基于差分进化算法的控制变量参数化方法及其在化工过程动态优化中的应用(英文)

Two general approaches are adopted in solving dynamic optimization problems in chemical processes, namely, the analytical and numerical methods. The numerical method, which is based on heuristic algorithms, has been widely used. An approach that combines differential evolution (DE) algorithm and control vector parameterization (CVP) is proposed in this paper. In the proposed CVP, control variables are approximated with polynomials based on state variables and time in the entire time interval. Region reduction strategy is used in DE to reduce the width of the search region, which improves the computing efficiency. The results of the case studies demonstrate the feasibility and efficiency of the proposed methods.     

Distillation optimization

The optimization of operating conditions for distillation processes is studied. The numerical performance of several equation-oriented, Newton-like methods are compared on a variety of example problems. Numerical results show that the thermodynamically constrained hybrid method provides a reliable and efficient way of solving distillation optimization problems.     

一种改进的差分进化算法及其在补料分批式生化反应器动态优化中的应用

动态优化是生物化工过程中的重要课题,求解动态优化问题通常有两种方法:解析法和数值法。基于智能进化算法的数值方法在动态优化中的应用越来越广泛,但是这些方法局部寻优能力不强,容易陷入局部最优,并且求解速度相对较慢。针对这些方法的不足,提出了一种改进的差分进化算法,设计了新的局部寻优算子来增强算法的局部寻优能力,并且采用一种新的控制策略表示方法来求解动态优化问题。通过求解补料分批式生化反应器的动态优化实例,证明了算法的有效性和鲁棒性。通过与其他几种方法进行对比,实验结果表明,所提出的方法在优化结果和计算代价方面都有优势。     

A control oriented study on the numerical solution of the population balance equation for crystallization processes

The population balance equation provides a well established mathematical framework for dynamic modeling of numerous particulate processes. Numerical solution of the population balance equation is often complicated due to the occurrence of steep moving fronts and/or sharp discontinuities. This study aims to give a comprehensive analysis of the most widely used population balance solution methods, namely the method of characteristics, the finite volume methods and the finite element methods, in terms of the performance requirements essential for on-line control applications. The numerical techniques are used to solve the dynamic population balance equation of various test problems as well as industrial crystallization processes undergoing simultaneous nucleation and growth. The time-varying supersaturation profiles in the latter real-life case studies provide more realistic scenarios to identify the advantages and pitfalls of a particular numerical technique.The simulation results demonstrate that the method of characteristics gives the most accurate numerical predictions, whereas high computational burden limits its use for complex real crystallization processes. It is shown that the high order finite volume methods in combination with flux limiting functions are well capable of capturing sharp discontinuities and steep moving fronts at a reasonable computational cost, which facilitates their use for on-line control applications. The finite element methods, namely the orthogonal collocation and the Galerkin's techniques, on the other hand may severely suffer from numerical problems. This shortcoming, in addition to their complex implementation and low computational efficiency, makes the finite element methods less appealing for the intended application.     

Part I: Dynamic evolution of the particle size distribution in particulate processes undergoing combined particle growth and aggregation

The present study provides a comprehensive investigation on the numerical problems arising in the solution of dynamic population balance equations (PBEs) for particulate processes undergoing simultaneous particle growth and aggregation. The general PBE was numerically solved in both the continuous and its equivalent discrete form using the orthogonal collocation on finite elements (OCFE) and the discretized PBE method (DPBE), respectively. A detailed investigation on the effect of different particle growth rate functions on the calculated PSD was carried out over a wide range of variation of dimensionless aggregation and growth times. The performance (i.e., accuracy and stability) of the employed numerical methods was assessed by a direct comparison of predicted PSDs or/and their respective moments to available analytical solutions. It was found that the OCFE method was in general more accurate than the discretized PBE method but was susceptible to numerical instabilities. On the other hand, for growth dominated systems, the discretized PBE method was very robust but suffered from poor accuracy. For both methods, discretization of the volume domain was found to affect significantly the performance of the numerical solution. The optimal discretization of the volume domain was closely related with the satisfactory resolution of the time-varying PSD. Finally, it was shown that, in specific cases, further improvement of the numerical results could be obtained with the addition of an artificial diffusion term or the use of a moment-weighting method to correct the calculated PSD.     

