Studies of Particulate System Dynamics in Rotating Drums using Markov Chains

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1 Preprints of the 8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Studies of Particulate System Dynamics in Rotating Drums using Markov Chains Javan D. Tjakra, Jie Bao 1, Nicolas Hudon and Runyu Yang School of Chemical Engineering, School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052 Australia. Abstract: This work aims to develop an approach to study and capture the collective dynamics of particulate systems, which are important for operation of those processes. The collective dynamics of particles in a horizontal rotating drum are modeled based on a stochastic approach, using the Markov chains operators developed from DEM simulations. Quantitative analysis of the features of collective dynamics, namely dynamic modes of oscillatory behavior and spatial particle distribution of particle movement, are performed based on eigenvalues and singular values analysis of Markov chains operator, respectively. Furthermore, the quantitative measures are shown that it can be linked to the qualitative flow regimes of particulate systems in a horizontal rotating drum. Keywords: collective dynamics, particulate systems, discrete element method (DEM), Markov chains, eigenvalues analysis, singular values analysis 1. INTRODUCTION Horizontal rotating drums are commonly used in pharmaceutical, mining and chemical industries for operations such as milling, mixing and granulation processes which involve vast number of particles. The particle dynamic trajectories can be modeled using differential equations, such as the discrete element method (DEM). The DEM can provide detailed microscopic information of each particle behaviors including particle location, velocity and forces acting on each particle. However, it is the collective (overall/macroscopic) dynamics of all particles that is important in their operation. The collective dynamic features of particulate systems in a horizontal rotating drums include particulate mixing, segregation, avalanche and flow regimes (Ristow [2000], Ottino and Khakhar [2000], Mellmann [2001]). Many industrial particulate processes are required to operate under a specific flow regime to meet product specificationandthismotivatesthestudyofflowregimes.thework that links flow regimes to Froud number, filling degree and wall-friction coefficient is presented in (Mellmann [2001], Henein et al. [1983]). The experimental analysis of rolling and slumping flow regimes in relation to the angle of respose for a horizontal rotating drum system can be found in (Liu et al. [2005]). The experimental study of particle behaviors in relation to drum rotational speed, fill level and particulate size using a digital recording device is presented in (Santomaso et al. [2003]). The approaches mentioned above provide useful qualitative and coarse quantitative analysis of steady-state overall particle be- 1 Corresponding author. Tel address: j.bao@unsw.edu.au (J.Bao). haviors. However, they are incapable to capture dynamical features of the collective particle motions. There are some works that used stochastic models, such as Markov chains, to describe the collective particulate behavior. Markov chains models were constructed based on experimental data to describe and predict particulate distributions in static mixers (Chen and Fan [1972]) and in a hoop mixer (Aoun-Habbache et al. [2002]). More recently, Markov chains models were constructed using data generated from DEM simulation and used to fast calculate particulate mixing in a horizontal rotating drum (Doucet et al. [2008]). There are only few studies on the analysis of Markov chains models reported in literature. The eigenvalues of Markov chain operator were used to study the rate of convergence of the overall system behavior, e.g., the time needed for the system to reach a stationary particle distribution (Rosenthal [1995], Boyd et al. [2004]). The objective of this paper is to provide a systematic approach to determine more detailed dynamic features of the particulate system collective behavior modeled using Markov chains. The Markov chain operators are constructed using the information generated from DEM simulations as in (Doucet et al. [2008]). The eigenvalues study on the Markov chain operator (Rosenthal [1995], Boyd et al. [2004]) is extended to obtain the magnitudes and frequencies of the key dynamic modes (components of the overall dynamics) and use these measures to develop a collective dynamic index (CDI) representing the system oscillatory behaviors. Additionally, singular value decomposition (SVD) is also used to study the spatial distributions of particle movements. The results from the proposed approaches are compared with the qualitative analysis of flow regimes reported in (Yang et al. [2008, 2003], McElroy et al. [2009a]). IFAC, All rights reserved. 487

