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Wiechert AI, Ladshaw AP, Kim YH, Tsouris C, Yiacoumi S. Simulation of radioactive plume transport in the atmosphere including dynamics of particle aggregation and breakup. JOURNAL OF ENVIRONMENTAL RADIOACTIVITY 2023; 263:107167. [PMID: 37023525 DOI: 10.1016/j.jenvrad.2023.107167] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2022] [Revised: 03/17/2023] [Accepted: 03/26/2023] [Indexed: 05/06/2023]
Abstract
Accurate prediction of the atmospheric transport of debris particles relies heavily on our knowledge of the size distribution of the particles within a debris cloud. Assuming a fixed particle size during simulations is not always viable since the size distribution of the debris can change during transport. Various microphysical processes, such as aggregation and breakup, influence debris particles and dictate any changes to the size distribution. To track those changes that can occur, a population balance model can be adopted and instituted within a model framework. Nonetheless, many of the models that simulate the transport of radioactive debris following a device-driven fission incident have historically neglected to consider these processes. As such, this work describes our effort to develop a modeling framework capable of simulating the transport and deposition of a radioactive plume generated from a fission incident with a dynamic population balance including particle aggregation and breakup. The impact of aggregation and breakup, individually and collectively, on the particle size distribution is explored using the developed framework. When simulating aggregation, for example, six mechanisms, including Brownian coagulation, the convective enhancement to Brownian coagulation, van der Waals-viscous force correction for Brownian coagulation, gravitational collection, turbulent inertial motion, and turbulent shear, are considered. Brownian coagulation and its corrections have, as one would expect, a large impact on relatively small aggregates. Aggregates with a diameter that is less than or equal to 1.0 μm, for instance, comprise 50.6 vol % of all aggregates in the absence of aggregation and 31.2 vol % when Brownian coagulation and its corrections are considered. Gravitational collection and, to a much lesser extent, turbulent shear and turbulent inertial motion are, conversely, of great importance to relatively large aggregates (i.e., diameter greater than 3.0 μm). Additionally, the individual effects of atmospheric and particle parameters, such as wind speed and particle density, are examined. Of the parameters examined, turbulent energy dissipation and aggregate fractal dimension (i.e., aggregate shape with lower values representing more irregular particles) were of substantial importance since both terms directly impact aggregate stability and, by extension, the breakup rate. Large-scale transport and deposition simulations in a dry atmosphere are also presented and discussed as a proof of concept.
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Affiliation(s)
| | | | - Yong-Ha Kim
- Louisiana State University, Baton Rouge, LA, 70803, USA
| | - Costas Tsouris
- Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
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DAS A, Paul J, Heinrich S, Kumar J. Development and analysis of moments preserving finite volume schemes for multi-variate nonlinear breakage model. Proc Math Phys Eng Sci 2023. [DOI: 10.1098/rspa.2022.0658] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/31/2023] Open
Abstract
Modelling and simulation of collisional particle breakage mechanisms are crucial in several physical phenomena (asteroid belts, molecular clouds, raindrop distribution etc.) and process industries (chemical, pharmaceutical, material etc.). This paper deals with the development and analysis of schemes to numerically solve the multi-dimensional nonlinear collisional fragmentation model. Two numerical techniques are presented based on the finite volume discretization method. It is shown that the proposed schemes are consistent with the hypervolume conservation property. Moreover, the number preservation property law also holds for one of them. Detailed mathematical discussions are presented to establish the convergence analysis and consistency of the multi-dimensional schemes under predefined restrictions on the kernel and initial data. The proposed schemes are shown to be second-order convergent. Finally, several numerical computations (one-, two- and three-dimensional fragmentation) are performed to validate the numerical schemes.
