Improving the accuracy of the moments method for solving the aerosol general dynamic equation

The General Dynamic Equation for aerosol evolution is converted into a set of ordinary differential equations for the moments M_m by multiplying by v^m and integrating over particle volume, v. Closure of these equations is achieved by assuming a functional form for the moments, instead of the usual assumption of a functional form for the size distribution itself. Specifically, it is assumed that In(M_m) can be expressed as a pth-order polynomial in m. The time-dependent coefficients in the polynomial are found by solving (p + 1) differential equations numerically. The case p = 2 corresponds to the assumption that the size distribution is always log-normal but comparison with accurate solutions shows that increasing p increases the accuracy of the method for all processes considered (removal, condensation and Brownian coagulation). Particle loss during evaporation and achievement of a self-preserving form for Brownian coagulation are also considered. Inversion of the moment expression to obtain the size distribution using the Mellin inversion formula is discussed.     

Automatic differentiation‐based quadrature method of moments for solving population balance equations

The quadrature method of moments (QMOM) is a promising tool for the solution of population balance equations. QMOM requires solving differential algebraic equations (DAEs) consisting of ordinary differential equations related to the evolution of moments and nonlinear algebraic equations resulting from the quadrature approximation of moments. The available techniques for QMOM are computationally expensive and are able to solve for only a few moments due to numerical robustness deficiencies. In this article, the use of automatic differentiation (AD) is proposed for solution of DAEs arising in QMOM. In the proposed method, the variables of interest are approximated using high‐order Taylor series. The use of AD and Taylor series gives rise to algebraic equations, which can be solved sequentially to obtain high‐fidelity solution of the DAEs. Benchmark examples involving different mechanisms are used to demonstrate the superior accuracy, computational advantage, and robustness of AD‐QMOM over the existing state‐of‐the‐art technique, that is, DAE‐QMOM. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

拟稳态离子交换过程操作行为的研究

本文对TPCIX塔的周期性操作行为进行了模拟研究.引用概率密度的概念,即转化率分布密度函数对连续离子交换过程进行分析.采用主导矩法对描述拟稳态操作的偏微分方程组进行数值解.计算结果与实验规律基本一致.     

The moment method for one-dimensional dynamic reactor models with axial dispersion

A polynomial approximation method for calculating state profiles for plug-flow reactors is extended to one-dimensional reactor models that include axial dispersion. The method is based on the conservation of reactor state profile moments along the spatial dimension. The moments are then transformed analytically into a polynomial approximation at each timestep. The boundary conditions of the parabolic partial differential equation are given special attention. It is shown that the Danckwerts boundary conditions are an appropriate set of boundary conditions for flow problems with axial dispersion in closed-closed geometries. A significant feature of the present method is that boundary conditions of the partial differential equation model to be solved are implicitly satisfied via the moment transformation, while the polynomial profile in the numerical approximation does not have to satisfy the boundary conditions exactly. The method is tested in two cases: startup of a tubular reactor and fixed-bed adsorber involving axial dispersion.     

An efficient numerical technique for solving one-dimensional batch crystallization models with size-dependent growth rates

In this work, an efficient numerical method is introduced for solving one-dimensional batch crystallization models with size-dependent growth rates. The proposed method consist of two parts. In the first part, a coupled system of ordinary differential equations (ODEs) for the moments and the solute concentration is numerically solved to obtain their discrete values in the time domain of interest. These discrete values are also used to get growth and nucleation rates in the same time domain. To overcome the issue of closure, a Gaussian quadrature method based on orthogonal polynomials is employed for approximating integrals appearing in the ODE system. In the second part, the discrete growth and nucleation rates along with the initial crystal size distribution (CSD) are used to construct the final CSD. The expression for CSD is obtained by applying the method of characteristics and Duhamel's principle on the given population balance model (PBM). The proposed method is efficient, accurate, and easy to implement in the computer. Several numerical test problems of batch crystallization processes are considered. For a validation, the results of the proposed technique are compared with those obtained using a high resolution finite volume scheme.     

The quadrature method of moments and its relationship with the method of weighted residuals

Population balance models are generally computationally intensive, so in many practical applications only a few moments of the density function are computed, minimizing the computational costs. Nevertheless, the moment formulation contains an excess of unknowns with respect to equations denoting a closure problem. One possible solution to this closure problem might be to apply a numerical quadrature approximation.In this work, the relationship between the quadrature approximation and the well-known method of weighted residuals (MWR) is discussed. An important result obtained in this work is that the problem of reconstruction of the density function is avoided using the MWR version of the quadrature approximation. Numerical experiments are performed in order to elucidate the advantages and disadvantages of the quadrature approximations.     

