LEGENDRE FUNCTION APPROXIMATIONS OF ORDINARY DIFFERENTIAL EQUATION AND APPLICATION TO CONTINUOUS CRYSTALLIZATION PROCESSES

The technique of the shifted Legendre integral transformation associated with its series expansion is applied to solve population balance equations describing the continuous crystallization processes. The ordinary differential equation of population balance is transformed into a series of algebraic equations of the expansion coefficients which can be easily solved. Illustrative examples for realistic cases are given to simulate the systems. Satisfactory computational results are obtained when are compared with those of the exact solutions. The method is straightforward and the computational algorithm is effective.     

LEGENDRE FUNCTION APPROXIMATIONS OF ORDINARY DIFFERENTIAL EQUATION AND APPLICATION TO CONTINUOUS CRYSTALLIZATION PROCESSES

The technique of the shifted Legendre integral transformation associated with its series expansion is applied to solve population balance equations describing the continuous crystallization processes. The ordinary differential equation of population balance is transformed into a series of algebraic equations of the expansion coefficients which can be easily solved. Illustrative examples for realistic cases are given to simulate the systems. Satisfactory computational results are obtained when are compared with those of the exact solutions. The method is straightforward and the computational algorithm is effective.     

MODELING OF CRYSTALLIZATION SYSTEMS VIA GENERALIZED ORTHOGONAL POLYNOMIALS METHOD

The population balance equation which is used to describe the dynamic of crystallization processes is simulated by the proposed generalized orthogonal polynomials (GOP). Examples of modeling include the ordinary differential equation for MSMPR crystallization processes, the partial differential equation for transient batch crystallization processes and the functional differential equation for breakage of crystals within the crystallizer. The key idea is to express the population density function by. a generalized orthogonal polynomials series. The ordinary (or partial, or functional) differential equation is then transformed into a set of algebraic (or ordinary differential, or algebraic) equations of expansion coefficients through the integration operation matrix. The advantage of using GOP approximation is that almost any kind of orthogonal polynomials can be used to approximate the system with very accurate results. Furthermore, it is very easy to calculate the expansion coefficients by using the present proposed operation matrix and the recursive formula.     

Simulation of discontinuous population balance equation with integral constraint of growth rate expression

An effective method based on the concept of continuous characteristics is developed to solve the continuous population equation with integral constraint of growth rate expression. This method can also be extended to solve a general form of a first order partial differential equation. A typical example of a class II MSMPR crystallization process at transient state is modelled and analyzed. The system which possesses originally a discontinuous population density function is transformed into a continuous one by appropriate treatment of the initial condition. The partial differential equation of the continuous population density function is solved by the shifted Legendre polynomials approximation and moments method simultaneously. The original discontinuous population density function is then transformed back from the calculated continuous one by the system characteristics. Very satisfactory computational results are obtained.     

NUMERICAL TECHNIQUE FOR SOLVING POPULATION BALANCES IN PRECIPITATION PROCESSES

A simple, accurate and efficient numerical technique for solving the dynamic population balance associated with precipitation/crystallization reactors is presented. The basic approach is to determine the solute concentration dynamics first using the method of moments and an efficient ordinary differential equation solver, and then to solve the dynamic population balance, a partial differential equation having coefficients which depend on the known solute concentration, by standard finite difference techniques. The method is illustrated by simulating the dynamics of MSMPR and semi-batch reactors for BaSO₄ precipitation. The technique should be applicable to a wide range of population balance models.     

Simulation of packed-bed separation processes using orthogonal collocation

The two-point boundary value problem resulting from the heat and material balance equations of a packed separation column are solved using polynomial approximation techniques. The model equations are based on the two-film theory of mass transfer. The resulting partial differential equations are first reduced to ordinary differential equations and then integrated using semi-implicit Runge-Kutta method of integration. Application of orthogonal collocation simplifies the solution of the two-point boundary value problem. For the examples studied, the algorithm is found to converge rapidly with respect to the number of collocation points used in the polynomial approximation.     

Approximate ODE models for population balance systems

We propose an approximate polynomial method of moments for a class of first-order linear PDEs (partial differential equations) of hyperbolic type, involving a filtering term with applications to population balance systems with fines removal terms. The resulting closed system of ODEs (ordinary differential equations) represents an extension to a recently published method of moments which utilizes least-square approximations of factors of the PDE over orthogonal polynomial bases. An extensive numerical analysis has been carried out for proof-of-concept purposes. The proposed modeling scheme is generally of interest for control and optimization of processes with distributed parameters.     

Solving population balance equations for two-component aggregation by a finite volume scheme

A conservative finite volume approach, originally proposed by Filbet and Laurençot [2004a. Numerical simulation of the Smoluchowski coagulation equation. SIAM Journal on Scientific Computing 25(6), 2004-2048] for the one-dimensional aggregation, is extended to simulate two-component aggregation. In order to apply the finite volume scheme, we reformulate the original integro-ordinary differential population balance equation for two-component aggregation problems into a partial differential equation of hyperbolic-type. Instead of using a fully discrete finite volume scheme and equidistant discretization of internal properties variables, we propose a semidiscrete upwind formulation and a geometric grid discretization of the internal variables. The resultant ordinary differential equations (ODEs) are then solved by using a standard adaptive ODEs-solver. Several numerical test cases for the one and two-components aggregation process are considered here. The numerical results are validated against available analytical solutions.     

