On the solution of diffusion controlled adsorption kinetics by means of orthogonal collocation

The problem of diffusion controlled adsorption kinetics is solved by orthogonal collocation for the Henry and the Langmuir isotherms. The dependence of the surface concentration of a surfactant on time is described by explicit formulae for the whole range of time. The results are compared with those obtained by power series expansions and finite difference methods and have a high accuracy for any time value. The method of orthogonal collocation has been shown to be very effective. It is suitable for applications to similar problems.     

Analysis of gas–solid noncatalytic reactions in porous particles: Finite volume method

Gas–solid noncatalytic (GSNC) reactions in porous particles are analyzed employing volume reaction model (VRM). Both single‐stage and two‐stage models have been used in the work. Earlier analysis for such reactions was mostly restricted to the first‐order reactions with respect to the gaseous component. The present work has used a range of reaction orders with respect to both gaseous and solid reactants. Front tracking method was employed for solving the moving boundary problem. Finite volume method (FVM) has been used for the first time for the analysis of GSNC reactions in porous particles, based on moving boundary zone models. The discretized equations are solved by tridiagonal matrix algorithm. The results for the first‐order reaction agree well with the analytical solution. FVM solution compares well with other numerical method (integral transformation followed by orthogonal collocation). Numerical results have also been validated with reported experimental data for reduction of Magnetite by CO. FVM is thus established to be a suitable numerical method to solve GSNC reactions in porous particles employing VRM. The effects of different process parameters on the progress of reaction have also been assessed.© 2003 Wiley Periodicals, Inc. Int J Chem Kinet 36: 1–11, 2003     

A Numerical technique for solving carrier-mediated transport problems

A numerical technique and a computer program based on the method of orthogonal collocation on finite elements for solving steady state carrier-mediated transport problems is presented. The method is especially suitable in solving boundary layer type of problems as arise in carrier-mediated transport. The method uses an efficient LU decomposition of a blockdiagonal matrix. A residual criterion is used for the placement of elements which can be easily implemented using an interactive feature in the computer program. Numerical results for three different systems; a system with simple kinetics and equal diffusivities for carrier and its complex, a system with simple kinetics and unequal diffusivities and a system with complex kinetics and equal diffusivities, indicate that the method is very versatile and has very good convergence properties. The method can be used very easily for all these systems by making minor modifications to the computer program.     

A new approach for modeling of RO membranes using MD-SF-PF model and CFD technique

The modified surface force-pore flow (MD-SF-PF) model is a successful and powerful model for prediction of reverse osmosis (RO) membrane separation. In the MD-SF-PF model, a suitable mass and momentum balances were made through the cylindrical micro porous of the membranes. A one-dimensional potential function in radial direction was used to attain the velocity profile. The obtained nonlinear equations were solved by orthogonal collocation method. In our model, extended MD-SF-PF (Ex-MD-SF-PF) model, the potential function of the MD-SF-PF model was developed as both radial and axial directions. Both Ex-MD-SF-PF and MD-SF-PF models were solved by finite volume and finite difference methods based on a computational fluid dynamics (CFD) technique. The profiles of velocity and concentration were achieved and thereby the values of separation factors and fluxes for both models were obtained and were compared accordingly. This approach affected only at the end of pore, mainly on the concentration profile and slightly on the velocity profile.     

The scaled Hermite–Weber basis in the spectral and pseudospectral pictures

Computational efficiencies of the discrete (pseudospectral, collocation) and continuous (spectral, Rayleigh–Ritz, Galerkin) variable representations of the scaled Hermite–Weber basis in finding the energy eigenvalues of Schrödinger operators with several potential functions have been compared. It is well known that the so-called differentiation matrices are neither skew-symmetric nor symmetric in a pseudospectral formulation of a differential equation, unlike their Rayleigh–Ritz counterparts. In spite of this fact, it is shown here that the spectra of matrix Hamiltonians generated by Hermite collocation method may be determined by way of diagonalizing symmetric matrices. Furthermore, the symmetric matrix elements do not require the evaluation of Hermite polynomials at the grid points. Surprisingly, the present numerical results suggest that the convergence rates of collocation and Rayleigh–Ritz methods are entirely the same.AMS subject classification: 65L60, 81Q05, 65L15, 34L40, 42C10     

