A Markov chain model to describe the Residence Time Distribution in a stirred bead mill

Handbook of Conveying and Handling of Particulate Solids A. Levy and H. Kalman (Editors) 9 2001 Elsevier Science B.V. All rights reserved.

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A M a r k o v chain m o d e l to describe the R e s i d e n c e T i m e D i s t r i b u t i o n in a stirred bead mill H. Berthiaux a, F. Espitalier a, J.C, Kiefer b, M. Niel a and V.E. Mizonov c a

Ecole des Mines d'Albi, campus Jarlard - route de Teillet - 81000 Albi (France)

b Rhodia- CRA, 52 rue de la Haie Coq - 93308 Aubervilliers (France) c Ivanovo State Power Engineering University, Rabfakovskaya 34 - 153003 Ivanovo (Russia) A Markov chain model is developed to describe fluid flow in a Netzsch stirred bead mill. On the basis of previous experience, a model comprising nine perfectly mixed cells interconnected, which corresponds to the stirring blades, was adjusted to experimental RTD curves through the determination of two recirculation ratios. It is also suggested that these are of a great value with respect to the grinding kinetics. 1. INTRODUCTION The interest for using the Markov chain theory to solve the many Chemical Engineering problems that combine kinetics and dynamics of systems has been recently pointed out by Tamir [1 ]. Examples of the simplicity, performance and "elegance" of this type of model are provided from many different fields in the literature such as: chemical reaction engineering (see Nassar et al. [2] or Too et al. [3]), gas adsorption (Raghuraman and Mohan [4]), or electrochemistry (Fahidy [5]). In the same way, a continuous grinding process can be viewed as a Markov chain combining the kinetics of grinding with the dynamics of powder or suspension flow through the mill. But while grinding mechanisms have been investigated from this stochastic point of view by various authors (Duggirala and Fan [6], Fan and Srivastava [7]), there seems to be no markovian study of the effect of the flow itself on the grinding performance. The objective of this work is to derive a Markov chain model for understanding the suspension flow behaviour through a complex horizontal stirred bead mill. This study is the very first step to the construction of an overall model of the mill as it only relates to pure fluid flow experiments, without transformation of the particles. It is also assumed that in such mills, the mixing effect of the blades is strong enough to hide local particle segregation with respect to the flow, so that in a first approach we will consider that fluid and suspension Residence Time Distributions (RTD) are the same.

2. RTD MODELLING IN STIRRED BEAD MILLS 2.1. Stirred bead mill modeling background Previous work (Heitzmann [8]) performed with coloured tracer experiments in a glass body version of a four blades Dyno mill showed that the action of the stirrer - beads system was first to delimit four perfectly mixed cells centered on each of the four blades. Further, it has been shown that classical models (plug flow, cascade of perfectly mixed cells, dispersion models) were unable to correctly represent RTD experiments. An internal recirculation loop model, with a single adjustable parameter R (see figure 1), was considered and gave very good results in continuous milling of suspensions of known grinding kinetics (Berthiaux et al. [9]). Another important conclusion of this work was concerned with the physical meaning of the recirculation ratio R which is undoubtedly linked with the local hydrodynamic conditions,

686 such as porosity, stirrer speed of rotation N, and perhaps mill flow-rate Q. It was also suggested that there exists an optimum value of R that leads to the best continuous grinding conditions (see figure 2). For example, low values of R benefits the flow itself as it approaches plug flow through tanks in series, while it clearly slows down the kinetics of grinding because the bead - particle collisions are of a lower intensity. In the absence of kinetic data, typical R-values should then range between 0.5 and 5. However, before generalising too much, these first conclusions need to be confirmed and the model perhaps adapted for the case of a stirred mill of different geometrical configuration. 2.2. Basis of stochastic RTD models

The procedure followed to obtain these RTD curves becomes tedious when dealing with a greater number of perfectly mixed cells, or better said a greater number of stirring blades, as it is the case for other types of mills. In general, the analytical or numerical derivation of the RTD from any ~ complex ~ model is in fact highly subjected to errors when done by the classical transfer function method. Particularly, many problems can be incurred when hypotheses are made to simplify the mathematical equations, which may lead to unrealistic dynamic behaviour (see Gibilaro et al. [10]). The advantage of using the Markov chain approach lies in the fact that it is systematic, and its application does not depend on the complexity of the flow scheme.

(a)

(b)

Fig. 1. Photograph of a glass body Dyno mill during a coloured tracer experiment showing the existence of perfectly mixed regions (a) ; corresponding Heitzmann's flow model (b).

Fig. 2. Influence of the recirculation ratio on the RTD (example of simulation for a Dyno mill).

