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.     

Simulation of transient gas flow using the orthogonal collocation method

Proper evaluation of the dynamics of the transmission system is the key element in the design and operation of natural gas pipelines. Basic equations describing the transient flow of gas in pipes are derived from the Euler equations. The orthogonal collocation technique is employed as the mathematical method for the numerical solution of the governing equations. This method leads to a set of non-linear ordinary differential equations which can be solved by the Runge–Kutta–Fehlberg method. The performance of the proposed method is tested using two practical examples. The predicted results clearly demonstrate that the proposed method can successfully simulate the isothermal and non-isothermal unsteady flow in gas transmission systems.     

Accuracy, temporal performance and stability comparisons of discretization methods for the numerical solution of Partial Differential Equations (PDEs) in the presence of steep moving fronts

In this paper, several competitive spatial discretization methods recently developed for the convection term are reviewed and analyzed in terms of accuracy, temporal performance, and stability. This analysis was performed using a prediction–correction DAE integrator within the framework of the MOL (Method Of Lines). The discretization methods are classified by Fixed Stencil (FS), Adaptive Stencil (AS) and Weighted Stencil (WS) approaches and their main characteristics are demonstrated via a number of bench marking tests. Of the 14 discretization methods tested, four have been shown to be the most reliable, when we consider the accuracy of the calculation and the computational time required. Application of commonly-used FS methods to convection-dominated problems containing a moving shock results in a relatively short calculation time. However, almost of the FS methods, except for the first-order upwind FS method, are unstable, providing spurious oscillatory profiles (termed the Gibbs phenomena) near the shock. Employing AS methods such as ENO (Essentially Non-Oscillatory) schemes, this Gibbs phenomena disappears. Therefore, AS methods enhance stability and accuracy but are somewhat prohibitive. WS methods (i.e. Weighted ENO schemes), which use a convex combination of candidate stencils weighted by the ENO idea, can reduce the calculation time, and are inheriting of the non-oscillation nature inherent in the AS methods. The ENO and WENO methods are efficient to track a shock and steep moving front and these methods are essential for numerical schemes which use relatively small number of mesh points.     

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.     

Constrained particle filtering methods for state estimation of nonlinear process

Increasingly in practical applications, nonlinearity, non‐Gaussianity, and constraint must be considered to obtain good state estimation. A constrained particle filter (PF) approach for state estimation, which involves three alternative strategies to impose the constraints on the prior particles, posterior particles, and state estimation has been proposed. First, to impose constraints on prior particles, a constrained Gibbs sampling method with a constrained inverse transform sampling is proposed to restrict sampling within the constraint region under cases of both univariate and coupling constraints. Second, to ensure validity of posterior particles, resampling is confined to the valid prior particles and the violated ones are discarded, which results in a similar formulation as the existing acceptance/rejection constrained PF method in literature. Third, if the state estimation violates the constraint, different from the existing methods that either discard all violated particles or accept all of them by projecting them onto the constraint region, the proposed method makes a balance between the prior and the likelihood function by adjusting the weights of violated and valid particles, respectively. Compared with the existing methods, the proposed method provides better physical interpretation and involves no restrictive assumptions about the distributions. Simulation results demonstrate effectiveness of the proposed methods. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2072–2082, 2014     

非光滑结构对轮胎抗滑水性能影响分析

借鉴流体领域中非光滑结构减阻特性,在轮胎纵向花纹沟底部引入V形沟槽结构,探索其对轮胎抗滑水性能的影响。以轮胎单个花纹沟作为分析对象,建立V形沟槽分布数学模型和轮胎滑水模型,运用CFD(计算流体力学)方法对轮胎滑水进行数值分析。借助正交试验法建立L16(45)正交表,分析V形沟槽结构相比光滑结构的壁面减阻率,得到减阻效果最佳的V形沟槽设计参数,接着将优选出的V形沟槽非光滑结构引入到轮胎纵向花纹沟槽底部进行滑水分析。结果表明：V形沟槽非光滑结构能有效降低轮胎花纹沟壁面阻力系数,提高轮胎的抗滑水性能。     

Static and Dynamic Interfacial Stresses Analysis for Three-Layer Beam Structures Using Spectral Element Method

As the structural integrity at the interfaces of a three-layer structure certainly depends on the shear and normal (peeling) stress concentrations near the free ends of the cover plate, it is very important to predict the static and dynamic interfacial stresses in an accurate and efficient way for successful designs of three-layer beam-type structures subject to various loading conditions. Thus, this paper presents a frequency-domain spectral element method by which both static and dynamic interfacial stresses can be accurately predicted. To that end, the governing equations of motion for three-layer beam-type structures are derived by using Hamilton's principle and then the spectral element model is formulated in the frequency domain by using the variational method. The high accuracy of the present spectral element model is verified in due course. Numerical studies are then conducted to investigate static and dynamic interfacial stresses for an example three-layer beam subjected to various loadings.     

