A comparative study of high resolution schemes for solving population balances in crystallization

Qamar, Shamsul; Elsner, M. P.; Angelov, I.; Warnecke, Gerald; Seidel‐Morgenstern, A.

doi:10.1016/j.compchemeng.2006.02.012

Cited by 149 publications

(118 citation statements)

References 15 publications

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“…It can be seen that MOC performed very well in resolving the sharp discontinuities of the step function and peak resolution of the Gaussian function. The MOC solution is comparable with the analytical solution and is superior over the solutions of finite volume schemes [12].…”

Section: Test Problemmentioning

confidence: 52%

“…Figure 3 shows the final crystal size distributions (CSDs). It can be clearly seen that MOC resolves the steep gradients in nucleation part much better than the second order finite volume scheme [12]. The zoomed plots in Figure 4 clearly indicate that MOC is superior over the finite volume scheme.…”

Section: Test Problemmentioning

confidence: 63%

“…The results of MOC are compared with analytical solutions and results of the finite volume scheme presented in Qamar et al [12].…”

Section: Numerical Test Problemsmentioning

confidence: 99%

“…Balancing the number of crystals in an infinitesimal interval of crystal length, a partial differential equation is obtained which explains the temporal evolution of CSD [12] [15] ( ) ( …”

Section: Batch Crystallization Modelmentioning

confidence: 99%

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Application of the Method of Characteristics to Population Balance Models Considering Growth and Nucleation Phenomena

Rehman¹,

Qamar²

2014

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The population balance modeling is regarded as a universally accepted mathematical framework for dynamic simulation of various particulate processes, such as crystallization, granulation and polymerization. This article is concerned with the application of the method of characteristics (MOC) for solving population balance models describing batch crystallization process. The growth and nucleation are considered as dominant phenomena, while the breakage and aggregation are neglected. The numerical solutions of such PBEs require high order accuracy due to the occurrence of steep moving fronts and narrow peaks in the solutions. The MOC has been found to be a very effective technique for resolving sharp discontinuities. Different case studies are carried out to analyze the accuracy of proposed algorithm. For validation, the results of MOC are compared with the available analytical solutions and the results of finite volume schemes. The results of MOC were found to be in good agreement with analytical solutions and superior than those obtained by finite volume schemes.

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Section: Test Problemmentioning

confidence: 52%

Section: Test Problemmentioning

confidence: 63%

“…The results of MOC are compared with analytical solutions and results of the finite volume scheme presented in Qamar et al [12].…”

Section: Numerical Test Problemsmentioning

confidence: 99%

“…Balancing the number of crystals in an infinitesimal interval of crystal length, a partial differential equation is obtained which explains the temporal evolution of CSD [12] [15] ( ) ( …”

Section: Batch Crystallization Modelmentioning

confidence: 99%

See 2 more Smart Citations

Application of the Method of Characteristics to Population Balance Models Considering Growth and Nucleation Phenomena

Rehman¹,

Qamar²

2014

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“…The flow solver, moreover, is based on the iterative SIMPLE scheme [49]. For the discretization in particle property space, we combine the EAGM with the κ = 1/3 high resolution finite volume scheme of Koren [30], also see Qamar et al [58,59] and Sewerin and Rigopoulos [66]. The EAGM features two parameters which control the grid quality: The first one represents the maximum admissible grid stretching, while the second one indicates the minimum permissible resolution in the interval in particle property space on which the nucleation source term is non-zero.…”

Section: Numerical Solution Schemementioning

confidence: 99%

An LES-PBE-PDF approach for modeling particle formation in turbulent reacting flows

Sewerin

Rigopoulos

2017

Physics of Fluids

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Many chemical and environmental processes involve the formation of a polydispersed particulate phase in a turbulent carrier flow. Frequently, the immersed particles are characterized by an intrinsic property such as the particle size and the distribution of this property across a sample population is taken as an indicator for the quality of the particulate product or its environmental impact.In the present article, we propose a comprehensive model and an efficient numerical solution scheme for predicting the evolution of the property distribution associated with a polydispersed particulate phase forming in a turbulent reacting flow. Here, the particulate phase is described in terms of the particle number density whose evolution in both physical and particle property space is governed by the population balance equation (PBE). Based on the concept of large eddy simulation (LES), we augment the existing LES-transported PDF approach for fluid phase scalars by the particle number density and obtain a modelled evolution equation for the filtered probability density function (PDF) associated with the instantaneous fluid composition and particle property distribution. This LES-PBE-PDF approach allows us to predict the LES-filtered fluid composition and particle property distribution at each spatial location and point in time without any restriction on the chemical or particle formation kinetics. In view of a numerical solution, we apply the method of Eulerian stochastic fields, invoking an explicit adaptive grid technique in order to discretize the stochastic field equation for the number density in particle property space. In this way, sharp moving features of the particle property distribution can be accurately resolved at a significantly reduced computational cost.As a test case, we consider the condensation of an aerosol in a developed turbulent mixing layer. Our investigation not only demonstrates the predictive capabilities of the LES-PBE-PDF model, but also indicates the computational efficiency of the numerical solution scheme.

