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.     

Detailed Dynamic Modeling of a Molten Carbonate Fuel Cell Stack with Indirect Internal Reformers

A detailed dynamic model of a molten carbonate fuel cell (MCFC) stack that can predict the distribution of process variables over the cell surface and along the stacking direction is presented. The model describes a repeating unit of the stack, which is composed of eight cells and one indirect internal reformer (IIR). The set of partial differential equations for the MCFC model is reduced to a set of ordinary differential equations through discretization using the cubic spline collocation method and the finite difference method. The steady‐state and dynamic behaviors of the eight‐cell stack model are compared with those of the one‐cell model, which has been commonly used in previous studies on MCFC dynamics, and three‐cell aggregate model. The distribution of the process variables along the stacking direction cannot be disregarded, and thus more detailed model should be used for the accurate reproduction of MCFC dynamics.     

Computer modelling and numerical analysis of hydrodynamics and heat transfer in non-porous catalytic reactor for the decomposition of ammonia

Chemical reactors exhibit very complex behaviours such as multiple steady states, oscillations, etc. resulting from complex linkage between the transport processes and the non-linear chemical reaction kinetics. Ammonia is a potential hydrogen source for a number of fuel cell applications for small scale power generation useful for portable equipments. In the present work, we analyse the fluid dynamics and heat transfer in catalytic microreactor systems for the decomposition of ammonia over a monolayer Ni non-porous catalyst. The overall model for this convective-diffusive-reactive system consists of a flow model, a mass transport model, an energy conservation model and a reaction kinetics model for ammonia decomposition. The flow model is described by the Stokes equation for a creeping flow regime. The mass transport and energy conservation models are based on convective-diffusion equations. The rate of ammonia decomposition can be measured as a function of the catalyst activity and ammonia concentration. A standard Galerkin finite element technique has been applied for the solution of the flow equations. A slightly perturbed form of the mass continuity equation is used to satisfy the Ladyzhenskaya-Babuška-Brezzi stability criterion. For the solution of convection-diffusion equations, a streamline inconsistent upwind finite element scheme has been chosen to avoid any spurious oscillations. C⁰-continuous 9-noded Lagrangian biquadratic isoparametric finite elements are used for the approximation of the field variables. A second-order Taylor-Galerkin time-stepping scheme has been chosen for the temporal discretisation of the flow equations whilst an implicit theta method has been used for convection-diffusion equations. The results are presented in the form of velocity vectors and concentration, temperature contours and are examined for stability, convergence and theoretical consistency.     

Third-order optical nonlinearity in single-wall carbon nanotubes

The third-order optical nonlinearity in carbon nanotubes (CNT) exposed to an intensive external electromagnetic field has been investigated. The analysis is based on the quantum kinetic equations for the density matrix of π-electrons in CNT. In the regime of weak driving field, the kinetic equations have been solved by the perturbation method and the third-order nonlinear polarizability of different achiral CNTs has been calculated. The theory elaborated has been used for the evaluation of nonlinear susceptibility of CNT-based composites. Comparison with available experimental data has been presented. In the case of high intensive driving field a nonperturbative numerical simulation of the process has been carried out in the time domain. The axial current density in CNT has been calculated. Excitation of π-plasmons in CNTs has been analyzed.     

多层次结构模型预测控制系统的层次间模型切换方法

引言实际的工业过程对象,大部分都呈现出很强的非线性特性,其控制过程十分复杂。虽然近年来,对非线性技术的研究已经取得了很多的成果。但是非线性系统精确建模困难[1]、非线性微分方程求解     

Solving complex chemical equilibria by a method of nested iterations

The equilibrium composition of a mixture of molecular species can be obtained by finding the solution of a set of simultaneous non-linear equations relating the concentrations of these species. Atom balance equations, wherein the species concentrations are related to each other through chemical equilibrium constants, form a many-variable set of simultaneous non-linear equations. It is shown how many-variable problems of this type can be treated as a nested set of single-variable equations. This treatment allows the use of recent, powerful methods for forcing convergence in single-variable iterations. The method developed does not require explicit matrix inversion. It is fast and suitable for a small computer. The method has been found particularly useful for routine computations. Up to two non-gaseous species can conveniently be included. An example is given of the use of the method.     

A simplified form of Holland's equations in the solution of multicolumn distillation systems

When the Holland convergence acceleration scheme has been extended to multicolumn sustems as by Nartker, Srygley and Holland a second set of unknowns which they term “systemΘ's” is required. An alternate formulation is presented here in which the Holland equations for multicolumn systmes are expressed without the introduction of any unknowns beyond those previously used in formulating the Hollad equations for a single column. This formulation permits the existence and uniqueness of a positive solution to be established when and arbitrary number of units are arranged in a single recycle loop. Furthermore a numerical difficulty encountered with the procedure of Nartker, et al is discussed and a modification is presented which elimiates the difficulty.     

