ORTHOGONAL COLLOCATION IN CHEMICAL REACTION ENGINEERING

The orthogonal collocation method is used to obtain approximate solutions to the differential equations modeling chemical reactors. There are two ways to view applications of the orthogonal collocation method. In one view it is a numerical method for which the convergence to the exact answer can be seen as the approximation is refined in successive calculations by using more collocation points, which are similar to grid points in a finite difference method. Another viewpoint considers only the first approximation, which can often be found analytically, and which gives valuable insight to the qualitative behavior of the solution. The answers, however, are of uncertain accuracy, so that the calculation must be refined to obtain useful numbers. However, with experience and appropriate caution, the first approximation is often sufficient and is easy to obtain. Thus it is very often useful in engineering work, where valid approximations are accepted. We present both viewpoints here: we use the first approximation to gain insight into a problem and we refine the calculations to obtain numerical convergence to the exact result. In this later view the method is similar to and in direct competition with finite difference methods, and some of the references listed in the next section discuss the relative advantages of the orthogonal collocation method.     

Analysis of wiped film reactors using the orthogonal collocation technique

Transport equation for ARB polymerization in wiped film reactors have been written. These have been reduced to the moment generation equations and using a suitable moment closure approximation, the zeroth and the second moments of the polymer have been numerically solved using the finite difference as well as the orthogonal collocation techniques. In the numerical solution by the finite difference technique, it is necessary to divide the dimensionless film thickness into at least 250 grid points to obtain stable results. The use of nine collocation points by the orthogonal collocation technique gives results close to those by the finite difference method and leads to considerable computational saving. The transport equations for the bulk and the film are found to involve four dimensionless parameters, and their effect upon the polymer formed at the end of the reactor has been studied.     

用正交配置法求解血液透析超滤的传质动力学模型

提出了血液透析、血液超滤和血液透析超滤过程传质的通用模型,并利用正交配置技术分析了影响传质速率的各种因素.结果表明,采用正交配置法进行上述传质过程的模拟时,简单的三点配置即与解析解的结果接近.采用正交配置法比有限差分法简单、快速和准确.     

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.     

联立法中全局和局部正交配置算法

化工过程的动态优化控制命题大多可以写成微分代数混合方程组(differential algebraic equations)的形式,联立法是求解该类命题的一种重要的数值方法。目前联立法中常用的离散方法是局部法,其中有限元正交配置(orthogonal collocation on finite elements)具有精度高、计算量小、稳定性好等优点。然而伪谱法(pseudo-spectral method)作为一种全局法,在离散中也有独特的优点,特别是具有指数级的收敛速度和较高精度,而且产生的NLP规模较小。本文分别以有限元正交配置法和伪谱法代表局部法和全局法比较其原理,并讨论离散配置点以及其在两种方法上的不同应用,针对离散后两种方法产生的NLP,分别提出判据以保证足够的自由度,最后用连续DAEs与不连续优化控制两个例子进一步比较这两种方法得出,如果命题平滑采用PS方法具有更好的收敛速度。     

Simulation of rapid pressure swing adsorption and reaction processes

A general model for non-isothermal adsorption and reaction in a rapid pressure swing process is described. Several numerical discretisation methods for the solution of the model are compared. These include the methods of orthogonal collocation, orthogonal collocation on finite elements, double orthogonal collocation on finite elements, and cells-in-series. Computationally, orthogonal collocation on finite elements is found to be the most efficient of these. The model is applied to air separation for oxygen production. Calculations confirm the formation of a concentration shock when an adsorbent bed is pressurised with air. The form and propagation of the shock over short times is found to be in excellent agreement with the exact similarity transformation solutions derived for an infinitely long bed. For air separation, novel experimental measurements, showing an optimum particle size for maximum product oxygen purity, are accurately described by the model. Calculations indicate that a poor separation results from ineffective pressure swing for beds containing very small particles, and from intraparticle diffusional limitations for beds containing very large particles. For adsorption coupled with reaction, finite rate and reversible reactions are considered. These include both competitive and non-competitive reaction schemes. For the test case of a dilute reaction A &.rlhar2; B + 3C, with B the only adsorbing species, bed pressurisation calculations are found to be in excellent agreement with the solutions obtained by the method of characteristics.     

