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Preprint

The effect of polydispersity on dust lifting behind shock waves Catalin G. Ilea, Pawel J. Kosinski and Alex C. Hoffmann Dept. of Physics and Technology, University of Bergen Allegt. 55, 5007 Bergen, Norway Abstract Knowledge-based modeling of dust lifting behind shock waves is a prerequisite for realistic simulation of dust explosions. Mostly numerical simulations of this process focus on dusts consisting of monodisperse particles, while real dusts are polydisperse. This article investigates the effect on the lifting process of the dust being polydisperse with a log-normal distribution of particle sizes. The spatial distribution of the various sizes in the rising layer is studied, and statistical results for the rise, the collision frequency and the particle kinetic energy are compared for polydisperse and monodisperse dusts. It is shown that a layer consisting of polydisperse particles rises significantly faster than a one consisting of monodisperse particles, all other parameters being the same. Key words: PACS: 02.60.Cb, 02.70.Bf, 47.11.Df, 47.40.Nm, 47.55.Kf Preprint submitted to Elsevier Powder Technol. 196 (2009), 194–201 1 Introduction Dust lifting is the process of dust entrainment behind pressure waves or shock waves which can be generated either intentionally or accidentally and has applications in process technology and process safety. Although this type of process has been the subject of experimental [1–3] and numerical studies, the mechanisms at work are still not well understood. Understanding dust entrainment from deposited dust layers is important in the fields of process safety and dust explosion mitigation. Commercial codes for the simulation of large-scale dust explosions depend on a fundamental understanding of the entrainment of deposited dust for knowledge-based dust entrainment algorithms in their Eulerian-Eulerian dust explosion models [4]. More generally, dust entrainment is also of interest in the field of geophysics, for example to understand dust entrainment in the air due to various types of atmospheric flow phenomena. The field is also receiving renewed interest in the study of dust entrainment in extra-terrestrial environments, for example in the martian atmosphere or of lunar dust due to the landing or take-off of spacecraft. Two main modelling techniques are used to simulate dust entrainment. One is the Eulerian-Eulerian technique, where both phases (i.e. the gas phase and the particle phase) are treated as fluids governed by the Euler equations. The second technique, increasingly used in numerical studies, is the EulerianLagrangian method. Here, particles are considered as point-like objects which are individually tracked inside the computational domain with the aid of Newton’s second law of motion. 2 One of the main advantages of using the Eulerian-Lagrangian simulation technique for multiphase flows is that it is able to treat particles individually, so that the model can be closer to the real physics of the multiphase flow. This makes it possible to account for a number of particle features, such as varying particle sizes and particle shapes. Real dusts do not consist of monodisperse particles. Nevertheless, published numerical studies of dust lifting behind shock waves have, till now, considered only dusts consisting of spherical, monodisperse particles [3, 5–15]. A few numerical studies of particle flow in pipes [16–18] do, however, consider polydisperse particles. Two of the most frequently used functional forms for particle size distributions are the log-normal distribution and the Rosin-Rammler distribution, both of which fit the size distributions of most real dusts quite well. The log-normal distribution is used in this work. 2 The mathematical model In this paper the gas phase is modelled using the Euler equations modified to account for interphase interactions: ∂ρ + ∇ · (ρu) = 0 ∂t " conservation of mass # ∂u + u · (∇u) + ∇p = SI ρ ∂t ∂E + ∇ · (u(E + p)) = SII ∂t conservation of momentum conservation of energy (1) (2) (3) The time and space dependent variables are: the gas pressure, p, the gas density, ρ, the gas velocity, u and the total energy of the gas per unit volume, 3 E. The source terms SI and SII are the coupling terms and can be written: SI = − Np X and fp,i i SII = − Np X i fp,i · vi (4) where Np is the number of particles per unit volume, fp,i is the force acting on particle i due to the fluid and vi is the velocity of particle i. We did not take into account heat transfer between the gas and the particles in the coupling. Exploratory simulations showed the effect of this to be small. We note that effects of heat transfer can be much more important if the gas and the particles have significantly different initial temperatures, for example if the gas is flue-gas from a primary