2 The paper is organized as follows. In Section 2, the DEM simulations for horizontal rotating drums and qualitative flow regimes are briefly reviewed. Markov chains theory used in the sequel is presented in Section 3. The proposed quantitative approaches to extract the features of collective dynamics from Markov chains operator are presented in Section 4. Conclusions and future works of the collective dynamical features of particulate systems are discussed in Section DISCRETE ELEMENT METHOD MODEL AND FLOW REGIMES In this work DEM simulations are used to provide microscopic information of each particle trajectory for the construction of the Markov chains operator. The DEM simulations used in the sequel follows (Yang et al. [2008]), where Newton s second law of motion is used to describe the translational and rotational motion of a particle of radius R i and mass m i, given as: (a) Rolling Regime (b) Cascading Regime m i dv i dt = n i,j=0 I i dθ i dt = n i,j=0 (F n ij +Fs ij +m ig) (1) R i F n ij µ rr i F n ij θ i, (2) where v i,θ i and I i are the transitional and angular velocities, and moment of inertia of particle i, respectively, and θ i represents a unit vector equal to θ i divided by its magnitude.thevectorr i isrunningfromthe centerofthe particle to the contact point with its magnitude equal to the particle radius R i. The first part of the right hand side in (2) representsthe torquedue to tangentialforcef s ij,the second part represents the rolling friction torque arising from the elastic hysteresis loss or viscous dissipation, where µ r is the coefficient of rolling friction. The quantities F n ij and F s ij represent, respectively, the normal contact force and the tangential contact force imposed on particle i by particle j. The reader is referred to (Yang et al. [2008]) for more details on DEM simulation. The DEM parameters used in the sequel are given below. Parameters Values Drum, D L (mm) Particle Size, d (mm) 3 Particle density, ρ (kg m 3 ) Loading (# particles, filling %), N p 2000(35%) Drum rotational speed, ω d (rpm) Number of cells, N s 88 The features of particulate collective dynamics in a horizontal rotating drums can be qualitatively classified into several types of flow regimes namely, rolling, cascading, cataracting and centrifuging regimes(henein et al.[1983]), shown in Figure 1. The main factors that affect the flow regimes are drum rotational speed, particle-drum size ratio and particle properties (size, density, etc.) Yang et al. [2003]. In this paper, we illustrate the proposed analysis by studying the effect of drum rotational speed. (c) Cataracting Regime (d) Centrifuging Regime Fig. 1. Particle flow regimes for various drum rotational speed, Ω, in a horizontal rotating drum with 2000 particles (Yang et al. [2008]) 3. MODELING OF PARTICULATE SYSTEM VIA MARKOV CHAINS 3.1 Markov Chains In this paper, the particle trajectories are modeled stochastically. Detailed information on stochastic processes can be found in (Hoel et al. [1986] and Bartlett [1966]). Markov processes are the simplest stochastic processes (Bremaud [2008]), where only the present states (but not the past states) affect the transition to future states (Chung [1967], Doucet et al. [2008], Behrends [2000]). Markov chains are Markov processes which operate at discrete time intervals and within a discrete state space which is used to model the trajectory of the particulate system. In this work, the computational domain Ψ of a horizontal rotating drum is decomposed into N cells based on the location, as shown in Figure 2. The state variables represent the fraction of number of particles in each cell (i.e., the number of particles in each cell divided by the total number of particles in the drum). As such, there are N states in the Markov chains model. For analysis purposes, the states are stacked to form a column vector called Markovianstates, S = [S 1 S 2... S N ] T, consistingelements which have a value between 0 to 1. Here, we consider how the states evolve with time after each drum revolution. Denote S k = [S 1 k S2 k... SN k ]T as the distribution of the number of particles in different cells after k revolutions (called k time steps in this paper), which are dependant only on the previous states of the system S k 1 mapped through a Markov chain operator (Doucet et al. [2008], Chung [1967], Behrends [2000]). 488