687 A Markov chain is a system which can occupy various states, and whose evolution is defined once an initial state and the probability transitions between the states are fixed. It can therefore be said that a Markov chain does not have "memory ". In the case of flow problems (Fan et al. [11 ]), the system is a fluid element, the states are the perfectly mixed cells of the flow model (as plug flow can be represented by a series of such cells), and the probability transitions are fixed by elementary mass balances. For example (figure 3), the probability pii of remaining in cell i is exp(-At/zi), where At is the time interval under which the system is observed, and zi is the geometric residence time corresponding to cell i. The other transitions Pij depend then on the flow rate ratios and on the value of 1- exp(-At/~i), which is the probability of getting out of cell i during At. All these information are then collected in a probability transitions matrix P, whose rows (i) and columns (j) are the pij's. Further, the initial state of the system is represented by a single row E0, being En the state of system after n transitions (steps of duration At), which is available from the following matrix product (Eq.1): En-- Eo pn

(1)

The last element of En, which is the collecting cell or outlet of the network, represents therefore the dynamic response of the system to a perturbation that may be a tracer impulse: E0=[1 0 ... 0]. Simulation of the RTD curve of the model is further performed by letting At become smaller and smaller until the stability of the solution is ensured. 3. EXPERIMENTAL

The mill under consideration here is a Netzsch laboratory stirred bead mill, which is used for various applications in the fine chemicals industry. The mill chamber (approximately 1.2 1) is filled with steel grinding beads, and the separation between the beads and the suspension is ensured by a fine slot at the outlet. A cooling jacket controls the internal temperature of the vessel at a constant value. It can be said that the complexity of the flow through such an apparatus is due to the high filling ratio of the beads (80 %), together with the particular geometry of the stirrer system, as it is constituted by nine eccentric slotted disk blades, which are fixed on a driven shaft (see figure 4). Residence Time Distribution experiments has been performed by pumping the fluid through the mill from a tank that contained a salt solution of known concentration until it reaches a stationary regime. Then a step change in composition is made by suddenly passing to a water flow instead of the salt. The decrease of the outlet conductivity in a cell placed after the mill was followed as the dynamic response to this negative step.

Fig. 3. Markov chain representation of fluid flow in a vessel. The collector is the absorbing state of the chain.

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Fig. 4. Sketch of the experimental set-up showing the particular disposition of the blades in the Netzsch mill. 4. RESULTS AND DISCUSSION 4.1. RTD modeling of the Netzsch mill In the case of the Netzsch mill, the adaptation of Heitzmann's model clearly indicates to use ten perfectly mixed cells, resulting then in an 11 * 11 transition matrix because both measuring and collecting cells are to be included. For each cell i, we have to calculate the active volume Vi which is effectively experienced by the tracer flow. Let us denote by Vgi the geometrical overall volume corresponding to cell i and that can easily be determined from the quotations of the mill itself. Further, if Vsi is the volume occupied by the stirrer and the blade, f the filling ratio and ~ the bed porosity at repose, then:

V, = [(1 - f ) + f c](Vg~ - Vs, )

(2)

In the actual case, and with the notations of figure 5, it was found that the active volumes V of the seven central cells were all identical (V = 53.2 ml), while the extreme cells near the inlet and the outlet were of equivalent volumes: V1 = 102.8 ml and V2 - 101.8 ml respectively. On the other hand, the volume V0 including that of the measuring cell together with that of the separation chamber was estimated from the experimental set up (V0 - 45 ml). Being all the active cell volumes under consideration known, the maximum number of parameters in the Markov chain model is eight, the number of possible recirculation ratios Ri. Of course, this is a too large number of independent parameters to be properly determined, so that the fitting strategy we will further follow will result in increasing the number of parameters from one up to eight. Let us further define the flow rate ratios ri, that are only function of the Ri's and represent the volumetric proportion of the fluid which is retumed to cell i-1. For example, when referring to cell 2:

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Fig. 5. Flow model in a Netzsch mill with eight recirculation ratios. r~ =

R~

(3)

I+R 2 +R 1

The probability of staying in a definite cell or moving away from it, depends then on these ratios, through the geometric characteristic times zi. Again for cell number 2: v r2=(l+R

(4)

2 +R1) Q

The transition matrix P can then be constructed and used for simulating the dynamics" Pl

1- Pl

0

r~(1- P2 )

Pz

(1- r~)(1- P2 )

0

r2(1- P3)

P3

(1- r2)(1- p3 )

P=

rs(1- P9) P9 9

0

(1-r8)(1- P9) Po

.

.

.

.

.

.

0

0 1 -

Po

1

Once the size of At is specified, the probabilities are only dependent on the recirculation ratios. In practice, values of At not greater than the tenth of the mean residence time are sufficient to give a good accuracy to the calculations. In the following two sections, a fixed value of 0.01 s was imposed so that the calculation time for a current computer, including parameter identification, never reached one minute. 4.2. O n e - p a r a m e t e r model As a first try, we may consider a single parameter approach, as used by Heitzmann, in which all the recirculation ratios are the same: Ri = R for every i. Figure 6-a shows an

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(a) (b) Fig. 6. Fit of the experimental hydrodynamic response to a negative step injection by: (a) the one-parameter model ; (b) the two-parameters model. example of the results obtained under the following operating conditions that are currently used in the industrial practice: Q = 7.7 1.h1 ; N = 2000 rpm. It can be seen that this model does not give a good fit to the experimental results in the first part of the curve, or more precisely the fluid spends more time in the mill during the first times than predicted. This can probably be interpreted by differences in the local hydrodynamic behaviour between the central and the extreme cells, resulting in the existence of different recirculation ratios. While the fit is not acceptable, the value found for the internal recirculation ratio is yet very high (R = 124).

