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Markov Models for Mixing of Powders in Static Mixers

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Markov Models for Mixing of Powders in Static Mixers THESE présentée à l’Ecole des Mines d’Albi-Carmaux pour obtenir LE TITRE DE DOCTEUR DE L’INSTITUT NATIONAL POLYTECHNIQUE DE TOULOUSE École doctorale : Science des Procédés Spécialité : Génie Procédés envirt Par M. PONOMAREV DENIS MODELES MARKOVIENS POUR LE MELANGE DES POUDRES EN MELANGEUR STATIQUE Soutenance prèvu le 09/11/2006 devant le jury composé de : THOMAS GÉRARD Rapporteur GUIGON PIERRE Rapporteur HEMATI MEHRDJI Membre DALLOZ BLANCHE Membre ZHUKOV VLADIMIR Membre GATUMEL CENDRINE Membre BERTHIAUX HENRI Directeur de thèse MIZONOV VADIM Co.directeur de thèse REMERCIEMENTS Je tiens ici à remercier deux principaux acteurs qui m’ont aidé dans ce travail et notamment Vadim Mizonov et Henri Berthiaux, mes directeurs de thèse, qui m’ont encadré durant ces trois dernière années et pour toute la confiance qu’ils ont su mettre en moi. Mes vifs remerciements vont aussi à monsieur Pierre Guigon, Professeur à l’Université de Technologie de Compiène, ainsi qu’à monsieur Gérard Thomas, Professeur de l’Ecole des Mines de Saint Etienne, pour avoir accepté d’être rapporteurs de ce travail. Je remercie également mesdames Blanche Dalloz et Cendrine Gatumel, messieurs Mehrdji Hemati et Vladimir Zhukov qui m’ont fait l’honneur de présider ce jury de thèse. Mes remerciements vont aussi à monsieur Janos Gyenis, Professeur de l’Université de Vezprem qui m’a donné quelques résultats d’une étude expérimentale des mélangeurs statiques. Enfin, mes pensées vont vers l’ensemble des personnes du centre RAPSODEE, qui ont permis la réalisation de ce travail dans une ambiance conviviale. Je tiens à dire mille mercis à ma famille qui a suivi le déroulement de la thèse et qui m’a témoigné un support moral. NOMENCLATURE b – transition probability of material going up; c – concentration; CV – coefficient of variation; D – diffusion coefficient; d – particle diameter [m]; d10 – accepted minimal diameter of particles [m]; d50 – average diameter of particles [m]; d90 – accepted maximal diameter of particles [m]; k – the number of mixer revolutions; L – the number of sections; m – the number of mixer cells; MA,B – the matrices of transition probabilities for the components A and B respec- tively; mph – mass of mixture after one cycle (a cycle here is considered to be the process time from charging to discharging) [kg]; mreq – required mass of final product [kg]; n – the number of samples taken; N – the number of sections in the mixing zone; P – the matrix of transition probabilities; pa –transition probability to the absorbing state; pb – transition probability from the further cell to the previous cell; pc – transition probability in the horizontal direction; pcA - probability of the component A to transit horizontally (2D model); pcB - probability of the component B to transit horizontally (2D model); pd – transition probability in vertical direction of the mixing zone; pf – transition probability to the further cell; pi j – transition probability from j-th to i-th state; ps – transition probability to remain in a cell after one transition; psA – probability of the component A to stay in a cell (2D model); psB - probability of the component B to stay in a cell (2D model); pА, рВ – transition probabilities to a further cell for the components A and B respec- tively; 1 qe(x,t) – function of source density; S – mean standard deviation; S – state vector; S i – elements of the state vector; 0 S 1 – initial state vector; 0 S z – state vector of material feed into the mixer; SA, SB – state vectors for the components A and B respectively; SfA1, SfB1 – state vectors of material feed into the first cell of the mixer the compo- nents A and B respectively; 2 Sopenflow – is the open flow area [m ]; SPAN - relative diversity of particle sizes with regard to d50; Sza, Szb – state column vectors for material in the columns of the loading container for the components A and B respectively; t – time; t1 - time of particle motion along the loading container till the first row of the screws [s]; t2 - time of particle motion through the mixing zone [s]; tcycle – total cycle time [s]; thandling – handling time [s]; tloading/unloading - the time of mixer loading and unloading [s]; tpassage – time of one passage [s]; u – transport coefficient; VRR – variance reduction ratio; X – state space; Xn – stochastic variable; z – number of transition; Greek letters σ 2 m – variance; α1, α2… αn – delay coefficients; µ – average concentration; 3 ρs- particle density [kg/m ]; υ – transport coefficient; ε – mixer porosity; 2 TABLE OF CONTENTS GENERAL INTRODUCTION……………………...………………………………… 5 CHAPTER I: STATE OF THE ART IN MIXING TECHNOLOGY AND ITS MATHEMATICAL SIMULATION…………………………………………….