Estimation of adsorption isotherm parameters with inverse method--possible problems

In recent years the inverse method (IM) has been frequently applied to estimate of isotherm parameters. The IM has been used for adsorption process modeling for one, two and even three component chromatography. This method requires only a few injections with various sample concentrations, so the solute consumption and time requirements are very modest. The successful estimation of isotherm parameters with IM depends on applied chromatography column model and a numerical method used to solve the model. For HPLC column the classical equilibrium-dispersive (ED) model can be used. This model is solved frequently with very fast Rouchon finite difference method. However, the accuracy of computations with Rouchon method is decreasing with increase of the number of analyzed components. The aim of this work is the comparison of the results obtained with inverse method when ED model was solved with Rouchon or orthogonal collocation on finite element (OCFE) scheme. Assuming that solution of ED model with OCFE method can be regarded as real a solution, it was found that the Rouchon scheme may not give satisfactory results even for column with 10,000 theoretical plates for three component chromatography. Moreover, the optimal conditions for separation, calculated with Rouchon method, can be remarkably different from that obtained with the OCFE method. The next aim of this work is the presentation of Craig method application to estimation of model parameters with IM.     

Two-dimensional chiral model for liquid crystals, bent hard needles: a Monte Carlo simulation

The liquid crystalline behavior of a two dimensional (2D) model of hard needles bent into a "zigzag shape" is studied. This model, originally designed to study two dimensional chiral segregation, also shows liquid crystalline behavior and has some anomalous features which are contrasted in relation to the following: (i) Most of the microscopical models used to study liquid crystals have a symmetry axis that coincides with a molecular axis; (ii) in three-dimensions, chiral molecules can form cholesteric instead of nematic phases; (iii) the smectic phase is usually found when attractions are present or at least when the molecules have finite volume. Despite the fact that the present 2D model does not have any of these characteristics, numerical evidence is found for the occurrence of nematic and smectic phases. Since these molecules are athermal, infinitely repulsive, and infinitesimally thin, the liquid crystalline characteristics are attributed to excluded volume effects. To determine the mesophases of the model, both nematic and smectic order parameters as well as distribution functions are computed.     

Numerical simulation of multicomponent ion exchange operations

A mathematical model of fixed bed ion exchange operations has been developed with reference to multicomponent systems with variable separation factors, such as those involved in the decalcification of multi-ion solutions.The model makes use of the orthogonal collocation method to solve the solid-phase diffusional equations.A preliminary test of the numerical procedure has been made by comparison with results on a binary exchange system, for which the values of the diffusivities inside the resin particle had been already determined. Experimental exhaustion breakthrough curves with sea water of a new tailor-made cationic resin and its regeneration with concentrated brines are predicted successfully by the model.     

Dispersive liquid–liquid microextraction based on ionic liquid in combination with high-performance liquid chromatography for the determination of bisphenol A in water

A method termed dispersive liquid–liquid microextraction (DLLME) coupled with high-performance liquid chromatography-variable wavelength detection (HPLC-VWD) was developed. DLLME-HPLC-VWD is a method for determination of bisphenol A (BPA) in water samples. In this microextraction method, several parameters such as extraction solvent volume, sample volume, disperser solvent, ionic strength, pH, and disperser volume were optimised with the aid of interactive orthogonal array and a mixed level experiment design. First, an orthogonal array design was used to screen the significant variables for the optimisation. Second, the significant factors were optimised by using a mixed level experiment. Under the optimised extraction conditions (extraction solvent: ionic liquid [C₆MIM][PF₆], 60 µL; dispersive solvent: methanol, 0.4 mL; and pH = 4.0), the performance of the established method was evaluated. The response linearity of the method was observed in a range of 0.002–1.0 mg L^?1 (three orders of magnitude) with correlation coefficient (R ²) of 0.9999. The repeatability of this method was 4.2–5.3% for three different BPA levels and the enrichment factors were above 180. The extraction recovery was about 50% for the three different concentrations with 3.4–6.4% of RSD. Limit of detection of the method was 0.40 µg L^?1 at a signal-to-noise ratio of 3. In addition, the relative recovery of sample of Songhua River, tap water and barrel-drain water at different spiked concentration levels was ranged 95.8–103.0%, 92.6–98.6% and 87.2–95.3%, respectively. Compared with other extraction technologies, there have been the following advantages of quick, easy operation, and time-saving for the present method.     

