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Discrete particle simulation of flow regimes in bulk solids mixing and conveying
Powder Technology 104 Ž1999. 248–257 www.elsevier.comrlocaterpowtec
Discrete particle simulation of flow regimes in bulk solids mixing and conveying J. Gyenis a
a,)
, Zs. Ulbert b, J. Szepvolgyi ´ ¨
c,1
, Y. Tsuji
d
Research Institute of Chemical and Process Engineering, Pannon UniÕersity of Agricultural Science, P.O. Box 125, Veszprem, ´ Egyetem u. 2, Hungary b UniÕersity of Veszprem, ´ Veszprem, ´ Egyetem u. 10, Hungary c National Institute for Resources and EnÕironment, Tsukuba, Ibaraki, 356, Japan d Department of Mechanical Engineering, Osaka UniÕersity, Suita, Osaka 565, Japan
Abstract Earlier experimental studies revealed that, depending on the feeding and discharging conditions, three distinct flow regimes can evolve during the gravity flow of particles through a vertical tube containing static mixer element. In this paper, the results obtained by discrete particle simulation ŽDPS. are reported. Applying this simulation method for gravity flow, the main characteristics of the flow regimes observed experimentally could be reproduced and explained theoretically. Some important features of gas–solids two-phase flows in tubes containing static mixer element were also revealed. By excavating a great amount of information from the data obtained by DPS, useful data and correlations become available, which are not directly measurable by experiments. Thus, DPS method can help to get useful explanations on particle level for the experimental observations. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Particulate solids; Particle; Gravity flow; Gas–solids flow; Flow resistance; Static mixer; Discrete particle simulation
1. Introduction During gravity flow of particulate solids, and in gas– solids two-phase flow through pipes, the bends, contractions, nozzles, or static mixer elements inserted into certain tube sections have great influence on the flow pattern and on the characteristics of the particle movements, such as on their trajectories, spatial distribution, velocities, residence time distribution and on the mixing characteristics of the solid phase. From experimental study of gravity flow of particles through vertical tubes containing helical static mixer elements, Gyenis w1x and Gyenis et al. w2x have found that, depending on the feeding and discharging conditions, three typical flow regimes can occur. The studied tube system consisted of three distinct sections: between an upper and a lower tube section there was a mixer section containing a series of static mixer elements.
Besides this experimental study, the authors carried out investigation using simulation methods to elucidate the main features of the flow and mixing characteristics of particulate solids in such tube systems. For this, various methods were used, starting from phenomenological description to stochastic modelling and DPS simulation. Regarding this latter method, it was recognized that the huge amount of data which can be obtained by DPS simulation gives much more possibility for the investigation of these systems. DPS simulation data give a good chance to find what is happening on the particle level. It made possible the study of various statistical features and microscopic phenomena inducting global consequences in the particle bed. The majority of the studies that have been carried out by several authors in the field of DPS simulation have not yet utilized entirely this great potential.
2. Experimental observations ) Corresponding author. Tel.: q36-88-425-206; fax: q36-88-424-424; E-mail: [email protected] 1 Note: Permanent position of J. Szepvolgyi is at Pannon University, ´ ¨ Research Institute of Chemical and Process Engineering, Veszprem, ´ Egyetem u. 2, H-8201, P.O. Box 125, Hungary.
Gyenis w1x and Gyenis et al. w2x have reported experimental observations in such systems. It was stated that in case of unlimited particle inflow into a static mixer tube, when the discharge rate was lower than the maximal
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 1 0 2 - 3
J. Gyenis et al.r Powder Technology 104 (1999) 248–257
possible throughput determined by the resistance of the static mixer elements against particle flow, a dense sliding particle bed was present in the whole tube showing almost uniform solids volume fraction along its length. It was named 1st flow regime. Increasing the discharge rate within a limit, the mean solids volume fraction in the tube has remained almost constant. But, at higher mass flow rates approaching the maximal throughput of the tube, there was a slight decrease in the mean solids volume fraction. In case of unlimited particle inflow at the top and free discharge of the particles from the tube, the mass flow became equal to the maximal throughput of the tube. This critical state was named 2nd flow regime. Under this conditions, there was a dense, sliding particle bed in the tube section above the mixer elements. In the mixer section, accelerating particle layers with gradually diminishing thickness were sliding down on the surface of the helical mixer elements. Below this mixer section, accelerating thread-like particle streams were twisting downwards near the tube wall, with decreasing solids volume fraction. In case of limited inflow at the top and free outlet at the bottom, a loose and accelerating particle bed was formed both above and below the mixer section. On the surface of the mixer elements, sliding particle layers were present, resulting in somewhat higher mean solids volume fraction in this tube section. A general state diagram was proposed w1,2x to describe the main characteristics of these flow regimes and to indicate the necessary conditions for their transformation into each other. Besides the quite different spatial distribution pattern of the solid particles characterizing these flow regimes, several quantitative characteristics, such as the mean solids volume fraction, the residence time distribution, the axial mixing characteristics of the particle bed, etc., were also determined experimentally. Introducing a streaming gas phase, together with the solid particles, the flow pattern and particle bed characteristics have changed significantly w3,4x. These studies were carried out in downward vertical gas–solids flow in a tube, similar to the simple gravity flow of particles mentioned above. The aim of these investigations was to study the influence of the gas phase on the flow characteristics of the particle bed. These conditions had led to the so called 4th flow regime characterized by lean particle flow along the static mixer elements. In case of low solids loading ratios, this flow regime was similar to the third flow regime, but, due to the higher acceleration of the particles caused by the streaming gas phase, the solids volume fraction was exponentially diminishing from the inlet, gradually approaching to the gas velocity. The maximal throughput of the tube was considerably higher relative to the gravity flow of particles, without choking the tube. In case of high solids mass flow rate, however, this flow regime became similar to the second flow regime, but with higher solids throughput. By applying static mixer elements, the residence time of the solid particles and the
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velocity difference between the gas and solid phases could be enhanced. In spite of the numerous quantitative results and observations obtained experimentally, it must be emphasized that studying the influences of all important design and process parameters by experiments in these flow systems are very time-consuming, tedious and expensive. Therefore, during the last few years, an attempt was made by the authors to apply discrete particle simulation ŽDPS. method to study the influence of several parameters on the particle flow in such mixer tubes. The recent results of these studies are reported below.
