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.     

Computing multiple steady states in homogeneous azeotropic and ideal two‐product distillation

Multiple steady states are typically discovered by tracing a solution path, including turning points. A new technique is presented here that does not follow this approach. The original problem is solved directly, without tracing a solution path. The proposed branch‐and‐prune algorithm is guaranteed to find all solutions automatically. Core components of the framework are affine arithmetic, constraint propagation, and linear programming. The C++ implementation is available as an open‐source solver and has an interface to the AMPL® modeling environment. In certain difficult cases, only continuation methods have been reported to find the unstable solution automatically. The proposed method seems to be the first published alternative method in those cases. Although this article focuses mainly on distillation, the presented framework is fairly general and applicable to a wide variety of problems. Further, computational results are given to demonstrate this. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A novel predictive homotopic path tracking algorithm to solve non-linear algebraic equations

In chemical engineering and other areas of mathematics and engineering sciences, systems of non-linear algebraic equations often must be solved, and a problem is that these types of systems often cannot be solved analytically. Homotopic continuation methods are globally convergent and can find several solutions to the analyzed algebraic system and consist of a predictive step followed by a corrective step, which can cause reversion of homotopic path tracking. Therefore, in this work, the use of a homotopy continuation method (HCM) based on a purely predictive methodology is proposed; that is, it only makes use of predictor vectors (without correction vectors) to find the solution to several systems of non-linear algebraic equations with a multiplicity of solutions, employing lower central proccessing unit (CPU) time in comparison with standard homotopic tracking algorithms.     

Development of a learning strategy to assist in explaining homotopy methods in the field of separation processes

In this work, a very simple exercise is proposed, which can be solved “by hand”, to facilitate the understanding of the application of homotopy methods as rigorous methods for the calculation of complex multicomponent distillation operations in the context of an advanced separation process course for final year students of Chemical Engineering. The problem involves calculating of the liquid–vapour equilibrium data of a binary mixture with a homogeneous maximum azeotrope. The results corresponding to an ideal mixture are considered as the starting point for the application of the homotopy principle, in order to achieve the results corresponding to the non-ideal mixture at the final point of the homotopy pathway. This procedure enables undergraduates to perfectly understand the characteristics of these methods, thus avoiding the mathematic complexity associated with more complex problems, which could need sophisticated software or some type of programming tools.     

Modified bounded homotopies to enable a narrow bounding zone

The aim of this paper is to introduce modifications that enhance the usability of the bounded homotopy methods proposed by Paloschi [1995. Bounded homotopies to solve systems of algebraic nonlinear equations. Computers and Chemical Engineering 19, 1243-1254; 1997. Bounded homotopies to solve systems of sparse algebraic nonlinear equations. Computers and Chemical Engineering 21, 531-541], especially in the area of chemical engineering. In modified bounded homotopies, the homotopy path is tracked by exploiting mapped variables instead of unmapped ones. Path tracking based on mapped variables makes it significantly easier to track the bounded homotopy path even though the bounding zone has to be narrow. Mapping also improves the bounding effect of bounded homotopies and makes it possible to avoid unreasonable variable values in homotopy path tracking. The performance of the modifications is illustrated with test cases. These examples clearly show that the modifications enlarge the capability and accuracy of bounded homotopies when solving both small- and large-scale sets of nonlinear equations describing chemical engineering problems.     

