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Peterson JD, Bagkeris I, Michael V. A new framework for numerical modeling of population balance equations: Solving for the inverse cumulative distribution function. Chem Eng Sci 2022. [DOI: 10.1016/j.ces.2022.117781] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
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2
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Kostoglou M, Petala M, Karapantsios T, Dovas C, Roilides E, Metallidis S, Papa A, Stylianidis E, Papadopoulos A, Papaioannou N. SARS-CoV-2 adsorption on suspended solids along a sewerage network: mathematical model formulation, sensitivity analysis, and parametric study. ENVIRONMENTAL SCIENCE AND POLLUTION RESEARCH INTERNATIONAL 2022; 29:11304-11319. [PMID: 34542818 PMCID: PMC8450709 DOI: 10.1007/s11356-021-16528-0] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2021] [Accepted: 09/08/2021] [Indexed: 05/24/2023]
Abstract
Accounting for SARS-CoV-2 adsorption on solids suspended in wastewater is a necessary step towards the reliable estimation of virus shedding rate in a sewerage system, based on measurements performed at a terminal collection station, i.e., at the entrance of a wastewater treatment plant. This concept is extended herein to include several measurement stations across a city to enable the estimation of spatial distribution of virus shedding rate. This study presents a pioneer general model describing the most relevant physicochemical phenomena with a special effort to reduce the complicated algebra. This is performed both in the topology regime, introducing a discrete-continuous approach, and in the domain of independent variables, introducing a monodisperse moment method to reduce the dimensionality of the resulting population balance equations. The resulting simplified model consists of a large system of ordinary differential equations. A sensitivity analysis is performed with respect to some key parameters for a single pipe topology. Specific numerical techniques are employed for the integration of the model. Finally, a parametric case study for an indicative-yet realistic-sewerage piping system is performed to show how the model is applied to SARS-CoV-2 adsorption on wastewater solids in the presence of other competing species. This is the first model of this kind appearing in scientific literature and a first step towards setting up an inverse problem to assess the spatial distribution of virus shedding rate based on its concentration in wastewater.
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Affiliation(s)
- Margaritis Kostoglou
- Laboratory of Chemical and Environmental Technology, Department of Chemistry, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
| | - Maria Petala
- Laboratory of Environmental Engineering & Planning, Department of Civil Engineering, Aristotle University of Thessaloniki, 54 124, Thessaloniki, Greece
| | - Thodoris Karapantsios
- Laboratory of Chemical and Environmental Technology, Department of Chemistry, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece.
| | - Chrysostomos Dovas
- Faculty of Veterinary Medicine, School of Health Sciences, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
| | - Emmanuel Roilides
- Infectious Diseases Unit and 3rd Department of Pediatrics, Aristotle University School of Health Sciences, Hippokration Hospital, 54642, Thessaloniki, Greece
| | - Simeon Metallidis
- Department of Haematology, First Department of Internal Medicine, Faculty of Medicine, AHEPA General Hospital, Aristotle University of Thessaloniki, 54636, Thessaloniki, Greece
| | - Anna Papa
- Department of Microbiology, Medical School, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
| | - Efstratios Stylianidis
- School of Spatial Planning and Development, Faculty of Engineering, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
| | - Agis Papadopoulos
- EYATH S.A., Thessaloniki Water Supply and Sewerage Company S.A., 54636, Thessaloniki, Greece
| | - Nikolaos Papaioannou
- Faculty of Veterinary Medicine, School of Health Sciences, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
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Wu S, Yang S, Tay KL, Yang W, Jia M. A hybrid sectional moment projection method for discrete population balance dynamics involving inception, growth, coagulation and fragmentation. Chem Eng Sci 2022. [DOI: 10.1016/j.ces.2021.117333] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Population balance modeling of flotation pulp: The route from process frequency functions to spatially distributed models. Comput Chem Eng 2021. [DOI: 10.1016/j.compchemeng.2021.107506] [Citation(s) in RCA: 3] [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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Ahamed F, Singh M, Song HS, Doshi P, Ooi CW, Ho YK. On the use of sectional techniques for the solution of depolymerization population balances: Results on a discrete-continuous mesh. ADV POWDER TECHNOL 2020. [DOI: 10.1016/j.apt.2020.04.032] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Jama MA, Zhao W, Ahmad W, Buffo A, Alopaeus V. Analytical time-stepping solution of the discretized population balance equation. Comput Chem Eng 2020. [DOI: 10.1016/j.compchemeng.2020.106741] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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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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8