3 Fig. 2. The side view of horizontal rotating drum. Computational domain Ψ is divided into N number of cells where N = 88. Each element in Markov chain operator refers to the transition probability of particle movement from cell i to cell j, p ij,k, after one drum revolution (from the time step of k 1 to k). Let φ p (i,k) denotes an indicator function which takes the value 1 if particle p is in cell i at time k, and zero otherwise. Then the elements of the Markov chain operators to be constructed in the sequel, denoted p ij,k, is constructed by using the probability of transition of a particle p from the cell i at iteration k 1 to cell j at iteration k, i.e., p ij,k = P{φ p (j,k) = 1 φ p (i,k 1) = 1}. (3) As in (Chen and Fan [1972], Berthiaux and Mizonov [2004], Doucet et al. [2008], Marikh et al. [2006]), the process is assumed to be stationary, so that the transition probability p ij,k is assumed to be constant for each transition time step, i.e., p ij,k = p ij. The Markov property implies that the state of the chain can be written as S k+1 =PS k or (4) S k =P k S 0 (5) where P is the Markov chain operator or the transition probability matrix and S 0 denotes as vector of initial states. The elements of P are probability measure of transition between two cells or within the same cell, given as p ii... p ji P = (6) p ij... p jj The matrix P has the property that the summation of the elements over each column vector is unity and the value of each element is always greater or equal to zero and less than or equal to one because these elements are probability measure of particle movement (Chen and Fan [1972], Aoun-Habbache et al. [2002], Berthiaux [2000], Berthiaux and Mizonov [2004]). 3.2 Construction of Transitions Probability Matrix In this work, stationary matrix P is constructed using data generated from DEM simulations and each element of matrix P is computed as follows. The total number of particles in cell i at time step k is first calculated as N p ϕ(i,k) = φ p (i,k), (7) p=1 where N p is the total number of particles in Ψ. Then the probability transition of particle movement from cell i to j, p ij, can be obtained by p ij = 1 N LT 1 φ p (j,k)φ p (i,k 1). (8) N LT ϕ(i,k 1) k=1 p=1 The number of time steps needed for the learning process tocomplete isn LT.In the sequel,the dynamicsofthe cells are represented as a function of the time steps (number of revolutions of the drum) rather than the actual time so that the results with different drum rotation speeds can be comparable. 4. ANALYSIS OF FEATURES OF COLLECTIVE DYNAMICS This section introduces two approaches to quantitatively study the features of collective dynamics of particulate systems modeled using Markov chains. In particular, to study the dynamic modes and spatial particle distributions of particle movements based on the eigenvalues and the singular value analysis of Markov chains operator, respectively. 4.1 Dynamic Modes Analysis Each particulate process, with certain drum rotational speed, consists of different level of oscillatory behavior related to different frequency, refers as dynamic modes. This section introduces an approach to analyze the dynamic modes via the magnitude and the frequency of oscillation obtained from eigenvalues of the P matrix to quantify the oscillatory behavior of particulate distributions. We start formulating the approach by presenting the following concepts. The eigendecomposition of the matrix P can be presented as, Pv i = λ i v i, i = 1,...,N, (9) where λ i is the i-th eigenvalue of the P matrix and v i is the corresponding eigenvector. Then, the initial particle distribution, S 0, can be represented as the linear combination of all eigenvectors: S 0 = a 1 v 1 +a 2 v a N v N, (10) wherea i C arecoefficientsassociatedto the eigenvectors v i, i = 1,...,N. Using (5), we have: N p S k =P k (a 1 v 1 +a 2 v a N v N ) =a 1 P k v 1 +a 2 P k v a N P k v N =a 1 λ k 1v 1 +a 2 λ k 2v a N λ k Nv N, (11) where λ k N corresponds to the N-th eigenvalues of the P matrix at time step k. Equation (11) describes the dynamics of the particle distribution, as the linear combination of 489

4 the terms λ k i v i, (i = 1,...,N). As such, λ k i, as a function of time steps, represents a dynamic mode of the collective dynamics of the system, on the direction of v i. The eigenvalues can be complex. Assuming λ i = α i +jβ i, where j = 1, we have: λ k i = (α i +jβ i ) k =r k ie jkωi =r k i(cos(kω i )+j sin(kω i )), (12) where r i = α 2 i +β2 i (13) ( ) ( ) ω i =tan 1 Im(λi ) = tan 1 βi. (14) Re(λ i ) α i The four-quadrant arctangent is used in (14) (e.g., the atan2 function in MATLAB) so that ω is in the interval of [ π, π]. A negative real eigenvalue is associated with a fastest possible frequency of π, meaning that the corresponding dynamic mode oscillates every time step, finishing a cycle in two time steps. Note that ω i is a normalized frequency, which is the ratio of the actual oscillationfrequency,denotedas ω i,tothedrumrotational angular frequency, denoted as ω d : ω i = ω i. (15) ω d The fastest dynamic mode has an actual frequency of ω d 2. Substituting (12) into(11), we can have the total dynamics of the particle distribution as follows: S k =a 1 r k 1(cos(kω 1 )+j sin(kω 1 ))v 1 + a 2 r k 2(cos(kω 2 )+j sin(kω 2 ))v a N r k N (cos(kω N)+j sin(kω N ))v N. (16) Since complex eigenvalues appear in pairs of complex conjugates, all imaginary parts cancel outs. A collective dynamics index (CDI) which describes the overall oscillatory behaviors can be constructed based