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 and Optimization of an Adiabatic Multi‐Bed Catalytic Reactor for the Oxidation of SO2

A software package was developed for the simulation and optimization of a multi‐bed adiabatic reactor for the catalytic oxidation of SO₂, using a heterogeneous plug flow model. The orthogonal collocation (OC) technique with up to eight collocation points was used for the solution of a nonlinear, two‐point boundary value differential equation for the catalyst particle, and it was shown that the use of the OC technique with two collocation points can describe the system well. Because of the nonlinear behavior of the effectiveness factor along the bed, optimal catalyst distribution between the beds and corresponding inlet temperatures were determined by two methods, including: the use of (1) intrinsic or (2) actual rate of reaction in the optimization criteria. The results showed that for the second case, the minimum amount of the catalyst can be reached at lower temperatures, the amount of catalyst required is always less, and the number of beds is greater than or equal to that of the first case.     

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.     

Wavelet-like collocation method for finite-dimensional reduction of distributed systems

A wavelet-like collocation method is proposed to approach the reduction of dissipative distributed systems, expressed by means of partial differential equations, applying the methods of inertial manifold theory. The collocation method proposed, based on localized trial functions, provides a convenient numerical framework to develop approximate inertial manifolds in the case of partial differential problems (e.g. reaction/diffusion models) containing nonpolynomial nonlinearities. The collocation method is based on the interpolation of concentration/temperature fields by means of Gaussian–sinc functions. As model systems, we consider reaction diffusion schemes such as the non-isothermal model for stockpile ignition and the Elezgaray–Arneodo diffusion model.     

用正交配置技术估计固定床内的传热参数

用正交配置技术与非线性最小二乘估计固定床内的传热参数.用多通道巡回检测装置同时测定沿床层径向分布的40个正交配置点上的温度.系统地考察了床高、Re_p、d_t/d_p、气体入口温度、壁温对K_(er)和h_w的影响,并提出了以下的关联式k_(er)=0.199+0.015d_t/d_p+0.0020/1+14.15(d_p/d_t)~2Re_ph_w=33.4+4.23d_t/d_p+0.331/1+8.45(d_p/d_t)~2Re_p(5.5相似文献   

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.     

MODIFIED METHODOLOGY FOR DISTILLATION MODELLING BY ORTHOGONAL COLLOCATION

A major shortcoming of polynomial approximation in the medelling of distillation columns isthe difficulty encountered while choosing the number and location of collocation points,which are usually doneby rule of the thumb,inevitably giving rise to high dimensionality and longer computation time for the resultingmodel.In order to take full advantage of polynomial approximation in the modelling of complicatedmulticomponent distillation columns,modifications must be made to the model reduction procedure originallyproposed by Cho.This is achieved by putting in special polynomials to each of the variable profiles.Furthermore,the number and location of the collocation points can be determined by the optimization of anappropriate objective function.This would bring about less dimensionality and less computation time for theresulting reduced--order model as compared with Cho's procedure while its accuracy is still kept excellent.Theeffectiveness of such modifications is illustrated by two simulation examples.Bot     

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.     

The use of cubic spline and far-side boundary condition for the collocation solution of a transient convection-diffusion problem

Cubic spline collocation method with the far-side boundary condition has been proposed as a numerical method for the convection-dominant convection-diffusion problem. It has been shown that the proposed method can give highly accurate result for very large Peclet number problems by effectively suppressing the undesired ripple that is commonly observed in ordinary orthogonal collocation method.     

Solution by triple collocation for periodic operation of heat regenerators

The problem of periodic, countercurrent operation of packed-bed heat regenerators is considered and solved using triple collocation. This novel method uses collocation in two space variables: along the bed length and along the radius of the solid particles, and collocation in time. The method reduces the problem to a solution of a set of linear algebraic equations and bypasses either lengthy integration of open methods or iterative schemes of closed methods of solution both of which have been used previously for problems of this type. It is shown that the accuracy achieved is good. The effect of various system parameters such as Stanton, Biot and Peclet number on exit temperature and regenerator efficiency are examined and discussed.     