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Process Systems Engineering, 9. Domain Engineering

Economou¹,

Pistikopoulos

Liu

et al. 2012

Ullmann's Encyclopedia of Industrial Chemistry

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The article contains sections titled: 1. Molecular Modeling and Simulation for Chemical Product and Process Design 1.1. Introduction 1.2. Elementary Statistical Mechanics 1.3. Major Molecular Simulation Methods 1.3.1. Molecular Dynamics (MD) 1.3.2. Metropolis Monte Carlo Simulation 1.4. Applications 1.4.1. Pharmaceuticals 1.4.2. Polymer Membranes for Gas Separation 1.4.3. Ionic Liquids for Sustainable Chemical Processes 1.5. Conclusions 2. Energy Systems Engineering 2.1. Introduction 2.2. Methods/Tools/Algorithm 2.2.1. Superstructure‐Based Modeling 2.2.2. Mixed‐Integer Programming (MIP) 2.2.3. Multiobjective Optimization 2.2.4. Optimization under Uncertainty 2.2.5. Life‐Cycle Assessment 2.3. Energy Systems Examples 2.3.1. Example 1–Polygeneration Energy Systems 2.3.2. Example 2–Hydrogen Infrastructure Planning 2.3.3. Example 3–Energy Systems in Commercial Buildings 2.4. Conclusions and Future Directions 3. Pharmaceutical Processes 3.1. Introduction 3.2. Pharmaceutical Process Development and Operation 3.2.1. Crystallization 3.2.2. Chromatography 3.3. Conclusion 4. Biochemical Engineering 4.1. Introduction 4.2. Industrial Biotechnology Processes 4.2.1. Fermentation Processes 4.2.2. Microbial Catalysis 4.2.3. Enzyme Processes 4.3. Modeling of Bioprocesses 4.3.1. Modeling of Bioprocesses–Mechanistic Models 4.3.2. Modeling of Bioprocesses–Data‐Driven Models 4.4. The Role of Process Systems Engineering 4.4.1. Evaluation of Process Options 4.4.2. Evaluation of Platform Chemicals 4.4.3. Process Integration 4.4.4. Biorefinery Design 4.4.5. Biocatalyst Design 4.5. Assessing the Sustainability of Bioprocesses 4.5.1. Life‐Cycle Inventory and Assessment 4.6. Future Outlook and Perspectives 5. Policies and Policy Making 5.1. Introduction 5.2. Policies and Policy Measures 5.3. Policy Making and the Systems Approach 5.4. Similarities between Policy Formulation and Conceptual Process Design 5.5. The Nature of Policy Formulation 5.6. The Nature of Sociotechnical Systems 5.7. Challenges for Modelers of Sociotechnical Systems 5.7.1. Multiple Stakeholders 5.7.2. Incommensurable Values 5.7.3. Externalities 5.7.4. Uncertainty 5.7.5. Emergent Behavior 5.7.6. Complexity of Causation 5.7.7. Objectivity in Policy Analysis 5.8. Types of Models used in the Analysis of Policies 5.8.1. Macroeconomic Models (Mainstream, Descriptive, Aggregated, Mechanistic) 5.8.2. Optimization Models (Mainstream, Normative, Aggregated, Mechanistic) 5.8.3. Control Models (Mainstream, Normative, Aggregated, Mechanistic) 5.8.4. Data‐Based Models 5.8.5. Game Theory (Descriptive) 5.8.6. System Dynamics (Aggregated, Mechanistic) 5.8.7. Network Theory (Descriptive) 5.8.8. Agent‐Based Approaches 5.8.9. Some Conclusions on Models for the Analysis of Policies 5.9. Synthesis of Policies 5.10. Future Directions 6. Acknowledgments

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A comparative study of high resolution schemes for solving population balances in crystallization

Cited by 149 publications

References 15 publications

Application of the Method of Characteristics to Population Balance Models Considering Growth and Nucleation Phenomena

Application of the Method of Characteristics to Population Balance Models Considering Growth and Nucleation Phenomena

An LES-PBE-PDF approach for modeling particle formation in turbulent reacting flows

Process Systems Engineering, 9. Domain Engineering

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