on the magnitudes, r = [r 1 r 2... r N ] T, and frequencies, ω = [ω 1 ω 2... ω N ] T, of all dynamical modes. Denote ˆω = [ ω 1 ω 2... ω N ] T. (17) The CDI can be quantified as follows: CDI = r, ˆω =r T ˆω (18) N = r i ω i. (19) i=1 Figures 3 show the values of CDIs against drum rotational speed for 2000 particles. It can be observed that with increasing rotational drum speed the CDI increases. A sudden drop to closed to zero can be observed and the index remains around zero with further increase (Ω 230rpm) in the drum rotational speed. This is to be expected, since the oscillatory behavior in particle distribution increases with increase in drum rotational speed, and as it reaches a certain limit, the centrifuging force dominate the system as a result the particle distribution in each cell remains relatively constant. It is interesting to see the CDIs are clearly related to the flow regimes reported in (McElroy CDI Rolling Regimes Cascading Regimes 2 Cataracting Regimes Centrifuging Regimes Drum rotational speed, Ω Fig. 3. r ˆω 1 plot against drum rotational speed et al.[2009a,b], Mellmann[2001]), summarized in the table below Particles Flow Regime Drum Speed (rpm) Rolling 0 Ω < 120 Cascading 120 Ω < 180 Cataracting 180 Ω < 230 Centrifuging 230 Ω 4.2 Spatial Distribution Analysis An approach to study the spatial distribution of particle movements based on SVD is presented in this section. This approach is useful to analyze the particulate movements and determine the locations where major particle movements occur. We start by computing the change of particle distributions between two time steps, from k to k + 1, described as follows: S k+1,k =S k+1 S k = PS k S k =(P I)S k (20) where I is an identity matrix. Denote ˆP = P I.Matrix ˆP can be understood as a mapping from the input S k to the output S k 1,k. It can be decomposed using SVD (Klema and Laub [1980]) as follows: ˆP = UΣX T (21) where U R N N and X T R N N are unitary matrices and Σ R N N is a diagonal matrix of singular values, where σ 1 > σ 2 >... > σ N 0. Therefore (20) can be written as S k+1,k =UΣX T S k σ 1 0 =[u 1 u N ]... [x 1 x N ] T S k 0 σ N =[σ 1 u 1 x T 1 + +σ N u N x T N]S k. (22) where u i = [u i1 u i2... u in ] T and x i = [x i1 x i2... x in ] T are left and right singular vectors, respectively, corresponding to the ith singular value, σ i. In the sequel, 490

5 N = 88 corresponding to the number of cells in the computational domain Ψ. Because U and X are unitary matrices, we have U T U = X T X = I. The set of vectors u i, i = 1,...,N form an orthonormal basis for the vector space of the output S k+1,k and the set of vectors x i, i = 1,...,N form an orthonormal basis for the vector space of the input S k. Any S k, can be represented as a linear combination of the x vectors, S k =α 1 x 1 + +α N x N =Xα, (23) where α = [α 1,...,α N ] T is the vector of the coefficients. The net changes in particle distribution after one time step shown in (22) can be represented as, S k+1,k =UΣX T Xα =UΣα =α 1 σ 1 u 1 + +α N σ N u N. (24) Therefore, S k+1,k is the linear combination of the left singular vectors u i with a coefficient of α i σ i, for i = 1,...,N. If S k is only related to one of the right singular vectors, e.g., S k = α i x i, then S k+1,k = α i σ i u i. If we quantify the magnitude of total particle distribution and its net change using the vector 2-norm: S k+1,k 2 = Then the i-th singular value Σ N q=1 ( Sq k+1,k )2, S k 2 = Σ N q=1 (Sq k )2. (25) σ i = S k+1,k 2 S k 2 (26) indicates how much net change in particle distribution will happen after one time step in relation to the previous step particle distribution (the gain of the mapping), in terms of their 2-norms. The change of particle distribution is proportionaltou i (withascalarmultiplierofα i σ i ).Therefore, u i is the typical spatial pattern of the distribution of net changes of particles in all cells. As σ 1 corresponds to the maximum singular value, u 1 shows the pattern of the changes of particle fractions in each cell related to the largest net change of particle distribution. Relocate each element of u 1 based on the cell geometry shown in Figure 2, the locations of the net changes in particle fractions (movement) can be determined. To illustrate the proposed approach, the u 1 (representing the largest net change of particle distribution) of a 2000 particles system operating on rolling (40 rpm), cascading (150rpm), cataracting (210 rpm) and centrifuging (230 rpm) flow regimes are used. The results are presented in Figure 4 with the legend shown in Figure 5. It can be seen that there are positive and negative elements corresponds to the percentage of particle moving in and out of the cells, respectively. The cells with high positive coefficients has high number of particles moving in while the cells with high negative coefficients has high number of particles moving out. The cells with zero coefficients has corresponds to zero net change in particles movement since there are no particle in these cells. The spatial patterns of net particle movement are very close to the reported flow regimes pattern in(mcelroy et al.[2009a,b]). (a) Rolling Regime (40 rpm) (b) Cascading Regime (150 rpm) (c) Cataracting Regime (210 rpm) (d) Centrifuging Regime (230 rpm) Fig. 4. Dominant spatial distribution for 2000 particles system 491