Successive linearisation analysis of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects

In this study, we investigate the application of the new successive linearisation method (SLM) to the problem of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects. The governing nonlinear momentum, energy and mass transfer equations are successfully solved numerically using the SLM approach coupled with the spectral collocation method for iteratively solving the governing linearised equations. Comparison of the SLM results for various flow parameters against numerical results and other published results, obtained using the homotopy analysis method (HAM) and Runge–Kutta methods, for related problems indicates that the SLM is a very powerful tool which is much more accurate and efficient than other methods. The SLM converges much faster than the traditional methods like the HAM and is very easy to implement. © 2011 Canadian Society for Chemical Engineering     

A novel algorithm for solving population balance equations: the parallel parent and daughter classes. Derivation, analysis and testing

A novel numerical method, the parallel parent and daughter classes (PPDC) technique, for solving population balance equations (PBEs) is presented in this paper. In many practical applications, the PBE of particles under investigation is coupled with the thermo-fluid dynamics of the surrounding fluid. Hence, the PBE needs to be implemented in a computational fluid dynamics (CFD) code, which leads to an additional computational load. The computational cost becomes intractable when techniques such as methods of classes (CM) or Monte Carlo method are used. Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) are accurate and require a relatively low additional computational cost when applied to CFD. The PPDC is shown to be as accurate as QMOM and DQMOM, and even more accurate in some cases, when the same number of classes is used. In the present work, the PPDC technique has been derived and tested. This technique can be used for solving a wide class of problems involving PBE such as polymerization, aerosol dynamics, bubble columns, etc. Numerical simulations have been carried out on aggregation processes with different kernels and on simultaneous aggregation and breakage processes. The numerical predictions are compared either with analytical solutions, when available, or with the numerical solutions obtained by methods of classes.     

流固耦合系统计算方法及应用软件概述

许多领域都会出现流固耦合的问题,尤其是化工机械行业中.流固耦合(Fluid-Structure Interaction FSI)即是热点又是难点,而难点在于它的计算方法.本文对化工机械充液系统流固耦合计算方法的发展进行了概述,阐述了各个阶段使用的数值研究方法,并介绍了常用的计算流固耦合的商业软件.最后对化工机械充液系统...     

Improved numerical inversion methods for the recovery of bivariate distributions of polymer properties from 2D probability generating function domains

The 2D probability-generating function technique is a powerful method for modeling bivariate distributions of polymer properties. It is based on the transformation of bivariate population balance equations using 2D probability generating functions (pgf) followed by a recovery of the distributions from the transform domain by numerical inversion. A key step of this method is the inversion of the pgf transforms. Available numerical inversion methods yield excellent results for pgf transforms of distributions with independent dimensions with similar orders of magnitude, for example bivariate molecular weight distributions in copolymerization systems. However, numerical problems are found for 2D distributions in which the independent dimensions have very different ranges of values, such as the molecular weight distribution-branching distribution in branched polymers. In this work, two new 2D pgf inversion methods are developed, which regard the pgf as a complex variable. The superior accuracy of these innovative methods makes them suitable for recovering any type of bivariate distribution. This enhances the capabilities of the 2D pgf modeling technique for simulation and optimization of polymer processes. An application example of the technique in a polymeric system of industrial interest is presented.     

序贯优化化工动态问题的蚁群算法

针对化工动态优化问题,分析现有数值解法的不足,提出序贯执行蚁群寻优操作,逐步寻找最佳解的策略,构建序贯蚁群算法.算法首先对时间区间和控制变量搜索域实施离散化,以一组整数编码的蚁群路径表示可行控制策略,进而应用蚁群寻优操作寻找离散问题的最优控制策略.逐步收缩控制搜索域并反复上述步骤,不断改善寻优结果.序贯蚁群算法简便快捷,用于化工动态优化问题效果良好,计算结果体现了算法的稳健性.     

The solution of continuous kinetic lumping models using the adaptive characterization method

The continuous kinetic lumping models are traditionally solved by methods that discretize the mixture into a large number of pseudo-components. This works proposes the usage of the adaptive characterization of continuous mixtures, grounded on the direct quadrature method of generalized moments, in the solution of kinetic lumping models, which allows a large reduction in the number of pseudo-components. Catalytic hydrogenation and hydrocracking problems were used to evaluate this methodology, comparing its results with analytical solutions or results from a classical numerical method. The results showed that the proposed methodology could accurately solve those continuous kinetic models using a small number of adaptive pseudo-components, leading to a large reduction in the computational cost of simulation when compared to the classical numerical method.     

承压含水层地下水稳定流的有限层分析

有限层法是一种对空间某一方向进行数值离散,而在其余两方向采用连续函数的半数值半解析方法.该方法能有效地将三维问题简化为一维问题求解,从根本上解决了常用数值分析方法在模拟三维地下水运动时存在的计算工作量大、占用内存多、耗时大等缺点.文中基于有限层法的优点,推导了以伽辽金法结合贝塞耳函数为基础的层状非均质各向异性承压含水层的稳定流有限层方程,并编制了相应的计算程序.通过对2个经典算例的数值解与解析解对比分析,验证了该方法的正确性.     