Microdroplet deposition under a liquid medium

An experimental and numerical study of the factors affecting the reproducibility of microdroplet depositions performed under a liquid medium is presented. In the deposition procedure, sample solution is dispensed from the end of a capillary by the aid of a pressure pulse onto a substrate with pillar-shaped sample anchors. The deposition was modeled using the convective Cahn-Hilliard equation coupled with the Navier-Stokes equations with added surface tension and gravity forces. To avoid a severe time-step restriction imposed by the fourth-order Cahn-Hilliard equation, a semi-implicit scheme was developed. An axisymmetric model was used, and an adaptive finite element method was implemented. In both the experimental and numerical study it was shown that the deposited volume mainly depends on the capillary-substrate distance and the anchor surface wettability. A critical equilibrium contact angle has been identified below which reproducible depositions are facilitated.     

A two-dimensional Haar wavelet approach for the numerical simulations of time and space fractional Fokker–Planck equations in modelling of anomalous diffusion systems

In this paper, the numerical solution for the fractional order Fokker–Planck equation has been presented using two dimensional Haar wavelet collocation method. Two dimensional Haar wavelet method is applied to compute the numerical solution of nonlinear time- and space-fractional Fokker–Planck equation. The approximate solutions of the nonlinear time- and space-fractional Fokker–Planck equation are compared with the exact solutions as well as solutions available in open literature. The present scheme is very simple, effective and convenient for obtaining numerical solution of the time and space-fractional Fokker–Planck equation.     

Application of Heart-Cutting 2D-LC for the Determination of Peak Purity for a Chiral Pharmaceutical Compound by HPLC

As an alterative to photodiode array or mass spectral analysis, a heart-cutting two-dimensional liquid chromatography technique (2D-LC) can be utilized to assess peak purity. In this work, the strengths of the heart-cutting 2D-LC technique are presented by highlighting a case study where a co-eluting impurity present at 0.8 % was non-detectable using traditional methods. However, when fractions taken across the main peak were re-assayed using an orthogonal method, this peak was readily observed. With modern switching valves and up-to-date chromatography software, implementing this technique into a validation protocol for assessing peak purity is simple and can be accomplished through only minor modifications to existing HPLC equipment.     

Determination of pressure in the extradendritic liquid area during solidification

Complex-shaped lamellar graphite iron castings are susceptible to casting defects related to the volume change during solidification. The formations of these recurring defects are caused by the flow of the liquid in the intradendritic area, between the austenite dendrite arms, and in the extradendritic area between the austenite grains. The conditions for the liquid flow, in turn, are determined by the solidification behavior. The present study suggests a new measurement method and a novel calculation algorithm to determine the pressure of the extradendritic liquid during solidification. The method involves a spherical sample suspended in a measurement device, where the temperature and the volume changes are measured during solidification. The calculation algorithm is based on the numerical interpretation of the Clausius–Clapeyron equation where the temperature variation, the volume change and the released latent heat are processed to determine the local pressure of the extradendritic liquid area during solidification.

     

Simulation of an ultrafiltration process of bovine serum albumin in hollow-fiber membranes

This work presents a numerical simulation of an ultrafiltration process of bovine serum albumin in solution, using hollow-fiber membranes. Such membranes are constituted of tiny polymer cylinders disposed in a tube-and-shell arrangement. The concentrate flows through the interior of the fibers and the pure solvent is recovered in the shell, assuming perfect solute rejection. In modeling the process, the flow of concentrate inside the fibers was considered to be laminar, with constant density, viscosity and solute diffusivity. Axial diffusion and angular effects were ignored. The model combines the effect of concentration polarization and adsorption, which are the two main limiting phenomena in ultrafiltration processes. The pressure on the shell side was considered constant and inside the fibers a linear pressure profile, dependent on the axial position, was adopted. The solution of the problem was achieved with the method of orthogonal collocation, with adequate choice of the weight function in the radial direction. In the axial direction, a finite-difference method was used. The numerical results were compared with experimental data available in the literature.     