explosion. The gas equation of state determines the density as a function of the pressure and closes the conservation of momentum equations: p = ρRg T (5) with Rg the specific gas constant and T is the temperature. The total energy, E is related to the temperature and the velocity by: E= ρRg T ρu2 + γ −1 2 (6) where γ is the adiabatic index of the gas, the ratio of heat capacities at constant pressure and constant volume. Particles are tracked in a Lagrangian frame of reference with the aid of Newton’s second law of motion. mi |u − vi |(u − vi ) dvi = CD Ap ρ + mi g dt 2 (7) with CD the drag coefficient and Ap the area of the particle projected on a plane normal to direction of the fluid flow relative to the particle. CD depends 4 on the particle Reynolds number: Re = ρdi |u − vi |/µ. An empirical formula due to Clift et al. [19] was used to calculate CD from Re. An empirical compressibility correction suggested by Crowe et al. [20] was applied to CD when the relative velocity became comparable to the velocity of sound. The voidage fraction in the initial layer was ǫ = 0.95 (see below), and the well-known drag correction factor for the packing: f (ǫ) = ǫ−2.65 equals 1.145 at this voidage. It was nevertheless chosen in this work not to apply a packing-correction to the drag, since the correction would be negligible in most of the flow during most of the simulation. Also the computational cost in determining the local voidage around each particle was rather high. The most important effect of the packing was through the effect on the gas flow, which was taken into account by the two-way coupling scheme. This study was mainly aimed at studying the effect of particle-particle and particle-wall interaction on the lifting process, and we did not consider Saffman or Magnus lift forces acting on the particles. Gosteev and Fedorov [8] and Thevand and Daniel [9] did consider the effect of lift forces in their studies, which did not consider particle collisions, while Zydak and Klemens [15] did not find them to be very significant in their simulations. Crowe et al. [20] write that if the relative particle Reynolds number is larger than the shear Reynolds number, Saffman forces are not likely to be very significant. This condition is satisfied in our study, which is limited to the initial stage of the lifting process where the relative gas-particle velocity is high. Observations during an earlier study of the authors [21] indicated that Magnus forces are not very significant either for the conditions we are studying here. We do acknowledge that lift forces may well play a significant role, particularly after the very initial period covered by the present work. 5 Linked to this is the issue of the aerodynamic torque. This is not taken into account here. The main role of aerodynamic torque in this work will probably be to attenuate the rotation caused by collisions. This will in turn affect lift forces and subsequent collision. Later in the process, where the particles are dilute, rotation induced by aerodynamic torque may influence the lift acting on the particles. Taking into account aerodynamic torque is a priority for subsequent work. A second-order flux corrected transport algorithm (LCPFCT) was used [22] to solve the fluid flow equations. The position of each particle and its velocity are calculated every time-step according to the following numerical scheme: mi vi n+1 − vi n (un+1 − vi n+1 )|un − vi n | = CD Ap ρ + mi g ∆t 2 n+1 rx,y,z;i = n rx,y,z;i n+1 n (vx,y,z;i + vx,y,z;i ) , + ∆t 2 (8) (9) where n is the time-step index, and (x, y, z) are the Cartesian coordinates. The angular velocity of the particles is only calculated as a consequence of the collisions, and is not updated between collisions due to the interaction with the surrounding fluid. The collision algorithm is based on the hard-sphere model in [20], which involves a coefficient of restitution and a Coulombian coefficient of friction. Since we are studying the effect of particle size, which may include segregation in the gravity field, particle gravity should be considered as it acts differently on every particle. Therefore, a gravity force term was added in addition to the drag force acting on the particles. A log-normal distribution function was used to generate values for the diameter of the particles. This function represents the probability distribution of a 6 random variable whose natural logarithm is normally distributed. If a random variable R has a normal distribution function, then exp(R) has a log-normal distribution; likewise, if R is log-normally distributed, then ln(R) is normally distributed. The functional form for the log-normal probability density distribution is: (ln dp − µ)2 f (dp ; µ, σ) = exp − , 2σ 2 dp σ 2π ! 