Predictive control of particlesize distribution of crystallization process using deep learning based image analysis

The challenges to regulate the particle-size distribution (PSD) stem from on-line measurement of the full distribution and the distributed nature of crystallization process. In this article, a novel nonlinear model predictive control method of PSD for crystallization process is proposed. Radial basis function neural network is adopted to approximate the PSD such that the population balance model with distributed nature can be transformed into the ordinary differential equation (ODE) models. Data driven nonlinear prediction model of the crystallization process is then constructed from the input and output data and further be used in the proposed nonlinear model predictive control algorithm. A deep learning based image analysis technology is developed for online measurement of the PSD. The proposed PSD control method is experimentally implemented on a jacketed batch crystallizer. The results of crystallization experiments demonstrate the effectiveness of the proposed control method.     

A numerical technique for the solution of integrodifferential equations arising from balances over populations of drops in turbulent flows

This paper describes a numerical technique designed to solve certain forms of partial differential equations. The method is applied to the partial integrodifferential population balance equations presented by Jairazbhoy [Jairazbhoy, V., Tavlarides, L. L., & Lewalle, J., (1995) A cascade model for neutrally buoyant two-phase homogeneous turbulence – part I. Model formulation. International Journal of Multiphase Flow, 21(3), 467] that describe the behavior of dense liquid dispersions of interacting drops in isotropic turbulence. In the successively contained semi-discretization scheme developed, the drop number density functions are discretized into non-uniform intervals corresponding to Gaussian quadrature points. The governing equations are assumed to hold identically at all the discretization points, generating a set of ordinary integrodifferential equations that are solved by an integrator package. The integrals in each function evaluation are calculated by Gaussian quadrature. The results show that, in some cases, as many as fifteen quadrature points are required to achieve grid independence. Each additional discretization point results in an additional ordinary integrodifferential equation. To achieve comparable accuracy with a uniform discretization scheme, many more discretization points would be required, resulting in an inordinately large number of ordinary integrodifferential equations. The computations also show that, in every run, there appears to be an optimum number of discretization intervals around which incremental increases in the resolution do not increase the CPU time or perceivable accuracy of the solution.     

Online inferential product attribute estimation for optimal operation of emulsion terpolymerisation: Application to styrene/MMA/MA

An advanced model for process design and control of emulsion terpolymerisation was developed. A test case of emulsion terpolymerisation of styrene (Sty), methyl methacrylate (MMA) and methyl acrylate (MA) was investigated on state of the art facilities for predicting, optimising and control end-use product properties including global and individual conversions, terpolymer composition, the average particle diameter and concentration, glass transition temperature, molecular weight distribution, the number- and weight-average molecular weights and particle size distribution.The model equations include diffusion-controlled kinetics at high monomer conversions, where transition from a ‘zero-one’ to a ‘pseudo-bulk’ regime occurs. Transport equations are used to describe the system transients for batch and semi-batch processes. The particle evolution is described by population balance equations which comprised a set of integro-partial differential and nonlinear algebraic equations. Backward finite difference approximation method is used to discretise the population equation and converts them from partial differential equations to ordinary differential equations. The model predictions were experimentally validated in the laboratory and were found to be in excellent agreement, thus paving the way for further application of the model.     

A Laplace transformation based technique for reconstructing crystal size distributions regarding size independent growth

This article introduces a technique for reconstructing crystal size distributions (CSDs) described by well-established batch crystallization models. The method requires the knowledge of the initial CSD which can also be used to calculate the initial moments and initial liquid mass. The solution of the reduced four-moment system of ordinary differential equations (ODEs) coupled with an algebraic equation for the mass gives us moments and mass at the discrete points of the given computational time domain. This information can be used to get the discrete values of size independent growth and nucleation rates. The discrete values of growth and nucleation rates along with the initial distribution are sufficient to reconstruct the final CSD. In the derivation of current technique the Laplace transformation of the population balance equation (PBE) plays an important role. The proposed technique has dual purposes. Firstly, it can be used as a numerical technique to solve the given population balance model (PBM) for batch crystallization. Secondly, it can be used to reconstruct the final CSD from the initial one and also vice versa. The method is very efficient, accurate and easy to implement. Several numerical test problems of batch crystallization processes are considered here. For validation, the results of the proposed technique are compared with those from the high resolution finite volume scheme which solves the given PBM directly.     

