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.     

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.     

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.     

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。     

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.     

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.     

Study of gas cavity size hysteresis in a packed bed using DEM

When a high velocity gas jet is introduced into a packed bed a cavity is formed. The size of the cavity shows hysteresis on increasing and decreasing gas flow rates. This hysteresis leads to different cavity sizes at same gas flow rate depending on the bed history. The size of cavity affects the gas flow profiles in the packed bed. In this study the cavity size hysteresis phenomenon has been modeled using discrete element method along with turbulent gas flow. A reasonable agreement has been found between computed and experimental results on cavity size hysteresis. The effect of various parameters, such as nozzle height from the bed bottom and packing height, on the cavity size hysteresis has been studied. It is found that inter-particle interaction forces along with gas drag and bed porosity play an important role in describing the cavity size hysteresis. The injection of gas flow allows the particles to go to an unconstrained state than they were previously in, and their ability to remain in that state, even under decreased gas drag force, leads to the phenomenon of cavity size hysteresis.     

毛细管气相色谱法测定甲基丙烯酸二甲氨基乙酯的含量

采用50mOV 1毛细管柱,FID检测器,正二十二烷做内标,测定甲基丙烯酸二甲氨基乙酯的含量,其变异系数为0.92%,结果准确、可靠。     

Transport and deposition of inertial aerosols in bifurcated tubes under oscillatory flow

The transport and deposition of aerosol particles in a single bifurcation tube under oscillatory airflow condition are investigated via experiments and numerical simulations. In the experimental study, a novel stabilized laser-photodiode measurement technique is used to quantify the effects of frequency of oscillatory airflow and airflow rate on surface deposition of particles within the bifurcation tube. Surface deposition in the parent and daughter tubes is measured by orientating the laser-photodiode device parallel and perpendicular to the bifurcation plane and calculated using a transmission loss model. In the numerical simulation study, the same bifurcation tube is constructed using a two- and three-dimensional computer mesh in COMSOL^® to model particle mobility and deposition characteristics, considering the simultaneous effects of inertial impaction, gravitational settling and interception. The effects of frequency of oscillatory airflow, airflow rate and particle size on the particle trajectory and spatial deposition pattern are examined.     

毛细管气相色谱法测定2-硝基对苯二甲酸二甲酯含量

研究了2 硝基对苯二甲酸二甲酯的气相色谱测定方法。在选定的试验条件下,线性关系良好,标准曲线的相关系数为0.9998,测定的相对偏差为1.05%,样品加标准品回收率为99.0%～100.3%。该方法操作简便,灵敏度高,定量准确。     

Finite element modeling of the laminar and transition flow of the Superblend dual shaft coaxial mixer on parallel computers

The macro-mixing mechanisms of the Superblend coaxial mixer consisting of a Maxblend impeller and a double helical ribbon agitator mounted on two independent coaxial shafts rotating at different speeds are numerically investigated. The simulations are based on the resolution of the Navier-Stokes equations with help of a parallel three-dimensional finite element solver exploiting the capabilities of high performance computers. To model the rotation of agitators a hybrid approach based on a novel finite element sliding mesh and fictitious domain method is used. The power consumption, the flow patterns, the shear rate distribution, the pumping capacity and the mixing time of the Superblend mixer are calculated from the simulated hydrodynamics. The simulations allow observing the flow as it evolves from deep laminar (Re=0.1) to transition (Re=520) regime. As Reynolds number increases, several recirculation zones above and below the middle of the tank are formed. It is found that operating the agitators in co-rotation mode requires less power consumption and exhibits equal or shorter mixing time than counter-rotation mode. The larger power consumption in counter-rotating mode is caused by the presence of high shear vortices generated between the two coaxial agitators. Furthermore it is shown that the shear distribution throughout the Superblend coaxial mixer operating in co-rotation mode is almost homogenous, which is highly desirable for shear sensitive products. In view of the results obtained in this work, the Superblend coaxial mixer is found as a good alternative for tough mixing applications.     

Simulation of freezing step in vial lyophilization using finite element method

Determination of heat transfer and temperature profile during freezing step is fundamental to predict the final structure of a lyophilized product. The aim of this study was to develop and optimize a dynamic model based on finite element method for simulation of freezing step in order to study final product morphology in both vertical and radial directions. Different factors have been taken into account: chamber pressure, shelf temperature, vial shape, initial solution temperature, nucleation temperature and phase changes. The dynamic axisymmetric model proposed could simulate temperature of each point in the vial and position of liquid-solid interface, without necessity of fitting parameters or questionable assumptions. In addition, the model was extended to predict the average crystal size in each element and the influence of different factors was examined.     

Dimensionless number for identification of flow patterns inside a T-micromixer

Results are presented from a numerical study examining the flow dynamics of the liquid phase inside T-type micromixers. The main aim of the study was to determine an identification number for the differentiation of the different flow regimes in the liquid phase in T-type micromixers. The critical value for the identification number at which the transition from vortex flow to engulfment flow occurs was obtained. The results were used to optimize the geometrical parameters and the operating conditions to achieve high mixing performance for the liquid phase in T-type micromixers. The model results were found to be consistent with experimental data for different T-mixers available in the literature.