1 √ (10) where dp is the particle size, while µ and σ are the mean and standard deviation of the normally distributed logarithm of dp . As mentioned, the hard-sphere model of Crowe et al. [20] was used to account for the inter-particle and particle-wall interactions. This model is one of the standard collision models used in the research literature. A detailed account of its implementation in the in-house developed software is given in [23]. According to this model, the post collisional values of the translational particle velocities, v1 and v2 , can in general be written as: m2 ; m1 + m2 m1 v2 = v2,(0) + A , m1 + m2 v1 = v1,(0) − A (11) where v1,(0) and v2,(0) are the precollisional velocities and A is a vector, the form of which depends on whether the particles stop sliding (and therefore start rolling) over each other during the collision or not. For the details of this model, and its derivation, we have to refer to the textbook of Crowe et al. [20]. Thus the difference between the pre-and post-collisional translational velocities depends on the product of the vector A, which is a function of the precollisional translational and rotational motion of the particles, and a term, m2 /(m1 + m2 ) or m1 /(m1 + m2 ), which depends on the relative sizes of the particles involved in the collision. 7 3 Simulation domain and parameters The simulations are three dimensional. The physical domain dimensions are 30 cm × 1 cm × 0.3 cm, with a uniform grid and a side length of 0.25 mm for the cubic computational cell. An air shock wave was generated by a pressure chamber separated from the main chamber containing the particles by an imaginary membrane, bursting at time t = 0. The boundary conditions at the front and end of the domain were of the zerogradient type by setting the velocity and pressure in the external boundary cells equal to the velocity and pressure in first/last internal cells. The boundary conditions at the side and top and bottom walls were zero slip and zero normal velocity by setting the tangential and normal velocities in the external boundary cells to mirror those in the internal ones. The particles were considered spherical with a particle density of 1000 kg/m3 . 5000 particles were simulated with sizes distributed using a log-normal random number generator [24]. The distribution parameters were: µ = −9.23 and σ = 0.2. This gives a number frequency size distribution with a mean of 100 µm and a standard deviation of 20.2 µm. An example of such distribution is shown in Figure 1a. The distribution is not very wide when considering real dusts; this was chosen to avoid problems due to the largest particles exceeding the size of the computational grid. This study aims to ascertain the nature of any effect of polydispersity on the lifting process and, as shown below, the width used was enough to elucidate that. 8 −7 a) 3 500 2.5 400 2 Mass [kg] Number frequency 600 300 1 100 0.5 1 Particle diameter [m] 0 0 2 b) 1.5 200 0 0 x 10 1 Particle diameter [m] −4 x 10 2 −4 x 10 Fig. 1. Particle size distribution for 5000 particles: a) Number distribution; b) Mass distribution. In order to compare simulations for this polydisperse powder with simulations of a monodisperse powder, a comparable particle size must be found for the latter. This was taken as the mass mean particle size, i.e. the mean size for the distribution shown in Figure 1b, which turned out to be 104.1 µm. We note that Loth et al. [25] define an effective mean diameter based on a drag/gravity balance. In the inertial range where the drag coefficient is constant, this reduces to the Sauter mean diameter, which for the present log-normally distributed powder is: exp[µ + 5σ 2 /2] = 108.4 µm. In our simulations the drag coefficient is constant due to the large relative velocity between the gas and the particles during the simulation. The initial positions of the particles in the layer were obtained using an algorithm that placed the particles at regularly spaced positions, which where subsequently altered by a small random value. The spacing was done so loose that initial overlaps between neighbouring particles was avoided, this restriction was more severe in the case of polydisperse particles. For this reason the volume fraction of the particles in the initial layer was approximately 5%. The monodisperse layer was initially placed using the same algorithm, so that 9 identical dimensions and volume fractions of the layers were obtained. With the present simulation method it was not possible to arrange the particles in a layer initially lying on the floor. To do this correctly would require simulations in which the particles are larger than the computational grid and the flow and pressure fields around the particles are resolved, the particles constituting moving boundaries in the computational domain. For the hard-sphere collision model, the value of the friction coefficient was set at 0.15 while the restitution coefficient was 0.8 in all simulations. Each type of simulation was repeated three times in order to check the reproducibility of the results. 