3. The applied modelling and simulation method
3.1. Mathematical description of the particle motions
3.1.1. GraÕity flow of particles For this study, the discrete particle simulation ŽDPS., also called distinct element method ŽDEM. based on Lagrangian approach w5,6x, was applied. This method well corresponds to the discontinuous nature of the particulate systems and can be used advantageously both for simple gravity flow of particles and for dense or lean gas–solids two-phase flows. By this approach, the particle trajectories, particle–particle and particle–wall interactions are considered distinctly for each particle existing within the studied system. As it has been written in fundamental papers describing this method w5–8x, the individual particles have two types of motion, namely translational and rotational movements. Translation motion is caused by contact forces, fluid forces and gravity force. As regards the rotation of the particles, only the contact forces were considered. The equations to determine the acceleration of the particles during their translation and rotation are as follows: ™
™
r¨ s fC q f D rm q ™ g,
™
ž
™
™ ˙ s TCrI, v
/
Ž 1. Ž 2.
where ™ r is the position vector of™the mass centre of the particle,™m is the particle mass, fC is the sum of contact forces, f D is ™ the fluid drag force, ™ g is the gravity acceleration vector, TC is the sum of torque induced by contact forces, and I is the moment of inertia of the particle. ™ The contact force fC is described by a mechanical model shown in Fig. 1, first proposed by Cundall and Strack w7x. This model expresses the contact forces by a spring, a dash-pot and a friction slider. Maximum 12 particles can be in contact with particle i at the same time. The total contact force acting on particle
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250
Fig. 1. Model of the contact forces between two particles, according to Cundall and Strack w7x Ža. normal forces, Žb. tangential forces.
™
i is obtained by taking the sum of the normal fCn i j and ™ tangential fCt i j forces with respect to j: ™
™
ž sÝ žr n
™
/ /.
fC i s Ý fCn i j q fCt i j , j
TC i
s
™ ™ i j = f Ct i j
j
Ž 3. Ž 4.
The details of calculation of contact forces caused by interparticle and particle–wall collisions and friction, influenced by the material properties of the particles and the equipment wall, are described in the literature w5,6x. After a time step D t, the velocity and position of a particle can be given as Õ s™ Õ 0 q™ r¨0 D t ,
™
Ž 5.
™
r s™ r 0 q™ ÕD t ,
Ž 6.
™ ™ ™ ˙ v sv 0 qv0Dt,
Ž 7.
length and diameter were equal to the inner diameter of the tube, thus their length to diameter ratio was 1:1. From the whole tube length, 0.150 m was occupied by the mixer elements. Thus, the lengths of the free tube sections above and below the mixer elements were 0.025 and 0.010 m, respectively. The particle properties, such as interparticle and wall friction coefficients, size, density, stiffness and damping coefficient of the particles, have great influences on the particle–particle and particle–wall interactions. For the simulation of gravity flow, spherical polymer particles were taken into consideration with uniform sizes of 0.003 m. The material properties were as follows: densitys 1190 kgrm3 , stiffnesss 75 Nrm, restitution coefficients 0.6, friction coefficient between the particless 0.43, friction coefficient between the particles and tube wall s 0.33. During the simulation of the 3rd flow regime, the particles were introduced into the tube with various mass flow rates, evenly distributed across the whole inlet crosssection. In case of the 1st and 2nd flow regimes, they were withdrawn from the bottom of the tube with various mass flow rates, with unlimited inlet into the tube. By this method, it was possible to reproduce the three different flow regimes observed experimentally w1,2x.
™
where Õ is the velocity vector and subscript 0 denotes the previous value. 3.1.2. Modelling of gas–solids two-phase flow For the simulation of gas–solids two-phase flow in a cylindrical tube section containing one helical static mixer element, the same mathematical equations described above were used. But, in addition to Eqs. Ž1. – Ž7., several additional correlation had to be taken into consideration, due to the interactions between the particles and the streaming gas. The details of this two-phase flow simulation method were described in earlier papers w5,9x. In the present work, the viscosity terms were also considered. 3.2. Conditions of the simulations 3.2.1. GraÕity flow in Õertical tube The schematic diagram of the studied tube with the inserted static mixer elements is shown in Fig. 2. The length and diameter of the tube were 0.185 and 0.050 m i.d., respectively. Three helical mixer elements, each twisted to right hand direction, were placed into the tube closely after each other, with an angle of 308 between the outlet and inlet edges of the successive elements. Their
Fig. 2. Schematic diagram of the tube used for the simulation of gravity flow of particles.
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The mass flow rate of the particles were varied from 468 to 2880 kgrh. The number of particles treated simultaneously by the simulation was also changing between 4000 and 12,000.
3.2.2. Gas–solids two-phase flow in horizontal tube To prove the application possibility of DPS simulation method in gas–solids two-phase flow in tubes containing static mixer elements, several studies were carried out in a short horizontal tube in case of non-steady-state conditions. The length and inner diameter of the tube were 0.6 and 0.2 m i.d., respectively. One helical static mixer element with 1:1 length to diameter ratio was placed into the tube at 0.2 m from the tube inlet. The orientation of its inlet and outlet edges was horizontal.
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The number of particles introduced into the tube in one time step was constant, corresponding to 1440 kgrh mass flow rate. The superficial gas velocity was 15 mrs, corresponding to 1695 m3rh volumetric flow rate and 2035 kgrh mass flow rate. Thus, the loading ratio Žkg solids per kg gas. was 0.7, corresponding to 0.5% volumetric particle concentration under the inlet conditions. In the present stage of the study, the inlet velocity of the particles was varied from 1 to 14 mrs. Due to the high slip velocity between the solid and gas phases at the inlet, the particles in the studied tube section were accelerating, except at certain regions of the mixer element, because of its strong interaction with the particles. The properties of the particles were similar to that used for the gravity flow. Only the particle diameter was higher, namely 0.004 m. As for the gas phase, the properties of air
X Fig. 3. Snapshots of gravity flow in three different flow regimes obtained by simulation Ža. 1st flow regime, G s 2880 kgrh, Žb. 2nd flow regime, X X G s 1150 kgrh, Žc. 3rd flow regime, G s 720 kgrh.
