Numerical approximation of a nonisothermal two-dimensional general rate model of reactive liquid chromatography

A nonlinear and nonisothermal two-dimensional general rate model is formulated and approximated numerically to allow quantitatively analyzing the effects of temperature variations on the separations and reactions in liquid chromatographic reactors of cylindrical geometry. The model equations form a nonlinear system of convection-diffusion-reaction partial differential equations coupled with algebraic equations for isotherms and reactions. A semidiscrete high-resolution finite volume method is modified to approximate the system of partial differential equations. The coupling between the thermal waves and concentration fronts is demonstrated through numerical simulations, and important parameters are pointed out that influence the reactor performance. To evaluate the precision of the model predictions, consistency checks are successfully carried out proving the accuracy of the predictions. The results allow to quantify the influence of thermal effects on the performance of the fixed beds for different typical values of enthalpies of adsorption and reaction and axial and radial Peclet numbers for mass and heat transfer. Furthermore, they provide useful insight into the sensitivity of nonisothermal chromatographic reactor operation.     

Numerical solution of nonlinear and non-isothermal general rate model of reactive liquid chromatography

Abstract

A nonlinear general rate model (GRM) of liquid chromatography is formulated to analyze the influence of temperature variations on the dynamics of multi-component mixtures in a thermally insulated liquid chromatographic reactor. The mathematical model is formed by a system of nonlinear convection–diffusion reaction partial differential equations (PDEs) coupled with nonlinear algebraic equations for reactions and isotherms. The model equations are solved numerically by applying a semi-discrete high-resolution finite volume scheme (HR-FVS). Several numerical case studies are conducted for two different types of reactions to demonstrate the influence of heat transfer on the retention time, separation, and reaction. It was found that the enthalpies of adsorption and reaction significantly influence the reactor performance. The ratio of density time heat capacity of solid and liquid phases significantly influences the magnitude and velocity of concentration and thermal waves. The results obtained could be very helpful for further developments in non-isothermal reactive chromatography and provide a deeper insight into the sensitivity of chromatographic reactor operating under non-isothermal conditions.     

丙烯在NiSO4催化剂上的液相齐聚—集总宏观动力学与反应器分析

以负载在γ－Ａｌ２Ｏ３上的ＮｉＳＯ４为催化剂，在微分反应器中进行了以壬烯与十二烯为目的产物的丙烯液相催化齐聚的研究。在２７３￣３１３Ｋ及高于丙烯蒸气压的压力条件下齐聚产物主要为己烯、壬烯及十二烯，十五烯以上的含量极少。根据反应的热力学性质可知齐聚反应为一组双分子不可逆反应。按集总原则回归实验数据，得到了丙烯齐聚过程的宏观动力学方程组及有关参数。由于齐聚反应的热效应较大，工业规模的丙烯齐聚只有在壁冷     

Analysis of two-dimensional models for liquid chromatographic reactors of cylindrical geometry

This work presents the analytical solutions of two-dimensional isothermal reactive general rate models for liquid chromatographic reactors of cylindrical geometry. Both irreversible and reversible reactions are considered. The model equations form a linear system of convection-diffusion-reaction partial differential equations coupled with algebraic equations for isotherms. Analytical solutions are derived by integrated implementation of finite Hankel transform, Laplace transform, eigen-decomposition technique, and conventional ordinary differential equations solution technique. To verify the analytical results, a high-resolution finite volume scheme is also applied to numerically approximate the model equations. The current results can be very useful to optimize and upgrade the liquid chromatographic reactors.     