Semianalytical method of lines for solving elliptic partial differential equations

A semianalytical method of lines is presented for solving elliptic partial differential equations, which are often used to describe steady-state mass and energy transport in solids. The method provides a semianalytical solution for linear equations and can be used to obtain explicit symbolic series solutions in one of the independent variables for non-linear equations.     

Shifted Legendre function approximation of differential equations; application to crystallization processes

An effective new method solves either the stretched functional ordinary differential equation describing a crystallization process using breakage models, or the partial differential equations for a population balance which represent the transient crystallization process. The population balance density function is first expanded into a series of shifted Legendre polynomial functions. The partial differential equation (or the ordinary differential equation) is first transformed into a series of ordinary differential equations (or of algebraic equations) for the expansion coefficients which are solved by numerical computation. The computational time is greatly reduced through a recursive algorithm for the integration of the triple product of the shifted Legendre polynomial functions. Illustrative examples are given and the results are compared with data available in the literature. The proposed method is powerful, accurate and more precise than previously documented ones.     

SOLUTION OF POPULATION BALANCE EQUATIONS IN EMULSION POLYMERIZATION USING METHOD OF MOMENTS

In this work, the method of moments is used for solution of population balance equations appearing in modeling of emulsion polymerization (EP). The zero-one model without coagulation effect and the pseudo-bulk model including coagulation effect are investigated as two common approaches for modeling EP processes. The fixed quadrature method is used to close the set of moment equations, and the maximum entropy approach is applied to reconstruct the particle size distribution from a finite number of its moments. Comparing the results with those obtained by the high-precision finite volume technique indicates that, despite the low computational load of the moment method, it has an acceptable accuracy. These features support use of the moment technique for other applications such as on-line control or optimization in particulate processes.     

Evaluation of the 1-point quadrature approximation in QMOM for combined aerosol growth laws

This work examines the applicability of various assumptions in implementation of the quadrature method of moments (QMOM) for solving problems in aerosol science involving simultaneous nucleation, surface growth and coagulation. The problem of aerosol growth and coagulation in a box and the problem of vapor condensation in a nozzle are reworked using quadrature method of moments. QMOM uses Gaussian quadrature to evaluate integrals appearing in the moment equations and therefore does not require any assumptions on the form of the size distribution function, the growth laws and coagulation kernels. Results are compared with calculations which assume a lognormal size distribution. The conditions for which one, two and higher quadrature points can be used in the quadrature formula and the issues regarding the accuracy are considered for combined aerosol nucleation, growth and coagulation processes. Results show that for these problems, the simplest 1-point quadrature scheme gives accuracy comparable with the lognormal calculations while using two and higher point quadrature gives highly accurate results. Some difficulties associated with the QMOM are discussed and some insights are provided.     

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.     

APPLICATIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS TO CHEMICAL-ENGINEERING PROBLEMS -AN INTRODUCTORY REVIEW

The underlying theoretical structure of stochastic differential equations requires an extension of the classical calculus, and this is described. These equations provide a convenient method for the construction of Markov processes that have pre-specified statistical properties and that are very useful as models for random disturbances in process plants. The utility as models for random systems is demonstrated by several examples.

Itôs lemma, the method of moments, and the method of partial moments are demonstrated as methods of solution of these equations. The theory provides an excellent framework for the design and analysis of control systems for randomly disturbed plants. Examples are given to show that classical design techniques can be completely inadequate for plants of this nature, whereas the stochastic theory leads to good design.     

Modeling the continuous melt transesterification of bisphenol-A and diphenyl carbonate in a continuous stirred tank reactor

A model of continuous melt transesterification of bisphenol-A and diphenyl carbonate in a continuous stirred tank reactor is developed using phase equilibria assumption and the method of molecular weight moments. The model equations can be simplified into a polynomial system that has 17 equations and 17 unknowns. Solution of the polynomial system gives out almost every aspects of the continuous transesterification process. Molecular weight and polydispersity index, end group ratio of hydroxyl to phenyl carbonate, contents of molecular species, and lost diphenyl carbonate fractions are studied in different operation parameters.     