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3
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Ultrasound frequency sonication facilitates high-throughput and uniform dissociation of cellular aggregates and tissues. SLAS Technol 2023; 28:70-81. [PMID: 36642327 DOI: 10.1016/j.slast.2023.01.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2022] [Revised: 12/08/2022] [Accepted: 01/01/2023] [Indexed: 01/15/2023]
Abstract
A sample preparation step involving dissociation of tissues into their component cells is often required to conduct analysis of nucleic acids and other constituents from tissue samples. Frequently, the extracellular matrix and cell-cell adhesions are disrupted via treatment with a chemical dissociating reagent or various mechanical forces. In this work, a new, high-throughput, multiplexed method of dissociating tissues and cellular aggregates into single cells using ultrasound frequency bath sonication is explored and characterized. Different operating parameters are evaluated, and a treatment protocol with potential for uniform, high-throughput tissue dissociation is compared to the existing best chemical and orbital plate shaking protocol. Metrics such as percent dissociation, cellular recovery, average aggregate size, proportion of various aggregate sizes, membrane circularity, and cellular viability are subsequently assessed and found to be favorable. In optimized conditions, 53 ± 8% of 1 mm biopsy cores are dissociated within 30 min using sonication alone, surpassing leading high-throughput orbital plate shaking techniques five-fold. Chemical digestion is also 2 times more effective when complexed with sonication rather than orbital plate shaking. RNA content, quality, and expression are found to be superior to the standard protocol in terms of transcriptional preservation.
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A Conservative Finite Volume Method for the Population Balance Equation with Aggregation, Fragmentation, Nucleation and Growth. Chem Eng Sci 2022. [DOI: 10.1016/j.ces.2022.117925] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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5
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Makoveeva EV, Alexandrov DV. The influence of non-stationarity and interphase curvature on the growth dynamics of spherical crystals in a metastable liquid. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200307. [PMID: 34275364 DOI: 10.1098/rsta.2020.0307] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 12/14/2020] [Indexed: 06/13/2023]
Abstract
This manuscript is concerned with the theory of nucleation and evolution of a polydisperse ensemble of crystals in metastable liquids during the intermediate stage of a phase transformation process. A generalized growth rate of individual crystals is obtained with allowance for the effects of their non-stationary evolution in unsteady temperature (solute concentration) field and the phase transition temperature shift appearing due to the particle curvature (the Gibbs-Thomson effect) and atomic kinetics. A complete system of balance and kinetic equations determining the transient behaviour of the metastability degree and the particle-radius distribution function is analytically solved in a parametric form. The coefficient of mutual Brownian diffusion in the Fokker-Planck equation is considered in a generalized form defined by an Einstein relation. It is shown that the effects under consideration substantially change the desupercooling/desupersaturation dynamics and the transient behaviour of the particle-size distribution function. The asymptotic state of the distribution function (its 'tail'), which determines the relaxation dynamics of the concluding (Ostwald ripening) stage of a phase transformation process, is derived. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.
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Affiliation(s)
- Eugenya V Makoveeva
- Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Ekaterinburg 620000, Russian Federation
| | - Dmitri V Alexandrov
- Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Ekaterinburg 620000, Russian Federation
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Nikishina MA, Alexandrov DV. Nucleation and growth dynamics of ellipsoidal crystals in metastable liquids. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200306. [PMID: 34275366 DOI: 10.1098/rsta.2020.0306] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 11/05/2020] [Indexed: 06/13/2023]
Abstract
When describing the growth of crystal ensembles from metastable solutions or melts, a significant deviation from a spherical shape is often observed. Experimental data show that the shape of growing crystals can often be considered ellipsoidal. The new theoretical models describing the transient nucleation of ellipsoidal particles and their growth with and without fluctuating rates at the intermediate stage of bulk phase transitions in metastable systems are considered. The nonlinear transport (diffusivity) of ellipsoidal crystals in the space of their volumes is taken into account in the Fokker-Planck equation allowing for fluctuating growth rates. The complete analytical solutions of integro-differential models of kinetic and balance equations are found and analysed. Our solutions show that the desupercooling dynamics is several times faster for ellipsoidal crystals as compared to spherical particles. In addition, the crystal-volume distribution function is lower and shifted to larger particle volumes when considering the growth of ellipsoidal crystals. What is more, this function is monotonically increasing to the maximum crystal size in the absence of fluctuations and is a bell-shaped curve when such fluctuations are taken into account. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.
