On the rate of heterogeneous catalytic reactions with ionic intermediates

A facile way to compute the rate law of a catalytic reaction with ionic intermediates is proposed, based on the theory of complex reactions. When the ionic nature of intermediates is taken into account it is necessary to add balance equations which correspond to the electroneutrality of the catalyst. Several mechanisms of the olefin hydrogenation were considered as examples.     

Numerical methodology for proton exchange membrane fuel cell simulation using computational fluid dynamics technique

To analyze the physical phenomena occurring in the Proton Exchange Membrane Fuel Cell (PEMFC) using Computational Fluid Dynamics (CFD) technique under an isothermal operating condition, four major governing equations such as continuity equation, momentum conservation equation, species transport equation and charge conservation equation should be solved. Among these governing equations, using the interfacial boundary condition is necessary for solving the water transport equation properly since the concept of water concentration in membrane/electrode assembly (MEA) and other regions is totally different. It was first attempted to solve the water transport equation directly in the MEA region by using interfacial boundary condition; and physically-meaningful data such as water content, proton conductivity, etc. were successfully obtained. A detailed problem-solving methodology for PEMFC is presented and result comparison with experimental data is also implemented in this paper.     

采用结构进化策略的Lagrange乘子法优化换热网络

针对罚函数法处理有约束问题时存在的不足,采用Lagrange乘子法优化换热网络.为求解Lagrange函数方程组,根据确定性方法,提出最速下降法求解策略以及Powell法求解策略.通过极小值判断机制,保证Lagrange函数方程组的解是原换热网络目标函数值的极小值.根据实际工况,提出结构进化策略,与Lagrange乘子法相结合,实现了换热网络全局最优化.通过经典算例验证了两种求解策略的有效性、准确性以及结构进化策略的通用性.与文献结果进行对比,结果表明本算法具有较强的局部搜索能力以及全局搜索能力,能够找到更优的换热网络结构,有利于在工业生产中节约成本.     

Inverse problems in population balances: Growth and nucleation from dynamic data

Particulate process modeling is critical for system design and control used widely in the chemical industly. Previous methods have focused on the assumption of appropriate models that can capture system behavior. A new technique presented is based on viewing the population balance from an inverse problem perspective that allows to determine appropriate models directly from experimental data. Under suitable assumptions (deterministic growth rate, no aggregation), the population balance equation may be solved by the method of characteristics, which associates the number density for any size at any time with a single point from the initial or boundary condition. The key to using this is the recognition that these characteristics correspond to the size history of individual particles and can be associated with constant cumulative number densities (quantiles) of the population. These quantiles are easily identifiable from experimental data. The variation of size and number density along these characteristics provides decoupled equations used to determine the growth rate. Validity of the determined growth law is checked by the collapse of the experimental data onto initial and boundary conditions.     

Diffuse layer effects on the current in galvanic cells containing supporting electrolyte

We study the effect of an inert supporting electrolyte on the steady-state ionic current through galvanic cells by solving the full Poisson-Nernst-Planck transport equation coupled to the generalized Frumkin-Butler-Volmer boundary equation for the electrochemical charge transfer at the electrodes. Consequently, the model presented here allows for non-zero space charge densities locally at the electrodes, thus extending the frequently used models based on the local electroneutrality condition by including diffuse layer (DL) effects. This extension is necessary since the DLs determine the ion concentration and electrical field at the reaction planes, which uniquely determine the charge transfer at the electrodes.In this work we present numerical results for systems which contain added inert supporting electrolyte using finite element discretization and compare those with semi-analytical results obtained using singular perturbation theory (limit of negligibly thin DLs). In case of negligibly thin DLs the presence of supporting electrolyte will introduce a limiting current below the classical diffusion-limiting current. Just as for systems without supporting electrolyte, the supporting electrolyte induced limiting current formally does not occur for systems having non-negligibly thin double DLs. For thin, however still finite, double layers this limit can still be seen as a steepening of the polarization curve for current vs. voltage.     

Overlimiting current by interactive ionic transport between space charge region and electric double layer near ion-exchange membranes

In electrodialysis, a theoretical understanding of overlimiting current (greater ionic transport beyond the electroneutrality limit) has not been clearly demonstrated, despite two hypothetical explanations: the water splitting reaction and electro-convection. To suggest another possible mechanism for this phenomenon, the Nernst-Planck-Poisson equations were solved via the Painlevé equation of the second kind. Numerical simulations found that the initiation of overlimiting current involves a substantial redistribution of the ionic charge near the membrane surface. To maintain a constant ionic charge in the aqueous phase near the membrane while the non-electroneutral space charge region grows, the shrinking electric double layer in the aqueous-phase releases ionic charges toward the membrane. Frequent repetition of this charge redistribution can contribute additional current above the limiting current without a large investment of electrical potential. A sensitivity study found that the relationship between the growth of the space charge region and applied potential is independent of the membrane's ion exchange capacity and the ionic diffusivity; however, the amount of the ionic charge created in the space charge region was greater with a greater bulk concentration or with an electrolyte solvent of a greater permittivity. These findings are consistent with previous experimental studies in the overlimiting current regime.     

