摘要 :
The population balance equation (PBE) has been used in engineering for more than four decades. However, several misconceptions and misuses can be found in the literature. The present work provides a review of the origins and the d...
展开
The population balance equation (PBE) has been used in engineering for more than four decades. However, several misconceptions and misuses can be found in the literature. The present work provides a review of the origins and the derivation of the PBE. The main focus is on the inconsistencies between the theoretical foundation and the usual practice, as well as identifying and addressing some of the issues that cause confusion and contradiction among authors.
收起
摘要 :
Drop breakup in high-energy emulsification is often modelled using a population balance equation (PBE) framework. For cases with high emulsifier to disperse phase concentration, the PBE is dominated by the fragmentation term. Sinc...
展开
Drop breakup in high-energy emulsification is often modelled using a population balance equation (PBE) framework. For cases with high emulsifier to disperse phase concentration, the PBE is dominated by the fragmentation term. Since there are no analytical solutions to the continuous PBE for physically reasonable fragmentation rate expressions, a discretization is often needed to evaluate the PBE, turning it into a set of ordinary differential equations that can be solved numerically. The fixed pivot technique (Kumar and Ramkrishna, 1996) is one of the most often applied class discretization methods. This contribution suggests an analytical solution to the fixed pivot technique fragmentation equation, that can be used instead of the traditional numerical approach provided that the fragmentation rate is constant over time. The proposed solution compares favorably to two special cases where analytical solutions for the continuous PBE are available, and to two more realistic PBE emulsification problems (with varying fragmentation intensity and varying number of fragments formed per breakup), while offering a substantial reduction in computational time compared to the traditional approach of solving the discretized equations numerically. (C) 2019 Elsevier Ltd. All rights reserved.
收起
摘要 :
In this simulation study, we compare the dynamics and thermal behavior of different ideal flow crystallizers. The first step in creating mathematical models for the crystallizers was the implementation of the population balance eq...
展开
In this simulation study, we compare the dynamics and thermal behavior of different ideal flow crystallizers. The first step in creating mathematical models for the crystallizers was the implementation of the population balance equation. The population balance equation was completed with mass balance equations for the solute and the solvent as well as in the case of non-isothermal crystallizers with an energy balance equation. The solution to the population balance equation, which is a partial differential equation, can only be performed numerically. Using the method of moments, which calculates the moments of the population density function, gives a mathematically simpler model for simulating and analyzing the crystallizers. All crystallizers studied are considered mixed suspension and mixed product crystallizers. In this simulation study, the investigated crystallizers are the batch mixed suspension and mixed product isothermal crystallizer, the batch mixed suspension and mixed product non-isothermal crystallizer, and the continuous mixed suspension and mixed product removal (CMSMPR) non-isothermal crystallizer equipped with a cooling jacket. We consider citric acid as the solid material to be crystallized, and a water–glycol system is used as a cooling medium. Considering the nucleation kinetics, we applied both primary and secondary nucleation. In the case of a crystal growth kinetic, we assumed a size-independent growth rate. The highest expected value and the variance of the crystal product occur in the isotherm batch case, which can be explained by the high crystallization rate caused by the high supersaturation. Contrary to this, in the non-isothermal batch case, the final mean particle size and variance are the lowest. In continuous mode, the variance and mean values are between the values obtained in the two other cases. In this case, the supersaturation is maintained at a constant level in the steady state, and the average residence time of the crystal particles also has an important influence on the crystal size distribution. In the case of non-isothermal crystallization, the simulation studies show that the application of the energy balance provides different dynamics for the crystallizers. The implementation of an energy balances into the mathematical model enables the calculation of the thermal behavior of the crystallizers, enabling the model to be used more widely.
收起
摘要 :
In this paper we improve the numerical schemes presented in a recent publication by Qamar and Warnecke [Chem.Eng.Sci., 31 (2007), pp. 1576-1589]. It is observed that the proposed numerical schemes do not preserve certain required ...
展开
In this paper we improve the numerical schemes presented in a recent publication by Qamar and Warnecke [Chem.Eng.Sci., 31 (2007), pp. 1576-1589]. It is observed that the proposed numerical schemes do not preserve certain required moments of size distributions. In this paper, slightly different formulations are given that preserve these moments. The new formulations are compared with the old formulations by the application of three analytically solvable problems. It is shown that the new formulations are very accurate for predict ing number density distributions as well as certain moments. While the old formulations are accurate enough for predicting the number density distribution, they show peculiar behavior for certain mo ments.
收起
摘要 :
In this paper,a model based on a population balance equation(PBE)is developed.It aims at reproducing experimental floe size distributions obtained at steady state in a jar-test.The objective is to develop a simple model,based on p...