Dynamic flexibility analysis of chemical reaction systems with time delay: Using a modified finite element collocation method

Chemical reaction systems are often complex dynamic time-delay systems that have to operate successfully in the presence of uncertainties. Under these circumstances, flexibility analysis comes to be much important to the design and operation of time-delay chemical reaction systems. In this work, a modified finite element collocation method was proposed to carry out flexibility analysis of chemical reaction systems with time delay. The proposed method is combined with the linear quadratic regulator (LQR) and Lagrange polynomial for the optimal solution of control variables and state variables respectively. The method is investigated by two typical chemical reaction systems with time delay. All the results demonstrate that the proposed modified finite element collocation method may provide a powerful tool for studying the dynamic flexibility of chemical reaction systems with time delay.     

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.     

基于综合平衡法的注塑工艺参数多目标优化设计

结合正交试验法和模流分析软件Moldflow,对不同工艺条件下的汽车轮轴盖塑件成型过程进行模拟分析,运用多目标综合平衡法,对塑件成型后的体积收缩率、翘曲变形量和表面气穴3个目标值进行综合评判。通过对各成型工艺参数的极差分析和方差分析,确定熔体温度、模具温度、保压压力、冷却时间等注塑工艺参数对目标值的影响程度分,析得出最优的注塑工艺参数组合方案。生产实践证明塑件质量得到了有效的改善。     

Model based on population balance for the simulation of bubble columns using methods of the least-square type

A novel model is presented which uses mass and momentum equations based on population balances to describe the dispersed phase of bubble columns. A set of integro-differential non-linear equations constitutes the model, which can be solved directly using the least-squares methods. The implementation for these is presented.The model presents a degree of flexibility, as different density functions (number, mass and volume) and different internal coordinates (bubble diameter, volume and mass) can be used.A simplified 1D model of a bubble column is solved using the least-squares spectral element method. The results obtained for two different gas volume fluxes at the inlet are compared with experiments by several authors. The simulations show good agreement with the measured data.     

乙炔氢氯化低汞催化剂的表征方法

介绍了乙炔氢氯化低汞催化剂表征中用到的4种分析方法,包括BET比表面积法、扫描电子显微镜、热分析技术和原子吸收光谱法。     

钢结构设计方法的研究进展与展望

计算机技术的飞速发展和结构分析理论的不断深入,为研究和发展结构分析设计方法提供了现实条件和理论基础,使人们对钢结构分析理论和设计方法的理解和应用也更加深入。文中主要评述了钢结构设计方法的发展与趋势,介绍了代表最新技术的结构高等分析方法的概念、特点及现状。     

Comparison of numerical methods for the simulation of dispersed phase systems

This contribution deals with a new numerical method for an accurate and efficient simulation of particulate processes. As an example for dispersed phase systems a detailed model for crystallization processes is considered. After the model derivation, which is based solely on physical principles, different techniques for the numerical simulation of the mathematical model are discussed. State-of-the-art finite volume schemes based on the ‘method of lines’ approach are compared to the recently published ‘Space-time conservation element and solution element method’. The presented simulation results show a strong dependence on the chosen numerical method. Guidelines for a proper selection of numerical methods for the treatment of population balance based models are given.     

Numerical methods for the simulation of the settling of flocculated suspensions

For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initial-boundary value problem for a second order partial differential equation of mixed hyperbolic–parabolic type. Due to the mixed hyperbolic-parabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical methods. In this paper we present and elaborate on numerical methods that can be used to correctly simulate this model, i.e. conservative methods satisfying a discrete entropy principle. Included in our discussion are finite difference methods and methods based on operator splitting. In particular, the operator splitting methods are used to simulate the settling of flocculated suspensions.     

Numerical simulation for diffusion of matter in compound round jet by three-dimensional particle method

A three-dimensional particle method for the diffusion of matter in a round jet is proposed. The flow field is calculated with a vortex method, whereas the concentration field is simulated through a method analogous to the vortex method. The particle method is based on the Lagrangian approach; thus no computational grids are needed. It is applied to simulate the diffusion of matter in a compound round jet. The large-scale eddies, appearing just downstream of the disappearing point of the potential core, cause the diffusion of matter in the radial direction. In the subsequent developed region of the velocity field, the diffusion proceeds to the smaller vortical structure. The mean concentration and the concentration fluctuation intensity are in the self-preservation state at x/D?8 and x/D?13, respectively. These numerical results are favorably compared with experimental ones, indicating that the present particle method is usefully employed to simulate the diffusion of matter in a round jet.     

制备方法对高CuO含量的Cuo/CeO_2催化剂水煤气变换性能的影响

采用机械混合法、固相合成法、水热法、沉积-沉淀法和共沉淀法制备系列CuO/CeO_2水煤气变换催化剂,采用XRD、N_2物理吸附和H_2-TPR等方法对样品进行表征,研究制备方法对CuO/CeO_2催化剂结构的影响以及催化剂结构与其催化性能的关系。结果表明,共沉淀法有利于获得较小的CuO和CeO_2晶粒、较大的比表面积、孔容和孔径以及较好的还原性能,制得的催化剂活性和稳定性较好。