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1 Available online at ScienceDirect Procedia Engineering 10 (015 ) The 7th World Congress on Particle Technology (WCPT7) Three-diensional Porosity Model Based on Volue Solver of Curved Top Cylinder Guorong Wu a,b, Jie Ouyang a, * a School of Science, Northwestern Polytechnical University, Xi an 71019, China b Departent of Matheatics, Gansu Noral University for Nationalities, Hezuo , China Abstract The ean porosity within a icro-unit should be properly calculated in discrete eleent siulation of gas-solid fluidization systes. This article presents a three-diensional porosity odel for fine grid calculation. All nine cases of particle-grid overlap are considered in the odel. The three-diensional porosity is analytically expressed, eploying a series of volue forulas for pertinent curved top cylinders. The double integrals involved in the volue expressions are nuerically calculated according to copound trapezoid forula. The adopted integral schee and nuerical ethod can be easily realized, which are suitable for coplicated three-diensional fine grid calculation The The Authors. Published Published by Elsevier by Elsevier Ltd. This Ltd. is an open access article under the CC BY-NC-ND license ( Selection and peer-review under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Selection and peer-review under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Acadey of Acadey Sciences of (CAS) Sciences (CAS). keywords: Siulation; DEM, Fine grid; Matheatical odelling; Nuerical integral * Corresponding author. Tel.: E-ail address: jieouyang@nwpu.edu.cn The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Selection and peer-review under responsibility of Chinese Society of Particuology, Institute of Process Engineering, Chinese Acadey of Sciences (CAS) doi: /j.proeng

1644 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) 1643 1649 1.

In traditional discrete eleent siulation, those structures saller than Eulerian grid are not properly resolved despite that particle trajectories are forally calculated individually.

[4] copared the siulated pressure fluctuations, porosity fluctuations, bed expansion and the visual representation of particle locations with experient results.

Nowadays, ore and ore researchers tend to use fine grid in fluidization siulations [5].

2 1644 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) Introduction The ean porosity within a icro-unit needs to be properly calculated in discrete eleent siulations [1-3] or Lagrangian-Eulerian siulations of gas-solid fluidization systes. In traditional discrete eleent siulation, those structures saller than Eulerian grid are not properly resolved despite that particle trajectories are forally calculated individually. Soe details of gas flow and particle otion history cannot be properly captured. Once van Wache et al. [4] copared the siulated pressure fluctuations, porosity fluctuations, bed expansion and the visual representation of particle locations with experient results. The ajor unconfority they found indicated that the unsolved structures within Eulerian grid did contribute uch to the bubbling behaviors. Nowadays, ore and ore researchers tend to use fine grid in fluidization siulations [5]. As can be easily iagined, fine grid is advantageous in reducing the aount of non-physical soothing for the estiated local quantities such as local porosity and local gas velocity. Wang et al. [6] suggested that the grid size of -4 particle diaeters be sufficient to fully resolve the flow structures in fluidized beds for various types of particles. Moreover, the issue of coputational liitation in discrete siulation ainly roots in large nuber of particles required rather than use of fine grid. Thus it is both necessary and possible to use fine grid in discrete eleent siulations. Although fine grid calculation has now becoe a ainstrea trend, the theoretical syste of fine grid siulation is far fro perfect and less attention has been paid to the coplety of fine grid siulation. One of the coplety factors lies in the calculation of grid porosity, the siulation sensitivity to which is very high. The grid porosity is ore often roughly estiated, especially in three-diensional siulations [6]. Our recent research [7] has given a precise area fraction odel for two-diensional siulation of fluidized bed. However, So far there has not been a clear and specific three-diensional porosity odel. The present conference paper presents this kind of odel for fine grid calculation.. Three-diensional porosity odel.1 Nine cases of particle-grid overlap To construct a three-diensional porosity odel, the first challenge is to divide all the cases of particle-grid overlap. Let us take just a particle to be a sphere and a grid to be a cube. According to the nuber of the cube adages and faces intersected with the spherical surface, there are totally nine cases of sphere-cube overlap. Fig. 1 gives the scheatic of all these cases. As should be pointed out, although there are three faces and three ages intersected with the spherical surface for both case 6 and case 9, the cube vertex closest to the sphere center is inside and out of the sphere, respectively. Case Fig.1 Nine cases of particle-grid overlap For Case 1, the solid volue fraction can be easily obtained by precisely calculated the volue fraction of the whole sphere. For Case, the solid volue fraction can be obtained by precisely calculated the volue fraction of the spherical segent. For Case 3 or 4, the solid volue can be obtained by precisely calculated the volue of the sphere inus the volue of two or three spherical segents, respectively. However, generally the solid volue is coplicated and can not be precisely calculated for the rest five cases. Fig. gives four intersected geoetries, the last three of which can frequently be regarded as curved top cylinders. The first geoetry is known to be a sphere segent. The second geoetry is called sei-segent because it is just the bisection of a sphere segent. The third geoetry is called quasi sei-segent. The last geoetry is called quasi quarter-segent.

Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) 1643 1649 1645 Sei-segent Quasi quarter-segent Sphere segent Quasi sei-segent Fig.

3 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) Sei-segent Quasi quarter-segent Sphere segent Quasi sei-segent Fig. Definition of intersected geoetries Consider the ways of volue calculation for all cases of particle-grid overlap. Although the counting processes vary fro different cases of particle-grid overlap, the solid volue can be uniforly expressed as V V V V V (1) solid k sphere l segent(s) quasisei segent(s) n quasiquatersegent where k and n are equal to 0 or 1, l and are equal to 0,1, or 3. For instance we can eploy (1) with k =1, l=3, =3 and n=0 to calculate V solid for case 6. As V sphere and V segent can be easily and precisely calculated, in the following sections. and.3, we ainly consider V quasi sei-segent and V quasi quarter-segent.. Volue of curved top cylinder The three-diensional porosity is calculated as V 1 solid () V cube where V solid is calculated by suing the volue of each particle overlapped with the target grid. Suppose there be only one particle intersected with the grid. According to (1), V solid can be expressed by V quasi sei-segent and V quasi quartersegent. Therefore, the three-diensional porosity odel ainly investigates the intersected volue for Cases 5 and 9. Let A 1 be the vertex closest to the sphere center. Since the volue fraction has the property of invariability to arbitrary rotation of the real graph, anyhow A 1 can be displayed as the lower-left vertex in the corresponding geoetric graph. Let in the geoetric graph the coordinates of the lower-left vertex A 1 and the sphere center O be (0, 0, 0) and (X,Y,Z), respectively, and the radius of the sphere be R. The spherical equation is x X y Y z Z R (3) Fig. 3 gives the real graph and geoetric graph for Case 5. That A 1 can be displayed as the lower-left vertex is iplied in the geoetric graph. As single integral ethod for volue calculation is very coplicated, here we adopt double integral ethod. For this case, there are two variations. For the first variation when Z 0, any point (x,y) in the integral doain D satisfies (4) and (5). 0 x X R Z (4) Y R Z ( x X ) y R Z ( x X ) Y (5) Define the integrand, i.e. the top surface function of the curved top cylinder, as y Y x z( x, y) Z R X (6) Then X R Z Y R Z xx V dx zx, ydy Y 0 R Z x X (7)

1646 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) 1643 1649 (a) real graph (b) geoetric graph Fig.

(a) real graph Fig. 4 Real graph and geoetric graph for case 9 (b) geoetric graph Fig. 4 gives the real graph and geoetric graph for Case 9. For this case, there are four variations.

4 1646 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) (a) real graph (b) geoetric graph Fig. 3 Real graph and geoetric graph for case 5 For the second variation when Z>0, V can be expressed as the following for. V V segent V (8) where V can be calculated by the way siilar to (7). (a) real graph Fig. 4 Real graph and geoetric graph for case 9 (b) geoetric graph Fig. 4 gives the real graph and geoetric graph for Case 9. For this case, there are four variations. For the first variation when Z 0 and Y 0, any point (x,y) in the integral doain D satisfies (9) and (10). 0 x X R Z (9) x X y 0 Y R Z (10) The integrand z(x,y) is siilar to that given in (6). Thus X R Z 0 V dx z x, y dy (11) 0 Y R Z x X For the last three variations when Z 0 and Y<0, Z>0 and Y 0, as well as Z>0 and Y<0, V can be expressed as V Vquasisei- segent V (1)

Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) 1643 1649 1647 where V can be calculated by the way siilar to (11).

5 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) where V can be calculated by the way siilar to (11). Although (7) and (11) can be transfored into repeated integral after polar coordinate transforation, there is no analytic expression for V. That is to say, the priitive of the reduced single integral with coplicated integrand cannot be explicitly expressed by eleentary function. Therefore, the solid volue cannot be precisely calculated for Cases 5-9. In the following, nuerical integration is used to obtain the approate answer for the definite integral in (7) or (11), which cannot be solved analytically..3 Nuerical integration We now approach the subject of nuerical integration [8]. The goal is to approate the definite integral of f(x,y) over an irregular two-diensional interval by evaluating f(x,y) at a finite nuber of saple points. (a) trapezoid forula Fig. 5 Scheatic of trapezoidal rule (b) copound trapezoid forula Firstly consider the definite integral of f( over the interval [a,b]. Fig. 5 gives the scheatic graph of trapezoidal rule for nuerical ethod of single integration. The integral is just the area under the curve y=f( for a x b, i.e., b a A f ( dx (13) In Fig. 5(a), A is approated to just by coputing the area of one trapezoid. In Fig. 5(b), each additional value of copound trapezoid forula is deterined by trapezoid forula. Copared to trapezoid forula, the copound trapezoid forula using a series saple points except a and b has higher degree of precision, which can be observably noticed in the figure. Then consider the definite integral of f(x,y) over a regular rectangle as I f x, ydxdy (14) D where the integral interval is D {( x, y) a x b, c y d} Since D is a direct product of two one-diensional intervals, the double integral can be transfored into repeated integral as d I dy f x, y dx (15) c b a According to the copound trapezoid forula of single integral, 1 n1 f (, y j ) f ( 1, y j ) f (, y j 1) f ( 1, y j 1) ( b a)( d c) I (16) 4 i0 j 0 where the intervals of x-as and y-as are divided into and n equal parts, respectively, and b a d c a i ( i 0,1,, ), y j c j ( j 0,1,, n) n Consider at last the nuerical ethod for the definite integral of f(x,y) over an irregular interval as

