A modified direct method for void fraction calculation ...

Advanced Powder Technology xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

Original Research Paper

A modified direct method for void fraction calculation in CFD–DEM simulations Zhengbiao Peng, Behdad Moghtaderi, Elham Doroodchi ⇑ Discipline of Chemical Engineering, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia

a r t i c l e

i n f o

Article history: Received 5 May 2015 Received in revised form 23 September 2015 Accepted 30 October 2015 Available online xxxx Keywords: CFD–DEM simulation Cell void fraction Particle meshing method Computational efficiency Particulate flow

a b s t r a c t The void fraction of computational cells in numerical simulations of particulate flows using computational fluid dynamics–discrete element method (CFD–DEM) is often directly (or crudely) calculated assuming that the entire body of a particle lies in the cell at which the particle centroid resides. This direct method is most inexpensive but inaccurate and may lead to simulation instabilities. In this study, a modified version of the direct method has been proposed. In this method, referred to as the particle meshing method (PMM), the particle is meshed and the solid volume in a fluid cell is calculated by adding up the particle mesh volume with the basic working principle being the same as that of the direct method. As a result, the PMM inherits the simplicity and hence the computational advantage from the direct method, whilst allowing for duplicating the particle shape and accurate accounting of particle volume in each fluid cell. The numerical simulation characteristics of PMM including numerical stability, minimum particle grid number, prediction accuracy, and computational efficiency have been examined. The results showed that for a specific cell-to-particle size ratio, there was a minimum particle grid number required to reach the stable simulation. A formula of estimating the minimum particle grid number was derived and discussed. Typically, a particle grid number of about 5 times the minimum number was suggested to achieve the best computational efficiency, which was comparable or even higher than that of simulations using the analytical approach. PMM also exhibited the potential to be applied for complex computational domain geometries and irregular shaped particles. Ó 2015 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

1. Introduction Computational fluid dynamics–discrete element method (CFD–DEM) simulation of particulate flows prevailing in industrial processes has become increasingly popular in the past few decades [see e.g., 1,2–5]. Generally, the CFD–DEM method has been proved to be effective in capturing the majority of macro- and micro-scale characteristics of the fluid–solid two-phase flow, whilst providing insight into the underlying science specifically at the particle scale. The quality of prediction results of the CFD–DEM simulation however is found to be strongly relying on the accurate account of the presence of particles in the two-phase flow. The presence of solid particles in CFD–DEM simulations of the two-phase flow is considered by incorporating the volume fraction of the solid phase in computational fluid cells into the governing ⇑ Corresponding author at: Priority Research Centre for Advanced Particle Processing & Transport, The University of Newcastle, Australia. Tel.: +61 2 4033 9066; fax: +61 2 4033 9095. E-mail address: [email protected] (E. Doroodchi).

equations. The solid volume fraction is commonly calculated by a direct or crude method assuming that the entire body of the particle lies in the fluid cell at which the particle centroid resides [1,2]. This direct method is computationally inexpensive as it only involves searching of the particle centroid host cell. However, the direct method may lead to large errors when the particle centroid is near the fluid cell boundaries. Such large errors might yield a significant fluctuation in the value of cell void fraction when the particle centroid moves in and out of the fluid cell leading to simulation instabilities and unrealistic particulate flow behaviours. Attempts have been made towards the accurate calculation of cell void fraction. Chen et al. [6] analytically calculated the void fraction in a one dimensional (1D) CFD–DEM simulation to solve classical soil mechanics problems. The 1D implementation of the analytical approach is quite straightforward. Li [7] analytically calculated the void fraction on two-dimensional (2D) structured meshes in the CFD–DEM simulation of flow structure formation and evolution in dense gas–solid flows. Freireich et al. [8] and Peng et al. [5] detailed the three-dimensional (3D) analytical solution of void fraction in structured rectangular cells. Wu et al. [9] derived a

http://dx.doi.org/10.1016/j.apt.2015.10.021 0921-8831/Ó 2015 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

Please cite this article in press as: Z. Peng et al., A modified direct method for void fraction calculation in CFD–DEM simulations, Advanced Powder Technology (2015), http://dx.doi.org/10.1016/j.apt.2015.10.021

2

Z. Peng et al. / Advanced Powder Technology xxx (2015) xxx–xxx

Nomenclature A A d fc ff fd fsg g I l0, l1, l2 m Np Nc,b p r Sc T Usf u V Ve v x

factor (–) face area factor diameter (m) collision contact forces acting on the particle (N) total fluid forces acting on the particle (N) drag force (N) local mean particle-fluid interaction forces (N) acceleration of gravity (m/s2) particle inertia (kg m2) domain dimension (m) particle mass (kg) particle number inside a cell total number of computational cells that cover the fluidised bed of particles pressure (Pa) radius (m) equivalent cell size (m) torque acting on the particle (N m) gas superficial velocity (m/s) fluid velocity vector (m/s) volume (m3) volume of particle grid (m3) particle velocity vector (m/s) coordinates of a point (m)

Greek symbols a fraction of particle volume divided by a fluid cell (–) b momentum exchanging coefficient (–) c scaling factor (–)

set of equations to analytically solve the cell void fraction on both 2D and 3D unstructured meshes. The analytical solution is computationally very expensive as evaluations of trigonometric functions need to be conducted at every time step [10]. Various non-analytical approaches have also been reported in the literature for the calculation of cell void fraction. For example, Link et al. [11] and Khawaja et al. [12] represented a particle as a porous cube. The size of the cube depended on the particle diameter and a constant scaling factor. Using the fictitious porous cube, the presence of particles was weakly felt by the fluid flow. As such, grid refinement did not lead to local extremes in the void fraction field, and hence the solution was grid independent. Lim et al. [13] calculated the void fraction for a fluid cell grouped together with its surrounding cells via the direct method. This method added stability by calculating the cell void fraction for a larger virtual cell that comprised several real fluid cells. However, this technique also added greater spatial smoothing to the averaging procedure since quantities such as velocity and pressure were calculated for the real fluid cells, but void fraction was calculated on a larger volume scale. Moreover, this method may lead to large errors if the local cell void fraction is vastly different from that of the larger virtual volume, e.g., in bubbly fluidised beds. Following the approach used in the local averaging of granular materials, Kuang et al. [14] proposed a more general method in this regard using virtual spherical cells containing local points of interest. The porosity and the source terms due to particle–fluid interactions were calculated for the virtual spherical cells and then mapped into CFD cells. Sun et al. [15] and Xiao and Sun [16] assumed that each computational cell accepts contributions from all of the particles in the system. Based on the assumption, the local cell void fraction was calculated through a statistically averaging approach using a weighting function that is similar to the estimation of probability density function

d

e Dt k

l q s w K u DVc

v H

x

#

contact overlapping (m) local void fraction time step (s) size magnification factor from template particle to real particle (–) shear viscosity (kg/(m s)) density (kg/m3) viscous stress tensor normalised net mass flow rate mass flow rate through cell faces (kg/s) cell-to-particle size ratio computational cell volume (m3) operator sign (–) granular temperature (m2/s2) particle angular velocity (rad/s) average deviation in the calculation of cell void fraction by PMM