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X
Fig. 4. Longitudinal distributions of solid volume fraction after different times in the 3rd flow regime, G s 448 kgrh.
were used for the simulation. The inlet pressure, as well as the local pressures across the tube, were calculated in every simulation step, in case of zero static pressure at the outlet.
estimated as the ratio of the sum of intersection areas of particles in the studied tube cross-section, relative to the total cross-sectional area of the tube. In Fig. 4, the longitudinal distributions of solids volume fractions, more exactly their cross-sectional averages ob-
4. Results and discussion 4.1. GraÕity flow Typical snapshots obtained by DEM simulation are shown in Fig. 3 for the three different flow regimes, using AVS visualization software. In this figure, similar to experimental observations w1,2x, the fundamental differences between these flow regimes can be clearly distinguished. From the analysis of the data obtained by simulation, several quantitative characteristics were also determined. Among them, the spatial distribution of solid particles within the studied tube is one of the most important feature to be studied. Generally, the local average of solids volume fraction in any subvolume of the tube can be determined according to Eq. Ž8., supposing uniform particle sizes by the term on the right side of the equation. N
Ý Õp Ž1y´ . s
is1
Vs
i
s
NÕp Vs
Ž 8.
where ´ is the voidage of the particle bed, and Ž1 y ´ . is the solids volume fraction in subvolume Vs . N and Õp are the number and volume of the particles, respectively, within the studied subvolume. By analogy, the average of solids volume fraction in a plain, e.g., in a given cross-section of the tube can be
Fig. 5. Variation of the solid volume fraction along a horizontal chord parallel to the mixer element surface; Ža. solids volume fraction vs. time, Žb. phase diagram.
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Fig. 6. Variation of the cross-sectional average of the solid volume fractions along the tube length for various mass flow rates.
tained after various times, are shown along the tube length, in case of a low solids mass flow rate Ž448 kgrh. corresponding to the 3rd flow regime. The onset of the quasisteady-state conditions can be clearly seen from the fact that after a given time, the curves are approaching each other, remaining in a rather narrow zone. After the onset of this quasi-steady-state situation, the differences between these curves represent only the stochastic variation and discrete nature of the particulate flow. Similar estimation was applied for the local average of solids volume fraction along a line lying in a given crosssection, by summing the lengths of the line sections falling
inside the particles and dividing this value by the total length of the line. These data are plotted in Fig. 5a in function of time from the start of simulation for a chord passing 1 mm above the outlet edge of the second mixer element and at 8 mm horizontally beside its surface. The onset of quasi-steady-state conditions during the process can be also clearly seen from this figure. After this, the variation of the solids volume fraction around a quasisteady-state average was caused by the discontinuous and stochastic nature of the particle flow. In the phase diagram shown in Fig. 5b, these solids volume fraction data were plotted versus the same data but
Fig. 7. Simulated state diagram for gravity flow of particles.
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J. Gyenis et al.r Powder Technology 104 (1999) 248–257
shifted by one D t time step. This figure clearly demonstrates the stochastic nature of the particle flow which could be revealed by this method of simulation. Fig. 6 shows the variation of the time averages of the solids volume fractions along the tube length, in case of various mass flow rates, after the onset of quasi-steady-state conditions. From this figure, it can be seen that in case of dense, sliding particle layer above the mixer section, i.e., in the 1st and 2nd flow regimes, after stepping into the static mixer section, the solids volume fraction was steadily decreasing downwards. In case of loose particle flow introduced into the upper tube section, i.e., in case of the 3rd flow regime, the particles were accelerating in this tube section. After arriving at the static mixer section, they slowed down, due to the resistance of the mixer elements against particle flow,
thus causing a temporary increase of the solids volume fraction. In the lower regions of the mixer elements, the particles were accelerating again, because they have found their optimal paths for flowing down. It has resulted in a decrease of the solids volume fractions again. In this figure, several local maximum can be seen along this tube section. The number of these maxima corresponds to the number of the mixer elements, due to the multiple impacts of particle flows coming from the previous mixer elements to the surface of the next ones. These results are in accordance with the experimental observations referred to above. It was also confirmed by the analysis of the simulated data, resulting in a simulated state diagram shown on Fig. 7. In this figure, the correlation between the mass flow rate and the average solids volume fraction are plotted. The feature of this diagram and the shape of the curves are similar to that obtained by
Fig. 8. Ža. Variation of the x-coordinates of five arbitrarily chosen particles in gravity flow. Žb. Variation of the x-directional velocities of the particles. Žc. Trajectories of the particles viewed from the flow direction.
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Fig. 8 Žcontinued..
the experiments. The lower curve, drawn by continuous line, corresponds to the conditions of the 3rd flow regime showing an increase of the solids volume fraction as a function of mass flow rate controlled by the feeding of particles at the top of the tube, in case of free outlet at the bottom. After arriving at a maximal value of the mass flow rate, which was about 2800 kgrh in this simulated case, the throughput of the tube with the given static mixer configuration could not be increased any more. At this point, the upper tube section has been choked, and the 3rd flow regime transformed to the 2nd flow regime. On the other hand, in case of limited outlet of the particles, there was a sliding, dense particle bed with maximal solids volume fraction both in the upper and bottom sections, as can be seen also in Fig. 3a. As regards the static mixer section, some distinct empty spaces or ‘‘air pillows’’ could be observed below the lower surface of the mixer elements even in the picture of the visualized simulation results, similar to experimental observations. In Fig. 7, this 1st flow regime is represented by the dotted upper curve. The only principal difference between this diagram and the state diagram measured by Gyenis w1x and Gyenis et al. w2x is, that in this simulated state diagram, the two curves touch each other at the point of the 2nd flow regime, while in the principal state diagram proposed on the basis of experimental observations, there was a broader gap between the right ends of the two curves.
The explanation of this difference between the simulated and experimental state diagram is obvious: the lengths of the upper and lower free tube sections used in the experiments were considerably higher. Therefore, in the experiments, the average solids volume fraction in the 3rd flow regime was highly depressed by the low solids volume fractions in the tube sections above and below the static mixer section. It is partly true also for the 2nd flow regime, due to the very low solids volume fraction in the lower tube section. The radial and longitudinal mixing effect of the static mixer tube during dry gravity flow also can be explained from the simulation data. In earlier studies w2,9x, the mixing effect of the static mixer elements have been estimated from the residence time distribution of the particles. In the present study, however, there was an additional possibility to trace the whole history of each individual particle during their pass through the mixer tube. It means that their trajectories, velocities, the occurrences of their collisions, their free path lengths, and a number of statistical characteristics of these data could be determined for any particle. These data gave additional information to clear up the mixing mechanism. As examples for these data, Fig. 8a and b show the variation of the x-coordinates and x-directional velocities of five arbitrarily chosen particles, respectively, in function of time while passing trough the mixer tube. Both plots
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Fig. 9. Snapshot of the simulated gas–solids two-phase flow.