A new homotopy for seeking all real roots of a nonlinear equation

A new continuation method, which applies a new homotopy that is a combination of the fixed-point and Newton homotopies (FPN), is developed for seeking all real solutions to a nonlinear equation, written as f(x) = 0, without having to specify a bounded interval. First, the equation to be solved is multiplied by (x − x⁰), where x⁰ is the starting value, which is set to zero unless the function does not exist at x⁰, in which case x⁰ becomes a tracking initiation point that can be set arbitrarily to any value where the function does exist. Next, the new function, (x − x⁰)f(x) = 0, is incorporated into the FPN homotopy. The initial step establishes a single bifurcation point from which all real roots can be found. The second step ensures a relatively simple continuation path that consists of just two branches that stem from the bifurcation point and prevents the formation of any isola. By tracking the two branches of the homotopy path, all real roots are located. Path tracking is carried out with MATLAB, using the continuation toolbox of CL_MATCONT, developed by Dhooge et al. (2006), based on the work of Dhooge, Govaerts, and Kuznetsov (2003), which applies Moore-Penrose predictor-corrector continuation to track the path, using convergence-dependent step-size control to negotiate turning points and other sharp changes in path curvature. This new method has been applied, without failure, to numerous nonlinear equations, including those with transcendental functions. As with other continuation methods, f(x)must have twice-continuous derivatives.     

An automatic initialization procedure in parameter estimation problems with parameter-affine dynamic models

This paper proposes an initialization approach for parameter estimation problems (PEPs) involving parameter-affine dynamic models. By using the state measurements, the nonconvex PEP is modified such that a convex approximation to the original PEP is obtained. The modified problem is solved by convex optimization methods yielding an approximate solution to the original PEP. The approximate solution can be further refined by linearizing the original problem around the obtained minimum. An assessment of the distance between the real solution and the one provided by the linearization of the problem around the convex approximation is presented. The optimum obtained by the convex approximation is used to subsequently initialize a simultaneous Gauss–Newton (SGN) approach on the original nonconvex PEP. Comparative results for the SGN with arbitrary initialization and with the proposed approach are presented using three benchmark examples in the chemical and biological fields.     

EXPERIENCE WITH ADOPTING ONE-PARAMETER IMBEDDING METHODS TOWARD CALCULATION OF COUNTERCURRENT SEPARATION PROCESSES

The one-parameter imbedding method (also called homotopy or continuation) was adopted toward solution of large sets of nonlinear algebraical equations describing counter-current separation processes. Different imbedding functions were tested on a spectrum of difficult distillation problems ranging from distillation of hydrocarbons to strongly nonideal distillation problems. For the one-parameter imbedding functions studied in this report the classical Newton-Raphson Formula can be easily generated after an appropriate selection of the control parameters. Two different approaches were used to solve the homotopy equations: i) marching integration, ii) sequential use of the Newton-Raphson method. The one-parameter imbedding technique represents a trade-off between robustness and computation time. The algorithm is more robust than the Newton-Raphson technique, however, the computational time is usually higher. A combination of the one-parameter imbedding and the Newton-Raphson approach seems to be a very efficient method, the solution is approached by the one-parameter imbedding technique and the Newton-Raphson method is used to finish the iteration process. Geometrical interpretation of convergence is presented.     

Homotopy continuation enhanced branch and bound algorithms for strongly nonconvex mixed-integer nonlinear optimization

Large-scale strongly nonlinear and nonconvex mixed-integer nonlinear programming (MINLP) models frequently appear in optimization-based process synthesis, integration, intensification, and process control. However, they are usually difficult to solve by existing algorithms within acceptable time. In this study, we propose two robust homotopy continuation enhanced branch and bound (HCBB) algorithms (denoted as HCBB-FP and HCBB-RB) where the homotopy continuation method is employed to gradually approach the optimum of the NLP subproblem at a node from the solution at its parent node. A variable step length is adapted to effectively balance feasibility and computational efficiency. The computational results from solving four existing process synthesis problems demonstrate that the proposed HCBB algorithms can find the same optimal solution from different initial points, while the existing MINLP algorithms fail or find much worse solutions. In addition, HCBB-RB is superior to HCBB-FP due to much lower computational effort required for the same locally optimal solution.     