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Liquid–liquid extraction in a rotating disc column: Solution of 2D population balance with HMMC. Chem Eng Res Des 2016. [DOI: 10.1016/j.cherd.2016.09.002] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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9
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Buffo A, Alopaeus V. Solution of bivariate population balance equations with high-order moment-conserving method of classes. Comput Chem Eng 2016. [DOI: 10.1016/j.compchemeng.2015.12.013] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Ho YK, Doshi P, Yeoh HK, Ngoh GC. Interlinked population balance and cybernetic models for the simultaneous saccharification and fermentation of natural polymers. Biotechnol Bioeng 2015; 112:2084-105. [DOI: 10.1002/bit.25616] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2014] [Revised: 03/02/2015] [Accepted: 04/06/2015] [Indexed: 11/06/2022]
Affiliation(s)
- Yong Kuen Ho
- Department of Chemical Engineering, Faculty of Engineering; University of Malaya; 50603 Kuala Lumpur Malaysia
| | - Pankaj Doshi
- National Chemical Laboratory; Dr. Homi Bhabha Road Pune 411008 India
| | - Hak Koon Yeoh
- Department of Chemical Engineering, Faculty of Engineering; University of Malaya; 50603 Kuala Lumpur Malaysia
| | - Gek Cheng Ngoh
- Department of Chemical Engineering, Faculty of Engineering; University of Malaya; 50603 Kuala Lumpur Malaysia
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Ho YK, Doshi P, Yeoh HK, Ngoh GC. Modeling chain-end scission using the Fixed Pivot technique. Chem Eng Sci 2014. [DOI: 10.1016/j.ces.2014.05.035] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Razzaghi K, Shahraki F. Improved recursive algorithm for calculation of probability density in breakup models. AIChE J 2010. [DOI: 10.1002/aic.12393] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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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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Energy and population balances in comminution process modelling based on the informational entropy. POWDER TECHNOL 2006. [DOI: 10.1016/j.powtec.2006.05.011] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Meimaroglou D, Roussos A, Kiparissides C. Part IV: Dynamic evolution of the particle size distribution in particulate processes. A comparative study between Monte Carlo and the generalized method of moments. Chem Eng Sci 2006. [DOI: 10.1016/j.ces.2006.05.001] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Li R, McCoy BJ, Diemer RB. Cluster aggregation and fragmentation kinetics model for gelation. J Colloid Interface Sci 2005; 291:375-87. [PMID: 15964583 DOI: 10.1016/j.jcis.2005.05.038] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2004] [Revised: 05/09/2005] [Accepted: 05/12/2005] [Indexed: 10/25/2022]
Abstract
Gelation can occur in polymer, hydrogel, and colloid systems that undergo reversible aggregation-fragmentation (crosslinking accompanied by breakage). Gelation, characterized by rapid divergence of weight-average molecular weight and viscosity due to initial network formation, can be reversed if conditions change. In this paper, reversible aggregation and fragmentation in the pre-gelation time period are modeled with distribution kinetics. Moment equations are obtained from the population balance equation, and solved for eight different rate kernels. We identify the cases for which gelation is possible and obtain the critical values for the rate constants that allow gelation. The model provides a good simulation of published experimental data for aggregation and degradation of plasticized wheat gluten during thermo-mechanical treatments. We also evaluate two closure approximations based on Gamma and log-normal distributions, and conclude that log-normal closure predicts all five possible steady states, in agreement with the Vigil-Ziff criterion, and Gamma closure predicts only three. However, Gamma closure approximates the steady state either closely or exactly, whereas log-normal closure only poorly approximates the steady-state distribution.
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Affiliation(s)
- Rujun Li
- School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
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