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Affiliation(s)
- Margarita A Nikishina
- Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Ekaterinburg, 620000, Russian Federation
| | - Dmitri V Alexandrov
- Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Ekaterinburg, 620000, Russian Federation
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Abstract
For many years, food engineers have attempted to describe physical phenomena such as heat and mass transfer in food via mathematical models. Still, the impact and benefits of computer-aided engineering are less established in food than in most other industries today. Complexity in the structure and composition of food matrices are largely responsible for this gap. During processing of food, its temperature, moisture, and structure can change continuously, along with its physical properties. We summarize the knowledge foundation, recent progress, and remaining limitations in modeling food particle systems in four relevant areas: flowability, size reduction, drying, and granulation and agglomeration. Our goal is to enable researchers in academia and industry dealing with food powders to identify approaches to address their challenges with adequate model systems or through structural and compositional simplifications. With advances in computer simulation capacity, detailed particle-scale models are now available for many applications. Here, we discuss aspects that require further attention, especially related to physics-based contact models for discrete-element models of food particle systems.
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Affiliation(s)
- Lennart Fries
- Nestlé Research Lausanne, Vers-Chez-les-Blanc, 1000 Lausanne 26, Switzerland;
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8
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Conservative Finite Volume Schemes for Multidimensional Fragmentation Problems. MATHEMATICS 2021. [DOI: 10.3390/math9060635] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In this article, a new numerical scheme for the solution of the multidimensional fragmentation problem is presented. It is the first that uses the conservative form of the multidimensional problem. The idea to apply the finite volume scheme for solving one-dimensional linear fragmentation problems is extended over a generalized multidimensional setup. The derivation is given in detail for two-dimensional and three-dimensional problems; an outline for the extension to higher dimensions is also presented. Additionally, the existing one-dimensional finite volume scheme for solving conservative one-dimensional multi-fragmentation equation is extended to solve multidimensional problems. The accuracy and efficiency of both proposed schemes is analyzed for several test problems.
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Abstract
We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer multicomponent mass is broken into fixed number of fragments and calculate the combinatorial multiplicity of all distributions in the set. We define random fragmentation by the condition that the probability of distribution be proportional to its multiplicity, and obtain the partition function and the mean distribution in closed form. We then introduce a functional that biases the probability of distribution to produce in a systematic manner fragment distributions that deviate to any arbitrary degree from the random case. We corroborate the results of the theory by Monte Carlo simulation, and demonstrate examples in which components in sieve cuts of the fragment distribution undergo preferential mixing or segregation relative to the parent particle.
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Population Balance Modeling with Coupled Agglomeration and Disintegration Processes for TiO2 Nanoparticles Formation and Experimental Validation. J CLUST SCI 2020. [DOI: 10.1007/s10876-020-01895-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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11
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12
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Bivariate extension of the moment projection method for the particle population balance dynamics. Comput Chem Eng 2019. [DOI: 10.1016/j.compchemeng.2018.12.011] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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13
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Saha J, Kumar J, Heinrich S. On the approximate solutions of fragmentation equations. Proc Math Phys Eng Sci 2018. [DOI: 10.1098/rspa.2017.0541] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A numerical model based on the finite volume scheme is proposed to approximate the binary breakage problems. Initially, it is considered that the particle fragments are characterized by a single property, i.e. particle’s volume. We then investigate the extension of the proposed model for solving breakage problems considering two properties of particles. The efficiency to estimate the different moments with good accuracy and simple extension for multi-variable problems are the key features of the proposed method. Moreover, the mathematical convergence analysis is performed for one-dimensional problems. All mathematical findings and numerical results are validated over several test problems. For numerical validation, we propose the extension of Bourgade & Filbet (2008
Math. Comput.
77
, 851–882. (
doi:10.1090/S0025-5718-07-02054-6
)) model for solving two-dimensional pure breakage problems. In this aspect, numerical treatment of the two-dimensional binary breakage models using finite volume methods can be treated to be the first instance in the literature.
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Affiliation(s)
- Jitraj Saha
- Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
| | - Jitendra Kumar
- Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
| | - Stefan Heinrich
- Institute of Solids Process Engineering and Particle Technology, Hamburg University of Technology, Hamburg 21073, Germany
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14
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Breschi G, Fontelos MA. A note on the self-similar solutions to the spontaneous fragmentation equation. Proc Math Phys Eng Sci 2017; 473:20160740. [PMID: 28588398 DOI: 10.1098/rspa.2016.0740] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2016] [Accepted: 04/10/2017] [Indexed: 11/12/2022] Open
Abstract
We provide a method to compute self-similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours. Our procedure is applied to specific cases: (i) the case of mitosis, where fragmentation results into two identical fragments, (ii) fragmentation limited to the formation of sufficiently large fragments, and (iii) processes with fragmentation kernel presenting a power-like behaviour.
