Simulation of Cefoselis hydrochloride adsorption on macroporous resin in a fixed-bed column using orthogonal collocation

Adsorption operation is of great importance for separation and purification of semi-synthetic cephalosporin compounds in pharmaceutical industry. The adsorption dynamics of Cefoselis hydrochloride (CFH) on XR 920C adsorbent in fixed bed was predicted by the model of modified film-pore diffusion (MFPD). The intraparticle diffusion equation and mass balance equation in fixed bed are discretized into two ordinary differential equations (ODEs) using the method of orthogonal collocation which largely improves the calculation accuracy. The MFPD model parameters including the pore diffusion coefficient (D_p), external mass-transfer coefficient (k_f), and the axial dispersion (D_L) were estimated. The k_f value was calculated by the Carberry equation, in which the effective diffusion coefficient D_e was fitted based on Crank Model through experimental data. Moreover, three key operating parameters (i.e., initial adsorbate concentration; flow rate of import feed, and bed height of adsorbent) and the corresponded breakthrough curves were systematically studied and optimized. Therefore, this research not only provides valuable experimental data, but also a successfully mathematical model for designing the continuous chromatographic adsorption process of CFH.     

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.     

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 particles and gas flow behavior in a riser using a filtered two-fluid model

Flow behavior of gas and particles is predicted by a filtered two-fluid model by taking into the effect of particle clustering on the interphase momentum-transfer account. The filtered gas–solid two-fluid model is proposed on the basis of the kinetic theory of granular flow. The subgrid closures for the solid pressure and drag coefficient (Andrews et al., 2005) and the solid viscosity (Riber et al., 2009) are used in the filtered two-fluid model. The model predicts the heterogeneous particle flow structure, and the distributions of gas and particle velocities and turbulent intensities. Simulated solids concentration and mass fluxes are in agreement with experimental data. Predicted effective solid phase viscosity and pressure increase with the increase of model constant c_g and c_s. At the low concentration of particles, simulations indicate that the anisotropy is obvious in the riser. Simulations show the subgrid closures for viscosity of gas phase and solid phase led to a qualitative change in the simulation results.     

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     

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.     

气液两相流段塞流持气率快关阀法优化设计

快关阀法(quick closing valve,QCV)是气液两相流流动实验中常用持气率标定手段。特别是由于段塞流中气塞与液塞表现为随机可变流动特性,不合理的快关阀间距及截取次数选择将会导致持气率测量误差增大。提出了一种持气率快关阀法优化设计方案。首先,采用环形电导传感器上下游阵列信号计算流体相关流速,根据相关测速结果提取上游传感器信号对应流动工况的气塞与液塞间隔长度序列,采用Maxwell方程提取液塞中含泡持气率;在此基础上,再依气塞在管道内占比模拟计算不同快关阀间距时捕获的持气率波动序列。通过分析持气率序列波动,从统计学角度指出了95%置信度及5%允许误差情况下所需最低截取次数。最后,在快关阀门间距为1.55 m的条件下对段塞流所需截取次数进行了实验验证。通过对快关阀法持气率测量误差进行统计分析,证明了设置两个快关阀门间距的充分条件。     

Numerical analysis of fixed bed adsorption kinetics using orthogonal collocation

Fixed bed adsorption kinetics is analyzed to test the validity of the simplified model based on the linear driving force approximation by comparison with the exact model by using the orthogonal collocation method. The axial dispersion, the external film diffusion, and the intraparticle diffusion are considered to be the major mass transfer phenomena involved with the fixed bed adsorption kinetics in this study. It is assumed that a local equilibrium is attained at the fluid-solid interface and the equilibrium can be described by the Langmuir isotherm. A homogeneous particle diffusion model is employed to describe the intraparticle diffusion.     

Spectral filtering for improved performance of collocation discretization methods

Spectral filtering methods are investigated for reducing the Gibbs oscillations that result when discontinuous functions are projected onto globally defined trial function expansions. Several physical-space filters are studied to examine which is best suited to high-degree, mixed collocation methods used for time integration of non-linear boundary value problems with piece-wise continuous, time-dependent boundary conditions. Accuracy of the filtered global collocation methods relative to the non-filtered approaches is both in terms of point-wise and norm-wise solution convergence, making the filtered global collocation approach a potential alternative to spline formulations for some time integration applications.     

基于文丘里管的页岩气试采期段塞流测量补偿方法

为探索页岩气段塞流气液两相流量的测量方法,以Chisholm模型为基础,利用量纲分析法建立了文丘里湿气虚高模型和液相求解方法,并用其预测段塞流液膜区的气液相流量测量。同时,假设段塞流的液塞区为均相流,基于段塞流闭合模型和文丘里均相流模型,建立了液塞区气液相流量与液膜区气相流量的关系。提出利用一种以管道压力信号、文丘里收缩段与扩张段差压信号判断液塞来临的新方法,并加入逻辑回归法提升对液塞来临判断的准确性。在间歇性段塞流某页岩气平台井口进行了19 d的在线测试比对,结果表明,与大型气液计量分离器相比,文丘里湿气测量系统在加入了段塞流补偿方法后,气相测量19 d累计偏差从14%改善至1.08%,液相累计偏差从-57%改善至-11%。     