Study on the consecutive reaction kinetics of synthesis of di(2‐ethylhexyl) terephthalate under nonisothermal conditions

Since more information concerning kinetic parameters can be obtained from a nonisothermal reaction, it was selected to investigate the consecutive esterification kinetics of terephthalic acid with 2‐ethylhexanol in the presence of tetrabutyl titanate catalyst. This is an equilibrium reaction that is carried out in industry to completion by removing the water formed. It results in an automatic rise in the esterifying temperature, from 453 to 519 K. Research shows that the first step of esterification carried out in a heterogeneous system has a slow reaction rate, but the second step of esterification in a homogeneous system has a relatively fast reaction rate. Based on the quasi‐homogeneous assumption, first the differential method is presented to deal with nonisothermal reaction data. Arrhenius equations of the two steps are established by using this method. It was found that the apparent activation energy of the first step of esterification was about 55 kJ/mol higher than that of the second step. The ratio (K) of reaction rate constants of the two steps decreases gradually with the increase in the reaction temperature. An equation of K vs. temperature is also derived from Arrhenius equations. Subsequently, integral expressions of components' concentrations are used to simulate experimental results of the nonisothermal reaction as well as a three‐stage isothermal reaction. The obtained simulations show that the determined kinetic equations and the parameters are reasonable. © 2006 Wiley Periodicals, Inc. Int J Chem Kinet 38: 577–584, 2006     

Numerical approximation of a two-dimensional model of non-isothermal reactive liquid chromatography involving slow rates of adsorption–desorption kinetics

Abstract

A two-dimensional general rate model of non-isothermal reactive column chromatography is formulated considering homogenous and heterogeneous reaction rates, slow rates of adsorption–desorption kinetics, and enthalpies of adsorption and reaction. The model is expressed by a system of six nonlinear partial differential equations (PDEs) coupled with algebraic expressions for the adsorption and reaction rates. The nonlinearity of adsorption isotherm and reaction term hinders the derivation of analytical solutions. For that reason, a flux-limiting high-resolution finite volume scheme is suggested to numerically approximate the model equations. The effects of several kinetic and thermodynamic parameters are rigorously analyzed on the reactant conversion and components separation.     

On the apparent compensation effect found for two parallel reactions

The authors present a critical analysis of the use of an overall single reaction rate equation instead of the true rate equations corresponding to the decomposition of a substance according to two parallel reactions. Isothermal as well as nonisothermal decomposition are considered. An apparent compensation effect has been evidenced in both cases. It has been assigned to the dependence of the kinetic parameters on temperature (for the isothermal case), conversion, and heating rate (for nonisothermal one). © 1998 John Wiley & Sons, Inc. Int J Chem Kinet: 30: 673–681, 1998     

Linearized non-equilibrium and non-isothermal two-dimensional model of liquid chromatography for studying thermal effects in cylindrical columns

A non-equilibrium and non-isothermal two-dimensional lumped kinetic model (2?D-LKM) is formulated and analytically solved to study the influence of temperature variations along the axial and radial coordinates of a liquid chromatographic column. The model includes convection-diffusion partial differential equations for mass and energy balances in the mobile phase coupled with differential equations for mass and energy in the stationary phase. The solutions are derived analytically through sequential implementation of finite Hankel and Laplace transformations using the Dirichlet inlet boundary conditions. The coupling between the thermal waves and concentration fronts is demonstrated through numerical simulations and important parameters are recognized that influence the column performance. For a more comprehensive study of the considered model, numerical temporal moments are obtained from the derived solutions. Several case studies are conducted and validity ranges of the derived analytical solutions are identified. The current analytical results will play a major role in the improvements of non-equilibrium and non-isothermal liquid chromatographic processes.     

Kinetic analysis of the transesterification of dimethyl terephthalate with ethylene glycol in a semibatch reactor

The kinetic analysis of the transesterification of ethylene glycol (EG) and dimethyl terephthalate (DMT) was studied in a semibatch reactor under nonisothermal conditions. The effect of the temperature profile, EG/DMT ratio, and catalyst concentration on the reaction rate was studied. Models were fitted by using an algorithm of parameter estimation in differential equations based on the Gauss–Newton method improved with the Marquardt extension. It was found that, unlike previous authors, the reaction rate was not a classical third order reaction of first order with respect to EG, DMT, and catalyst concentration, but the EG reaction order was 0.6 and a hyperbolic function was found to fit the relationship between reaction rate and catalyst concentration. Also, it was found that, at high conversions, the effect of the reverse reaction should not be neglected.     