Decomposition of systems of nonlinear algebraic equations

A new method for decomposing irreducible subsets, in the solution of systems of nonlinear algebraic equations, is presented. This method consists of two steps: (1) elimination of the nonlinearity from some of the equations by replacing nonlinear expressions by new variables; and (2) formulation of a problem of smaller dimension by tearing the linear subset of equations. It is shown that these modifications do not change considerably the convergence rate of the Newton-Raphson and Broyden's methods while reducing the problem's dimension. Computer time reduction up to 80% is demonstrated in the examples solved. An algorithm for elimination of nonlinear expressions, which uses Boolean matrices instead of formula manipulation, is also presented.     

联立方程法模拟多循环流程的新解算策略

<正>化工过程的模拟算法有三类:序贯模块法、联立方程法和联立模块法.序贯模块法已很成熟,应用颇广,国内外通用软件多用此法.但它也有局限性,如难以用于多循环复杂流程的模拟,难以处理设计型、控制型模拟问题等.为克服这些不足,展开了联立方程法等的研究.如进行了SPEEDUP、ASCEND、QUASILIN等软件的开发.由于涉及大型非线性方程组,其求解算法的优劣是此类算法成败的关键.目前人们已对一些典型问题考察了牛顿法及各种拟牛顿法的适用性,在算法取舍和改进方面取得了一些进展,但要获得有效而通用的算法还有许多工作要做.目前,用联立方程法进行实际过程模拟的例子较少,在微机上进行有关研究未见报道.作者从我国微机较为普及这一实情出发,在微机上进行了联立方程法的研究.针对一个具体的多循环复杂流程的模拟与分析问题,提出了一个新的联立方程法解算策略:部分解耦加速牛顿法.     

计算机辅助现有化工过程操作最佳化

应用计算机技术来改进现有化工过程操作,对提高化工企业的经济效益有重要的意义.模拟调优通过计算机辅助过程优化技术寻求过程系统的最佳条件,可以使过程达到最佳运行.本文对基于序贯二次规划的两层法作了改进,在二次规划中引入额外变量,构造了新的二次规划问题,并提出了此规划问题的分解策略,从而保证分解矩阵为奇异时仍能对其进行分解.本文提出的两层次-序贯二次规划法,数值稳定性好,收敛较快.用此算法对合成甲醇过程进行优化计算得到了满意的结果,巳在工业装置上实现.     

Modified method of mathematical modeling of processes of the dynamics of thermotechnical units

A modified method of lumped parameters that gives a dynamic model of thermotechnical objects used for studying control systems is presented. The dynamic equations having constant coefficients that characterize static and dynamic properties of an object are transformed into modified equations with variables determined by the size and direction of deviation of the process parameters. A system of equations for a drying unit is used to show that the modified method reduces the computation error by a factor of 5 –10 and can describe important features of the behavior of a thermotechnical object, which could not be done earlier. The volume of computations by the modified method more than doubles     

A DISCRETE RELAXATION METHOD FOR SOLVING MULTI-COMPONENT DISTILLATION PROBLEMS

A computational method for solving multi-component distillation problems is presented, where the steady state composition profiles of a column are determined by solving the vapor-liquid equilibrium problem for each stage sequentially to convergence. The Rachford-Rice equation is solved for each stage and the calculations are continued until the column temperature profile converges to the steady state solution. The computational efficiency of this method is found to be comparable to the relaxation method in which the differential equations describing the transient behaviour of the column are solved. The advantages of the method presented here are its conceptual simplicity and its ease to program.     

Problem-oriented modules for process simulation. resolution strategies for simulation and optimization

A new resolution strategy is presented to overcome short-comings of the sequential modular process flowsheeting. Following a logic flow of information, different from the material flow of streams in the flowsheet, changes the mathematical structure of the system of equations that models the units. New programs have been generated, very different from traditional modules. Deciding an appropriate resolution strategy usually leads to a dramatic reduction in required CPU times and to an increased flexibility to handle different problems. The simulation convergence and process optimization can be approached from a different perspective.     

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.     