6 1648 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) y x b I dy y x f x, y dx (17) 1 a As can be seen, the double integral is over D {( x, y) a x b, y1( y y( } which is not a regular rectangle. The nuerical ethod for this condition is relatively unfailiar to researchers. Here we define the new rectangle as R {( x, y) a x b,in y1( y ax y( } Furtherore, define the new integrand as f (, y j ), if y1( ) y j y( ) F(, y j ) (18) 0, else Then we have ax y x b I dy y x F x, y dx (19) in 1 a According to (16), 1 n1 x in y1 x ) F (, y j ) F( 1, y j ) F(, y j1) F( 1, y j1 ( b a)(ax y I 4 ) (0) i0 j0 where the intervals of x-as and y-as are divided into and n equal parts, respectively, and b a ax y( in y1( a i ( i 0,1,, ), y j in y1( j ( j 0,1,, n) n To su up, the three-diensional porosity odel coes down to (7) and (11), both of which can be nuerically calculated eploying (0). 3. Conclusion A atched three-diensional porosity odel for fine grid DEM siulation of fluidized bed was proposed in this conference article. Totally nine cases of particle-grid overlap was outlined which was divided according to the nuber of the cube adages and faces intersected with the spherical surface. It was shown that the intersected solid volue cae down to the volue of two kinds of special curved top cylinders called quasi sei-segent and quasi quarter-segent. The volue of either curved top cylinder was precisely expressed in a for of double definite integral. The double integrals involved in the volue expressions were nuerically calculated according to copound trapezoid forula. It was indicated that one could directly calculated the intersected solid volue fraction for an arbitrarily particle just by use of a single atheatical forula, eploying the double definite integral and the nuerical forat. Therefore, the proposed three-diensional porosity odel is suitable for coplicated three-diensional fine grid DEM siulation. Acknowledgeent The authors are grateful to the Presidential Foundation of Gansu Noral University for Nationalities (01301) for financial support of this work. References [1] Y. Tsuji, T. Kawaguchi, T. Tanake. Discrete particle siulation of two-diensional fluidized bed, Powder Technol. 77 (1993) [] B.P.B. Hooans, J.A.M. Kuipers, W.J. Briels, W.P.M. van Swaaij, Discrete particle siulation of bubble and slug foration in a two diensional gas-fluidized bed: a hard-sphere approach, Che. Eng. Sci. 51 (1996) [3] B.H. Xu, A.B. Yu, Nuerical siulation of the gas-solid flow in a fluidized bed by cobing discrete particle ethod with coputational fluid dynaics, Che. Eng. Sci. 5 (1997) [4] B.G.M. van Wache, J. van der Schaaf, J.C. Schouten, R. Krishna, C.M. van den Bleek, Experient validation of Lagraingian-Eulerian siulation of fluidized beds, Powder Technol. 116 (001) [4] Q.C. Sun, G.Q. Wang, An introduction to the echanics of granular aterials, WIT Press, Southapton, UK, 011.

7 Guorong Wu and Jie Ouyang / Procedia Engineering 10 ( 015 ) [5] J.M. Link, L.A. Cuypers, N.G. Deen, J.A.M. Kuipers, Flow regies in a spout-fluidized bed: A cobined experient and siulation study, Che. Eng. Sci. 60 (005) [6] J.W. Wang, M.A. van der Hoef, J.A.M. Kuipers, Why the two-fluid odel fails to predict the bed expansion characteristics of Geldart A particles in gas-fluidized beds: A tentative answer, Che. Eng. Sci. 64 (009) [7] G.R. Wu, J. Ouyang, Fine grid DEM siulation of bed layer height in bubbling fluidized bed, Journal of Cheical Industry and Engineering, 65 (014) [8] S.Z. Cai, M. Yang, Y.J. Lei, Nuerical ethod, National Defense Industry Press, Beijing, 011.

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