Discrete solution of the breakage equation using markov chains / Muammer Catak in Industrial & engineering chemistry research, Vol. 49 N° 17 (Septembre 1, 2010)  

Public ISBD [article]  inIndustrial & engineering chemistry research > Vol. 49 N° 17 (Septembre 1, 2010) . - pp 8248–8257 Titre : Discrete solution of the breakage equation using markov chains Type de document : texte imprimé Auteurs : Muammer Catak, Auteur ; Nursin Bas, Auteur ; Kevin Cronin, Auteur Année de publication : 2010 Article en page(s) : pp 8248–8257 Note générale : Chimie industrielle Langues : Anglais (eng) Mots-clés : Markov chains Breakage equation. Résumé : Analytical solution of population balance equations (PBEs) may be impossible except for some simple cases. In the literature there are a number of methods to solve PBEs including discrete methods, Monte Carlo simulation, and method of moments. In this paper, the Markov chain is presented as a discrete solution for a population balance equation of a breakage process for determining the particle size distribution (PSD) over time. The transition matrix P, which is the key operator of a Markov chain, is built using breakage equations. Thereafter, from calculating transition matrix, P, the particle size distribution of the system is easily evaluated using the Markov chain. According to simulation results, if the size range of the system is divided into a sufficient number of states and an appropriate transition time step was chosen, then results from the Markov chain are in agreement with the analytical solution of PBEs governed by the same breakage functions. In addition to theoretical illustration, the Markov theory was employed to model the breakage process of aggregated food products passing through a pneumatic conveying pipeline rig. DEWEY : 660 ISSN : 0888-5885 En ligne : http://pubs.acs.org/doi/abs/10.1021/ie100216g [article] Discrete solution of the breakage equation using markov chains [texte imprimé] / Muammer Catak, Auteur ; Nursin Bas, Auteur ; Kevin Cronin, Auteur . - 2010 . - pp 8248–8257. Chimie industrielle Langues : Anglais (eng) in Industrial & engineering chemistry research > Vol. 49 N° 17 (Septembre 1, 2010) . - pp 8248–8257 Mots-clés : Markov chains Breakage equation. Résumé : Analytical solution of population balance equations (PBEs) may be impossible except for some simple cases. In the literature there are a number of methods to solve PBEs including discrete methods, Monte Carlo simulation, and method of moments. In this paper, the Markov chain is presented as a discrete solution for a population balance equation of a breakage process for determining the particle size distribution (PSD) over time. The transition matrix P, which is the key operator of a Markov chain, is built using breakage equations. Thereafter, from calculating transition matrix, P, the particle size distribution of the system is easily evaluated using the Markov chain. According to simulation results, if the size range of the system is divided into a sufficient number of states and an appropriate transition time step was chosen, then results from the Markov chain are in agreement with the analytical solution of PBEs governed by the same breakage functions. In addition to theoretical illustration, the Markov theory was employed to model the breakage process of aggregated food products passing through a pneumatic conveying pipeline rig. DEWEY : 660 ISSN : 0888-5885 En ligne : http://pubs.acs.org/doi/abs/10.1021/ie100216g

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Markov chain modeling of fluidized bed granulation incorporating simultaneous aggregation and breakage / Muammer Catak in Industrial & engineering chemistry research, Vol. 50 N° 18 (Septembre 2011)  

Public ISBD [article]  inIndustrial & engineering chemistry research > Vol. 50 N° 18 (Septembre 2011) . - pp. 10811-10823 Titre : Markov chain modeling of fluidized bed granulation incorporating simultaneous aggregation and breakage Type de document : texte imprimé Auteurs : Muammer Catak, Auteur ; Kevin Cronin, Auteur ; Dario Medina-Tellez, Auteur Année de publication : 2011 Article en page(s) : pp. 10811-10823 Note générale : Chimie industrielle Langues : Anglais (eng) Mots-clés : Aggregation Granulation Fluidized bed Fluidization Modeling Résumé : Fluidized bed granulation to achieve particle size enlargement is a significant process in chemical engineering. A key aim of the modeling of such a system is prediction of the evolution of the particle size distribution with time. The traditional tool to achieve this is through the use of population balances. This Article proposes an alternative modeling approach of Markov chain, which is a well-known tool in stochastic theory. Specifically, in this Article, it is employed to model bottom spray Wurster type fluidized bed granulation and a one-dimensional chain where the particle size is the state variable was used. Three important elements of Markov chains are the initial state vector a(t), the transition time step τ, and the transition matrix P. The transition matrix was assembled from consideration of breakage and aggregation dynamics. Experiments were conducted to validate the approach using glass beads as the elementary particles and polyethylene glycol as the liquid binder. From the results, it is shown that the Markov chain method is an efficient and accurate tool to analyze the particle size evolution of a size enlargement process. DEWEY : 660 ISSN : 0888-5885 En ligne : http://cat.inist.fr/?aModele=afficheN&cpsidt=24523898 [article] Markov chain modeling of fluidized bed granulation incorporating simultaneous aggregation and breakage [texte imprimé] / Muammer Catak, Auteur ; Kevin Cronin, Auteur ; Dario Medina-Tellez, Auteur . - 2011 . - pp. 10811-10823. Chimie industrielle Langues : Anglais (eng) in Industrial & engineering chemistry research > Vol. 50 N° 18 (Septembre 2011) . - pp. 10811-10823 Mots-clés : Aggregation Granulation Fluidized bed Fluidization Modeling Résumé : Fluidized bed granulation to achieve particle size enlargement is a significant process in chemical engineering. A key aim of the modeling of such a system is prediction of the evolution of the particle size distribution with time. The traditional tool to achieve this is through the use of population balances. This Article proposes an alternative modeling approach of Markov chain, which is a well-known tool in stochastic theory. Specifically, in this Article, it is employed to model bottom spray Wurster type fluidized bed granulation and a one-dimensional chain where the particle size is the state variable was used. Three important elements of Markov chains are the initial state vector a(t), the transition time step τ, and the transition matrix P. The transition matrix was assembled from consideration of breakage and aggregation dynamics. Experiments were conducted to validate the approach using glass beads as the elementary particles and polyethylene glycol as the liquid binder. From the results, it is shown that the Markov chain method is an efficient and accurate tool to analyze the particle size evolution of a size enlargement process. DEWEY : 660 ISSN : 0888-5885 En ligne : http://cat.inist.fr/?aModele=afficheN&cpsidt=24523898

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