4 Validation of the code Before we present the simulation results, we briefly describe the way in which the code was validated. The ability to simulate a shock wave correctly was tested by simulating conditions similar to those in a simple, one-phase shock tube and compare the results with theory. Three shock waves with Mach numbers 1.5, 2.0 and 2.5, respectively, were generated by suddenly removing a membrane between high-pressure and lowpressure sections of a 20 × 3 × 3 cm computational domain. The physical properties of air were used, and the temperature set to 293.15 K. Both the ends of the domain were open. The problem of shock wave propagation in a shock tube can be solved analytically and the velocity of the shock wave and the gas velocity and pressure behind it can be calculated directly [26]. The comparison is shown in Table 1. 10 The agreement is satisfactory. Table 1 Comparison of shock tube simulations with theory Simulation Theory Wave Mach number 1.5 2.0 2.5 1.5 2.0 2.5 Gas velocity behind wave [m/s] 239 430 602 238.3 429.0 600.6 Pressure behind wave [bar] 2.45 4.48 7.06 2.46 4.50 7.13 The correctness of the implementation of the particle drag law (including the correction for compressibility) was ascertained by carrying out simulations of a single particle in a shock tube. In a more severe test of the code, two experiments in the paper of Boiko et al. [27] were reproduced in simulations. Boiko et al. performed experiments studying the interaction between a shock wave and clouds of bronze and acrylic-plastic particles. Two of these experiments were reproduced here, both involving clouds of acrylic-plastic particles with a diameter of 300 µm and a density of 1200 kg/m3 and with a shock wave with a Mach number of 2.8. The initial particle concentration in the clouds were 0.1% and 3.0%, respectively. Figure 2 shows a comparison between the results from the simulations and the experiments. It can be seen that the agreement is very good for both initial concentrations. The motion of the particle cloud is due to the action of drag on the particles, which is influenced both by two-way coupling and particle collisions. This comparison is therefore a severe test of the validity of the model and its implementation. Also the detailed shape of the shock wave as it passes the 11 50 experimental computations experimental computations 45 Displacement [mm] 40 0.1% (vol.) 0.1% (vol.) 3% (vol.) 3% (vol.) 35 30 25 20 15 10 5 0 0 50 100 150 200 Time [s] Fig. 2. Comparison between experiment [27] and simulation of the motion due to a 2.8 Mach shock wave of the front of two clouds of 300 µm plastic particles with different initial volume concentrations particle cloud was found [28] to agree with the equivalent visualizations of the wave in the paper of Boiko et al. as shown in Figure 3. 5 Simulation results The presence of the particles has an effect on the shock wave. This is illustrated in Figure 4, where the pressure profile just after the shock wave has passed over the layer is shown as a color-coded 3-D plot. A high-pressure region just before the layer and a low-pressure region just after are indicated by arrows. Figures 5 and 6 present the vertical positions of the particles and their vertical velocity component as functions of particle size at four points in time during the simulation. At the beginning of the simulation, after 0.2 ms, the vertical particle position 12 a) b) Fig. 3. a) Pressure profile obtained from the simulations; b) snapshot of wave structure obtained by Boiko et al., reproduced from Figure 6b in Boiko et al. [27] with kind permission from Springer Science+Business Media Fig. 4. Profile of the pressure just after the passing of the shock wave. A high-pressure region just before and a low-pressure region just after the layer are indicated with arrows. 13 a1) a2) 1500 Vertical velocity [cm/s] Vertical position [cm] 1 0.8 0.6 0.4 0.2 0 50 100 150 1000 500 0 −500 −1000 200 50 Particle size [microns] 100 b1) 1500 Vertical velocity [cm/s] Vertical position [cm] 200 b2) 1 0.8 0.6 0.4 0.2 0 150 Particle size [microns] 50 100 150 200 1000 500 0 −500 −1000 Particle size [microns] 50 100 150 200 Particle size [microns] Fig. 5. 