4.3. Two-parameters model Due to the equality of the volumes of the cells n~ to n~ is seems reasonable to assume the "regularity" of the flow and the equality of the re-circulation ratios: Ri = R for i = 1.... ,7. For the last cell (cell n~ the re-circulation ratio R8 may be affected by many factors, such as the difference in the corresponding active volume (which is twice that of the intermediate cell volumes), or a different impeller pumping action due to the proximity of the mill outlet. The resulting two-parameter (R, Rs) model gives a much better fit of the experimental RTD curve than the first model, as it can be seen from figure 6-b. As expected, the value of this final re-circulation ratio is much lower, while the central value is higher than for the onparameter model: R - 249 and R8 = 0.4. The values obtained for the central recirculation ratios are extremely high when compared to those taken into account for the simulation in figure 2. This indicates that the action of the stirrer results in a perfectly mixed central zone which is favourable to the grinding kinetics but not to the overall conversion in continuous regime. A flow model consisting of three perfectly mixed cells (including the measuring cell) with one single recirculation ratio would then give the same accuracy of the fit as the present model. It can be concluded that such a mill, driven under typical operating conditions, is well designed for batch operation but not for continuous grinding of suspensions. With respect to the Dyno mill, an increase of the number of the stirring blades is only efficient if the stirrer speed of rotation can be significantly reduced. This should be taken into account when scaling up of stirred mills. 5. CONCLUDING REMARKS From our own experience, with two stirred bead mills of different design, it can be said that fluid flow (and probably suspension flow) can be represented by internal recirculation loop models, and that a Markov chain is a very elegant way of developing and systemising these

691 models. The next step of this work may include the establishment of the link between these model parameters and the operating conditions. This may be performed by the establishment of correlations between the recirculation ratios and Froude, Reynolds and the Power Number through the help of a rigourous dimensional analysis However, for the case of some mills, such as the Netzsch mill, the great number of blades is only efficient in continuous milling if the flow conditions are gentler than, say, in a Dyno mill ... but in this case, the general flow can be affected by the presence of the particles, and the Residence Time Distribution be influenced by particle properties, such as particle size. From this would result an additional non-linear problem, as the markovian transitions become state dependent. NOMENCLATURE

N speed of rotation t time V, Vi cell volumes R, Ri re-circulation ratios P Transition matrix

rad.s l s m3 -

Q At ri pi, pij Ei

fluid flow-rate time interval recirculation ratios transitions state vector

m3.s -i s -

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

A. Tamir, Applications of Markov Chains in Chemical Engineering, Elsevier, 1998. R. Nassar, L.T. Fan, J.R. Too, L.S. Fan, A Stochastic Treatment of Unimolecular reactions in an Unsteady State Continuous Flow System, Chemical Engineering Science, 36, pp 1307-1317, 1981. J.R. Too, L.T. Fan, R. Nassar, Markov Chain Models of Complex Chemical Reactions in Continuous Flow Reactors, Computers and Chemical Engineering, 7 (1), pp 1-12, 1983. J. Raghuraman, V. Mohan, A Markov Chain Model for Residence Time Distribution and Contact Time Distributions in Packed Beds, Chemical Engineering Science, 30, pp 549-553, 1975. T.Z. Fahidy, Modelling of Tank Electrolysers via Markov Chains, Journal of Applied Electrochemistry, 17, pp 841-848, 1987. S.K. Duggirala, L.T. Fan, Stochastic Analysis of Attrition - A General Cell Model, Powder Technology, 57, pp 1-20, 1989. L.S. Fan, R.C. Srivastava, A Stochastic Model for Particle Disintegration - I : Grinding Mechanism, Chemical Engineering Science, 36, pp 1091-1096, 1981. D. Heitzmann, Caractdrisation des Opdrations de Dispersion-Broyage - Cas d'un Broyeur/~ Billes Continu pour la Dispersion des Pigments, PhD Thesis, INPL Nancy, 1992. H. Berthiaux, D. Heitzmann, J.A. Dodds, Validation of a Model of a Stirred Bead Mill by Comparing Results Obtained in Batch and Continuous Mode Grinding, International Journal of Mineral Processing, 44-45, pp 653-661, 1996. L.G. Gibilaro, H.W. Kropholler, D.J. Spikins, Solution of a Mixing Model due to Van de Vusse by a Simple Probability Method, Chemical Engineering Science, 22, pp 517523, 1967. L.T. Fan, J.R. Too and R. Nassar, Stochastic Simulation of Residence Time distribution Curves, Chemical Engineering Science, 40, 9, p 1743, 1985.