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.     

Packed bed reactor analysis by orthogonal collocation

The equations governing a packed bed reactor with radial temperature and concentration gradients are solved using the orthogonal collocation method. The method is shown to be faster and more accurate than finite difference calculations. Using the orthogonal collocation method it is straightforward to extend one-dimensional (lumped parameter) models to the two-dimensional models needed when radial temperature and concentration gradients are important.     

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.     

Solution of final stages of polyethylene terephthalate reactors using orthogonal collocation technique

The formation of polyethylene terephthalate (PET) has been modeled to have reactions with monofunctional compounds, redistribution, and cyclization reactions in addition to the usual polycondensation step. In the final stages, the overall polymerization is mass-transfer controlled and solution of the reactor performance equations have been determined through the orthogonal collocation technique. This technique is found to be considerably more efficient for PET reactors compared to the finite difference method; the use of ten collocation points gives results which are close to the exact solution.     

Orthogonal collocation for the analysis of immobilized enzyme systems

Orthogonal collocation method is applied to the analysis of nonlinear ordinary differential equations containing Michaelis—Menten kinetics. The solution is expanded in a series of Lagrange interpolation polynomials and Gauss—Jacobi quadratures are used in calculating effectiveness factors. A set of nonlinear algebraic equations resulting from collocation approximation is conveniently solved by the Gauss—Seidel iterative method, but its convergence path is not monotonous. Although the rate of convergence depends on system parameters, orthogonal collocation is more efficient than the Runge—Kutta method in solving boundary value problems even at a high Thiele modulus.     

Studies in the control of tubular reactors-I General Considerations

In this part the dynamical model of a jacketed non-adiabatic tubular reactor is developed and the effect of the wall heat capacity is briefly examined. The distributed stability and feed-back control problems are defined and the method of orthogonal collocations is used to obtain a discretized model. The design of the cooling section around the reactor is examined and the number of collocation points and their location is determined for an accurate approximation of the distributed model.     

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.     

用正交配置法求算双重孔催化剂的有效因子

本文讨论了基于正交配置法的双重孔催化剂有效因子计算法;成功地求解了具有耦合边界条件的二阶微分方程组:考察了用Legendre多项式与Jacobi多项式作为配置法的试验函数,以及在催化剂颗粒和微粒内配置点组合格式对计算的收敛速度和精确度的影响.文中给出了不可逆反应、可逆反应条件下双重孔催化剂的有效因子值.一级不可逆反应的计算值与精确解吻合;二级不可逆反应的计算值与Jayaraman的试差解相当接近.文中还给出了Langmuir-Hinshelwood动力学模型有效因子曲线组.     

On Green’s function methods to solve nonlinear reaction–diffusion systems

Recent studies have shown that the usage of classical discretization techniques (e.g., orthogonal collocation, finite-differences, etc.) for reaction–diffusion models cannot be stable in a wide range of parameter values as required, for instance, in model parameter estimation. Oriented to reduce the adverse effects of numerical differentiation, integral equation formulations based on Green’s function methods have been considered, in the chemical engineering fields. In this paper, a further exploration of this approach for nonlinear reaction–diffusion transport is carried out. To this end, the Green’s function problem is presented and solved for three geometries (i.e., rectangular, cylindrical and spherical), and three representative examples are worked out to illustrate the ability of the method to describe accurately the phenomena with respect to analytical and numerical solutions via finite-differences. Our results show that: (i) by avoiding numerical differentiation, the round-off error propagation is significantly reduced, (ii) boundary conditions are exactly incorporated without approximation order reduction and (iii) more accurate calculations are performed making use of less mesh points and computer time.     