A general dynamic model for multiphase electrolytic systems

The study and characterization of aqueous solutions are becoming an important research topic because of their central role in many processes. The modeling of these solutions is complicated by different factors: the important number of ionic and nonionic species, the nonideality of aqueous solutions, the appearance and disappearance of thermodynamic phases (solid, vapor) depending on operating conditions. This article presents a general model in dynamic state of aqueous electrolytic system involving liquid, vapor, and solid phases. The mathematical formulation includes: physical and chemical equilibria, mass and energy balances, electroneutrality equation and evaporation equation. The set of equations is solved by the Gear's method. The model is validated by the comparison between modeled results (activity coefficients, solubility, evaporite sequence) and experimental data. One of the purposes of the model will be the simulation of a production process of salt crystals and waters concentrated in minerals of interest. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

Estimation of excess enthalpy for binary systems

Based on the author’s molecular thermodynamic model for solution structure, a new equation with three parameters for correlating the excess enthalpy of binary solutions is proposed. We make use of the group statistical theory, Lagrange undetermined multiplier method and a pseudo-chemical reaction mechanism. Tested by 6059 experimental data in 282 kinds of binary systems, the new model shows the characteristics of convenience and wide applicability. The accuracy of the new equation has an advantage over the Redlich–Kister equation [J. Chem. Ind. Eng. (China) 49 (1998) 103] and the Shen–Liu equation proposed recently [J. Chem. Ind. Eng. (China) 51 (2000) 181], both of which have three parameters too. As for the SSF equation with four parameters, the accuracy of it and the new equation is equivallent, some times the latter is more accurate. The new equation is still tenable in the limiting condition of x→0 and 1.     

Application of the method of weighted residuals to mixed boundary value problems: Dual-series relations

Mixed boundary value problems arise in a number of chemical engineering models for heat conduction, simultaneous diffusion and reaction, and fluid flow whenever there is a change in the type of boundary condition along the same boundary. In this paper a general computer algorithm based on the method of weighted residuals (MWR) is developed for determining approximate solutions of all problems of the above type which lead to dual series equations. The solution procedure reduces the determination of the series coefficients to the solution of a large system of algebraic equations. This technique offers advantages over finite difference or artificial interface methods in the accuracy which can be obtained, the type of problem which may be treated, and the simplicity of calculations to be performed. Solutions obtained by the application of MWR are compared and analysed for three example problems which are of interest in diffusion, reaction and conduction.     

On boundary conditions to the film model of mass transfer in chemical absorption

The problem of appropriate boundary conditions for mass transfer within the liquid film in chemical absorption ih reconsidered. It has been shown that the most general solution to the problem of mass transfer in the liquid film can be obtained if a fixed bulk composition is taken as a film-bulk boundary condition to the differential balance equations of the film. This method leads to solutions which can be incorporated into absorber balance equations of any type. As an illustrative example, the simultaneous absorption of two gases which react together in a semi-batch absorber is discussed. A new, approximate but very simple and accurate solution to this problem is proposed.     

High-resolution method for numerically solving PDEs in process engineering

Abrupt phenomena in modelling real-world systems indicate the importance of investigating systems with steep gradients. However, it is difficult to solve such systems either analytically or numerically. In 1993, Koren developed a high-resolution numerical computing scheme to deal with compressible fluid dynamics with Dirichlet boundary condition. Recently, Qamar adapted this scheme to numerically solve population balance equations without diffusion terms. This paper extends Koren’s scheme for partial differential equations (PDEs) that describe both nonlinear propagation and diffusive effects, and for PDEs with Cauchy or Neumann boundary condition. Accurate and convergent numerical solutions to the test problems have been obtained. The new results are also compared to those obtained by wavelet-based methods. It is shown that the method developed in this paper is more efficient.     

A technique for analysing integral rate data in non-ideal flow reactors to estimate unknown parameters

A technique is described for analysing integral rate data taken from a non-ideal flow chemical reactor characterised by the axial dispersion model, using a weighted residual approach. This approach avoids the formidable computational difficulties involved with the solution of the boundary value problem associated with the model differential equations. The measured concentration profile is approximated by a series of orthogonal polynomials. Model parameters are estimated by specifying that the fitting polynomial satisfy certain moment relations derived from the mass balance equation and boundary conditions. If the form of the rate equation is known, a priori, the estimation of the rate constant and the dispersion coefficient becomes explicit. In the more general case, in which the reaction order is also unknown, the estimation problem reduces to a uni-dimensional search for the reaction order parameter (or the adsorption parameter in the case of complex kinetics). The method is applied to both linear and non-linear systems and conclusions are drawn regarding the sensitivity of the chemical kinetic and transport parameters to experimental error.     

Variational model for ionomeric polymer–metal composite

Governing equations for one-dimensional ionic polymer–metal composites are reviewed and then cast into the framework of a variational statement. Starting from a trial-Lagrangian, a generalized functional is derived through a systematic procedure of the semi-inverse method proposed by Ji-Huan He. All field equations and boundary conditions are cast into Euler equations of the obtained functional, leading to much convenience in incorporating analytical and numerical approaches.     