demonstrate that the residence times of the individual particles can be rather different. These curves, showing the place and velocity of the particles, were interrupted after different time intervals, at the very time when the given particle left the tube. This is a clear graphical explanation of the broad residence time distribution obtained earlier by experiments or simulation w2,9x. Fig. 8a,b also provides an explanation for the good radial and longitudinal mixing effect of the static mixer elements. The radial mixing effect is even more obvious observing the trajectories of the particles from the direction of the particle flow, i.e., perpendicularly to the tube cross-section, as is shown for a few selected particles in Fig. 8c. From this figure, it can be seen that, in addition to the broad cross-sectional displacements of particles, their relative positions to each other have also changed, thus causing effective radial mixing. 4.2. Gas–solids two phase flow Using the simulation method and conditions described above, the movement of the particles, the fluid pressure and velocity field have been studied in a relatively short horizontal tube section containing one helical mixer element in gas–solids two-phase flow. Because of the short length of the tube and the high slip velocity between the gas and solid phases at the inlet, the steady-state conditions and the equilibrium slip velocity have not been approached in this study. But the results obtained till the present stage of this work have already demonstrated the applicability of DPS method for gas–solids two-phase flow, even in the case of very complex shape of flow channels formed between the surface of the helical mixer element and the cylindrical tube wall. Some interesting features of gas–solids two-phase flow were revealed by this simulation. In Fig. 9, a snapshot of the visualized results is shown, with clear signs of the settlement tendency of the particles along the inlet tube section before arriving at the mixer element. It can be more clearly seen from the particle trajectories in the left side section of Fig. 10. However, the special velocity field
evolved in the flow channels along the helical surface of the mixer element proved to be able to re-disperse the particles again. Another interesting observation is that the particles arriving to the upper half cross-section above the inlet edge of the mixer element have passed through this tube section more rapidly than those which were arriving below the inlet edge. This observation can be explained energetically. The particles that arrived at the lower half cross-section have lost some of their kinetic energy when the gas flow forced them to be lifted up to the upper half crosssection while passing through the mixer element. On the contrary, the other part of the particles were accelerating during their downward fly along the other side of the mixer element. This phenomenon, in addition to the mixing caused by the interaction with the helical surface, gives further enhancement to the axial mixing effect of the static mixer elements. In addition to this advantageous effect, the periodic variation of slip velocities between the solid and gas phases, due to the abovementioned difference in the particle velocities in the two sides of the mixer elements, can be used advantageously to improve the heat and mass transport during pneumatic transportation.
Fig. 10. Trajectories of the particles in gas–solids two-phase flow, side view.
J. Gyenis et al.r Powder Technology 104 (1999) 248–257 ™
fCt i j ™
fD ™ g GX I m ni j N ™
Fig. 11. Normalized residence time distributions of the particles.
The particle trajectories obtained by the simulation, shown in Fig. 10, and the broad residence time distribution of the particles caused by the mixer element plotted in Fig. 11 confirm these explanations. For the sake of comparison, the residence time distributions obtained in dry gravity flow and in gas–solids two-phase flow, both normalized to the respective mean residence time Žtrt ., are plotted in this figure. It seems that the normalized residence time distribution is significantly broader in gas–solids two-phase flow, indicating better axial mixing.
r
rs D t ™ TC ™ Õ Õp Vs ´ Ž1 y ´ . © t t
257
Tangential component of the contact force ŽN. Fluid drag force ŽN. Gravity acceleration vector Žmrs 2 . Mass flow rate of the solid phase Žkgrh. Moment of inertia of the particle ŽNm. Particle mass Žkg. Unit vector drawn from particle i to particle j Number of the particles in the reference volume Žarm3 . Position vector of the mass centre of the particle Žm. Radius of spheres Žm. Time step of the simulation Žs. Sum of torque caused by contact forces ŽNm. Velocity vector Žmrs. Volume of the particles Žm3 . Reference volume of the particle bed Žm3 . Voidage of the particle bed in the reference volume Solids volume fraction in the reference volume Angular velocity vector Ž1rs. Residence time Žs. Mean residence time Žs.
5. Conclusions In this study, carried out to simulate gravity flow of particles and gas–solids two-phase flow in tubes containing static mixer elements, it was confirmed that the great computational potential of the applied DPS method can be advantageously utilized even for very complex flows, evolving in the vicinity of helical static mixer elements. In case of gravity flow, the DPS simulation method proved to be a suitable tool to reproduce theoretically the main characteristics of the distinct flow regimes observed experimentally. Some important features of the gas–solids two-phase flows were also revealed by this simulation, mainly regarding the re-dispersing effect of the static mixer elements, their potential to improve axial mixing or the efficiency of other transport processes during pneumatic conveying. By excavating great amount of information from the data sets obtained by DPS simulation, such useful data and correlations can be produced which are not, or hardly measurable by experiments, helping us to get reasonable explanations for practical observations.
6. List of symbols d™ fC ™ fCn i j
Particle diameter Žm. Sum of the contact forces ŽN. Normal component of the contact force ŽN.
Acknowledgements The authors would like to acknowledge the support of the Hungarian National Foundation for Fundamental Researches ŽOTKA T16242, OTKA F16732., as well as the support given by JSPS ŽJapanese Society for Promotion of Sciences. and the Hungarian Academy of Sciences.
References w1x J. Gyenis, Chem.-Ing.-Tech. 64 Ž1992. 306. ´ w2x J. Gyenis, J. Arva, L. Nemeth, in: E.L. Gaden, G.B. Tatterson ŽEds.., ´ Industrial Mixing Technology, AIChE Symp. Ser., Vol. 90, No. 299. AIChE, New York, 1994, p. 144. w3x J. Gyenis, L. Nemeth, Proc. First Particle Technology Forum, Part III, ´ Denver, USA Ž1994. p. 454. w4x J. Nemeth, J. Gyenis, L. Nemeth, Proc. Partec ’95, 3rd Eur. Symp. ´ ´ Storage and Flow of Particulate Solids, Nurnberg, Germany Ž1995. p. ¨ 459. w5x Y. Tsuji, KONA 11 Ž1993. 57. w6x Y. Tsuji, T. Tanaka, T. Ishida, Powder Technol. 71 Ž1992. 239. w7x P.A. Cundall, O.D. Strack, Geotechnique 29 Ž1979. 47. w8x J.W. Prichett, T.R. Blake, S.K. Garg, in: Numerical Model of Gas Fluidized Beds AIChE Symp. Ser., Vol. 176, AIChI, New York, 1978, p. 134. w9x Zs. Ulbert, J. Szepvolgyi, J. Gyenis, Y. Tsuji, in: J. Bertrand, J. ´ ¨ Villermaux ŽEds.., Recent Progres en Genie ´ des Procedes, ´ Vol. 11, No. 51, Lavoisier, Paris, 1997, p. 299.