1) Distribution of the vertical position of the particles as a function of particle size. 2) distribution of the vertical component of the velocity as a function of particle size. a) Results at 0.2 ms. b) Results at 0.5 ms. can be seen to have changed only little. However, the smaller particles already possess significant values of the vertical velocity. Thus collision processes take place within the layer, but have not yet resulted in significant lifting. At 0.5 and 1.0 ms the elevation and the vertical velocity of the particles both increase successively, both with a clear bias for the smaller particles. However at the last time of 1.6 ms the vertical velocity can be seen to have decreased compared to that at 1.0 ms, especially for the small particles. The elevation, however, has increased. Thus it would seem that the collision intensity has decreased, as one would expect since the voidage in the layer is increasing, 14 a1) a2) 1500 Vertical velocity [cm/s] Vertical position [cm] 1 0.8 0.6 0.4 0.2 0 50 100 150 1000 500 0 −500 −1000 200 50 Particle size [microns] 100 b1) 1500 Vertical velocity [cm/s] Vertical position [cm] 200 b2) 1 0.8 0.6 0.4 0.2 0 150 Particle size [microns] 50 100 150 200 1000 500 0 −500 −1000 Particle size [microns] 50 100 150 200 Particle size [microns] Fig. 6. 1) Distribution of the vertical position of the particle as a function of particle size; 2) Distribution of the vertical component of the velocity as a function of particle size. a) Results at 1.0 ms; b) Results at 1.6 ms. and that drag has reduced the vertical velocity of the particles, especially the small ones. Figures 5 and 6 thus show clear increases in the spreads in vertical position and the vertical velocity with decreasing particle size. The effect on particle position might be thought to be due to particle segregation in the gravity field, but a solution of the particle equation of motion shows that a 100 µm particle will only move 0.001 mm in 0.5 ms and 0.005 mm in 1.0 ms when starting from rest, so that gravity is unlikely to have an observable effect. This was confirmed by simulations where gravity was neglected. The two Equations (11) 15 do, however, offer a clue. They show that the collisional change of velocity for e.g. particle 1: v1 −v1,(0) is proportional to the ratio of the mass of particle 2 to the sum of the particle masses, i.e. the change of velocity of the lighter particle will, as one intuitively knows, be the larger, giving rise to more displacement of the lighter particles, although this will be tempered by the higher drag force acting on them relative to their inertia. Figure 7 compares the number average height of the particles as a function of time: h(t) = N 1 X zi (t) N i=1 (12) for the polydisperse and monodisperse powders. In this and in the following figures the three repeat simulations mentioned earlier in which different random displacements of the particles in the initial layer were used, are all shown, such that the reproducibility of the results can be gleaned. Significantly higher values for the average height are obtained for the polydisperse powder over the entire simulated time frame. The lifting effect begins earlier and with an increased intensity for the polydisperse powder. 0.16 Average height [cm] 0.14 Distributed sizes Uniform sizes 0.12 0.1 0.08 0.06 0.04 0 0.5 1 Time [s] 1.5 2 −3 x 10 Fig. 7. Average height of particles as a function of time. 16 The influence of gravity is visible during the early part of the simulation when the drag force acting on the particle is weak. This can be observed in Figure 8 where a drop in the average height of particles, consistent with what was mentioned above, is noticed. However, for the remainder of the simulations, gravity has, as mentioned, a negligible effect. Average height [cm] 0.056 0.0559 Distributed sizes Uniform sizes 0.0558 0.0557 0.0556 0.0555 0 1 2 3 Time [s] 4 5 −4 x 10 Fig. 8. Average height of particles as a function of time.Detail showing the initial behaviour and the effect of gravity. Figure 9a and b present the evolution of the cumulative number of collisions and the collision frequency, respectively, as functions of simulated time. Here it can be seen that, the polydisperse particles start to collide early and continue to collide with a frequency that is nearly constant for much of the time. Toward the end of the simulation, the number of collisions begins to settle towards a constant value, due to the increased voidage in the layer. This is consistent with what was found when following the evolution in time of the vertical particle velocity. In contrast with this, the collision process starts very slowly in the monodisperse particles, followed by a sharp increase. In the end the collision process settles down for this powder too. 17 Comparing the above-mentioned charts, it can be seen that the longer delay in observing the lifting effect of the monodisperse particles is related to the collision process. As soon as the collision frequency increases drastically, the particles begin to exhibit higher average height values. However, the lifting effect for the monodisperse particles is not as strong as for the polydisperse ones, where the constant collision frequency lead to higher average heights. 