for Core-Material Relocation in Core Disruptive Accidents” entrusted to the Japan Atomic En-ergy Agency by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). The computation was mainly carried out using computer facilities at the Re-search Institute for Information Technology, Kyushu Univer-sity, Japan.
Disclosure statement
No potential conflict of interest was reported by the authors.
Nomenclature
A: torque (N·m)
Cd: model coefficient of drag force
dp: particle diameter (m)
E: Young’s modulus (Pa)
Fc: contact force (N)
Ff: drag force (N)
Fg: gravitation force (N)
g: gravitational acceleration (m/s2)
G: shear modulus (Pa)
h: overlap length (m)
H: bed height (m)
I: moment of inertia (kg·m2)
l: vector specifying the position of the contact point of particle i (m)
m: mass (kg)
n: normal unit vector
p: pressure (Pa)
r: position vector (m)
R: radius (m)
Rep: particle Reynolds number
Sd: momentum exchange term (kg·m/s)
t: tangential unit vector
t: time (s)
v: velocity (m/s)
V: volume (m3)
Greek symbols
α: volume fraction β: drag force coefficient γ : viscous damping coefficient
δi j: integrated tangential displacement vector
ς: friction coefficient θ: angular displacement μ: viscosity coefficient ν: Poisson’s ratio ρ: density (kg ·m3) ˆ
τf: viscous stress tensor
ω: angular velocity (s−1)
Superscripts and subscripts f : fluid
i, j: particle identity k: DEM step number nor: normal direction
p: particle
tan: tangential direction
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