COMPARISON OF APPROXIMATE METHODS FOR SOLVING NON-LINEAR CONVECTIVE DIFFUSION PROBLEMS

A comparison is made of the approximate methods of solving a high Schmidt number non-linear convective diffusion equation and boundary conditions similar to those which arise in various separation problems including reverse osmosis and directional solidification. The series expansion method gives the best agreement with exact numerical solutions. The film theory, which provides very simple results, yields surprisingly good estimates which are second in accuracy only to the series expansion. Several more sophisticated techniques including rapidly varying boundary conditions, the integral method and local non-similarity yield results of somewhat disappoinling accuracy. The linear problem, B₂ = 0, for which exact analytical and numerical solutions exist, provides a discriminating test of the accuracy of the methods.     

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.     

SIMULATION VIA ORTHOGONAL COLLOCATION ON FINITE ELEMENT OF A CHROMATOGRAPHIC COLUMN WITH NONLINEAR ISOTHERM

The orthogonal collocation on finite element method is implemented for simulating a liquid chromatographic column. This procedure was retained from a comparative study of various numerical methods described in the first part of the paper. The modelling equations represent a general method for multicomponent systems, with linear or nonlinear equilibrium isotherms. The numerical procedure is illustrated in the second part by the simulation of single and binary systems. In the case of nonlinear isotherm the Langmuir isotherm is chosen.

The numerical results show that the orthogonal collocation on finite elements is an efficient tool for solving liquid chromatography problems, even if high Peclet numbers are considered. When a non-competitive Langmuir isotherm is considered for the separation of binary systems, the retained numerical method reaches the convergence within low CPU times. For the case of competitive isotherm, the simulation of binary systems was carried out successfully but with larger CPU time.     

COMPARISON OF APPROXIMATE METHODS FOR SOLVING NON-LINEAR CONVECTIVE DIFFUSION PROBLEMS

A comparison is made of the approximate methods of solving a high Schmidt number non-linear convective diffusion equation and boundary conditions similar to those which arise in various separation problems including reverse osmosis and directional solidification. The series expansion method gives the best agreement with exact numerical solutions. The film theory, which provides very simple results, yields surprisingly good estimates which are second in accuracy only to the series expansion. Several more sophisticated techniques including rapidly varying boundary conditions, the integral method and local non-similarity yield results of somewhat disappoinling accuracy. The linear problem, B ₂ = 0, for which exact analytical and numerical solutions exist, provides a discriminating test of the accuracy of the methods.     

Computation of the gradient and sensitivity coefficients in sum of squares minimization problems with differential equation models

In parameter estimation problems where the system model consists of differential equations, methods for minimizing a sum of squares of residuals objective function require derivatives of the residuals with respect to the parameters being estimated (sensitivity coefficients) or the gradient of the objective function (depending on the numerical optimization method). This paper considers two methods for generating such derivatives: (1) the adjoint equation — gradient formula; and (2) complimentary sensitivity coefficient differential equations. Particular attention is given to the consistency between the method used to solve the model equations and the proper formulation of the additional equations required by the two methods. Two example problems illustrate computational experience using a modified quasi-Newton method with the adjoint method used to generate gradients and applying a modified Gauss-Newton approach with the sensitivity coefficient equations to calculate both the Gauss-Newton matrix and the objective function gradient. Results indicate the superiority of the sensitivity coefficient approach. When comparing the computational effort required by the two methods and the results from the simple examples, it appears that the use of complimentary sensitivity coefficient equations is much more efficient than using only the gradient of the sum of squares function.     

Solution of hyperbolic PDEs using a stable adaptive multiresolution method

An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. The grid is dynamically adapted during the integration procedure so that only the relevant information is stored. The convection terms are discretised with high-resolution methods, thus ensuring boundedness. The proposed method is general, but is particularly useful for highly convective problems involving sharp moving fronts, a situation that frequently occurs in many chemical engineering problems, and where standard procedures may lead to unphysical oscillations in the computed solution.Numerical results for five test problems are presented to illustrate the efficiency and robustness of the method. The adaptive strategy is found to significantly reduce the computation time and memory requirements, as compared to the fixed grid approach.     

Two Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process

Various particulate systems were modeled by the population balance equation (PBE). However, only few cases of analytical solutions for the breakage process do exist, with most solutions being valid for the batch stirred vessel. The analytical solutions of the PBE for particulate processes under the influence of particle breakage in batch and continuous processes were investigated. Such solutions are obtained from the integro‐differential PBE governing the particle size distribution density function by two analytical approaches: the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). ADM generates an infinite series which converges uniformly to the exact solution of the problem, while HPM transforms a difficult problem into a simple one which can be easily handled. The results indicate that the two methods can avoid numerical stability problems which often characterize general numerical techniques in this area.