General subscripts d drag f face g gas phase i, j, k particle index min minimum max maximum p particle phase pg particle grid r relative x, y, z direction

from discrete points. Inevitable statistical error exists in this approach and strongly depends on the form of the weighting functions and the parameters (e.g. bandwidth). Moreover, the implementation of this approach has a computational complexity of the order of O(NpNc), as all particles and cells need to be visited and examined at each time step. Gui et al. [17], Hilton et al. [18] and Hobbs [19] used regular squares or cubes to subdivide the particles. Apparently, the regular squares or cubes cannot provide a smooth representation of particle boundaries and thus unavoidably introduces error. Hilton et al. [18] minimised the error by linearly approximating the particle boundary, but the method is limited to spherical or regular non-spherical particles (e.g. ellipsoids and cuboids) and simple geometries (only in which the recursive approach and the linear approximation can be implemented). Boyce et al. [10] proposed a square-grid method, in which the distribution of particle volume on a square grid was calculated and then mapped on to the computational fluid cells that can be of arbitrarily complex shape. The method has an obvious source of inaccuracy as it is very likely that a fraction of the particle volume can be registered as being in a fluid cell in which the particle is not located, considering the squared cell is larger and contains several fluid cells. As a whole, the simplistic nature of the current non-analytical approaches can compromise the accuracy of the calculation for the flexibility to cope with complex domain geometries. However, as repeatedly indicated in the literature [see e.g., 4,5], the accurate calculation of cell void fraction is a must in CFD–DEM simulations to ensure the numerical stability and the reliability of prediction results. Moreover, in most of practical problems the simulation needs to deal with complex geometries and/or irregularly shaped particles. The implementation of the analytical approach becomes extremely difficult and very computationally demanding.


3


Therefore, there is a need to seek for an effective alternative of the analytical approach for the accurate void fraction calculation in such simulations. In this study, a modified version of the conventional direct method for the calculation of cell void fraction in CFD–DEM simulations is proposed. The proposed method relies on the versatile meshing technique, by which the particles are meshed into a number of small particle grids, henceforth referred to as the particle meshing method (PMM). The solid volume in a fluid cell is calculated by counting the number of particle grids in the cell and adding up their volumes. Therefore, the PMM inherits the basic working principle, thus the numerical simplicity and computational advantage of the direct method. With meshing techniques, the PMM allows for duplicating the particle shape and accurate accounting of particle volume in each fluid cell specifically for particles at the fluid cell boundaries. The numerical simulation characteristics of PMM including numerical stability, minimum particle grid number, prediction accuracy, and computational efficiency have been examined. Furthermore, the potential of the application of PMM in scenarios where complex domain and particle geometries are involved has also been discussed.

the particles in the systems. A particle frame with the origin lying at the particle mass centroid was also introduced, as depicted in Fig. 2(a). The template particle has the same mesh structure with that meshed on the particles. The data of particle grids of the template particle, i.e., centroid coordinates and volume in the particle frame, were calculated only once prior to the CFD–DEM simulation and stored in arrays of x[Npg] and V[Npg]. During each time step of the simulation, the data stored in the arrays of x [Npg] and V[Npg] were simply mapped to those of the particle grids of each particle that was moving around in the inertial frame, as shown in Fig. 2(b). For a spherical particle, the centroid coordinates of particle grid i can be calculated by the translational transformation of coordinates from the particle frame to the inertial frame,

xe;i ¼ xp þ kðxe0;i x0 Þ

ð1Þ

where xe is the global centroid coordinates of particle grid i of a real particle in the inertial frame. k is the magnification factor and defined as k = rp/r0, where rp is the real particle size. For non-spherical particles, the rotational motion of the particle needs to be considered. Therefore, the centroid coordinates of particle grid i were calculated by,

xe;i ¼ xp þ RðtÞkðxe0;i x0 Þ 2. Models and methods 2.1. Governing equations of particulate flows The interactions between fluid–particle, particle–particle and particle–wall were fully considered and solved in the CFD–DEM model, developed previously by the authors [5,20]. The fluid flow was modelled by the local averaged continuity and momentum equations of the continuum proposed by Anderson and Jackson [21]. The motion of each particle was tracked individually based on Newton’s second law. Particle–particle contact mechanics were modelled using a linear spring model with velocity-dependent damping (i.e., the spring-dashpot interaction) and static friction [22–25]. The implementation of the coupling of interphase interactions between the fluid and the solid phases followed the pressure gradient force (PGF) model, as extensively described by Kafui et al. [3]. For brevity, the governing equations are listed in Table 1. 2.2. Particle meshing method for cell void fraction calculation As detailed above, approaches of cell void fraction calculation can be generally classified into three categories: (i) direct (or crude); (ii) non-analytical; (iii) analytical. The PMM, proposed in this work falls into the second category. In PMM, the particle is meshed into small particle grids that maintain the particle shape. Then each particle grid is affiliated with a cell based on the position of their centroids. The solid volume in a fluid cell is then calculated by counting the number of particle grids in the cell and adding up their volumes with the basic working principle being the same as that of the direct method. Existing advanced meshing techniques (e.g., commercial software including ANSYS Meshing and ICEM CFD [26]; open-source codes including DistMesh [27] and GMSH [28]) can be used to mesh the particles prior to the CFD–DEM simulations. The ANSYS Meshing was used in the present study to mesh the particles and the TGrid scheme was applied to obtain a high uniformity of particle grids [26]. Fig. 1 displays the meshing of a particle into 20, 52, 251, 547, and 1214 particle grids. Similar to the direct method, the global centroid coordinates and the volume of each particle grid need to be known a priori to calculate the total solid volume in a fluid cell. To save the computation time, a template particle was introduced with a radius of r0 and the same shape with that of

ð2Þ

where R(t) is the rotational operator, which were commonly expressed by three frameworks: Euler angles, rotation matrices, and unit quaternions. More details on the rotation dynamics of a rigid body and the unit quaternions can be readily found in the literature, e.g., Eberly [29] and Hoffmann [30]. The magnification factor k can be calculated as the ratio of any dimension of the real particle to that of the template particle. The volume of particle grid i was calculated by,

V e;i ¼ k3 V e0;i

ð3Þ

where Ve are the local volume of particle grid i of a real particle and Ve0 is the volume of particle grid i in the template particle. As the curved surface of a particle is meshed into a number of small grids with planar surfaces, the sum of the volume of particle grids is less than the volume of the real particle. For this reason, the volume of each particle grid is modified by multiplying a scaling factor of,

PNpg

c¼

i¼0 V e;i

ð4Þ

Vp

Similar to the direct method, the computational expense using PMM is mainly associated with searching the host cell that contains the centroid of each particle grid. For structured Cartesian meshes, it is straightforward to find the host cell based on the particle grid centroid (i.e., xe) and the cell dimensions by,

2 3 2 xe;0 i 6 7 6 4 j 5 ¼ ðintÞ4 0 0 k

0 xe;1 0

32 1 3 l0 7 76 6 7 0 54 l1 1 5 1 xe;2 l 0

ð5Þ

2

where i, j, and k are the host cell index along x, y and z directions, respectively; l0, l1, and l2 are the cell dimensions along x, y and z directions, respectively. For unstructured meshes, we have developed an efficient searching algorithm in which a virtual structured mesh is introduced that encloses the computational domain. As shown in Fig. 3(a), a virtual structured mesh (coloured in blue1) is introduced to cover the entire computational unstructured mesh (coloured in 1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.