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Affiliation(s)
- Giancarlo Breschi
- Instituto de Ciencias Matemáticas (ICMAT, CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 15, 28049 Madrid, Spain
| | - Marco A Fontelos
- Instituto de Ciencias Matemáticas (ICMAT, CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 15, 28049 Madrid, Spain
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15
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Saha J, Kumar J, Heinrich S. A volume-consistent discrete formulation of particle breakage equation. Comput Chem Eng 2017. [DOI: 10.1016/j.compchemeng.2016.11.013] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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16
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Kostoglou M, Fuentes-Garí M, García-Münzer D, Georgiadis MC, Panoskaltsis N, Pistikopoulos EN, Mantalaris A. A comprehensive mathematical analysis of a novel multistage population balance model for cell proliferation. Comput Chem Eng 2016. [DOI: 10.1016/j.compchemeng.2016.02.012] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
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17
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Saha J, Kumar J, Bück A, Tsotsas E. Finite volume approximations of breakage population balance equation. Chem Eng Res Des 2016. [DOI: 10.1016/j.cherd.2016.02.012] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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18
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Lin S, Yu W, Wang X, Zhou C. Study on the Thermal Degradation Kinetics of Biodegradable Poly(propylene carbonate) during Melt Processing by Population Balance Model and Rheology. Ind Eng Chem Res 2014. [DOI: 10.1021/ie404049v] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Shanshan Lin
- Advanced
Rheology Institute, Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
| | - Wei Yu
- Advanced
Rheology Institute, Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
| | - Xianhong Wang
- Key
Laboratory of Polymer Ecomaterials, Changchun Institute of Applied
Chemistry, Chinese Academy of Sciences, Changchun 130022, P. R. China
| | - Chixing Zhou
- Advanced
Rheology Institute, Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
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19
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Bari AH, Pandit AB. Ultrasound-facilitated particle breakage: Estimation of kinetic parameters using population balance modelling. CAN J CHEM ENG 2014. [DOI: 10.1002/cjce.22072] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Affiliation(s)
- Atul H. Bari
- Department of Chemical Engineering; Institute of Chemical Technology; Mumbai 40019 India
| | - Aniruddha B. Pandit
- Department of Chemical Engineering; Institute of Chemical Technology; Mumbai 40019 India
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20
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Kumar R, Kumar J, Warnecke G. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. ACTA ACUST UNITED AC 2014. [DOI: 10.3934/krm.2014.7.713] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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21
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Saveliev VL, Gorokhovski MA. Renormalization of the fragmentation equation: exact self-similar solutions and turbulent cascades. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:061112. [PMID: 23367898 DOI: 10.1103/physreve.86.061112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2012] [Indexed: 06/01/2023]
Abstract
Using an approach developed earlier for renormalization of the Boltzmann collision integral [Saveliev and Nanbu, Phys. Rev. E 65, 051205 (2002)], we derive an exact divergence form for the fragmentation operator. Then we reduce the fragmentation equation to the continuity equation in size space, with the flux given explicitly. This allows us to obtain self-similar solutions and to find the integral of motion for these solutions (we call it the bare flux). We show how these solutions can be applied as a description of cascade processes in three- and two-dimensional turbulence. We also suggested an empirical cascade model of impact fragmentation of brittle materials.