Reduced-order models for separation columns—V. Selection of collocation points

In the earlier parts of this paper, a method for developing reduced-order models for separation columns was presented. This article presents some guidelines for determining the degree of order reduction achievable for a given column. An index called ORP (Order Reduction Parameter) is developed to predict a priori, the extent of order reduction achievable and to provide guidelines for choosing a reduced-order model of the appropriate size. This, combined with a comparison with a steady state solution, can provide the engineer with a good estimate of the number and location of collocation points to be used in the reduced-order model. The method is developed using a simple binary absorber and illustrated by application to a more complex, multicomponent distillation column.     

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.     

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.     

Dynamic optimization for grade transition processes using orthogonal collocation on molecular weight distribution

To meet the demands of a competitive market, an industrial plant often produces several grades of polymer product through the same process in an economical way. As molecular weight distribution (MWD) is a crucial quality index of polymers, dynamic optimization for grade transition based on MWD is highly important, but challenging. This study considers the development of optimization models for MWD-based grade transition. An MWD reconstruction method using orthogonal collocation in two dimensions is developed to capture the dynamic feature of MWD in time and the distributive feature in chain length. The simultaneous collocation approach is adopted to discretize the model. Two optimization formulations are proposed to describe minimizing the transition time as well as off-spec production. Both formulations inherit the advantages of the simultaneous collocation approach. The numerical results show that the proposed methods can efficiently solve the grade transition problem with MWD specification, and obtain high performance control profiles to reduce the production cost. © 2019 American Institute of Chemical Engineers AIChE J, 65: 1198–1210, 2019     

Optimal control of diffusion-convection-reaction processes using reduced-order models

Two approaches for optimal control of diffusion-convection-reaction processes based on reduced-order models are presented. The approaches differ in the way spatial discretization is carried out to compute a reduced-order model suitable for controller design. In the first approach, the partial differential equation (PDE) that describes the process is first discretized in space and time using the finite difference method to derive a large number of recursive algebraic equations, which are written in the form of a discrete-time state-space model with sparse state, input and output matrices. Snapshots based on this high-dimensional state-space model are generated to calculate empirical eigenfunctions using proper orthogonal decomposition. The Galerkin projection with the computed empirical eigenfunctions as basis functions is then directly applied to the high-dimensional state-space model to derive a reduced-order model. In the second approach, a continuous-time finite-dimensional state-space model is constructed directly from the PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. In both cases, optimal controllers are designed based on the low-order state-space models using discrete-time and continuous-time linear quadratic regulator (LQR) techniques. The effectiveness of the proposed methods are illustrated through applications to a diffusion-convection process and a diffusion-convection-reaction process.     

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.     

GAS-LIQUID REACTORS WITH VOLATILE LIQUID REACTANTS: APPROXIMATE SOLUTION FOR CONSECUTIVE BIMOLECULAR REACTIONS OF GENERAL ORDER

An approximate solution of the film model in presence of volatile liquid reactants is presented for the case of a general van de Vusse-like reaction scheme and general power-law reaction rates. The solution is obtained in a closed-form and simply requires the numerical solution of a system of nonlinear equations. The developed approximate solution is valid for any reaction regime and any number of reactions taking place within the liquid him. The application to a CSTR is discussed; this also serves as a test of the accuracy of the approximate solution carried out by comparison with the complete film model solved via the orthogonal collocation method. The numerical studies show that the approximate solution can be used with confidence for the usual encountered rate laws and that the volatility of the liquid reactants affects the performance of the CSTR mostly at the intermediate value of the Halt a number.     

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.     

Simulation of an adiabatic packed bed reactor

A model has been developed for adiabatic packed bed reactors which includes all the intra-particle and interphase effects plus the additional effect of axial diffusion. The computer time required for the solution was considerably less than in earlier models. The technique is based on the application of the orthogonal collocation method to both the single particle and the external field, thereby reducing the whole problem into one of solving a number of simultaneous first order ordinary differential equations.     

过程动态优化问题的自热式求解策略

分析了基于微分代数方程（DAE）的动态优化问题的联立求解原理,提出了基于Lobatto配置的全离散模型的简洁描述形式。根据离散化模型的最优解具有结构相似性的特点,利用低密度离散的解来近似高密度离散的解,并且配合内点法求解的暖启动技术与障碍参数初值设定方法,提出了能实现动态优化问题快速求解的自热式策略。最后通过求解一个结晶过程的动态优化算例,证实了所提出的自热式策略能够将求解速度提高6倍左右。