An implementation variant of the polynomial finite difference method with orthogonal collocation and adjustable element length

The polynomial finite difference method, an easy-to-use variant of the finite difference method for the numerical solution of differential and differential–algebraic equations, has been recently presented [Wu, B., & White, R.E. (2004). Computers & Chemical Engineering, 28, 303–309]. In this work, it is shown that the polynomial finite difference method can be seen as a collocation method with finite elements of equal size with uniform distribution of collocation points within each element. We show that the same type of implementation can be improved if one uses orthogonal distribution of collocation points, without significantly affecting the computational effort. The suggested method is further improved with the use of Michelsen's technique for step-size adjustment to solve stiff differential equations with a semi-implicit third order method. Several examples that show improvements of one or two orders of magnitude of the proposed approach over the implementation by Wu and White are presented.     

A dynamic model for plug flow reactor state profiles

A dynamic model for state profiles of a plug flow reactor is developed, including multiple fluid and solid phases. The model is based on conservation of reactor state profile moments along the spatial dimension of the reactor. These moments are transformed analytically into a polynomial approximation at each time step. The method is flexible, and low as well as high order numerical schemes are resulted in by appropriate choice of parameters. A significant advantage of the present method is that boundary conditions of the partial differential equation reactor model are implicitly satisfied via the moment transformation, whereas the polynomial profile in the numerical solution does not have to be forced to satisfy the boundary conditions. The method is tested numerically against analytical solutions in three numerically challenging benchmark cases: prediction of breakthrough curve in packed bed adsorbers; simulation of chromatographic separation; and feeding a step impulse in a plug flow dimerization reactor. It is shown that the high resolution methods result in considerably smaller numerical errors than a simple low-order assumption of piecewise continuous solution.     

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.     

Modal Aerosol Dynamics Modeling

ABSTRACT

The derivation of the governing equations for modal aerosol dynamics (MAD) models is presented. MAD models represent the aerosol size distribution as an assemblage of distinct populations of aerosol, where each population is distinguished by size or chemical composition. The size distribution of each population is approximated by an analytical modal distribution function; usually by a lognormal distribution function. By substituting the MAD representation of aerosol size distributions into the governing equation for aerosol processes, the governing differential equations for MAD models are derived. These differential equations express the time dependence of the moments of the aerosol size distribution and are called Moment Dynamics Equations (MDEs). The MDEs for Continuously-Stirred Tank Aerosol Reactors (CSTARs) are also derived.     

Feedback control of hyperbolic distributed parameter systems

Hyperbolic distributed parameter systems (DPS) represent a large number of industrial processes with spatially nonuniform operating variable profiles. Research has been conducted to develop high-performance control strategies for these systems by exploiting their high-fidelity models. In this paper, a feedback control method that yields improved performance is proposed for DPS modelled by first-order hyperbolic partial differential equations (PDEs) using the method of characteristics. Simulation results show that this method can provide effective control for the systems modelled by a scalar PDE as well as a system of PDEs. Further, it can efficiently compensate the effect of model-plant mismatch and effectively reject the disturbances.     

PRISM: Piecewise Reusable Implementation of Solution Mapping. An Economical Strategy for Chemical Kinetics

In a chemical kinetics calculation, a solution-mapping procedure is applied to parametrize the solution of the initial-value ordinary differential equation system as a set of algebraic polynomial equations. To increase the accuracy, the parametrization is done piecewise, dividing the multidimensional chemical composition space into hypercubes and constructing polynomials for each hypercube. A differential equation solver is used to provide the solution at selected points throughout a hypercube, and from these solutions the polynomial coefficients are determined. Factorial design methods are used to reduce the required number of computed points. The polynomial coefficients for each hypercube are stored in a data structure for subsequent reuse, since over the duration of a flame simulation it is likely that a particular set of concentrations and temperature will occur repeatedly at different times and positions. The method is applied to H₂–air combustion using an 8-species reaction set. After N₂ is added as an inert species and enthalpy is considered, this results in a 10-dimensional chemical composition space. To add the capability of using a variable time-step, time-step is added as an additional dimension, making an 11-dimensional space. Reactive fluid dynamical simulations of a 1-D laminar premixed flame and a 2-D turbulent non-premixed jet are performed. The results are compared to identical control runs which use an ordinary differential equation solver to calculate the chemical kinetic rate equations. The resulting accuracy is very good, and a factor of 10 increase in computational efficiency is attained.