Identification of parameters in systems of ordinary differential equations using quasilinearization and data perturbation

Given a set of observed data for a particular physical phenomenon, the problem of computing the “best fit” parameters for the mathematical model describing the phenomenon is a common problem in process or reaction mechanism identification. If the mathematical model comprises a set of non-linear ordinary differential equations, this leads to a non-linear boundary value problem. A very powerful way of attacking this class of problem uses an adaptation of the Newton-Raphson-Kantorovich procedure, called quasilinearization, which regards the non-linear problem as the limit of a sequence of linear problems. Starting from an initial trial solution, convergence if it does occur, occurs rapidly; further, convergence is assured if the initial guess is “close enough” to the true solution. The difficulty of making a good initial guess, a serious limitation of the method in the past, can in principle be overcome by the algorithm proposed. When a given vector may not be within the domain of convergence of the original problem, it must be within the domain of convergence of some other derived problem. The latter may then be perturbed towards the original problem in a finite number of steps. In the case of process identification, new data points are derived; these are subsequently adjusted until they coincide with the original data. The algorithm has been successfully applied to several examples from recent chemical engineering literature.     

Analysis of the Behavior of the Simulated Moving Bed Reactor in the Sucrose Inversion Process

Abstract

The simulated moving bed reactor (SMBR) is a device in which reaction and separation processes take place simultaneously. The separation of products allows higher conversion and high‐purity product can be also obtained. In this work, a mathematical model has been presented to predict the behavior of the SMBR in the sucrose inversion process. For this process, the triangular region which defines operating conditions to recover high‐purity products in SMBR has been obtained using two modeling strategies. The set of partial differencing equations is solved by finite volume method. The influence of some operation conditions on the reactor performance is analysed for the sucrose inversion process.     

A robust technique for process flowsheet optimization using simplified model approximations

Recently, embedded simplified process models have been shown to be very efficient for process simulation. When compared to the direct use of rigorous models, this approach has the potential to reduce the computational effort of process simulation by up to an order of magnitude or more. Application of this approach to process optimization should therefore lead to similar savings in computational effort as well as substantial improvement of the process.

However, current simplified model embedding schemes applied to process optimization cannot, in general, converge to the optimum defined by the more rigorous process models. Consequently, they require an expensive rigorous model optimization starting from the solution of the simplified model optimum to guarantee convergence.

In this paper we develop a framework that incorporates simplified models into an optimization algorithm and guarantees convergence to the rigorous model optimum. Here rigorous process models are evaluated only when necessary to insure progress toward the optimal solution. A theoretical justification of the algorithm is presented and several process examples are solved to demonstrate the effectiveness of this approach.     

基于稳定性的循环物流系统流程模拟——以催化裂化反应-再生系统为例

过程流程模拟中广泛应用的序贯模块法处理循环物流系统存在很多困难,如断裂流股的选择、迭代方程的收敛性等。基于稳定性理论解决循环物流系统的流程模拟问题,首先根据化工单元装置的模型方程将其分为正向模型和反向模型,并将循环物流中的变量定义为迭代变量和收敛变量;然后在收敛变量处将循环物流的流股断裂,迭代变量分别通过正向模型和反向模型计算收敛变量,二者的偏差通过增益系数对迭代变量进行修正,进而得到迭代方程;最后,利用控制理论中的稳定性理论来确定迭代方程的增益系数,将迭代方程线性化,采用劳斯判据确定增益系数的稳定范围,当增益系数位于稳定范围以内时,迭代方程必然收敛。催化裂化装置反应-再生系统因催化剂循环的存在而成为典型的循环物流系统,本文将反应器作为正向模型,再生器作为反向模型,以再生温度和再生催化剂含碳量作为迭代变量,构造了催化裂化装置反应-再生系统流程模拟的迭代方程,利用稳定性理论找到增益系数的稳定范围,保证流程模拟计算必定达到收敛,验证了该方法的可行性和有效性。     

Accelerated successive substitution schemes for bubble-point and dew-point calculations

The accelerated successive substitution (SS) algorithms developed by Mehra et al. for flash calculations have been extended to the prediction of saturation points. A transformation matrix which is used to calculate the acceleration parameter has been rewritten in a form that is applicable at the saturation conditions. Simple equations for estimating the initial values and recursive formulae according to which the iterates can be updated are presented. The proposed schemes were compared with the conventional SS method and a multivariate Newton's method. The comparison suggests that the accelerated SS schemes are more tolerant of poor initial values and sometimes more efficient than Newton's method.     

Solving partial differential equations of gas–solid reactions by orthogonal collocation

In this work the solution of the coupled partial differential equations for noncatalytic gas–solid reactions has been considered by orthogonal collocation. First of all, by an integral transformation and then by applying the orthogonal collocation method, these partial differential equations are converted to the ordinary differential equations. Then the equations are solved and the conversion–time profiles are obtained. The solution of the equations for volume reaction model, grain model and grain model with product layer resistance, modified grain model, random pore model, nucleation model and reaction of two gas with one solid has been presented in this work. The orthogonal collocation is a rapid method for solving of these equations and shows a good accuracy with respect to other solution techniques in the literature.