Generalized initialization for the dynamic simulation and optimization of grade transition processes using two-dimensional collocation

Optimization modeling tools are essential to determine optimal design specifications and operation conditions of polymerization processes, especially when quality indices based on molecular weight distributions (MWDs) must be enforced. This study proposes a generalized MWD-based optimization strategy using orthogonal collocation in two dimensions, which can capture the dynamic features of MWDs accurately. To enable the strategy, this study considers generalized initialization methods for large-scale simulation and optimization. Here, a homotopy method based on intermediate solutions is adopted to generate initial values for general steady-state simulation models, starting from an arbitrary known solution for any steady-state simulation model. For dynamic simulation models, the response of a first-order linear system is adopted to initialize the state variables. Case studies show the effectiveness of this procedure to enable systematic, reliable, and efficient solution of the optimization problem.     

Ternary Diffusion with Two Moving Boundaries

The problem of unidirectional diffusion with 2 moving boundaries is pertinent to the study of reactions between condensed phases. The solution for binary diffusion is presented, and an eigenvector approach is outlined for the solution of the problem in a ternary diffusion system. The complete solution is given for some specific situations. Typically, the diffusion path deviates from the series of simple mixtures and tends to follow the direction of the minor eigenvector. This behavior is also true for cases where linear approximations are applicable.     

Studies on optimal control approach in a fed-batch fermentation

Fed-batch operation of fermentation processes has been receiving a great deal of interest as it offers the possibility to control a substrate concentration at a desired condition. However, control of a fed-batch fermentation reactor has been known to be a difficult task due to its highly nonlinear and complicated behavior. This work addresses an optimization-based control strategy for a fed-batch bioreactor where an ethanol fermentation process is chosen as a case study. The optimal control problem is formulated to determine the optimal feeding rate policy giving the highest product yield. The resulting optimization problem is solved by using an efficient sequential approach with a piecewise constant control parameterization. Due to the limitation of the sequential approach to cope with inequality path constraints, comparative studies of the methods for handling such constraints are carried out. Furthermore, the impact of time interval and switching time on the solution of the optimal control is investigated.     

A Differential Homotopy Continuation Method for Interlinked Solvent Extraction Cascades

Abstract

A robust differential homotopy procedure is developed that is suitable for solving the bordered block tridiagonal systems of nonlinear equations that result from liquid-liquid extraction models. The procedure is capable of readily utilizing newton, fixed-point, and custom homotopy mapping. It is shown to have an efficiency far superior to Kubicek's method and comparable to that of Rheinboldt. Application of the method to liquid-liquid extraction cascade models revealed that the newton homotopy would converge to a solution as rapidly as a simultaneous correction procedure and that use of a custom homotopy resulted in faster convercrence.     

基于微粒群优化算法的不确定性调和调度

Blending is an important unit operation in process industry. Blending scheduling is nonlinear optimization problem with constraints. It is difficult to obtain optimum solution by other general optimization methods. Particle swarm optimization （PSO） algorithm is developed for nonlinear optimization problems with both continuous and discrete variables. In order to obtain a global optimum solution quickly, PSO algorithm is applied to solve the problem of blending scheduling under uncertainty. The calculation results based on an example of gasoline blending agree satisfactory with the ideal values, which illustrates that the PSO algorithm is valid and effective in solving the blending scheduling problem.     

网格最优化信息反馈模型研究

针对网格计算中信息反馈机制或者信息传递机制因网格结点多、弧数量及考虑重点变化大、反馈路径和信息传递路径选择难度大的问题,提出一种基于优势比及路径期望的方法,对网格下的信息反馈问题进行建模,利用求解算法解出最优值。实验表明,该模型及求解算法在大规模网格信息反馈或者信息传递过程中能求得最优解,而且迭代次数与时间开销将趋于最优。     

Integration and dynamic inversion of population balance equations with size-dependent growth rate