Efficient Simulation of Batch Distillation Processes by Using Orthogonal Collocation

Orthogonal collocation on finite elements is applied to discretize the DAE system for the simulation of multiple-fraction batch distillation processes. A detailed dynamic tray-by-tray model is used to describe batch columns more accurately which, however, leads to a set of model equations composed of nonlinear DAEs with a fairly high dimension. In addition, batch distillation operation usually takes a long period of time and therefore it costs large computational expense to simulate such processes. The use of orthogonal collocation is demonstrated to obtain a stable and highly accurate algebraic representation of the differential equations so as to improve the computational efficiency significantly. Because of the orthogonality of the polynomials introduced to approximate the state variables within a time interval, large integration steps can be taken with the collocation approximation without reducing the computational accuracy. Through simulation of two real batch distillation processes it is found that with this discretization approach 50% CPU time can be saved in comparison to the implicit Euler method normally used.     

Studies in approximation methods—V: An optimal Hlavacek-Kubicek approximation

A means of selecting the optimal mesh points in the Hlavacek-Kubicek approximation to linear operators is proposed. It is based upon the use of orthogonal collocation, of which the Hlavacek-Kubicek approximation is an extension. Numerical results on two examples verify the utility of the present optimal mesh selection.     

Evaluation of weighted residual methods for the solution of the pellet equations: The orthogonal collocation,Galerkin, tau and least-squares methods

In recent years a number of publications have adopted the least-squares method for chemical reactor engineering problems such as the population balance equation, fixed bed reactors and pellet equations. Evaluation of the performance of the least-squares method compared to other weighted residual methods is therefore of interest. Thus, in the present study, numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau, and least-squares methods, have been adopted to solve a non-linear comprehensive and highly coupled pellet problem. The methanol synthesis and the steam methane reforming process have been adopted for this study.Based on the residual of the governing equations, the orthogonal collocation method obtains the same accuracy as the Galerkin and tau methods. Moreover, the orthogonal collocation method is associated with the simplest algebra theory and thus holds the simplest implementation. Another benefit of the orthogonal collocation method is that the technique is more computational efficient than the other methods evaluated. The least-squares method does not obtain the same accuracy as the other weighted residual methods. In particular, the least-squares method is not suitable for strongly diffusion limited systems that give rise to steep gradients in the pellet. On the other side, considering the spectral–element framework, the least-squares method is less sensitive to the placement of the element boundaries than observed for the orthogonal collocation, Galerkin and tau methods.The present paper outlines the algebra of the weighted residual methods and illustrate the numerical solution techniques through a simplified pellet problem.     

FLASH CALCULATIONS FOR A CRUDE OIL BY CONTINUOUS THERMODYNAMICS

Continuous thermodynamics is a suitable concept for performing phase equilibrium calculations of complex multi-component mixtures. In contrast to the traditional pseudo-component method, the continuous distribution density function obtained by the characterization experiment is used directly for the thermodynamic calculations.

This paper describes the application of continuous thermodynamics to flash calculations for a crude oil. In the simple version the coexisting phases are assumed to behave ideally. In the refined version the real behavior is accounted for describing non-aromatics and aromatics as two different continuous ensembles.

It proves possible to separate in the calculations the problem of the compositions in the coexisting vapor and liquid phases from the problem of the equilibrium temperatures. For the compositions exact explicit equations are provided.

To obtain the equilibrium temperatures the numerical solution of only one equation (in the simple version) or of a system of only three equations (in the refined version) is required. The refined as well as the simple version lead to good coincidence with experimental data     

Use of orthogonal collocation methods for the modeling of catalyst particles—I. Analysis of the multiplicity of the solutions

Different collocation methods are compared in their ability to predict the effectiveness factor for the general case of a catalyst particle with both internal and external resistance to mass and heat transfer. The accuracy of the methods in determining the bifurcation points which separate regions of uniqueness and multiplicity of the steady states is tested. First order orthogonal collocation and the linearization method, shown to be almost equivalent for all cases, give poor approximations. Even high order orthogonal collocation is sometimes inaccurate in predicting the low concentration steady states. An excellent alternative is a variation of the Paterson—Cresswell technique which is a combination of the low and high reactivity models. This modified technique is able to predict the existence of five steady states first demonstrated by Hatfield and Airs.