Discrete particle simulation of flow regimes in bulk solids mixing and conveying J. Gyenis a
a,)
, Zs. Ulbert b, J. Szepvolgyi ´ ¨
c,1
, Y. Tsuji
d
Research Institute of Chemical and Process Engineering, Pannon UniÕersity of Agricultural Science, P.O. Box 125, Veszprem, ´ Egyetem u. 2, Hungary b UniÕersity of Veszprem, ´ Veszprem, ´ Egyetem u. 10, Hungary c National Institute for Resources and EnÕironment, Tsukuba, Ibaraki, 356, Japan d Department of Mechanical Engineering, Osaka UniÕersity, Suita, Osaka 565, Japan
Abstract Earlier experimental studies revealed that, depending on the feeding and discharging conditions, three distinct flow regimes can evolve during the gravity flow of particles through a vertical tube containing static mixer element. In this paper, the results obtained by discrete particle simulation ŽDPS. are reported. Applying this simulation method for gravity flow, the main characteristics of the flow regimes observed experimentally could be reproduced and explained theoretically. Some important features of gas–solids two-phase flows in tubes containing static mixer element were also revealed. By excavating a great amount of information from the data obtained by DPS, useful data and correlations become available, which are not directly measurable by experiments. Thus, DPS method can help to get useful explanations on particle level for the experimental observations. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Particulate solids; Particle; Gravity flow; Gas–solids flow; Flow resistance; Static mixer; Discrete particle simulation
1. Introduction During gravity flow of particulate solids, and in gas– solids two-phase flow through pipes, the bends, contractions, nozzles, or static mixer elements inserted into certain tube sections have great influence on the flow pattern and on the characteristics of the particle movements, such as on their trajectories, spatial distribution, velocities, residence time distribution and on the mixing characteristics of the solid phase. From experimental study of gravity flow of particles through vertical tubes containing helical static mixer elements, Gyenis w1x and Gyenis et al. w2x have found that, depending on the feeding and discharging conditions, three typical flow regimes can occur. The studied tube system consisted of three distinct sections: between an upper and a lower tube section there was a mixer section containing a series of static mixer elements.
Besides this experimental study, the authors carried out investigation using simulation methods to elucidate the main features of the flow and mixing characteristics of particulate solids in such tube systems. For this, various methods were used, starting from phenomenological description to stochastic modelling and DPS simulation. Regarding this latter method, it was recognized that the huge amount of data which can be obtained by DPS simulation gives much more possibility for the investigation of these systems. DPS simulation data give a good chance to find what is happening on the particle level. It made possible the study of various statistical features and microscopic phenomena inducting global consequences in the particle bed. The majority of the studies that have been carried out by several authors in the field of DPS simulation have not yet utilized entirely this great potential.
2. Experimental observations ) Corresponding author. Tel.: q36-88-425-206; fax: q36-88-424-424; E-mail: [email protected] 1 Note: Permanent position of J. Szepvolgyi is at Pannon University, ´ ¨ Research Institute of Chemical and Process Engineering, Veszprem, ´ Egyetem u. 2, H-8201, P.O. Box 125, Hungary.
Gyenis w1x and Gyenis et al. w2x have reported experimental observations in such systems. It was stated that in case of unlimited particle inflow into a static mixer tube, when the discharge rate was lower than the maximal
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 1 0 2 - 3
J. Gyenis et al.r Powder Technology 104 (1999) 248–257
possible throughput determined by the resistance of the static mixer elements against particle flow, a dense sliding particle bed was present in the whole tube showing almost uniform solids volume fraction along its length. It was named 1st flow regime. Increasing the discharge rate within a limit, the mean solids volume fraction in the tube has remained almost constant. But, at higher mass flow rates approaching the maximal throughput of the tube, there was a slight decrease in the mean solids volume fraction. In case of unlimited particle inflow at the top and free discharge of the particles from the tube, the mass flow became equal to the maximal throughput of the tube. This critical state was named 2nd flow regime. Under this conditions, there was a dense, sliding particle bed in the tube section above the mixer elements. In the mixer section, accelerating particle layers with gradually diminishing thickness were sliding down on the surface of the helical mixer elements. Below this mixer section, accelerating thread-like particle streams were twisting downwards near the tube wall, with decreasing solids volume fraction. In case of limited inflow at the top and free outlet at the bottom, a loose and accelerating particle bed was formed both above and below the mixer section. On the surface of the mixer elements, sliding particle layers were present, resulting in somewhat higher mean solids volume fraction in this tube section. A general state diagram was proposed w1,2x to describe the main characteristics of these flow regimes and to indicate the necessary conditions for their transformation into each other. Besides the quite different spatial distribution pattern of the solid particles characterizing these flow regimes, several quantitative characteristics, such as the mean solids volume fraction, the residence time distribution, the axial mixing characteristics of the particle bed, etc., were also determined experimentally. Introducing a streaming gas phase, together with the solid particles, the flow pattern and particle bed characteristics have changed significantly w3,4x. These studies were carried out in downward vertical gas–solids flow in a tube, similar to the simple gravity flow of particles mentioned above. The aim of these investigations was to study the influence of the gas phase on the flow characteristics of the particle bed. These conditions had led to the so called 4th flow regime characterized by lean particle flow along the static mixer elements. In case of low solids loading ratios, this flow regime was similar to the third flow regime, but, due to the higher acceleration of the particles caused by the streaming gas phase, the solids volume fraction was exponentially diminishing from the inlet, gradually approaching to the gas velocity. The maximal throughput of the tube was considerably higher relative to the gravity flow of particles, without choking the tube. In case of high solids mass flow rate, however, this flow regime became similar to the second flow regime, but with higher solids throughput. By applying static mixer elements, the residence time of the solid particles and the
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velocity difference between the gas and solid phases could be enhanced. In spite of the numerous quantitative results and observations obtained experimentally, it must be emphasized that studying the influences of all important design and process parameters by experiments in these flow systems are very time-consuming, tedious and expensive. Therefore, during the last few years, an attempt was made by the authors to apply discrete particle simulation ŽDPS. method to study the influence of several parameters on the particle flow in such mixer tubes. The recent results of these studies are reported below.