4


Table 1 Governing equations of the fluid–solid two phase flow. Solid phase: [5,20,25] Translational motion: mi ddtvi ¼ f c;i þ f f;i þ mi g xi ¼ Tc;i þ Tr;i Rotational motion: Ii ddt where f f;i ¼ ½V i rp þ V i ðr sf Þ þ ef d;i ;

f t ¼ kt dt gt ðv r tÞt;

n¼

xi xj

jxi xj j

;

f c;i ¼ f n;i þ f t;i ;

f n ¼ kn dn n gn ðv r nÞn,

v r;ij ¼ v i v j þ ðwi Li þ wj Lj Þ n,

v t ¼ v t;ij ; dt ¼ v t;ij min vdrnn ; Dt , j t;ij j dt ðt þ DtÞ ¼ dt ðt þ DtÞ ðdt ðt þ DtÞ nÞn, 8 t–0 < lf jf n jt If jf t j > lf jf n j; f t ¼ lf jf n j jff tt j t ¼ 0; jf t j–0 , : 0 otherwise P P x Tc;i ¼ j ðn f t ÞRi ; Tr;i ¼ j lr jf n j jxijij j r i

v t;ij ¼ v r;ij ðv r;ij nÞn; dt ðt þ DtÞ ¼ dt ðtÞ þ v t Dt;

Gas phase: [3,21] Continuity:

@ðeqg Þ @t

þ r ðeqg ug Þ ¼ 0 @ðeq u Þ

Momentum conservation: @tg g þ r ðeqg ug ug Þ ¼ rp þ r sf þ f sg þ eqg g h i where sf ¼ lf ðrug Þ þ ðrug Þ1 þ k 23 lf ðr ug ÞI Interphase interactions: [5,20] Fluid–particle interaction force: f sg ¼

PNp i¼1

ðaf f;i Þ=DV c

Vi where f f;i ¼ ½V i rp þ V i ðr sf Þ þ ef d;i ; f d;i ¼ ð1 eÞ bðug v i Þ 8 150ð1eÞ2 lg 1:75ð1eÞqg jug v i j > < þ ð e 6 0:8Þ 2 dp dp e b¼ , > : 0:75C d qg eð1eÞjug v i j e2:65 ðe > 0:8Þ dp ( 0:44 ðRep 6 1000Þ ; Rep ¼ qg dp ejug vi j C d ¼ 24 0:687 lg Þ Rep > 1000 Rep ð1 þ 0:15Rep

Fig. 1. A meshed particle with different numbers of particle grids: (a) 20; (b) 52; (c) 251; (d) 547; (e) 1214.

pink). To reduce the searching time, the cell size of the virtual structured mesh is selected to be comparable with that of the computational mesh. As the centroid coordinates of the cells on the virtual and computational meshes do not change throughout the simulation, the geometric relationship between the computational unstructured cells and the virtual structured cells can be established prior to the simulation. Specifically, each virtual structured cell is related to a number of computational unstructured cells if any node of the unstructured cell is lying within the structured cell. As depicted in Fig. 3(b), the structured cell (i, j, k) is related to 12 unstructured cells, labelled as (i, j, k, 0), (i, j, k, 1), . . ., and (i, j, k, 11). During every time step, the index of the virtual structured cell (i.e., i, j, k) that contains the centroid of a particle grid is calculated by Eq. (5). The search of the computational unstructured cell that contains the centroid of the particle grid is conducted only amongst the unstructured cells that are related to the virtual structured cell (i, j, k). To judge if an unstructured cell contains the centroid of particle grid, the following relationship must be satisfied,

vAf;i ðxe xf;i Þ > 0

ð6Þ

where Af,i and xf,i are the area vector and the centroid of face i of the unstructured cell; v is the operator sign, i.e., ‘‘+” if Af,i is pointing inside of the cell and ‘‘” otherwise, as depicted in Fig. 3(b). However, care needs to be taken when applying Eq. (6) to map particle grids into CFD cells that have non-planar faces, particularly for three dimensional complex geometries. The problem of non-

planar cell faces associated with the point locating has been detailed and addressed by Kuang et al. [31]. 2.3. Numerical strategy and methodologies The Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was used to solve the pressure–velocity coupling equations of the fluid flow, namely the continuity and momentum conservation equations. The quadratic upwind interpolation of convective kinematics (QUICK) scheme was employed for the spatial discretization of the convection term. The diffusion term was discretized by a central-differenced scheme that is always of second-order accuracy. The Green-Gauss Node Based method was employed to calculate the variable gradients for constructing values of a scalar at cell faces and also for computing secondary diffusion terms and velocity derivatives. The interpolation of pressure values at the cell faces was carried out using a standard pressure interpolation scheme. When calculating the fluid forces acting on the particles, the values of Eulerian variables at the particle position were needed which were calculated based on their spatial gradient distribution [5,32] in this work. Subsequently, the particle–fluid forces was calculated in a semi-implicit way by treating the particle velocity (vi) implicitly, as detailed by Peng et al. [5]. The experimental data of van Wachem et al. [33] was used to validate the prediction results. For the purpose of consistent


5


z

Table 2 Simulation conditions and parameters.

a xe0, i

y

Ve0, i

r0 x

x0

z

b

xe, i rp y

Ve, i

xp

x O Fig. 2. Acquisition of centroid coordinate and volume of particle grids of each particle in particle meshing method: (a) particle frame and template particle; (b) inertial frame and real particle.

comparisons, the geometry of the simulated domain and the particle (polystyrene) properties were kept exactly the same as those used in the experimental work of van Wachem et al. [33]. The detailed conditions and parameters used in the simulations are listed in Table 2. The values of restitution coefficient and the friction coefficient were taken from literature [33,34]. A value of 10,000 for the spring stiffness has been chosen for all computational cases in this study, corresponding to a solid time step of 5 106 s and a maximum particle normal overlap of 0.0013 rp (averaging over 10 s simulation) under the inlet velocity Usf = 0.9 m/s. The total torque was generated only by the tangential contact

Fluid phase Domain size (mm3) Gas density (kg/m3) Gas viscosity (Pa s) Gas superficial velocity (m/s) Fluid time step (s)

90 8 500 1.125 1.8e5 0.9 1 104

Solid phase Particle diameter (mm) Particle density (kg/m3) Particle mass (kg) Normal spring-stiffness (N/m) Restitution coefficient (–) Friction coefficient (–) Solid time step (s)

1.545 1150 0.039 104 0.9 0.3 5 106

force, and the contribution by the rolling resistance [35,36] was ignored. In each time step, the globally scaled residual of 104 has been set as the convergence criteria for solving gas phase equations. For all simulations, 10 s of physical time was completed and the data in the last 5 s was used for the statistical averaging calculation to avoid start-up transients. Multi-thread parallel computation (OpenMP) based on the shared memory was conducted to improve the computational efficiency. For a typical parallel simulation with 4000 meshes on two computational nodes (clock speed 2.66 GHz, Smart Cache 12 M and QPI speed 6.4GT/s), it expends around 58.5 h to complete the simulation of 10 s gas–solid flow.