Viscous fingering in a horizontal flow through a porous medium induced by chemical reactions under isothermal and adiabatic conditions

In this work we analyze the viscous fingering instability induced by an autocatalytic chemical reaction in a liquid flowing horizontally through a porous medium. We have analyzed the behavior of the system for isothermal as well as adiabatic conditions. The kinetics of the reaction is chosen so that the rate depends on the concentration of only a single species. Since the reaction is autocatalytic the system admits a traveling wave solution. For endothermic reactions the concentration wave and temperature wave are mirror images, whereas for an exothermic reaction they are similar or parallel. The viscosity of the fluid is assumed to depend strongly on the concentration of the product and temperature of the medium. The dependence of viscosity on concentration (decrease with concentration) can destabilize the traveling wave resulting in the formation of viscous fingers. We have performed a linear stability analysis to determine the stability of the base traveling wave solution. The stability predictions have been confirmed by nonlinear simulations of the governing equations based on a finite difference scheme. We observe that including the temperature dependency of viscosity stabilizes the flow for an endothermic reaction, i.e., regions which exhibited viscous fingering now demonstrate stable displacement. For exothermic systems, however, the system exhibits less stable behavior under adiabatic conditions, i.e., it is destabilized by both concentration and temperature dependencies of viscosity.     

Modeling and experimental studies on combustion characteristics of porous coal char: Volume reaction model

A generalized single‐particle model for the prediction of combustion dynamics of a porous coal char in a fluidized bed is analyzed in the present work using a volume reaction model (VRM). A fully transient nonisothermal model involving both heterogeneous and homogeneous chemical reactions, multicomponent mass transfer, heat transfer with intraparticle resistances, as well as char structure evolution is developed. The model takes into account convection and diffusion inside the particle pores, as well as in the boundary layer. By addressing the Stefan flow originated due to nonequimolar mass transfer and chemical reactions, this work enables a more realistic analysis of the combustion process. The model, characterized by a set of partial differential equations coupled with nonlinear boundary conditions, is solved numerically using the implicit finite volume method (FVM) with a FORTRAN code developed in‐house. The use of a FVM for solving such an elaborate char combustion model, based on the VRM, was not reported earlier. Experiments consisting of fluidized‐bed combustion of a single char particle were carried out to determine the internal surface area of a partially burned char particle and to enable model validation. Predicted results are found to compare well with the reported experimental results for porous coal char combustion. The effects of various parameters (i.e., bulk temperature and initial particle radius) are examined on the dynamics of combustion of coal char. The phenomena of ignition and extinction are also investigated. © 2010 Wiley Periodicals, Inc. Int J Chem Kinet 42: 299–315, 2010     

Mathematical modeling of self-oscillations in ethane oxidation over nickel

The methodology of constructing a phenomenological model for complex heterogeneous catalytic reactions is described in detail. The proposed approach is applicable to development of mathematical models describing the onset of self-oscillations in hydrocarbon oxidation on the transition metal surface. The approach is based on construction of a microkinetic scheme taking into account the formation of main reaction products and intermediates, on estimation of the heat of reaction, activation energy, and preexponential factor for elementary steps and includes development and a subsequent analysis of the corresponding mathematical model. Catalytic reactions are considered in the ideal adsorption layer approximation without taking into account the relationship between coverages and spatial coordinates. Accordingly, the mathematical model is an independent system of ordinary differential equations. This methodology is used to develop a point (lumped) model for ethane oxidation over nickel, which is based on a 36-step microkinetic scheme taking into account the oxidation and reduction of nickel and the formation of total (CO₂ and H₂O) or partial (CO and H₂) ethane oxidation products, as well as the dehydrogenation of ethane into ethylene. The proposed model predicts the onset of self-oscillations in this reaction at atmospheric pressure in the temperature range from 850 to 1400 K. The kinetic oscillations are caused by the cyclic oxidation and reduction of nickel. The self-oscillations of the reaction rate are accompanied by oscillations of the catalyst temperature. The results of modeling are compared with experimental data.     