4 7 x 10 a) 12 Collision frequency [collisions/s] 8 Number of collisions 7 6 5 4 3 2 Distributed sizes Uniform sizes 1 0 0 0.5 1 1.5 Time [s] x 10 Distributed sizes Uniform sizes 10 8 6 4 2 0 0 2 b) 0.5 1 Time [s] −3 x 10 1.5 2 −3 x 10 Fig. 9. a) Cumulative number of collisions as a function of time; b) Collision frequency as a function of time. Figures 10–12 present the evolution of the particle averaged kinetic energy in the three coordinate directions, x, y and z. Obviously, the displacement is by far the largest in the direction of the fluid flow, x. Figure 10 shows that during the first half of the process, up to 1 ms, only little particle displacement takes place, although Figure 9 shows that this is the period of higher collision frequency. This is the period during which the layer compresses as the particles facing the flow behind the shockwave are dragged by the flow, while the particles further in the layer are largely unaffected by the flow due to the shielding effect. This suggests that displacement of particles only takes place after the shielding effect is overcome and all the particles are 18 brought into motion, partly by the forces created by the collisions in the layer. Collisions are the main cause for change in particle kinetic energy in the vertical and lateral directions. Both of the charts in Figures 11 and 12 exhibit similar trends. It can be seen that at the time at which the collisions frequency starts to decrease, so does the the kinetic energy of the particles in the plane perpendicular to the direction of the flow. Also the collision process is more effective in transferring particle kinetic energy to these directions in the case of polydisperse particles, in spite of the fact that the collision frequency is not always the higher in this case. −6 Kinetic energy along x axis [J] 1.4 x 10 1.2 Distributed sizes Uniform sizes 1 0.8 0.6 0.4 0.2 0 0 0.5 1 Time [s] 1.5 2 −3 x 10 Fig. 10. Particle averaged kinetic energy along the x-axis as a function of time. The contention that the collisions are more effective in the case of polydiserse particles can be supported by analyzing the evolution in time of the mechanical energy losses of the particles due to collisions (Figure 13). The calculation of the mechanical energy losses due to dissipation in collisions is described in detail—at least for two identical particles—in [23]. Even though the number of collisions is smaller for the polydisperse particles, the collisions are more intense and give rise to the loss of more mechanical energy. It therefore ap19 −10 Kinetic energy along y axis [J] 8 x 10 7 6 5 4 3 2 Distributed sizes Uniform sizes 1 0 0 0.5 1 1.5 Time [s] 2 −3 x 10 Fig. 11. Particle averaged kinetic energy along the y-axis as a function of time. −9 Kinetic energy along z axis [J] 1.2 x 10 1 0.8 0.6 0.4 Distributed sizes Uniform sizes 0.2 0 0 0.5 1 Time [s] 1.5 2 −3 x 10 Fig. 12. Particle averaged kinetic energy along the z-axis as a function of time. pears that the polydisperse particles interact more vigorously with the solid environment around them. 6 Conclusions It can be concluded that simulations of a layer containing polydisperse particles rendered results that are significantly different from the case of monodis20 −4 Energy lost in collisions [J] 1 x 10 0.8 Distributed sizes Uniform sizes 0.6 0.4 0.2 0 0 0.5 1 Time [s] 1.5 2 −3 x 10 Fig. 13. Total lost energy by particles in collisions. Values are cumulated. perse particles. The particles with smaller sizes than the mean value of the distribution were found to contribute the most to the lifting effect, as shown by the evolution in time of their vertical position. The overall lifting effect appears more intense in the case of polydisperse particles. The main disparity between the two powders, which may give rise of this difference in lifting, was in the evolution of the collision frequency, which for the polydisperse case exhibited an almost constant value, while in the monodisperse powder it exhibited a delayed spike, taking on significantly higher values. Even though the number of collisions remained at higher values for the monodisperse powder until the end of the simulations, these did not contribute significantly to the lifting effect. This difference in the nature and intensity of collisions between the two powders is confirmed when comparing data both for the average mechanical energy of particles in the plane perpendicular to the direction of the flow and for lost mechanical energy due to collisions. 