3. Results and discussion 3.1. Numerical stability and minimum particle grid number Fig. 4 depicts the contours of void fraction distribution of an instantaneous solid distribution field (at t = 10 s for Usf = 0.9 m/s and Sc/dp = 2.38), calculated using PMM with different particles grid numbers and the analytical approach. A wide range of particle grid numbers from 1 (i.e., the direct method) to 4550 have been used in the PMM to calculate the cell void fraction. It can be seen that for the direct method (i.e., at Npg = 1, Fig. 4(b)), the resolved

a

b

2

1

3

11 xf,i

xe

10

4

(i, j, k)

12

Af,i

9

5

8 7

6

Fig. 3. Schematic of the efficient searching algorithm for the host cell that contains the centroid of a particle grid in PMM: (a) virtual structured mesh and computational unstructured mesh; (b) a structured virtual cell and its related multiple unstructured cells.


6

Z. Peng et al. / Advanced Powder Technology xxx (2015) xxx–xxx 0.15

0.15

a

0.15

b

0.15

c

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

0 0

0.45

0.15

0.9

0

0

0.45

0.15

e

0

0.9

0

0.45

0.15

f

0.9

0

d

0

g

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

0

0

0.45

0.15

0.9

0

0

0.45

0.15

i

0

0.9

0

0.45

0.15

j

0.45

0.15

0.9

0

h

0

0.45

0.15

k

0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

0.9

0.9

l

Void fraction, (-) 1 0.97 0.94 0.91 0.88 0.85 0.82 0.79 0.76 0.73 0.7 0.67 0.64 0.61 0.58 0.55 0.52 0.49 0.46 0.43 0.4

z x

0

0

0 0

0.45

0.9

0

0.45

0.9

0

0.45

0.9

0

0

0.45

0.9

Fig. 4. Contour of void fraction distribution calculated using PMM with different particle grid numbers and the analytical approach for the solid distribution at t = 10 s for Usf = 0.9 m/s and Sc/dp = 2.38: (a) instantaneous solid distribution; (b) Npg = 1 (direct method); (c) Npg = 10; (d) Npg = 12; (e) Npg = 20; (f) Npg = 33; (g) Npg = 52; (h) Npg = 251; (i) Npg = 547; (j) Npg = 1214; (k) Npg = 4550; (l) analytical solution.

void fraction distribution appears vastly different from the analytical solution (Fig. 4(l)). After the particle grid number increases to 10 (Fig. 4(c)), a significant improvement has been observed in the solution of the void fraction distribution. As the particle grid number keeps increasing, the void fraction distribution calculated using the non-analytical approach PMM becomes gradually closer to the solution using the analytical approach. In the case of Npg = 4550, the calculated void fraction distribution by PMM can be regarded as the same as the analytical solution. Table 3 quantitatively shows the deviation of the calculated cell void fraction using PMM with different particle grid numbers from those calculated using the analytical approach. The deviation is the average error in the calculation of cell void fraction by PMM over

all of the computational cells that cover the entire fluidised bed of particles, and was calculated by,

#¼

N c;b ei;PMM ei;analytical 1 X N c i¼1 ei;analytical

ð7Þ

The results demonstrate that as the particle grid number increases, the deviation in the calculation of cell void fraction using PMM gradually decreases. Specifically the deviation is 7.8% for the direct method with Npg = 1, and decreases to values below 1% when Npg P 52. Indubitably, it is impossible for a non-analytical approach including PMM to achieve exactly the same solution of cell void


7

Z. Peng et al. / Advanced Powder Technology xxx (2015) xxx–xxx Table 3 Deviation of the calculated cell void fraction using PMM with different particle grid numbers from those calculated using the analytical approach. Npg (–) Deviation, # (%)

1 7.73

10 2.84

12 2.79

20 1.46

33 1.3

52 0.89

2 0 -2 -4 -6 -8

fraction as that using the analytical approach. However, when the particle grid number of PMM is greater than a certain value and the cell-to-particle grid volume ratio is large enough, the error in the calculation of solid volume contained in the cell becomes negligible [5]. As a result, the numerical stability of the solution and the high accuracy of prediction results could be achieved, as to be discussed extensively in the following sections. The normalised net mass flow rate of the gas flow was monitored during the simulation as a criterion of solution convergence and a measure satisfying the mass conservation law to reflect the numerical stability of the solution. The normalised net mass flow rate is defined as the mass flow rate difference between inlet and outlet, normalised by the inlet mass flow rate, expressed as,

X

X Kf;out = Kf;in

ð8Þ

where Ʌf = qgugAfef is the mass flow rate through the cell face. Subscripts ‘‘in” and ‘‘out” denote the inlet and outlet of the computational domain, respectively. ef is the face void fraction, interpolated from the values of cell void fraction. Fig. 5 shows the time series of the normalised net mass flow rate in simulations using different particle grid number (Npg) for a cell-to-particle size ratio of Sc/dp = 2.38. The value of Npg was increased gradually from 1, for which the single particle grid represents the real particle and the PMM actually is the direct method. The process of the simulation showed that the solution diverged with Npg 6 12, where the net mass flow rate reached over 4% of the inlet mass flow rate. It simply suggests that for Npg 6 12 the solution does not even satisfy the mass conservation law. This is due to the error in the calculation of solid volume in fluid cells. The error is related to the particle grid volume, which is too large due to the insufficient particle grid number and the small cell size. The inaccurate cell void fraction leads to the unrealistic solution of the fluid flow, which gives rise to the imbalanced flow flux through the faces of fluid cells. As the particle grid number was increased further and Npg P 20, the net mass flow rate immediately decreased to a value less than 0.0005% of the inlet mass flow rate, implying the mass conservation law was strictly satisfied in the simulation of the gas–solid twophase flow. The minimum cell void fraction (emin) was also monitored throughout the simulation process for the scrutiny of the numerical stability of the solution. Fig. 6 demonstrates the variation of minimum cell void fraction as a function of particle grid number for two different fluid cell sizes, i.e., Sc/dp = 2.38 and Sc/dp = 2.9. It can be seen that for Sc/dp = 2.38, emin is unrealistically low with a value even under 0.2. The unrealistic local void fraction explains the imbalanced mass flow and the divergence of the solution shown in Fig. 6. After Npg increases over 12, the value of emin becomes reasonable, i.e., above 0.3 and the solution becomes stable afterwards. Similarly, for Sc/dp = 2.9, the value of emin becomes reasonable when Npg P 10. The results suggest that a larger fluid cell size requires a smaller number of particle grids to achieve a stable solution. In other words, there is a minimum particle grid number (Npg,min) required for the stable solution and the value of Npg,min strongly depends on the cell size used for the simulation. For simplicity, we assume a particle is uniformly meshed into Npg particle grids. Therefore, each particle grid has a volume of Vpg = Vp/Npg and thus an equivalent size of,

(%)

Kf;in þ

m

X

1 0 -1 -2 -3 -4 -5

547 0.2

1214 0.12

4550 0.05

a 0

0.002

0.004

0.006

0.008

0.01

b 0

0.01

0.02

0.03

0.04

0.05

0.06

c 0

0.01

0.02

0.03

0.04

0.05

5E-4

d

0E+0 -5E-4 5

6

7

8

9

10

5E-4

e

0E+0 -5E-4 5

6

7

8

9

10

Time (s) Fig. 5. Normalised net mass flow rate of the system with different particle grid number: (a) Npg = 1 (direct method); (b) Npg = 10; (c) Npg = 12; (d) Npg = 20; (e) Npg = 33 at Usf = 0.9 m/s for Sc/dp = 2.38.