不规则管道中气体流量的测定方法探讨

探讨了不规则管道中气体流量的测定方法,通过工程实践和理论分析,提出采用测定静压及温度来计算气体流量的新方法。该方法适用于形状不规则的各种管道系统,测试简单且结果可靠。该方法同样适用于形状规则的管道系统的气体流量的计算。     

Liquid–gas flow patterns in a narrow electrochemical channel

The flow in a narrow (3 mm wide) vertical gap of an electrochemical cell with gas evolution at one electrode is modeled by means of the two-phase Euler–Euler model. The results indicate that at certain conditions an unsteady type of flow with vortices and recirculation regions can occur. Such flow pattern has been observed experimentally, but not reported in previous modeling studies. Further analysis establishes that the presence of a sufficient amount of small (∼10 μm) bubbles is the main factor causing this type of flow at high current densities.     

Simulation of reverse flow operation for manipulation of catalyst surface coverage in the selective oxidation of ammonia

A mathematical model has been developed for the simulation of the ammonia oxidation in a reverse flow reactor. Computer simulations were carried out with a kinetic scheme, based upon elementary reaction steps. Aim was to explore the potential of a reverse flow reactor for selective oxidation of NH₃ to produce either N₂, NO, or N₂O via a dedicated operation procedure. Therefore, the conversion of NH₃ and the selectivity toward N₂, NO, or N₂O, were compared for a reverse flow operation and a steady state, once-through operation.A new operation concept of reverse flow operation in combination with a periodically lower feed concentration is proposed. The novel reactor concept shows a better performance compared to normal reverse flow operation and to steady state, once-through operation. The results indicate that reverse flow operation can be applied for manipulation of conversion and selectivity. A periodically lower feed concentration may increase the conversion, even up to levels that exceed the steady state value.     

Z法气体超声波流量计湿气测量的虚高预测模型

引言湿气通常是指气相为连续相,液相为离散相的气液两相流。美国机械工程师学会ASME将其界定为Lockhart-Martinelli参数(L-M参数)X<0.3的气液两相流[1]。湿气普遍存在于自然界与工业现场中,如常规天然气田井口产出气(湿天然气)、煤层气、湿饱和蒸汽以及页岩气等均属湿气。随着工业生产的发展,对湿气不分离计量的要求越     

A numerical method for solving the transient multidimensional population balance equation using an Euler-Lagrange formulation

A numerical method was developed to solve the population balance equation for transient multidimensional problems including particle-particle interactions. The population balance equation was written in a mixed Euler-Lagrange formulation which was solved using the discretization method that represents the number density function by impulse functions, an operator splitting method and a remeshing procedure for the internal variable that conserves the mass and the number of particles.This method was successfully tested against analytical and semi-analytical solutions for pure breakage, pure coalescence, breakage and coalescence, pure advection, advection with absorption, advection with binary uniform breakage and with constant or linear absorption. The method was also applied to a free-boundary transient one-dimensional gas-phase model in a bubble column reactor with simplified hydrodynamics. Accurate solutions were obtained for several simulation conditions for the bubble column, including gas absorption, bubble breakage, bubble coalescence and variable gas density effects. The results showed that the numerical method is adequate and robust for solving transient population balance problems with spatial dependence and particle-particle interactions.     

正交实验法确定乙丙装置水洗单元的最佳工艺条件

在乙丙橡胶后处理步骤的水洗过程中胶液难于分相,乙丙橡胶己烷溶液相与水相分界边缘较浑浊,胶液提取困难。作者通过水凝聚正交实验法进行了包括温度、水相pH值、m（胶）：m（水）、搅拌速度等因素条件的正交筛选实验,确定了最佳的水洗条件。     

The transient response technique for the study of hydrocarbon oxidation on metal oxides: Simulation of the responses for various propene oxidation mechanisms

Partial differential equations, describing the consumption of reactants and the formation of products, have been solved numerically for a step change from inert to reactant feed considering different variants of propene oxidation mechanisms. The response features and their dependency on the rate limiting step have been rationalised in terms of conclusions, which can serve as general guidelines for mechanistic interpretation of experimental transients from hydrocarbon oxidation over metal oxides. Though the response pattern is characteristic of the rate limiting step, it is demonstrated that different variants of mechanisms can give identical response features. In that case complementary investigation with other techniques, or more detailed mathematical analysis is necessary.     

An improved model for bubble formation using the boundary-integral method

A model for bubble formation from a single submerged orifice is developed using the boundary-integral method. Since the flow field is assumed to be irrotational, potential-flow theory is used to predict the growth of the bubble. The effects of the surface tension and the liquid circulation around the bubble are included in the calculation. Predictions of bubble shape, chamber pressure and the effect of surface tension are presented to compare with reported experimental data.     

Simulation in electrochemistry using the finite element method part 2: scanning electrochemical microscopy

The previously introduced adaptive finite element (AFE) algorithm for use in electrochemistry is applied to the simulation of selected multidimensional problems: steady state simulation, chronoamperometric simulation, cyclic voltammetry at microelectrodes, and simulation of arbitrarily shaped scanning electrochemical microscope (SECM) tips. It is shown that the algorithm is suitable for this kind of problems and can be easily extended to the simulation of many types of electrochemical experiments.