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Affiliation(s)
- V L Saveliev
- Institute of Ionosphere, National Center of Space Research and Technology, 050020 Almaty, Kazakhstan
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22
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23
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Chakraborty J, Ramkrishna D. Population Balance Modeling of Environment Dependent Breakage: Role of Granular Viscosity, Density and Compaction. Model Formulation and Similarity Analysis. Ind Eng Chem Res 2011. [DOI: 10.1021/ie200038z] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Jayanta Chakraborty
- School of Chemical Engineering, Purdue University, West Lafayette, Indiana, United States
| | - Doraiswami Ramkrishna
- School of Chemical Engineering, Purdue University, West Lafayette, Indiana, United States
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24
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Catak M, Bas N, Cronin K, Fitzpatrick JJ, Byrne EP. Discrete Solution of the Breakage Equation Using Markov Chains. Ind Eng Chem Res 2010. [DOI: 10.1021/ie100216g] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Muammer Catak
- Department of Process & Chemical Engineering, University College, Cork, Ireland
| | - Nursin Bas
- Department of Process & Chemical Engineering, University College, Cork, Ireland
| | - Kevin Cronin
- Department of Process & Chemical Engineering, University College, Cork, Ireland
| | - John J. Fitzpatrick
- Department of Process & Chemical Engineering, University College, Cork, Ireland
| | - Edmond P. Byrne
- Department of Process & Chemical Engineering, University College, Cork, Ireland
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26
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Disintegration process of surface stabilized sol–gel TiO2 nanoparticles by population balances. Chem Eng Sci 2009. [DOI: 10.1016/j.ces.2009.09.015] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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27
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Lee S, Jeon YP. Effects of mixing on electrical properties of carbon nanofiber and polymer composites. J Appl Polym Sci 2009. [DOI: 10.1002/app.30381] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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28
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Bruns M, Koo J, Ezekoye O. Population-based models of thermoplastic degradation: Using optimization to determine model parameters. Polym Degrad Stab 2009. [DOI: 10.1016/j.polymdegradstab.2009.02.007] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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29
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Yeow YL, Guan B, Wu L, Yap TM, Leong YK. Obtaining the evolving concentration distribution curves during binary disintegration of macromolecules. AIChE J 2008. [DOI: 10.1002/aic.11590] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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30
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A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles. Chem Eng Sci 2008. [DOI: 10.1016/j.ces.2008.04.054] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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31
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Adjustable discretized population balance equations: Numerical simulation and parameter estimation for fractal aggregation and break-up. Colloids Surf A Physicochem Eng Asp 2007. [DOI: 10.1016/j.colsurfa.2006.06.020] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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32
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Kostoglou M, Karabelas AJ. A study of the collisional fragmentation problem using the Gamma distribution approximation. J Colloid Interface Sci 2006; 303:419-29. [PMID: 16949600 DOI: 10.1016/j.jcis.2006.08.005] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2006] [Revised: 07/10/2006] [Accepted: 08/03/2006] [Indexed: 11/19/2022]
Abstract
The nonlinear fragmentation population balance formulation has been elevated in recent years from a prototype for studying nonlinear integro-differential equations to a vehicle for analyzing and understanding several physicochemical processes of technological interest. The so-called pure collisional fragmentation, which is the particular mode of nonlinear fragmentation induced by collisions between particles, is studied here. It is shown that the corresponding population balance equation admits large time asymptotic (self-similarity) solutions for homogeneous fragmentation and collision functions (kernels). The self-similar solutions are given in closed form for some simple kernels. Based on the shape of the self-similar solutions the method of moments with Gamma distribution approximation is employed for transient solution (from initial state to establishment of the asymptotic shape) of the collisional fragmentation equation. These solutions are presented for several sets of parameters and their behavior is discussed rather extensively. The present study is similar to the one has already been performed for the case of the much simpler linear fragmentation equation [G. Madras, B.J. McCoy, AIChE J. 44 (1998) 647].
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Affiliation(s)
- M Kostoglou
- Division of Chemical Technology, Department of Chemistry, Aristotle University, 54124 Thessaloniki, Greece.
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33
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Browarzik D, Koch A. Application of Continuous Kinetics to Polymer Degradation. JOURNAL OF MACROMOLECULAR SCIENCE PART A-PURE AND APPLIED CHEMISTRY 2006. [DOI: 10.1080/10601329608010928] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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34
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Kostoglou M. The equal-size binary breakage problem: evolution toward a steady shape or periodic behavior? J Colloid Interface Sci 2006; 299:703-12. [PMID: 16529764 DOI: 10.1016/j.jcis.2006.02.019] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2005] [Revised: 02/07/2006] [Accepted: 02/11/2006] [Indexed: 11/22/2022]
Abstract
During the last few years, the self-similar particle size distribution for a particle population undergoing breakage in equal size fragments has been derived using approximating, numerical, and analytical means. But very recently it was shown [N.V. Mantzaris, J. Phys. A Math. Gen. 38 (2005) 5111] through transient simulation of the breakage process that the particle size distribution in case of breakage in two equal fragments, never attains a steady shape, i.e., a self-similar form. The new results give rise to questions about the real meaning and utility of the previously derived self-similar distributions for these systems. The scope of the present work is to answer these questions and it is attempted using only analytical (exact) means for the solution of the transient breakage problem. In doing so, the very interesting and rich underlying structure and properties of the solutions of the equal size breakage problem (seemingly, very simple) are revealed. It appears that the utility of the known self-similar distributions for this particular problem has to be redefined but yet not entirely abandoned.