0.4 0.35 0.3 0.25 min (-)

wm ¼

8 6 4 2 0 -2

251 0.31

Sc/dp (-):

0.2

2.38

0.15

2.9

0.1 0.05 0 0

5

10

15

20

25

30

35

Npg (-) Fig. 6. Minimum cell void fraction over the computational domain using different particle grid number at Usf = 0.9 m/s for Sc/dp = 2.38 and Sc/dp = 2.9.


8

dpg


rffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi 1 3 6V pg 3 6 Vp 3 ¼ dp ¼ ¼ Npg p p Npg

ð9Þ

Therefore, the ratio of the inaccurately calculated particle grid volume to the cell volume becomes much smaller than that in cases using the direct method, in which the volume of the particles are considered. As a convention, we use the term ‘‘solid individuals” to denote the particles in the direct method and the particle grids in the PMM. It was found by Peng et al. [5] that when applying the direct method the cell size needs to 3.82 times larger than the particle diameter (i.e., Sc/dp P 3.82) so as to render the error generated in the calculation of particle volume negligible compared to the cell volume. Considering the PMM is in principle the same as the direct method, the requirement of cell-to-solid individual size ratio should be applicable in PMM. However, it should be noted that the error in the calculation of solid volume in a fluid cell is only related to the error generated in summing up the volume of particle grids that are divided by cell boundaries. Also, as indicated by Eq. (4), there is a discrepancy between the sum of particle grid volume and the particle volume. For this reason, a factor A is introduced to account for the above inconsistencies. As such, the impact of the inaccurately calculated particle grid volume becomes negligible only when,

Sc P 3:82A dpg

ð10Þ

Substituting Eqs. (9) into (10) and after rearrangement, Eq. (10) gives,

Npg P Npg;min ¼

3 3:82A Sc =dp

ð11Þ

where Npg,min is the theoretical minimum particle grid number for the application of PMM. Eq. (11) indicates, for a specific system with the cell size and the particle size known, the PMM with a particle grid number of Npg > Npg,min satisfies the requirements of the numerical stability of the solution. For the two cases shown in Fig. 6, the factor A equals to 1.43 and 1.49 for Sc/dp = 2.38 and Sc/ dp = 2.9, respectively. The value of A might vary slightly in other simulations with a different cell-to-particle size ratio. However, a value around 1.5 is suggested to start with and a gradually increased value can be used if the simulation is not stable. 3.2. Dependence of simulation results on Npg As indicated above, if Npg > Npg,min, simulations using PMM is stable. However, it is unclear that whether or not a larger Npg improves the quality of prediction results or is there a need to use an even larger Npg. To this end, a wide range of particle grid number (Npg = 12–1214) above Npg,min have been used in the simulation with Sc/dp = 2.9 to investigate the effect of particle grid number on the simulation outputs. As the fluid dynamics and bubble characteristics in a gas–solid fluidised bed is typically characterised by the pressure signals of the bed [37], the power spectral densities (PSD) of local pressure fluctuation at 45 mm above the distributor have been processed by Fast Fourier Transform (FFT) analysis. Fig. 7 demonstrates the PSD of the pressure fluctuation resulting from different values of Npg. It can be seen that for all Npg, there is only one protruding dominant frequency in the pressure fluctuation. The dominant frequency keeps constant at 3.13 Hz over the entire range of Npg investigated in this study. It suggests that the statistical fluid dynamics and bubble characteristics of the gas–solid flow modelled using the PMM remained the same when different values of Npg were used.

In addition to the PSD of local pressure fluctuation, simulation results in terms of time-averaged maximum granular temperature (Hmax,ave), time series fluctuation of local bed voidage (e0.045) and local pressure (p0.045) at 45 mm above the distributor, total pressure drop of the fluidised bed, time-averaged maximum overlap of particle–particle (p–p) and particle–wall (p–w) contacts, and time-averaged minimum cell void fraction, have also been compared and analysed when different values of Npg were used, as shown in Table 4. Essentially identical results have been observed when the particle grid number is altered. The value of particle–particle overlap (i.e., dmax,p–p) has the maximum variation of 7.82% of the average value (0.3% rp) when Npg is altered, which is very minor. The results indicate that the value of Npg has a marginal effect on the simulation outputs once Npg becomes greater than Npg,min. This is because beyond Npg,min the part of particle volume that is incorrectly calculated becomes negligible in comparison to the cell volume. A further increase in Npg thus only increases the computational cost with no improvement on the quality of prediction results. The computational efficiency of PMM was evaluated against that of the analytical approach. A range of particle grid numbers (i.e., Npg = 12–1214) have been used for Sc/dp = 2.9. The computational efficiency, denoted by g, is defined as the time consumed using PMM normalised by the time consumed using the analytical approach under the same simulation conditions and computer configuration. Fig. 8 shows the computational efficiency of PMM as a function of particle grid number (Npg). The horizontal dot line denotes the normalised computational efficiency using the analytical approach. It can be seen that for Npg 6 52, simulations using PMM have an even higher computational efficiency than those using the analytical approach. With a particle gird number up to 1214, the computational time using PMM increases only by 36.19% compared to that using the analytical approach. With the efficient algorithm developed in the present study to search the host cell of particle grids, the computational efficiency with PMM has been significantly improved. The advantage of PMM in the computational efficiency is supposed to be more pronounced on unstructured meshes, considering the complicity in the implementation of the analytical approach [38]. It should be also noted that with Npg = 12, the simulation has an unexpectedly lower efficiency by 7.94% than those using Npg = 20 and Npg = 33. Though a small particle grid number just above Npg,min can yield a stable solution, it may lead to difficulties in the solution convergence, which requires more steps of iterations in each fluid step. As a result, the simulation with a particle grid number near Npg,min inversely has a lower computational efficiency. The above result suggests that a particle grid number that is about 5 times greater than Npg,min is preferred in the application of PMM to achieve the best computational performance. 3.3. Prediction accuracy The accuracy of prediction results using the PMM with the required minimum particle grid number has been examined against the experimental data by van Wachem [33] and the prediction results using the analytical approach. The fluid cell size (i.e., Sc = 2.38 dp) was selected based on the findings of Peng et al. [5] to ensure the cell size is fine enough to resolve all major smallscale flow structures and also large enough to satisfy the assumptions used to derive the governing equations of the CFD–DEM model [21]. In gas fluidised beds, the gas–solid flow exhibits periodic features owing to the evolution of the bubbles, i.e., from formation at the bottom to the eruption at the top. Therefore, snapshots that show the complete evolution of the gas–solid flow in a period are demonstrated and compared in Fig. 9 at Usf = 0.9 m/s. The top row


9


900

1500

a

fp = 3.13 Hz Pressure PSD (-)

600

300

900 600 300

0

1

6

11

0

16

1

6

1500

fp = 3.13 Hz

600 300

d

fp(1) = 3.13 Hz

1500

Pressure PSD (-)

Pressure PSD (-)

900

1200 900 600 300

1

6

11

0

16

1

6

11

16

Frequency (1/s)

Frequency (1/s) 1800

1800

e

fp = 3.13 Hz (1)

1500

f

fp(1) = 3.13 Hz

1500

1200

Pressure PSD (-)

1200

900 600 300 0

16

1800

c

(1)

1200

0

11

Frequency (1/s)

Frequency (1/s)

Pressure PSD (-)

b

fp(1) = 3.13 Hz

1200

Pressure PSD (-)

(1)

1

6

11

900 600 300 0

16

Frequency (1/s)

1

6

11

16

Frequency (1/s)

1800

Pressure PSD (-)

g

fp(1) = 3.13 Hz

1500 1200 900 600 300 0

1

6

11

16

Frequency (1/s) Fig. 7. Influence of particle grid size (Npg) on the predicted local pressure fluctuation processed by FFT analysis for Sc/dp = 2.9 at Usf = 0.9 m/s using the PMM for cell void fraction calculation with a particle grid number of: (a) Npg = 12; (b) Npg = 20; (c) Npg = 33; (d) Npg = 52; (e) Npg = 251; (f) Npg = 547; (g) Npg = 1214.