3. The applied modelling and simulation method
3.1. Mathematical description of the particle motions
3.1.1. GraÕity flow of particles For this study, the discrete particle simulation ŽDPS., also called distinct element method ŽDEM. based on Lagrangian approach w5,6x, was applied. This method well corresponds to the discontinuous nature of the particulate systems and can be used advantageously both for simple gravity flow of particles and for dense or lean gas–solids two-phase flows. By this approach, the particle trajectories, particle–particle and particle–wall interactions are considered distinctly for each particle existing within the studied system. As it has been written in fundamental papers describing this method w5–8x, the individual particles have two types of motion, namely translational and rotational movements. Translation motion is caused by contact forces, fluid forces and gravity force. As regards the rotation of the particles, only the contact forces were considered. The equations to determine the acceleration of the particles during their translation and rotation are as follows: ™
™
r¨ s fC q f D rm q ™ g,
™
ž
™
™ ˙ s TCrI, v
/
Ž 1. Ž 2.
where ™ r is the position vector of™the mass centre of the particle,™m is the particle mass, fC is the sum of contact forces, f D is ™ the fluid drag force, ™ g is the gravity acceleration vector, TC is the sum of torque induced by contact forces, and I is the moment of inertia of the particle. ™ The contact force fC is described by a mechanical model shown in Fig. 1, first proposed by Cundall and Strack w7x. This model expresses the contact forces by a spring, a dash-pot and a friction slider. Maximum 12 particles can be in contact with particle i at the same time. The total contact force acting on particle
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Fig. 1. Model of the contact forces between two particles, according to Cundall and Strack w7x Ža. normal forces, Žb. tangential forces.
™
i is obtained by taking the sum of the normal fCn i j and ™ tangential fCt i j forces with respect to j: ™
™
ž sÝ žr n
™
/ /.
fC i s Ý fCn i j q fCt i j , j
TC i
s
™ ™ i j = f Ct i j
j
Ž 3. Ž 4.
The details of calculation of contact forces caused by interparticle and particle–wall collisions and friction, influenced by the material properties of the particles and the equipment wall, are described in the literature w5,6x. After a time step D t, the velocity and position of a particle can be given as Õ s™ Õ 0 q™ r¨0 D t ,
™
Ž 5.
™
r s™ r 0 q™ ÕD t ,
Ž 6.
™ ™ ™ ˙ v sv 0 qv0Dt,
Ž 7.
length and diameter were equal to the inner diameter of the tube, thus their length to diameter ratio was 1:1. From the whole tube length, 0.150 m was occupied by the mixer elements. Thus, the lengths of the free tube sections above and below the mixer elements were 0.025 and 0.010 m, respectively. The particle properties, such as interparticle and wall friction coefficients, size, density, stiffness and damping coefficient of the particles, have great influences on the particle–particle and particle–wall interactions. For the simulation of gravity flow, spherical polymer particles were taken into consideration with uniform sizes of 0.003 m. The material properties were as follows: densitys 1190 kgrm3 , stiffnesss 75 Nrm, restitution coefficients 0.6, friction coefficient between the particless 0.43, friction coefficient between the particles and tube wall s 0.33. During the simulation of the 3rd flow regime, the particles were introduced into the tube with various mass flow rates, evenly distributed across the whole inlet crosssection. In case of the 1st and 2nd flow regimes, they were withdrawn from the bottom of the tube with various mass flow rates, with unlimited inlet into the tube. By this method, it was possible to reproduce the three different flow regimes observed experimentally w1,2x.
™
where Õ is the velocity vector and subscript 0 denotes the previous value. 3.1.2. Modelling of gas–solids two-phase flow For the simulation of gas–solids two-phase flow in a cylindrical tube section containing one helical static mixer element, the same mathematical equations described above were used. But, in addition to Eqs. Ž1. – Ž7., several additional correlation had to be taken into consideration, due to the interactions between the particles and the streaming gas. The details of this two-phase flow simulation method were described in earlier papers w5,9x. In the present work, the viscosity terms were also considered. 3.2. Conditions of the simulations 3.2.1. GraÕity flow in Õertical tube The schematic diagram of the studied tube with the inserted static mixer elements is shown in Fig. 2. The length and diameter of the tube were 0.185 and 0.050 m i.d., respectively. Three helical mixer elements, each twisted to right hand direction, were placed into the tube closely after each other, with an angle of 308 between the outlet and inlet edges of the successive elements. Their
Fig. 2. Schematic diagram of the tube used for the simulation of gravity flow of particles.
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The mass flow rate of the particles were varied from 468 to 2880 kgrh. The number of particles treated simultaneously by the simulation was also changing between 4000 and 12,000.
3.2.2. Gas–solids two-phase flow in horizontal tube To prove the application possibility of DPS simulation method in gas–solids two-phase flow in tubes containing static mixer elements, several studies were carried out in a short horizontal tube in case of non-steady-state conditions. The length and inner diameter of the tube were 0.6 and 0.2 m i.d., respectively. One helical static mixer element with 1:1 length to diameter ratio was placed into the tube at 0.2 m from the tube inlet. The orientation of its inlet and outlet edges was horizontal.
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The number of particles introduced into the tube in one time step was constant, corresponding to 1440 kgrh mass flow rate. The superficial gas velocity was 15 mrs, corresponding to 1695 m3rh volumetric flow rate and 2035 kgrh mass flow rate. Thus, the loading ratio Žkg solids per kg gas. was 0.7, corresponding to 0.5% volumetric particle concentration under the inlet conditions. In the present stage of the study, the inlet velocity of the particles was varied from 1 to 14 mrs. Due to the high slip velocity between the solid and gas phases at the inlet, the particles in the studied tube section were accelerating, except at certain regions of the mixer element, because of its strong interaction with the particles. The properties of the particles were similar to that used for the gravity flow. Only the particle diameter was higher, namely 0.004 m. As for the gas phase, the properties of air
X Fig. 3. Snapshots of gravity flow in three different flow regimes obtained by simulation Ža. 1st flow regime, G s 2880 kgrh, Žb. 2nd flow regime, X X G s 1150 kgrh, Žc. 3rd flow regime, G s 720 kgrh.