Mathematical modelling of the transient behaviour of cstrs with reactive particulates: Part 1 — The population balance framework

The population balance model appears to be the best approach to model particulate systems where multiple heterogeneous reactions occur. This work demonstrates a mathematical formulation that is based on the population balance model, and aims at simulating the non steady-state behaviour of a single-stage CSTR under isothermal operation. The chemical reaction system is a typical example from the field of hydrometallurgy with two parallel reactions, one being leaching, the other precipitation with simultaneous reactant regeneration. The solution of the resulting system of the partial and ordinary differential equations is achieved by combining the moment transformation of the population balance equations with the numerical method of lines, using the Mathematica® software. Finally, examples are given for a reactor startup in two cases: a single leaching reaction, and simultaneous leaching and precipitation reactions. In the first case, the difference between simultaneous and sequential feeding in achieving steady-state is also discussed.     

Numerical solution of multi-variable cell population balance models. II. Spectral methods

Several Galerkin, Tau and Collocation (pseudospectral) approximations have been developed for the solution of the multi-variable cell population balance model in its most general formulation, i.e. for any set of single-cell physiological state functions. Time-explicit methods were found to be more efficient than time-implicit methods for the time integration of the system of ordinary differential equations that results after the spectral approximation in space. The Legendre and Tchebysheff polynomials that were used in Tau algorithms were shown to have significantly worse convergence and stability properties than the Galerkin and collocation algorithms that were applied with sinusoidal trial functions. The collocation method that was implemented with discrete fast Fourier transforms was found to be the most efficient from all the Galerkin and Tau algorithms that were developed. However, the method was inferior to the best finite difference algorithm that was presented in our earlier work.     

Analytical solutions to the semi-discrete form of the conduction equation for non-homogeneous media

The partial differential equation describing transient conduction (or diffusion) in non-homogeneous media may be approximated by a set of first order linear ordinary differential equations if the derivatives involving the space variables are replaced by finite difference expressions. A general method of obtaining a closed form solution of these equations is presented, using some operational methods of linear algebra. The solution is given in terms of a matrix, which describes the spatial distribution of physical properties in the media, and vectors describing the initial and boundary conditions.     

Numerical technique for solving partial differential equations with applications to adsorption process

A new computational scheme of partial differential equations which describe the characteristics of an adsorption process in a packed bed is developed. The solution for the concentration and temperature profiles is obtained by first transforming the partial differential equations into ordinary differential equations along characteristic paths and then solving them with a combination of finite difference methods. The computed temperature and concentration distributions are compared with those available from the literature. The computational algorithm is found more efficient and more precise than previously documented ones.     

Adaptive finite difference method for the simulation of batch crystallization

The modeling of batch crystallization involves a population balance to get the crystal size distribution which is one of the most important properties in this process. Thus, this model leads to a system of integral, partial differential and algebraic equations (IPDAE). This system can be easily solved by finite difference methods with uniform discretization. However to increase the calculation efficiency and the crystal size distribution accuracy, an adaptive finite difference method with a non-uniform discretization was developed. Results of both methods are compared in term of efficiency and are confronted to experimental data in term of accuracy.     

Interfacial area density in bubbly flow

The interfacial area per unit volume is one of the key parameters in bubbly flow. Momentum, mass and energy transfer occur through the interface between the phases. The functionality of two phase reactors with bubbly flow depends mainly on these three transfer processes. Thus, the design process of a reactor requires the prediction of interfacial area density. In the present work a simple equation for the interfacial area density is derived from the population balance, taking into account the events of coalescence and bubble break-up for each bubble fraction. The system of partial integro-differential equations is simplified. Since the integrals in these equations complicate a numerical treatment. This reduces the balance to one single partial differential equation. An approximate analytical solution is given. If the resulting equation is applied to large gas fluxes, the instability of the coalescence process causes large bubbles and gas plugs to develop. From the instability the volume fraction of the large bubbles and gas plugs may be predicted. Additives may hinder the coalescence process. Experiments show that coalescence hindrance changes the coalescence kernel only by a factor. Calculations are done for bubble columns and vertical pipe flow.     

Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods

In this paper, we analyse some properties of the orthogonal collocation in the context of its use for reducing PDE (partial differential equations) chemical reactor models for numerical simulation and/or control design. The approximation of the first order derivatives is first considered and analysed with respect to the transfer of the stability properties of the transport component from the PDE model to its approximated ODE (ordinary differential equations) model. Then the choice of the collocation points as zero of Jacobi polynomial is analysed and interpreted as an optimal choice with respect to a weighted norm. Finally, some guidelines for the use of orthogonal collocation are proposed and the results are illustrated on a simulation example.     

Nonlinear model reduction for dynamic analysis of cell population models

Transient cell population balance models consist of nonlinear partial differential-integro equations. An accurate discretized approximation typically requires a large number of nonlinear ordinary differential equations that are not well suited for dynamic analysis and model based controller design. In this paper, proper orthogonal decomposition (also known as the method of empirical orthogonal eigenfunctions and Karhunen Loéve expansion) is used to construct nonlinear reduced-order models from spatiotemporal data sets obtained via simulations of an accurate discretized yeast cell population model. The short-term and long-term behavior of the reduced-order models are evaluated by comparison to the full-order model. Dynamic simulation and bifurcation analysis results demonstrate that reduced-order models with a comparatively small number of differential equations yield accurate predictions over a wide range of operating conditions.