(i.e., Fig. 9(a)) shows the snapshots taken from experiments [33]; the middle row (i.e., Fig. 9(b)) is the simulation results using the PMM and the bottom row (i.e., Fig. 9(c)) shows the simulation results using the analytical approach. It can be seen the simulation using the PMM produces very similar results to that using the analytical approach. Both the simulations using the PMM and the analytical approach reproduce the gas–solid bubbling flow behaviour captured in the experiments. Specifically, the bubble size, shape and evolution including bubble formation, growth, coalescence and eruption, agree very well with those observed in experiments.

Moreover, the overall particle fluidisation characteristics such as bed expansion and oscillation frequency are visually the same. The accuracy of prediction results using the PMM was also quantitatively examined by analysing the local pressure signal. Fig. 10 shows the time series data of local relative pressure at 45 mm above the distributor obtained in the simulation using PMM (i.e., Fig. 10(b)), the analytical approach (i.e., Fig. 10(c)), and experiments [33] (i.e., Fig. 10(a)). It can be seen that the predicted pressure fluctuation characteristics in terms of frequency and amplitude by the PMM show an excellent agreement with


10


Table 4 Comparison between PMM and the analytical approach at Usf = 0.9 m/s for Sc/dp = 2.9.

Hmax,ave (m2/s2)

12 20 33 52 251 547 1214

0.186 0.1815 0.1812 0.1827 0.184 0.1844 0.1862

= me

mePMM

Analycal

(-)

Npg

e0.045 (–)

Dp (Pa)

p0.045(Pa)

Mean (–)

SD (–)

Mean (Pa)

SD (–)

0.5680 0.5643 0.5691 0.5671 0.5712 0.5673 0.5663

0.1023 0.1037 0.1040 0.1047 0.1049 0.1045 0.1046

279.5816 283.139 284.5444 280.904 280.5552 281.3199 281.8566

74.3156 67.9194 71.3224 71.9458 73.281 75.7035 74.0768

539.5113 542.8743 545.3407 542.9111 539.626 539.4363 541.2638

emin,ave (–)

dmax (rp%) p–p

p–w

0.3144 0.3254 0.3275 0.2639 0.2926 0.3094 0.2818

0.1506 0.1556 0.1566 0.137 0.1336 0.151 0.1357

0.324 0.344 0.343 0.350 0.357 0.360 0.360

1.4

3.4. Further discussion

1.3

As indicated above, in order to ensure the validity of simulation results, it is desirable to accurately calculate the cell void fraction. However, in most of practical problems where the simulation needs to handle complex computational domains and irregularly shaped particles, the implementation of the analytical approach becomes extremely difficult and computationally very expensive. To circumvent this problem, the PMM can be used an alternative to the analytical approach to correctly calculate the cell void fraction in such cases, as demonstrated above. Since the PMM calculates local void fraction based on the centroid of particle grids and the computational cell centroid, it suits arbitrarily complex computational domain geometries. The match between the particle grid centroid and the cell centroid can be established efficiently based on the searching algorithm developed in this work and hence allows for a substantial save in computational time. Moreover, with the aid of advanced meshing techniques, the arbitrarily complex particle shape can be duplicated to fully consider the presence of the particles in the fluid flow. The centroids of particle grids can be directly obtained when meshing the particle. Therefore, PMM possesses potential to handle arbitrarily complex particle geometries as well, for which there is no analytical solution so far for the calculation of cell void fraction. As an example, the PMM has been applied on unstructured meshes in the present study. With respect to the application of PMM to handle complex particle geometries, as the current version of the CFD–DEM code was developed for spherical particles, no relevant simulation data of irregular particles are available. However, the implementation of the methodology has been included above, intending to provide the researcher with an option when dealing with the complex particle geometries. For the purpose of consistency, the same computational domain as the above was considered. In total, 20,770 tetrahedral hybrid meshes have been generated for the solution of the fluid flow, as shown in Fig. 12. Since unstructured meshes were used, the third-order monotone upstream cantered schemes for conservation laws (MUSCL) was adopted for the discretization of the convection term of momentum governing equations, and the pressure staggering option (PRESTO!) scheme was employed to obtain the face values of pressure by the interpolation of the values at cell centres. The numerical strategies for other terms and simulation conditions are the same as those used for the structured mesh, as detailed in Section 2.3 and Table 2. The net mass flow rate of the system was monitored to reflect the stability of the solution, as shown in Fig. 13 for the simulation using PMM on unstructured meshes. It can be seen the net mass flow rate stays within 4 104% of the inlet mass flow rate, which is indicative of the convergence of the solution and the conservation of the mass flow throughout the system. It is also implying that the implementation of the PMM for the calculation of cell void fraction on unstructured meshes is accurate and hence allows for a stable solution. It is worth noting that the majority of transient

1.2 1.1 Npg = 52 1.0 Analytical approach

0.9 0.8 10

Npg (-)

Fig. 8. Computational efficiency of simulations with PPM normalised by the time consumed by simulations using the analytical approach at Usf = 0.9 m/s for Sc/ dp = 2.9.

the experimental data and the prediction results using the analytical approach. It is worth mentioning that in each set of data (i.e., Fig. 10(a)–(c)) the shape and peaks of the fluctuation line of the pressure signals vary with time. This is attributed to the heterogeneous nature of the gas–solid fluidised bed due to particle–particle collisional energy dissipation and the non-linearity of gas–particle interactions. The heterogeneous flow of the gas–solid flow leads to the irregular shape and non-uniform or asymmetric distribution of bubbles, as demonstrated later. As the local pressure corresponds to the evolution of bubbles, the shape and peaks of the fluctuation of local pressure varies with time. Fig. 11 shows the details of the simulated gas–solid flow characteristics at t = tb + 0.28 s in Fig. 9 using PMM (Fig. 11(a)) and those obtained using the analytical approach (Fig. 11(b)). The left part displays the distribution of vorticity of the gas flow field, and the right shows the solid distribution and gas flow vector. Similar results were obtained using the PMM and analytical approach. Specifically, the small eddies have been adequately resolved and the distribution of eddies appears very similar using both approaches. The small eddies drive the heterogeneous motion of discrete particles and as a result the big bubble that is about to burst tilts to one side of the bed in both simulation results. The above similar gas–solid flow characteristics simply suggest that the cell void fraction has been calculated accurately by the PMM. Only with the accurate cell void fraction, the fluid flow, interphase interactions and the resultant solid flow can be resolved accurately. The results discussed above imply that the PMM can achieve the same level of accuracy as that of the analytical approach. Hence, the PMM can be considered as an alternative to the analytical approach for calculating the cell void fraction.