Global optimization for parameter estimation of differential-algebraic systems

The estimation of parameters in semi-empirical models is essential in numerous areas of engineering and applied science. In many cases, these models are described by a set of ordinary-differential equations or by a set of differential-algebraic equations. Due to the presence of non-convexities of functions participating in these equations, current gradient-based optimization methods can guarantee only locally optimal solutions. This deficiency can have a marked impact on the operation of chemical processes from the economical, environmental and safety points of view and it thus motivates the development of global optimization algorithms. This paper presents a global optimization method which guarantees ɛ-convergence to the global solution. The approach consists in the transformation of the dynamic optimization problem into a nonlinear programming problem (NLP) using the method of orthogonal collocation on finite elements. Rigorous convex underestimators of the nonconvex NLP problem are employed within the spatial branch-and-bound method and solved to global optimality. The proposed method was applied to two example problems dealing with parameter estimation from time series data.     

分散液液微萃取-柱前衍生-高效液相色谱法测定水样中双酚A

建立了分散液液微萃取-柱前衍生-高效液相色谱法测定水样中双酚A的分析方法.通过交互正交试验和混合型优化实验设计对影响因素(萃取剂体积、分散剂类型及其体积、水样体积、pH值及离子强度)进行了优化.优化后的分散液液微萃取条件为:60 μL萃取剂,0.4 mL分散剂(甲醇),pH 4.0;优化后的柱前衍生化条件:0.1 mL 2.0 g/L衍生剂(对硝基苯甲酰氯)、衍生化时间30 min;方法的线性范围:0.002～0.2 mg/L(r=0.9997),检出限0.007 μg/L(S/N=3);不同浓度双酚A的萃取率为59.0%～63.0%,相对标准偏差(RSD)2.5%～9.2%(n=5);水样中双酚A的加标率为86.5%～107.1%,RSD为4.0%～11.9%(n=5),其它雌激素(雌酮、雌二醇、雌三醇和17α-乙炔基雌二醇)对双酚A的测定无干扰.本方法可以对水环境中的痕量BPA进行检测,具有操作简便、快速等优点.     

基于详细机理动力学的费-托合成单颗粒催化剂模拟 Ⅰ.颗粒模型化与数值计算方法

 基于详细机理动力学的研究成果,讨论了费－托合成单颗粒催化\r\n剂在液蜡填充情形下的详细模型化问题,建立了针对这一复杂多重反应\r\n体系的多组分扩散－反应的等温和非等温颗粒模型．在此基础上,从催\r\n化剂设计灵活定位考虑,进一步建立了模型方程的有限元正交配置数值\r\n计算方法．     

微波辅助-顶空液相微萃取/高效液相色谱分析水样中的邻硝基苯酚

建立了基于微波辅助-顶空液相微萃取在线联用、高效液相色谱法测定水样中邻硝基苯酚的分析方法。采用L16(45)正交实验设计对影响萃取的各种因素,如萃取有机溶剂、微波辐射功率、萃取时间、离子强度、样品液体积,进行了优化。优化后萃取条件为,以乙酸丁酯作为萃取溶剂,功率和时间分别为100W和12min条件下,离子强度为0的样品溶液体积为20mL。在优化萃取条件下,邻硝基苯酚的检出限LOD(S/N=3)为0.94μg/L,萃取富集倍数为30,实际水样的加标回收率为85.2%。理论分析和实验结果表明,微波辅助-顶空液相微萃取在线联用方法具有简便、快速、高效、节省溶剂、选择性好、应用范围广的特点。     

Modeling flows of confined nematic liquid crystals

The flow of nematic liquid crystals in tightly confined systems was simulated using a molecular theory and an unsymmetric radial basis function collocation approach. When a nematic liquid crystal is subjected to a cavity flow, we find that moderate flows facilitate the relaxation of the system to the stable defect configuration observed in the absence of flow. Under more extreme flow conditions, e.g., an Ericksen number Er=20, flows can alter the steady-state defect structure observed in the cavity. The proposed numerical method was also used to examine defect annihilation in a thin liquid crystal film. The flows that arise from shear stresses within the system result in a higher velocity for s = +1∕2 defect than for the defect of opposing charge. This higher velocity can be attributed to reactive stresses within the deformed liquid crystal, which result in a net flow that favors the motion of one defect. These two examples serve to illustrate the usefulness of radial basis functions methods in the context of liquid crystal dynamics both at and beyond equilibrium.