……… 9 1 Sampling size. Criteria of the mixture state. Mixture quality……………. 9 1.1 Sampling size……………………………………………………….……….. 9 1.2 Criteria of mixture quality………………………………………….………… 10 2 Mechanisms of mixing…………………………………………………………. 12 3 Mechanisms of segregation…………………………………………………... 15 4 Classification of mixing process and types of mixers according to different criteria…………………………………………………………………….… 18 4.1 General characteristics of particulate solid mixing……………………….. 18 4.2 Classification of mixing and mixer types…..…………………………….… 20 4.3 Mixers with agitating force……………………..……..…………………..... 22 4.4 Mixers with gravity force. Static mixers………………………………….… 22 5 Approaches to simulation of mixing. Basic classification..……..….....… 26 5.1 The scale of modeling. Lagrangian and Eulerian approach………..…… 26 5.2 Models of local and integral characteristics.…………………………….... 27 5.3 Current situation in mathematical modeling of particulate solid mixing.. 28 6 Modeling by means of Markov chain theory……………………………….. 33 6.1 General information about the Markov chains..………………………….. 33 6.2 Simulation of continuous flows…………………………………………….. 37 6.3 Simulation of batch mixing………………………………………………….. 41 6.4 Modeling flow in fluidized beds…………………………………….………. 44 6.5 Modeling segregation……………………………………………………….. 45 7 Conclusions on chapter 1 …………………………………………………….. 46 CHAPTER II : LABORATORY EQUIPMENT AND METHODS OF EXPERIMENTATION………………………………………………………………… 47 1 The laboratory static mixer (the “lab-made” mixer)…...………..………. 47 1.1 The mixer concept……….………………………………………………. 47 1.2 Methods of experimentation………………………………….…………. 48 2 Sulzer static mixer…………….…………………..…………………………… 49 2.1 The mixer concept……….………………………………………………. 49 2.2 Methods of experimentation……………………………………….……. 51 3 Comparing volumes of the mixers………………………………………….. 51 4 SysMix alternately revolving static mixer…………………………………. 53 4.1 The mixer concept……….………………………………………………. 53 4.2 Methods of experimentation………………………………….…………. 54 5 Materials……….……….……………………………………………………...… 55 CHAPTER III : ONE AND TWO DIMENSIONAL MARCOV CHAIN MODELS FOR STATIC MIXERS………………………………………………………………... 59 1 ONE dimensional Markov chain models for static mixers……..…………. 59 1.1 Model description. Algorithms of its numerical representation…………….. 59 3 1.1.1 A model scheme………….………………………………….……………. 59 1.1.2 The matrix of transition probabilities…………………………………….. 60 1.1.3 Transformation of microstates to macrostates…………..………….….. 63 1.1.4 Realization of mixer feeding in the model………………………………. 66 1.1.5 Consecutive and reversed material loading of static mixers…………. 67 1.2 Evolution of the mixture state and its numerical characteristics………….. 68 1.3 Results of mathematical modeling. Optimal number of revolutions.……… 71 1.3.1 Reversed loading…………………………………………………..……… 72 1.3.2 Consecutive loading……………………………………………….……… 77 1.4 Choice of the mixer……………………………………………………..……… 79 1.4.1 Mixing time definition……………………………………………………... 79 1.4.2 Comparison of reversed and consecutive loading in a static mixer…. 80 1.5 The length of the loading container………………………………………….. 83 2 A two-dimensional Markov chain model of axial-crosswise mixing in alternately revolving static mixers………………………………….…………….. 87 2.1 Model description. Algorithms of its numerical representation…...……….. 87 2.1.1 A model scheme…………..…………………………………….…………. 87 2.1.2 Realization of mixer feeding in the model...…………………………….. 87 2.1.3 Matrix of transition probabilities……………………………….………….. 89 2.2 Evolution of the mixture state and its numerical characteristics..…………. 94 2.2.1 Some words about the model and its parameters……………………... 94 2.2.2 Study of initial distribution influence on mixture quality……….……….. 94 2.2.3 Study of probability influence on mixture quality……………………….. 98 3 Conclusions on chapter III……………….….…………………………..……… 102 CHAPTER IV : EXPERIMENTAL STUDY OF STATIC MIXERS.…..…………… 104 1 Laboratory alternately revolving static mixer……………….………..……… 104 1.1 Identification of the model parameters…………………….………..…….. 104 1.2 Description of the experimental work…......………………………………. 106 1.3 Experimental and calculated results…...………………………………….. 107 2 Sulzer static mixer……………………………………………………….……..... 110 2.1 Installation scheme. Particularities of mixer feeding…….……………..… 110 2.2 Description of the experimental work…...…………………………………. 112 2.3 Experimental and calculated results………………...…………………….. 114 2.3.1 Non segregating mixtures……………………………….……………… 114 2.3.2 Segregating mixtures…………………………………….……………… 120 3 SYSMIX alternately revolving static mixer…….......…………….….………. 122 3.1 Identification of the model parameters ….…….………………………….. 122 3.2 Description of the experimental work…......………………………………. 123 3.3 Experimental and calculated results…...………………………………….. 124 4 Conclusions on chapter IV……………….…..………………………………… 129 General conclusions……………………………………………….………………... 130 References…………………………………………………………………………….
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