11


a

b

c

tb

tb + 0.04 s tb + 0.08 s

tb + 0.12 s

tb + 0.16 s

tb + 0.20 s tb + 0.24 s

tb + 0.28 s tb + 0.32 s tb + 0.36 s

Fig. 9. Snapshots of particles location over time at Usf = 0.9 m/s for Sc/dp = 2.38: (a) experiment; (b) simulation using PMM; (b) simulation using the analytical approach. tb is the beginning time of the series of snapshots showing the complete period of a bubble evolution in the statistically steady state of the simulation.

200

a

Local relave pressure (Pa)

0

200

3

200

4

5

6

4

5

6

5

6

b

0

200 200

3

c

0

200

3

4

Time (s)

Fig. 10. Local relative pressure taken at 45 mm above the distributor at Usf = 0.9 m/ s for Sc/dp = 2.38: (a) experiment; (b) simulation using PMM; (b) simulation using the analytical approach.

values of the normalised net mass flow rate appear a little greater than those in the solution with structured meshes (as shown in Fig. 5). This is primarily due to the use of the unstructured meshes. The complexity of the discretization and interpolation schemes used to solve the governing equation increases the difficulty to reach the convergence of the solution. Fig. 14 demonstrates the time series of local pressure signals monitored at the 45 mm above the distributor obtained using PMM on the unstructured meshes. The fluctuation features of the

local pressure are affected by and hence indicative of the fluid flow dynamics and bubble characteristics solved by the model. The amplitude falls within 200 Pa around the mean value, which agree well with those obtained in experiments by van Wachem [33] and in the simulation using the structured mesh and the analytical approach (as shown in Fig. 10). The dominant frequency of the pressure fluctuation obtained by the FFT analysis, i.e., 2.93 Hz as shown in Fig. 15, is very close to the dominant frequency of the pressure fluctuation measured in experiments and that obtained in the simulation using structured mesh and the analytical approach (i.e. 2.74 Hz). The subtle difference might be attributed to the use of the tetrahedral hybrid unstructured mesh, which is numerically unsuitable for the regular computational domain (i.e., rectangular). As shown in Fig. 15, some lower level peaks appear in the pressure fluctuation, which indicates the generation and evolution of small bubbles in the gas–solid fluidized bed. The presence of these small bubbles decreases the size but increases the generation rate of the primary bubble. As a result, the dominant frequency of the pressure fluctuation appears slightly greater than those in experiments and in the simulation using the structured mesh (Fig. 10).

4. Conclusion A modified version of the conventional direct method for the calculation of cell void fraction in CFD–DEM simulations has been proposed by meshing the particles into a number of small particle grids. The numerical stability, minimum particle grid number, prediction accuracy, and computational efficiency of the PMM have been demonstrated and discussed in detail. Furthermore, the


12


a

0.35

0.20

0.30 0.15

0.25

0.20 0.10 0.15

0.10 Vorticity (s-1):

0.05

0

b

0.05

100 75 50 25 0 -25 -50 -75 -100

0.35

| ug | (m/s):

0

2.8 2.1 1.4 0.7 0

0.20

0.30 0.15

0.25

0.20 0.10 0.15

0.10 Vorticity (s-1):

0.05

0

0.05

100 75 50 25 0 -25 -50 -75 -100

| ug | (m/s):

0

2.8 2.1 1.4 0.7 0

Fig. 11. Gas and solid flow at tb + 0.28 s (in Fig. 8) at Usf = 0.9 m/s for Sc/dp = 2.38: (a) simulation using PMM; (b) simulation using the analytical approach. In (a) and (b), the left is the distribution of vorticity of gas flow, and the right is the gas flow vector and the solid distribution.

potential of the application of PMM in scenarios where complex domain and particle geometries are involved has also been discussed. The PMM can also achieve the same level of numerical stability and prediction accuracy as those of the analytical approach. There was a minimum particle grid number (Npg,min) required to achieve the stable solution. The value of Npg,min strongly depended on the

fluid cell size that was used for the simulation. A further increase in the particle grid number beyond the required minimum number only increased the computational cost with no appreciable improvements on the quality of prediction results. A particle grid number that is about 5 times the minimum number is preferred in the application of PMM. Simulations using the PMM showed comparable or even higher computational efficiency than those


13


450 400

fp(1) = 2.93

Pressure PSD (-)

350 300 250 200 150 100 50 0

1

6

11

16

Frequency (1/s) Fig. 15. FFT analysis of the local pressure fluctuation predicted at Usf = 0.9 m/s using PMM on unstructured meshes.

and prediction accuracy. It is also suggested that the PMM has the potential to be applied in simulations where arbitrarily shaped particles are involved. However, relevant modelling work is warranted for justification and further understanding of the application of PMM to calculate the cell void fraction with the complex particle geometries. Acknowledgements Thanks to R. Dear for his assistance with the usage of high performance cluster (HPC) facilities in the University of Newcastle. The authors wish to acknowledge the financial support of the Australian Research Council for the work presented in this paper. Fig. 12. Tetrahedral hybrid unstructured meshes for the solution of the gas flow using the PMM at Usf = 0.9 m/s: (a) computational domain; (b) perspective and close-up view of the top part of the computational domain.

0.0002 0

m

(%)

0.0004

-0.0002 -0.0004

5

6

7

8

9

10

Time (s)

Local relative pressure (Pa)

Fig. 13. Normalised net mass flow rate of the system at Usf = 0.9 m/s using PMM on unstructured meshes.

200 0 -200

3

4

5

6

Time (s) Fig. 14. Local relative pressure taken at 45 mm above the distributor at Usf = 0.9 m/ s using PMM on unstructured meshes.

using the analytical approach. The PMM has also been successfully applied on unstructured meshes with excellent numerical stability

References [1] Y. Tsuji, T. Kawaguchi, T. Tanaka, Discrete particle simulation of twodimensional fluidized bed, Powder Technol. 77 (1993) 79–87. [2] B.H. Xu, A.B. Yu, Numerical simulation of the gas–solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chem. Eng. Sci. 52 (1997) 2785–2809. [3] K.D. Kafui, C. Thornton, M.J. Adams, Discrete particle–continuum fluid modelling of gas–solid fluidised beds, Chem. Eng. Sci. 57 (2002) 2395–2410. [4] N.G. Deen, M. van Sint Annaland, M.A. van der Hoef, J.A.M. Kuipers, Review of discrete particle modeling of fluidized beds, Chem. Eng. Sci. 62 (2007) 28–44. [5] Z. Peng, E. Doroodchi, C. Luo, B. Moghtaderi, Influence of void fraction calculation on fidelity of CFD–DEM simulation of gas–solid bubbling fluidized beds, AIChE J. 60 (2014) 2000–2018. [6] F. Chen, E.C. Drumm, G. Guiochon, Coupled discrete element and finite volume solution of two classical soil mechanics problems, Comput. Geotech. 38 (2011) 638–647. [7] J. Li, Euler–Lagrange simulation of flow structure formation and evolution in dense gas–solid flows, Ph.D. Thesis, University of Twente, The Netherlands, 2003. [8] B. Freireich, M. Kodam, C. Wassgren, An exact method for determining local solid fractions in discrete element method simulations, AIChE J. 56 (2010) 3036–3048. [9] C.L. Wu, J.M. Zhan, Y.S. Li, K.S. Lam, A.S. Berrouk, Accurate void fraction calculation for three-dimensional discrete particle model on unstructured mesh, Chem. Eng. Sci. 64 (2009) 1260–1266. [10] C.M. Boyce, D.J. Holland, S.A. Scott, J.S. Dennis, Novel fluid grid and voidage calculation techniques for a discrete element model of a 3D cylindrical fluidized bed, Comput. Chem. Eng. 65 (2014) 18–27. [11] J.M. Link, L.A. Cuypers, N.G. Deen, J.A.M. Kuipers, Flow regimes in a spout–fluid bed: a combined experimental and simulation study, Chem. Eng. Sci. 60 (2005) 3425–3442. [12] H. Khawaja, S.A. Scott, S. Virk, M. Moatamedi, Quantitative analysis of accuracy of voidage computations in CFD–DEM simulations, Int. J. Multiphase Flow 4 (2012) 183–192.