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X
Fig. 4. Longitudinal distributions of solid volume fraction after different times in the 3rd flow regime, G s 448 kgrh.
were used for the simulation. The inlet pressure, as well as the local pressures across the tube, were calculated in every simulation step, in case of zero static pressure at the outlet.
estimated as the ratio of the sum of intersection areas of particles in the studied tube cross-section, relative to the total cross-sectional area of the tube. In Fig. 4, the longitudinal distributions of solids volume fractions, more exactly their cross-sectional averages ob-
4. Results and discussion 4.1. GraÕity flow Typical snapshots obtained by DEM simulation are shown in Fig. 3 for the three different flow regimes, using AVS visualization software. In this figure, similar to experimental observations w1,2x, the fundamental differences between these flow regimes can be clearly distinguished. From the analysis of the data obtained by simulation, several quantitative characteristics were also determined. Among them, the spatial distribution of solid particles within the studied tube is one of the most important feature to be studied. Generally, the local average of solids volume fraction in any subvolume of the tube can be determined according to Eq. Ž8., supposing uniform particle sizes by the term on the right side of the equation. N
Ý Õp Ž1y´ . s
is1
Vs
i
s
NÕp Vs
Ž 8.
where ´ is the voidage of the particle bed, and Ž1 y ´ . is the solids volume fraction in subvolume Vs . N and Õp are the number and volume of the particles, respectively, within the studied subvolume. By analogy, the average of solids volume fraction in a plain, e.g., in a given cross-section of the tube can be
Fig. 5. Variation of the solid volume fraction along a horizontal chord parallel to the mixer element surface; Ža. solids volume fraction vs. time, Žb. phase diagram.
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Fig. 6. Variation of the cross-sectional average of the solid volume fractions along the tube length for various mass flow rates.
tained after various times, are shown along the tube length, in case of a low solids mass flow rate Ž448 kgrh. corresponding to the 3rd flow regime. The onset of the quasisteady-state conditions can be clearly seen from the fact that after a given time, the curves are approaching each other, remaining in a rather narrow zone. After the onset of this quasi-steady-state situation, the differences between these curves represent only the stochastic variation and discrete nature of the particulate flow. Similar estimation was applied for the local average of solids volume fraction along a line lying in a given crosssection, by summing the lengths of the line sections falling
inside the particles and dividing this value by the total length of the line. These data are plotted in Fig. 5a in function of time from the start of simulation for a chord passing 1 mm above the outlet edge of the second mixer element and at 8 mm horizontally beside its surface. The onset of quasi-steady-state conditions during the process can be also clearly seen from this figure. After this, the variation of the solids volume fraction around a quasisteady-state average was caused by the discontinuous and stochastic nature of the particle flow. In the phase diagram shown in Fig. 5b, these solids volume fraction data were plotted versus the same data but
Fig. 7. Simulated state diagram for gravity flow of particles.
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shifted by one D t time step. This figure clearly demonstrates the stochastic nature of the particle flow which could be revealed by this method of simulation. Fig. 6 shows the variation of the time averages of the solids volume fractions along the tube length, in case of various mass flow rates, after the onset of quasi-steady-state conditions. From this figure, it can be seen that in case of dense, sliding particle layer above the mixer section, i.e., in the 1st and 2nd flow regimes, after stepping into the static mixer section, the solids volume fraction was steadily decreasing downwards. In case of loose particle flow introduced into the upper tube section, i.e., in case of the 3rd flow regime, the particles were accelerating in this tube section. After arriving at the static mixer section, they slowed down, due to the resistance of the mixer elements against particle flow,
thus causing a temporary increase of the solids volume fraction. In the lower regions of the mixer elements, the particles were accelerating again, because they have found their optimal paths for flowing down. It has resulted in a decrease of the solids volume fractions again. In this figure, several local maximum can be seen along this tube section. The number of these maxima corresponds to the number of the mixer elements, due to the multiple impacts of particle flows coming from the previous mixer elements to the surface of the next ones. These results are in accordance with the experimental observations referred to above. It was also confirmed by the analysis of the simulated data, resulting in a simulated state diagram shown on Fig. 7. In this figure, the correlation between the mass flow rate and the average solids volume fraction are plotted. The feature of this diagram and the shape of the curves are similar to that obtained by
Fig. 8. Ža. Variation of the x-coordinates of five arbitrarily chosen particles in gravity flow. Žb. Variation of the x-directional velocities of the particles. Žc. Trajectories of the particles viewed from the flow direction.
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Fig. 8 Žcontinued..
the experiments. The lower curve, drawn by continuous line, corresponds to the conditions of the 3rd flow regime showing an increase of the solids volume fraction as a function of mass flow rate controlled by the feeding of particles at the top of the tube, in case of free outlet at the bottom. After arriving at a maximal value of the mass flow rate, which was about 2800 kgrh in this simulated case, the throughput of the tube with the given static mixer configuration could not be increased any more. At this point, the upper tube section has been choked, and the 3rd flow regime transformed to the 2nd flow regime. On the other hand, in case of limited outlet of the particles, there was a sliding, dense particle bed with maximal solids volume fraction both in the upper and bottom sections, as can be seen also in Fig. 3a. As regards the static mixer section, some distinct empty spaces or ‘‘air pillows’’ could be observed below the lower surface of the mixer elements even in the picture of the visualized simulation results, similar to experimental observations. In Fig. 7, this 1st flow regime is represented by the dotted upper curve. The only principal difference between this diagram and the state diagram measured by Gyenis w1x and Gyenis et al. w2x is, that in this simulated state diagram, the two curves touch each other at the point of the 2nd flow regime, while in the principal state diagram proposed on the basis of experimental observations, there was a broader gap between the right ends of the two curves.
The explanation of this difference between the simulated and experimental state diagram is obvious: the lengths of the upper and lower free tube sections used in the experiments were considerably higher. Therefore, in the experiments, the average solids volume fraction in the 3rd flow regime was highly depressed by the low solids volume fractions in the tube sections above and below the static mixer section. It is partly true also for the 2nd flow regime, due to the very low solids volume fraction in the lower tube section. The radial and longitudinal mixing effect of the static mixer tube during dry gravity flow also can be explained from the simulation data. In earlier studies w2,9x, the mixing effect of the static mixer elements have been estimated from the residence time distribution of the particles. In the present study, however, there was an additional possibility to trace the whole history of each individual particle during their pass through the mixer tube. It means that their trajectories, velocities, the occurrences of their collisions, their free path lengths, and a number of statistical characteristics of these data could be determined for any particle. These data gave additional information to clear up the mixing mechanism. As examples for these data, Fig. 8a and b show the variation of the x-coordinates and x-directional velocities of five arbitrarily chosen particles, respectively, in function of time while passing trough the mixer tube. Both plots
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Fig. 9. Snapshot of the simulated gas–solids two-phase flow.