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Affiliation(s)
- Margaritis Kostoglou
- Division of Chemical Technology, Department of Chemistry, Aristotle University, Univ. Box 116, 54124 Thessaloniki, Greece.
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39
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Madras G, McCoy BJ. Molecular-weight distribution kinetics for ultrasonic reactions of polymers. AIChE J 2006. [DOI: 10.1002/aic.690471018] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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40
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Rigopoulos S, Jones AG. Finite-element scheme for solution of the dynamic population balance equation. AIChE J 2006. [DOI: 10.1002/aic.690490507] [Citation(s) in RCA: 88] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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41
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Kim J, Kramer TA. Improved orthokinetic coagulation model for fractal colloids: Aggregation and breakup. Chem Eng Sci 2006. [DOI: 10.1016/j.ces.2005.01.044] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Diemer RB, Spahr DE, Olson JH, Magan RV. Interpretation of size reduction data via moment models. POWDER TECHNOL 2005. [DOI: 10.1016/j.powtec.2005.04.015] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Kostoglou M, Karabelas AJ. On the self-similar solution of fragmentation equation: Numerical evaluation with implications for the inverse problem. J Colloid Interface Sci 2005; 284:571-81. [PMID: 15780296 DOI: 10.1016/j.jcis.2004.10.029] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2004] [Accepted: 10/19/2004] [Indexed: 10/26/2022]
Abstract
It is well known that the fragmentation equation admits self-similar solutions for evolving particle-size distributions (PSD); i.e., if the shape of PSD is independent of time after an initial transient period. Although an analytical derivations of the self-similar PSD cases have been studied extensively, results for cases requiring numerical solutions are rare. The aim of the present work is to fill this gap for the case of homogeneous breakage functions. The known analytical and approximate solutions for the self-similar PSD are reviewed and a general algorithm for the numerical solution is proposed. Results for a broad range of breakage functions (kernel and rate) are presented. Further, the work is focused on the sensitivity of the relation between self-similar PSD and breakage kernel and its influence on the inverse breakage problem, i.e., that of estimating the breakage kernel from experimental self-similar PSDs. Useful suggestions are made for tackling the inverse problem.
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Affiliation(s)
- M Kostoglou
- Chemical Process Engineering Research Institute, P.O. Box 1517, 54006 Thessaloniki, Greece.
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Cheng H, Wu C, Winnik MA. Kinetics of Reversible Aggregation of Soft Polymeric Particles in Dilute Dispersion. Macromolecules 2004. [DOI: 10.1021/ma0496201] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- He Cheng
- The Open Laboratory of Bond Selective Chemistry, Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui, China; Department of Chemistry, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China; and Department of Chemistry, University of Toronto, Canada M5S 3H6
| | - Chi Wu
- The Open Laboratory of Bond Selective Chemistry, Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui, China; Department of Chemistry, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China; and Department of Chemistry, University of Toronto, Canada M5S 3H6
| | - Mitchell A. Winnik
- The Open Laboratory of Bond Selective Chemistry, Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui, China; Department of Chemistry, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China; and Department of Chemistry, University of Toronto, Canada M5S 3H6
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Kostoglou M, Karabelas A. Optimal low order methods of moments for solving the fragmentation equation. POWDER TECHNOL 2004. [DOI: 10.1016/j.powtec.2004.04.020] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Sivalingam G, Agarwal N, Madras G. Distributed midpoint chain scission in ultrasonic degradation of polymers. AIChE J 2004. [DOI: 10.1002/aic.10185] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
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J. McCoy B, Madras G. Analytical solution for a population balance equation with aggregation and fragmentation. Chem Eng Sci 2003. [DOI: 10.1016/s0009-2509(03)00159-3] [Citation(s) in RCA: 59] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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