21 Results for the mean squared displacement and the particle averaged kinetic energy in the flow (x) direction showed that particles were significantly displaced only after the collision process had reduced in intensity and most of the particles had been lifted. This observation is valid for both cases. The monodisperse particle layer exhibited slightly less displacement due to the reduced lifting effect that lead to a slightly more compact layer. Overall, it can be said that the difference in the simulation results of a model for polydisperse particles is significant enough to warrant the inclusion of polydispersity in detailed dust lifting simulations. Acknowledgments The authors gratefully acknowledge funding from the Research Council of Norway during this project. References [1] B. Fletcher, The interaction of a shock with a dust deposit, Journal of Physics D: Applied Physics 9 (1976) 197–202. [2] K. Lebecki, K. Cybulski, J. Sliz, Z. Dyduch, P. Wolanski, Large scale grain dust explosions-research in Poland, Shock Waves 5 (1995) 109–114. [3] R. Klemens, P. Zydak, M. Kaluzny, D. Litwin, P. Wolanski, Dynamics of dust dispersion from the layer behind the propagating shock wave, Journal of Loss Prevention in the Process Industries 19 (2006) 200–209. 22 [4] R. K. Eckhoff, Dust Explosions in the Process Industries, Gulf Professional Publishing, 2003. [5] G. Ben-Dor, Dust entrainment by means of a planar shock induced vortex over loose dust layers, Shock Waves 4 (1995) 285–288. [6] R. Klemens, P. Kosinski, P. Wolanski, V. P. Korobeinikov, V. V. Markov, I. S. Menshov, I. V. Semenov, Numerical study of dust lifting in a channel with vertical obstacles, Journal of Loss Prevention in the Process Industries 14 (2001) 469–473. [7] A. V. Fedorov, N. N. Fedorova, I. A. Fedorchenko, V. M. Fomin, Mathematical simulation of dust lifting from the surface, Journal of Applied Mechanics and Technical Physics 43 (2002) 877–887. [8] Y. A. Gosteev, A. V. Fedorov, Calculation of dust lifting by a transient shock wave, Combustion, Explosion, and Shock Waves 38 (2002) 322–326. [9] N. Thevand, E. Daniel, Numerical study of the lift force influence on two-phase shock tube boundary layer characteristics, Shock Waves 11 (2002) 279–288. [10] R. Klemens, P. Zydak, M. Kaluzny, D. Litwin, P. Wolanski, Mechanism of dust dispersing from the layer by propagating shock wave in the flow without obstacles, Proceedings of the Fifth International Symposium on Hazards, Prevention and Mitigation of Industrial Explosions (2004) 189–198. [11] P. Kosinski, A. C. Hoffmann, Modelling of dust lifting using the Lagrangian approach, International Journal of Multiphase Flow 31 (2005) 1097–1115. [12] P. Kosinski, A. C. Hoffmann, R. Klemens, Dust lifting behind shockwaves: Comparison of two modelling techniques, Chemical Engineering Science 60 (2005) 5219–5230. [13] A. V. Fedorov, I. A. Fedorchenko, Computation of dust lifting behind a shock 23 wave sliding along the layer. Verification of the model, Combustion, Explosion, and Shock Waves 41 (2005) 336–345. [14] A. V. Fedorov, I. A. Fedorchenko, I. V. Leontev, Mathematical modeling of two problems of wave dynamics in heterogeneous media, Shock Waves 15 (2006) 453–460. [15] P. Zydak, R. Klemens, Modelling of dust lifting process behind propagating shock wave, Journal of Loss Prevention in the Process Industries 20 (2007) 417–426. [16] M. Sommerfeld, Modelling of particle-wall collisions in confined gas-particle flows, International Journal of Multiphase Flow 18 (1992) 905–926. [17] N. Huber, M. Sommerfeld, Modelling and numerical calculation of dilute-phase pneumatic conveying in pipe systems, Powder Technology 99 (1998) 90–101. [18] S. Lain, M. Sommerfeld, J. Kussin, Experimental studies and modelling of fourway coupling in particle-laden horizontal channel flow, International Journal of Heat and Fluid Flow 23 (2002) 647–656. [19] R. Clift, W. H. Gauvin, The motion of particles in turbulent gas streams, Proceedings Chemeca ’70 1 (1970) 14. [20] C. Crowe, M. Sommerfeld, Y. Tsuji, Multiphase flows with droplets and particles, CRC Press LLC, 1998. [21] P. Kosinski, A. C. Hoffmann, An Eulerian-Lagrangian model for dense particle clouds, Computers & Fluids 36 (2007) 714–723. [22] J. P. Boris, A. M. Landsberg, E. S. Oran, J. H. Gardner, LCPFCT - FluxCorrected Transport algorithm for solving generalized continuity equations, NRL Memorandum Report 6410-93-7192, Washington, DC, 1993. 24 Naval Research Laboratory, [23] C. G. Ilea, P. Kosinski, A. C. Hoffmann, Three-dimensional simulation of a dust lifting process with varying parameters, International Journal of Multiphase Flow DOI: 10.1016/j.ijmultiphaseflow.2008.02.007. [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical recipes in FORTRAN 77: The art of scientific computing. Second edition, Cambridge University Press, 1992. [25] E. Loth, T. O’Brien, M. Syamlal, M. Cantero, Effective diameter for group motion of polydisperse particle mixtures, Powder Technology 142 (2004) 209– 218. [26] J. D. Anderson Jr., Modern Compressible Flow with Historical Perspective, 3rd Edition, McGraw–Hill, Inc., 2002. [27] V. M. Boiko, S. P. Kiselev, A. N. Papyrin, S. V. Poplavsky, V. M. Fomin, Shock wave interaction with a cloud of particles, Shock Waves 7 (1997) 275–285. [28] C. G. Ilea, Numerical simulations of dust lifting, Ph.D. thesis, University of Bergen, Dept. of Physics and Technology (2008). 25