Open source simulation tool for electrophoretic stacking,focusing, and separation

We present the development, formulation, and performance of a new simulation tool for electrophoretic preconcentration and separation processes such as capillary electrophoresis, isotachophoresis, and field amplified sample stacking. The code solves the one-dimensional transient advection-diffusion equations for multiple multivalent weak electrolytes (including ampholytes) and includes a model for pressure-driven flow and Taylor–Aris dispersion. The code uses a new approach for the discretization of the equations, consisting of a high resolution compact scheme which is combined with an adaptive grid algorithm. We show that this combination allows for accurate resolution of sharp concentration gradients at high electric fields, while at the same time significantly reducing the computational time. We demonstrate smooth, stable, and accurate solutions at current densities as high as 5000 A/m² using only 300 grid points, and a 75-fold reduction in computational time compared with equivalent uniform grid techniques. The code is available as an open source for free at http://microfluidics.stanford.edu.     

高效液相色谱手性流动相法拆分甲状腺素对映体

运用高效液相色谱手性流动相法（ＨＰＬＣ－ＣＭＰ）对影响甲状腺素对映体（Ｄ－,Ｌ－Ｔ４）分离方法的因素：三乙胺（ＴＥＡ）浓度,流动相ｐＨ值,铜离子（Ｃｕ２＋）浓度,Ｌ－脯氨酸（Ｌ－ｐｒｏ）浓度,柱温以及流动相的流速进行了系统的研究。同时,考察了色谱方法分离Ｔ４对映体的线性关系,精密度和准确度。线性响应范围为０．６～３．２　ｎｍｏｌ　（Ｄ－,Ｌ－Ｔ４）,线性相关系数为ｒＤ－Ｔ４＝０．９９８０,ｒＬ－Ｔ４＝０．９９９０,日内和日间的精密度分别为ＲＳＤ＜２．３％（ｎ＝６）,ＲＳＤ＜３．１５％（ｎ＝５）。结果表明本实验所得的色谱条件较文献报道的优越,分离条件简单,重现性好。ＨＰＬＣ－ＣＭＰ法测定甲状腺素对映体其意义在于该方法可为定量测定药品及人体血液中Ｄ－,Ｌ－Ｔ４两种异构体,为治疗药物监测（ＴＤＭ）和药物不良反应监测（ＡＤＲｓ）提供了依据。     

Mechanistic modeling of reversion phenomenon in sulphur cured natural rubber vulcanization kinetics

A novel kinetic model of natural rubber sulphur vulcanization is proposed. The modeling approach takes into account current knowledge on the different polysulfidic structures present during vulcanization, and the associated individual reactions. A simplified scheme is proposed, giving a mechanistic view of the reversion phenomenon, which results in a decrease of the elastic modulus (related to the sulphur crosslink density) for long vulcanization times at high temperature. The resulting set of differential equations is solved by an appropriate numerical method to predict the evolution of the degree of vulcanization for isothermal cure conditions.     

Group additivity methods without group values

The conventional group additivity (GA) formalism may be identically reduced to a stoichiometric and thermochemical analysis of a special class of reactions referred to as GA reactions, that is, reactions that preserve the type and number of groups. Within this approach, the performance (error) of a GA scheme is determined by the stoichiometry and enthalpy changes of the GA reactions. That is, the lower the enthalpy changes of the GA reactions, the better the performance of a GA scheme. Ideally, an exact GA scheme would imply any conceivable GA reaction to be precisely thermoneutral, that is, have a zero enthalpy change. A somewhat surprising result is that, additionally, the performance of GA methods is influenced by a purely stoichiometric factor of GA reactions. These findings do not improve the performance of a given GA scheme. Rather, it is an interpretation that leads to a deeper understanding of the performance of a GA scheme and may be used in designing more accurate GA schemes.     