14


[13] E.W.C. Lim, C.H. Wang, A.B. Yu, Discrete element simulation for pneumatic conveying of granular material, AIChE J. 52 (2006). [14] S.B. Kuang, K.W. Chu, A.B. Yu, Z.S. Zou, Y.Q. Feng, Computational investigation of horizontal slug flow in pneumatic conveying, Ind. Eng. Chem. Res. 47 (2008) 470–480. [15] J. Sun, F. Battaglia, S. Subramaniam, Hybrid two-fluid DEM simulation of gas– solid fluidized beds, J. Fluid Eng. – Trans. ASME 129 (2007) 1394–1403. [16] H. Xiao, J. Sun, Algorithms in a robust hybrid CFD–DEM solver for particleladen flows, Commun. Comput. Phys. 9 (2011) 297–323. [17] N. Gui, J.R. Fan, K. Luo, DEM–LES study of 3-D bubbling fluidized bed with immersed tubes, Chem. Eng. Sci. 63 (2008) 3654–3663. [18] J.E. Hilton, L.R. Mason, P.W. Cleary, Dynamics of gas–solid fluidised beds with non-spherical particle geometry, Chem. Eng. Sci. 65 (2010) 1584–1596. [19] A. Hobbs, Simulation of an aggregate dryer using coupled CFD and DEM methods, Int. J. Comput. Fluid Dyn. – Min. Miner. Proc. 23 (2009) 199–207. [20] Z. Peng, S.V. Ghatage, E. Doroodchi, J.B. Joshi, G. Evans, M. Moatamedi, Forces acting on a single introduced particle in a solid–liquid fluidised bed, Chem. Eng. Sci. 116 (2014) 49–70. [21] T.B. Anderson, R. Jackson, A fluid mechanical description of fluidized beds – equations of motion, Ind. Eng. Chem. Fund. 6 (1967) 527–539. [22] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1979) 47–65. [23] O.R. Walton, R.L. Braun, Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks, J. Rheol. 30 (1986) 949–981. [24] L.E. Silbert, D. Ertaz, G.S. Grest, T.C. Halsey, D. Levine, S.J. Plimpton, Granular flow down an inclined plane: Bagnold scaling and rheology, Phys. Rev. E 64 (2001) 051302. [25] Z. Peng, E. Doroodchi, Y. Alghamdi, B. Moghtaderi, Mixing and segregation of solid mixtures in bubbling fluidized beds under conditions pertinent to the fuel reactor of a chemical looping system, Powder Technol. 235 (2013) 823– 837.

[26] ANSYS Inc., Meshing User’s Guide, ANSYS 15.0 Help, ANSYS Inc., USA, 2015. [27] P.O. Persson, G. Strang, A simple mesh generator in MATLAB, SIAM Rev. 46 (2004) 329–345. [28] C. Geuzaine, J.F. Remacle, GMSH: a three dimensional finite element mesh generator with built-in pre- and post processing facilities, Int. J. Numer. Meth. Eng. 79 (2009) 1309–1331. [29] D. Eberly, Quaternion Algebra and Calculus, Magic Software, Inc, 2002. [30] G. Hoffmann, Application of quaternions. Tech. Rep. February, University of Brunswick – Institute of Technology, 1978. [31] S.B. Kuang, A.B. Yu, Z.S. Zou, A new point-locating algorithm under three-dimensional hybrid meshes, Int. J. Multiphase Flow 34 (2008) 1023– 1030. [32] C.L. Wu, A.S. Berrouk, K. Nandakumar, Three-dimensional discrete particle model for gas–solid fluidized beds on unstructured mesh, Chem. Eng. J. 152 (2009) 514–529. [33] B.G.M. van Wachem, J. van der Schaaf, J.C. Schouten, R. Krishna, C.M. van den Bleek, Experimental validation of Lagrangian–Eulerian simulations of fluidized beds, Powder Technol. 116 (2001) 155–165. [34] G.R. Wu, J. Ouyang, B.X. Yang, Q. Li, Use of compromise-based local porosities for coarse grid DEM simulation of bubbling fluidized bed with large particles, Adv. Powder Technol. 24 (2012) 68–78. [35] J. Ai, J.F. Chen, J.M. Rotter, J.Y. Ooi, Assessment of rolling resistance models in discrete element simulations, Powder Technol. 206 (2011) 269–282. [36] Y.C. Zhou, B.D. Wright, R.Y. Yang, B.H. Xu, A.B. Yu, Rolling friction in the dynamic simulation of sandpile formation, Phys. A 269 (1999) 536– 553. [37] S. Sasic, B. Leckner, F. Johnsson, Characterization of fluid dynamics of fluidized beds by analysis of pressure fluctuations, Prog. Energ. Combust. 33 (2007) 453–496. [38] C.L. Wu, J.M. Zhan, Y.S. Li, K.S. Lam, A.S. Berrouk, Accurate void fraction calculation for three-dimensional discrete particle model on unstructured mesh, Chem. Eng. Sci. 64 (2009) 1260–1266.


A modified direct method for void fraction calculation ...

A modified direct method for void fraction calculation ...

Suggest Documents

a simple capacitance sensor for void fraction

Direct Calculation Method for Excited-state

Modified Degree-Hour Calculation Method

Early Fraction Calculation Ability

capacitance sensor for void fraction measurement in a natural ...

A New Void Fraction Correlation for Simulation of Two

The Modified Direct Method: A Solution for Obtaining Accurate Coal ...

Analysis Of Void Fraction In Microchannels - CiteSeerX

Experimental Studies on Void Fraction ... - ScienceDirect.com

Pressure drop and average void fraction ...

Noise Calculation Method for Industrial Premises with ... - Science Direct

multi-scale void fraction analysis for cell density

Development of void fraction-quality correlation for ...

Design of capacitance sensor system for void fraction measurement

Measurements of void fraction and bubble properties on a stepped

A Simple Method for Calculation of thePermeability

A Calculation Method for Voltage - IEEE Xplore

Modified Jendrassik-Grof Method for Measurement of Direct Bilirubin

Evaluation of a modified direct agar proportion method ... - CyberLeninka

Modified equipartition calculation for supernova remnants. Cases

A modified subgradient extragradient method for

RESEARCH ARTICLE A Modified Iterative Method for

Validation of a modified cryopreservation method for

Phase fraction calculation of microstructure using ImageJ