6 0.1 < <= < <= < <= < <= = = < <= < 0.1 < <= <= < <= 0.1 Fig. 5. Color table: percentage particle movement It is interesting to note that the left singular vector coefficients, u 1, for centrifuging regimes (230 rpm), shown in Figure 4d, are very close to zero. This is consistent with the observation of centrifuging flow regime where the particle distribution is constant. 5. CONCLUDING REMARKS This work proposed a systematic approaches to quantify the features of collective dynamics of particulate systems modeled using Markov chains. The Markov chains operators are constructed using data generated from DEM simulations. The operators are then analyzed using eigenvalues and singular values analysis to study the dynamic modes and spatial distribution of particle movements, respectively. The results from the proposed approaches are then validated using qualitative analysis of flow regimes reported in the literature. Current research focuses on extending the above results to dynamical modeling for varying drum operating conditions and control design of the features of particulate systems collective dynamics. REFERENCES Aoun-Habbache, M., Aoun, M., Berthiaux, H., and Mizonov, V. (2002). An experimental method and a Markov chain model to describe axial and radial mixing in a hoop mixer. Powder Technology, 128, Bartlett, M. (1966). An Introduction to Stochastic Processes, with Special Reference to Methods and Applications. Cambridge University Press. Behrends, E. (2000). Introduction to Markov Chains : With Special Emphasis on Rapid Mixing. Friedrich Vieweg and Sohn Verlag, Berlin. Berthiaux, H. and Mizonov, V. (2004). Applications of Markov chains in particulate process engineering: A review. The Canadian Journal of Chemical Engineering, 82, Berthiaux, H. (2000). Analysis of grinding processes by Markov chains. Chemical Engineering Science, 55(19), Boyd, S., Diaconis, P., and Xiao, L. (2004). Fastest mixing Markov chain on a graph. SIAM Review, 48, Bremaud, P. (2008). Markov Chains: Gibbs Fields, Monte Carlo Simulation and Queues. Springer, New York. Chen, S. and Fan, L. (1972). The mixing of solid particles in a motionless mixed - A stochastic approach. AIChE Journal, 18, Chung, K. (1967). Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin. Doucet, J., Hudon, N., Bertrand, F., and Chaouki, J. (2008). Modeling of the mixing of monodisperse particles using a stationary DEM-based Markov process. Computer and Chemical Engineering, 32, Henein, H., Brimacombe, J., and Watkinson, A. (1983). Experimental study of transverse bed motion in rotary kilns. Metallurgical and Materials Transaction B, Hoel, P., Port, S., and Stone, C. (1986). Introduction to Stochastic Processes. Waveland Press Inc, Boston. Klema, V. and Laub, A. (1980). The singular value decomposition: Its computation and some applications. IEEE Transactions on Automatic Control, 25, Liu, X., Specht, E., and Mellmann, J. (2005). Experimental study of the lower and upper angles of repose of granular materials in rotating drums. Powder Technology, 154, Marikh, K., Berthiaux, H., Mizonov, V., Barantseva, E., and Ponomarev, D. (2006). Flow analysis and Markov chain modelling to quantify the agitation effect in a continuous power mixer. Chemical Engineering Research and Design, 84, McElroy, L., Bao, J., Yang, R., and Yu, A. (2009a). A soft-sensor approach to flow regime detection for milling processes. Powder Technology, 188, McElroy, L., Bao, J., Yang, R., and Yu, A. (2009b). Softsensors for prediction of impact energy in horizontal rotating drums. Powder Technology, 195, Mellmann, J. (2001). The transverse motion of solids in rotating cylinders - Forms of motion and transition behavior. Powder Technology, 118, Ottino, J. and Khakhar, D. (2000). Mixing and segregation of granular materials. Annual Review of Fluid Mechanics, 32, Ristow, G. (2000). Pattern Formation in Granular Materials - Introduction, Pattern Formation in Granular Materials. Springer-Verlag, Berlin. Rosenthal, J.(1995). Convergence rates for Markov chains. SIAM Review, 37, Santomaso, A., Ding, Y., Lickiss, J., and York, D. (2003). Investigation of the granular behaviour in a rotating drum operated over a wide range of rotational speed. Chemical Engineering Research and Design, 81, Yang, R., Yu, A., McElroy, L., and Bao, J. (2008). Numerical simulation of particle dynamics in different flow regimes in a rotating drum. Powder Technology, 188, Yang, R., Zou, R., and Yu, A. (2003). Microdynamic analysis of particle flow in a horizontal rotating drum. Powder Technology, 130,