Nonisothermal decomposition kinetics and computational studies on the properties of 2,4,6,8-tetranitro-2,4,6,8-tetraazabicyclo[3,3,1]onan-3,7-dione (TNPDU)

The thermal decomposition and the nonisothermal kinetics of the thermal decomposition reaction of 2,4,6,8-tetranitro-2,4,6,8-tetraazabicyclo[3,3,1]onan-3,7-dione (TNPDU) were studied under the nonisothermal condition by differential scanning calorimetry (DSC) and thermogravimetry-derivative thermogravimetry (TG-DTG) methods. The kinetic model function in differential form and the value of Ea and A of the decomposition reaction of TNPDU are f(alpha) = 3(1 - alpha)[-ln(1 - alpha)](2/3), 141.72 kJ mol(-1), and 10(11.99) s(-1), respectively. The critical temperature of thermal explosion of the title compound is 232.58 degrees C. The values of DeltaS(++), DeltaH(++), and DeltaG(++) of this reaction are -15.50 J mol(-1) K(-1), 147.65 kJ mol(-1), and 155.26 kJ mol(-1), respectively. The theoretical investigation on the title compound as a structure unit was carried out by the DFT-B3LYP/6-311++G** method. The IR frequencies and NMR chemical shift were performed and compared with the experimental results. The heat of formation (HOF) for TNPDU was evaluated by designing isodesmic reactions. The detonation velocity (D) and detonation pressure (P) were estimated by using the well-known Kamlet-Jacobs equation, based on the theoretical densities and HOF. The calculation on bond dissociation energy suggests that the N-N bond should be the trigger bond during the pyrolysis initiation process.     

The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry

In this paper, we have applied an accurate and efficient wavelet scheme (due to Legendre polynomial) to find the numerical solutions for a set of coupled reaction–diffusion equations. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of non linear terms appearing in the governing differential equations. The highest derivative in the differential equation is expanded into wavelet series, this approximation is then integrated while the boundary conditions are applied by using integration constants. With the help of operational matrices, the nonlinear reaction–diffusion equations are converted into a system of algebraic equations. Finally, some numerical examples to demonstrate the validity and applicability of the method have been furnished. The use of Legendre wavelets is found to be accurate, efficient, simple, and computationally attractive. This wavelet method can be used for obtaining quick solution in many chemical Engineering problems.     

Specific Features of a Two-Phase Bimolecular Chemical Reaction in the Case of the Association of Both Reactants

The effect of the association of both reactants on the kinetics of their bimolecular reaction in the liquid phase is studied. The mathematical modeling of chemical reactions that are described by nonlinear differential equations is performed. The steady states, the conditions for the emergences of intermediates, and the nature of their concentration oscillations in the reaction system are described. It is found that the concentration of the intermediates has two types of oscillations (harmonic and relaxation oscillations) characterized by significantly different times. The relationship between the observed rate constant of the process, the rate constants for the elementary stages, and the reactant concentrations is found.     

Nonisothermal thermogravimetry of polymers

The model of weight loss taking place in each step of a scheme of consecutive reactions was applied to nonisothermal thermogravimetric records of bromomethylated poly(2,6-dimethyl-1,4-phenylene oxide). The release of HBr was found to be faster in air than in nitrogen. A significant reduction in the apparent activation energy of dehydrobromination as compared with decomposition of the nonbrominated polymer was ascribed not only to a reduction in the activation energy of the initiation reaction, but also to an increase in the order of the termination reaction (from 1 for the nonbrominated polymer to 2 for the highly brominated polymer).