Two novel solution schemes for integration and dynamic inversion of a class of population balance equations with size-dependent growth rate are contributed in this article. The proposed methods are developed for population balance systems that, in addition to an advective and a birth rate term, include an external variable which may be fixed (referring to the integration problem), or is to be computed for some prespecified evolution of the population system (referring to the dynamic inversion problem). A unique diffeomorphism for the independent time and internal (property) coordinates is introduced, which transforms the original nonlinear partial differential equation into a linear one with straight lines as equation characteristics. The evolution of the density function in the time and internal property coordinates is then computed by transporting the initial and boundary density functions in the transformed domain. For the integration of the temporal behavior of the boundary density function, a generalization of the standard method of moments is introduced, resulting in a closed integro-differential structure driven by convolution and correlation integrals. While the correlation integrals refer to the given initial density function and, hence, can be a priori computed, the convolution integrals involve the boundary density function and have to be integrated a posteriori online. The solution of the dynamic inversion problem, on the other hand, turns out to become simpler, as it converts to an algebraic equation after pre-computation of the correlation/convolution integrals for the given initial and boundary density function. In a next step, we introduce the concept of internal or eigenmoments which is useful for the representation of the original physical moments in terms of an infinite series. The dynamics of eigenmoments exhibits a closed ODE structure, which we refer to as the internal model. From the perspective of the systems theory this turns out to be an infinite dimensional flat system. Hence, in addition to providing a highly simple structure – basically, a chain of integrators – for the integration of the moments and the density function, the internal model allows for a direct solution of the dynamic inversion problem due to its flatness property. However, the ease and elegance of the method, in general, come at the price of an approximation of the infinite dimensional problem by a finite one. The usability of both proposed solution methods is illustrated on a batch crystallization process with size-dependent growth rate kinetics. The proposed methods are compared in terms of efficiency and accuracy with a state-of-the-art high-resolution finite volume scheme using a numerical example.     

Rapid and robust resolution of Underwood equations using convex transformations

In this work, a new method is proposed for solving Underwood's equations. Newton methods cannot be used without interval control, and may require many iterations or experience severe convergence problems if the roots are near poles and the initial guess is poor. It is shown that removing only one adjacent asymptote leads to convex functions, while removing both asymptotes leads to quasi convex functions which are close to linearity on wide intervals. Using a change of variable, a pair of convex functions is defined; at each point within the search interval one of the two functions is guaranteed to satisfy the monotonic convergence condition for Newton methods. The search interval is restricted to narrow solution windows (simple and costless) and a simple high quality initial guess can be obtained using their bounds. Two solution algorithms are presented: in the first one, Newton (including higher-order) methods can be safely applied without any interval control using the appropriate convex function; in the second one, Newton iterations are applied to a quasi-convex function, and convex functions are used only if an iterate goes out of its bounds. The algorithms are tested on several numerical examples, some of them recognized as very difficult in the literature. The proposed solution methods are simple, robust, very rapid (quadratic or super-quadratic convergence) and easy to implement. In most cases, convergence is obtained in 2–3 Newton iterations, even for roots extremely close to a pole.     

APPLICATION OF THE HOMOTOPY ANALYSIS METHOD (HAM) TO THIN FILM FLOW OF A GENERALIZED SECOND-GRADE FLUID ON A VERTICALLY MOVING BELT

We formulate and apply the homotopy analysis method (HAM) to determine the solution of the nonlinear differential equation governing thin film flow of a generalized second-grade fluid on a vertically moving belt. This problem is also known as the Landau–Levich or drag-out problem, which is the problem of withdrawal of a plate or fiber from a liquid bath. We make use of a modified model that has shear-dependent viscosity and that can predict normal stress differences. The results obtained exhibit the effectiveness and reliability of HAM.     

用于多产品间歇化工过程排序的模拟退火算法

提出了一种适用于多产品间歇化工过程排序的改进模拟退火算法 ,该算法采用了非平衡、多次退火策略及多个相邻解产生方法 .通过对无限中间储罐 (UIS)类问题的研究表明 ,同已有的方法相比 ,该方法求解精度高 ,计算速度快 ,适合于求解多产品间歇化工过程的排序问题 .