展开
In this paper,a model based on a population balance equation(PBE)is developed.It aims at reproducing experimental floe size distributions obtained at steady state in a jar-test.The objective is to develop a simple model,based on physical phenomena,and that does not contain any adjustable parameters.Floe size distributions obtained using a part of a particle image velocimetry(PIV)device and image analysis are used to develop mathematical expressions for the aggregation and breakage kernels.A critical volume beyond which breakage is of significant importance is identified and related to the hydrodynamics.Hydrodynamic sequencing allows the distribution of the daughter particles resulting from a breakage event to be established.The model is finally successfully validated against experimental results.
收起
摘要 :
This paper deals with the dynamic modeling of a batch crystallizer. A complete model taking into account primary and secondary nucleations, crystal growth, agglomeration and attrition mechanisms is established. The proposed model ...
展开
This paper deals with the dynamic modeling of a batch crystallizer. A complete model taking into account primary and secondary nucleations, crystal growth, agglomeration and attrition mechanisms is established. The proposed model is not restricted to binary agglomeration and breakage phenomena. From markovian considerations, continuous kernel functions are built and the basic balance equations are then presented. The complete model is solved using a finite difference method for the discretization of the size variable. As to distinguish agglomeration and breakage parameters from the others, on line measurement of the Crystal Size Distribution is necessary, a new on line measurement strategy is proposed. Finally, simulations of the crystal size distribution are compared with experimental results at different times. It appears that simulated curves are in good agreement with the experimental data. (C) 2007 Elsevier Ltd. All rights reserved.
收起
摘要 :
This paper introduces a method to solve the inverse problem of determining an unknown rate function in a partial differential equation (PDE) based on discrete measurements of the modeled quantity. The focus is put on a size-struct...
展开
This paper introduces a method to solve the inverse problem of determining an unknown rate function in a partial differential equation (PDE) based on discrete measurements of the modeled quantity. The focus is put on a size-structured population balance equation (PBE) predicting the evolution of the number distribution of a single cell population as a function of the size variable. Since the inverse problem at hand is ill-posed, an adequate regularization scheme is required to avoid amplification of measurement errors in the solution method. The technique developed in this work to determine a rate function in a PBE is based on the approximate inverse method, a pointwise regularization scheme, which employs two key ideas. Firstly, the mollification in the directions of time and size variables are separated. Secondly, instable numerical data derivatives are circumvented by shifting the differentiation to an analytically given function. To examine the performance of the introduced scheme, adapted test scenarios have been designed with different levels of data disturbance simulating the model and measurement errors in practice. The success of the method is substantiated by visualizing the results of these numerical experiments.
收起
摘要 :
We consider the numerical solution of the multivariate aggregation population balance equation on a uniform tensor grid. We introduce a multidimensional fast Fourier transformation for the efficient evaluation of the aggregation i...
展开
We consider the numerical solution of the multivariate aggregation population balance equation on a uniform tensor grid. We introduce a multidimensional fast Fourier transformation for the efficient evaluation of the aggregation integrals leading to a reduction in the complexity order of the algorithm compared to the direct evaluation approach. We illustrate the new evaluation algorithm for two discretizations, an FEM approach as well as the sectional method. We discuss the conservation of moments for these methods and provide numerical comparisons illustrating the superior performance of FFT-based algorithms. We also discuss and numerically illustrate their potential for parallelization. (C) 2018 Elsevier B.V. All rights reserved.
收起
摘要 :
Crosslinking radical polymerisation in a continuously stirred tank reactor has been studied by means of a four-dimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiracli...
展开
Crosslinking radical polymerisation in a continuously stirred tank reactor has been studied by means of a four-dimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiraclicals as dimensions. The model covers both pre-gel and gel regimes in a straightforward manner. Approximations on radial basis functions have been employed to reduce the size of the system with minimal information loss. The comparison with Monte Carlo simulations shows interesting and unexpected features. (C) 2014 Elsevier Ltd. All rights reserved.
收起
摘要 :
A general framework is developed, using the least squares method (LSM), for the solution of a generalized population balance equation. The basic idea in the LSM is to minimize the integral of the square of the residual over the co...
展开
A general framework is developed, using the least squares method (LSM), for the solution of a generalized population balance equation. The basic idea in the LSM is to minimize the integral of the square of the residual over the computational domain. The capability of the method for solving the PBE is evaluated by using case problems involving coalescence and breakage kernels having analytical solutions which allow the analysis of the method to be performed in a general way. By using the LSM to solve the PBE, the error in the properties of the distribution function depends on the order of the expansion, thus avoiding the introduction of heuristic rules to obtain sufficient accuracy in the values of a few of the physical moments. An interesting characteristic of the LSM applied to PBE is that a low number of equations are required to solve the problem.
收起
原文获取