demonstrate that the residence times of the individual particles can be rather different. These curves, showing the place and velocity of the particles, were interrupted after different time intervals, at the very time when the given particle left the tube. This is a clear graphical explanation of the broad residence time distribution obtained earlier by experiments or simulation w2,9x. Fig. 8a,b also provides an explanation for the good radial and longitudinal mixing effect of the static mixer elements. The radial mixing effect is even more obvious observing the trajectories of the particles from the direction of the particle flow, i.e., perpendicularly to the tube cross-section, as is shown for a few selected particles in Fig. 8c. From this figure, it can be seen that, in addition to the broad cross-sectional displacements of particles, their relative positions to each other have also changed, thus causing effective radial mixing. 4.2. Gas–solids two phase flow Using the simulation method and conditions described above, the movement of the particles, the fluid pressure and velocity field have been studied in a relatively short horizontal tube section containing one helical mixer element in gas–solids two-phase flow. Because of the short length of the tube and the high slip velocity between the gas and solid phases at the inlet, the steady-state conditions and the equilibrium slip velocity have not been approached in this study. But the results obtained till the present stage of this work have already demonstrated the applicability of DPS method for gas–solids two-phase flow, even in the case of very complex shape of flow channels formed between the surface of the helical mixer element and the cylindrical tube wall. Some interesting features of gas–solids two-phase flow were revealed by this simulation. In Fig. 9, a snapshot of the visualized results is shown, with clear signs of the settlement tendency of the particles along the inlet tube section before arriving at the mixer element. It can be more clearly seen from the particle trajectories in the left side section of Fig. 10. However, the special velocity field
evolved in the flow channels along the helical surface of the mixer element proved to be able to re-disperse the particles again. Another interesting observation is that the particles arriving to the upper half cross-section above the inlet edge of the mixer element have passed through this tube section more rapidly than those which were arriving below the inlet edge. This observation can be explained energetically. The particles that arrived at the lower half cross-section have lost some of their kinetic energy when the gas flow forced them to be lifted up to the upper half crosssection while passing through the mixer element. On the contrary, the other part of the particles were accelerating during their downward fly along the other side of the mixer element. This phenomenon, in addition to the mixing caused by the interaction with the helical surface, gives further enhancement to the axial mixing effect of the static mixer elements. In addition to this advantageous effect, the periodic variation of slip velocities between the solid and gas phases, due to the abovementioned difference in the particle velocities in the two sides of the mixer elements, can be used advantageously to improve the heat and mass transport during pneumatic transportation.
Fig. 10. Trajectories of the particles in gas–solids two-phase flow, side view.
J. Gyenis et al.r Powder Technology 104 (1999) 248–257 ™
fCt i j ™
fD ™ g GX I m ni j N ™
Fig. 11. Normalized residence time distributions of the particles.
The particle trajectories obtained by the simulation, shown in Fig. 10, and the broad residence time distribution of the particles caused by the mixer element plotted in Fig. 11 confirm these explanations. For the sake of comparison, the residence time distributions obtained in dry gravity flow and in gas–solids two-phase flow, both normalized to the respective mean residence time Žtrt ., are plotted in this figure. It seems that the normalized residence time distribution is significantly broader in gas–solids two-phase flow, indicating better axial mixing.
r
rs D t ™ TC ™ Õ Õp Vs ´ Ž1 y ´ . © t t
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Tangential component of the contact force ŽN. Fluid drag force ŽN. Gravity acceleration vector Žmrs 2 . Mass flow rate of the solid phase Žkgrh. Moment of inertia of the particle ŽNm. Particle mass Žkg. Unit vector drawn from particle i to particle j Number of the particles in the reference volume Žarm3 . Position vector of the mass centre of the particle Žm. Radius of spheres Žm. Time step of the simulation Žs. Sum of torque caused by contact forces ŽNm. Velocity vector Žmrs. Volume of the particles Žm3 . Reference volume of the particle bed Žm3 . Voidage of the particle bed in the reference volume Solids volume fraction in the reference volume Angular velocity vector Ž1rs. Residence time Žs. Mean residence time Žs.
5. Conclusions In this study, carried out to simulate gravity flow of particles and gas–solids two-phase flow in tubes containing static mixer elements, it was confirmed that the great computational potential of the applied DPS method can be advantageously utilized even for very complex flows, evolving in the vicinity of helical static mixer elements. In case of gravity flow, the DPS simulation method proved to be a suitable tool to reproduce theoretically the main characteristics of the distinct flow regimes observed experimentally. Some important features of the gas–solids two-phase flows were also revealed by this simulation, mainly regarding the re-dispersing effect of the static mixer elements, their potential to improve axial mixing or the efficiency of other transport processes during pneumatic conveying. By excavating great amount of information from the data sets obtained by DPS simulation, such useful data and correlations can be produced which are not, or hardly measurable by experiments, helping us to get reasonable explanations for practical observations.
6. List of symbols d™ fC ™ fCn i j
Particle diameter Žm. Sum of the contact forces ŽN. Normal component of the contact force ŽN.
Acknowledgements The authors would like to acknowledge the support of the Hungarian National Foundation for Fundamental Researches ŽOTKA T16242, OTKA F16732., as well as the support given by JSPS ŽJapanese Society for Promotion of Sciences. and the Hungarian Academy of Sciences.
References w1x J. Gyenis, Chem.-Ing.-Tech. 64 Ž1992. 306. ´ w2x J. Gyenis, J. Arva, L. Nemeth, in: E.L. Gaden, G.B. Tatterson ŽEds.., ´ Industrial Mixing Technology, AIChE Symp. Ser., Vol. 90, No. 299. AIChE, New York, 1994, p. 144. w3x J. Gyenis, L. Nemeth, Proc. First Particle Technology Forum, Part III, ´ Denver, USA Ž1994. p. 454. w4x J. Nemeth, J. Gyenis, L. Nemeth, Proc. Partec ’95, 3rd Eur. Symp. ´ ´ Storage and Flow of Particulate Solids, Nurnberg, Germany Ž1995. p. ¨ 459. w5x Y. Tsuji, KONA 11 Ž1993. 57. w6x Y. Tsuji, T. Tanaka, T. Ishida, Powder Technol. 71 Ž1992. 239. w7x P.A. Cundall, O.D. Strack, Geotechnique 29 Ž1979. 47. w8x J.W. Prichett, T.R. Blake, S.K. Garg, in: Numerical Model of Gas Fluidized Beds AIChE Symp. Ser., Vol. 176, AIChI, New York, 1978, p. 134. w9x Zs. Ulbert, J. Szepvolgyi, J. Gyenis, Y. Tsuji, in: J. Bertrand, J. ´ ¨ Villermaux ŽEds.., Recent Progres en Genie ´ des Procedes, ´ Vol. 11, No. 51, Lavoisier, Paris, 1997, p. 299.