Numerical solution of stiff boundary valued problems in kinetics and diffusion

Two new numerical methods for the solution of stiff boundary valued ordinary differential equations are presented and compared. The specific problem solved is that of diffusion and reaction in a char pore, where eight species diffuse and react through ten free radical combustion reactions. A new variable mesh finite difference algorithm is presented, and it is shown how existing techniques for treating stiff initial value problems can be adapted for the boundary valued case. The numerical techniques presented have general application in problems involving diffusion and stiff chemical reactions, such as occur in combustion, gasification, and reactor design.     

Solving Systems of Fourth-Order Emden–Fowler Type Equations by the Variational Iteration Method

In this paper, we investigate systems of nonlinear fourth-order Emden–Fowler type equations. We employ the variational iteration method (VIM) for solving these systems of equations. The newly developed fourth-order Emden–Fowler equation is characterized by three types, where the shape factor appears three times, twice, and once for the first type, second type, and the third type respectively. We use distinct Lagrange multipliers for these specific types to overcome the singularity at the origin. We solve several numerical examples obtaining a rapidly convergent sequence of approximations. In all examples investigated, we achieved approximations with a high level of accuracy, thus confirming that we can get accurate approximations with few variational iterations.     

求解大规模优化问题的修改增广Lagrange乘子法

Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studied in this paper. The Lagrange function contains the penalty terms on equality and inequality constraints and the methods can he applied to solve a series of bound constrained sub-problems instead of a series of unconstrained sub-problems. The steps of the methods are examined in full detail. Numerical experiments are made for a variety of problems, from small to very large-scale, which show the stability and effectiveness of the methods in large-scale problems.     

Thermodynamic calculation of ideal and nonideal detonation

Conclusion A method has been proposed for calculating ideal and nonideal detonation, the thermodynamic basis of which is the method of characteristic function extrema, with the integral equations of conservation of the hydrodynamic model of steady-state flow of a continuous medium supplemented by the Chapman-Jouguet condition in the form of a minimum in normal detonation rate.The method has been realized in the form of a package of thermodynamic calculation programs which allow calculation of the composition and thermodynamic parameters of complex chemically reacting systems in the six basic problems of thermodynamics and in steady-state gas dynamic problems under conditions of equilibrium and specified chemical., thermal, and mechanical nonequilibrium of the products with arbitrary equations of state of the phases.Generalized results have been presented from calculations of ideal detonation of a large number of CHNO explosives and compared to experimental data. A new set of coefficients has been proposed for the BKW equation of state called the BKW-RR coefficients, which in combination with proper consideration of the phase state of the condensed carbon in the detonation products permits significant improvement in predicting the detonation rate.Results of nonideal detonation calculations have been presented, with the detonation products being under conditions of chemical, thermal, and mechanical nonequilibrium. It has been shown that the mass fraction of inert condensed phase in the detonation products not in mechanical equilibrium with the remaining products can differ significantly at the Chapman-Jouguet point from the inert content of the original mixture. Therefore, in the thermodynamic calculation of nonideal detonation one should use the equations of conservation of mass fluxes of the chemical elements in place of the equations of mass balance of chemical elements. It has been noted that in the case of nonideal detonation it is possible to use the Chapman-Jouguet condition in the form of equality of the flux velocity of the deoration product mixture as a whole relative to the front to the local speed of sound calculated with consideration of nonequilibrium conditions.Moscow. Translated from Fizika Goreniya i Vzryva, Vol. 23, No. 4, pp. 75–84, July–August, 1987.     

Boundary value problems in transport with mixed or oblique derivative boundary conditions—I: Formulation of equivalent integral equations

When an internally heated body is cooled along its boundary by a peripherally flowing fluid that is continually replenished from an external source, a differential energy balance on the boundary leads to unfamiliar boundary conditions. Such boundary conditions involve mixed second derivatives with respect to spatial variables, which under additional assumptions (such as an infinite heat transfer coefficient) lead to oblique derivative boundary conditions; i.e. at the boundary an oblique derivative of the temperature is specified. Classical attempts at solution of elliptic partial differential equations with oblique derivative boundary conditions have been through the establishment of equivalent singular integral equations, using complex analytic continuation. The theory of singular integral equations is complicated, however.Using appropriate Green's functions, the boundary value problems of interest have been reduced to equivalent integral equations in this work. While oblique derivative boundary value problems are shown to lead to singular integral equations, the mixed derivative boundary value problem is shown to yield Fredholm integral equations directly. This surprising finding is mathematically significant, because Fredholm integral equations are solved more easily, and physically significant because the mixed derivative boundary condition is the more realistic condition in the present context. A method of solution of Fredholm integral equations is discussed.More complicated boundary conditions in which axial conduction in the coolant fluid is important have also been shown to lead to Fredholm integral equations. Finally a transient problem has been formulated.     

Inductive behaviour by charge-transfer and relaxation in solid-state electrochemistry

We discuss a model system of electroactive material containing heterogeneity or energy disorder, in which charge accumulation and transfer is governed mainly by chemical potentials of species in the solid material. The equivalent circuit for impedance spectroscopy is composed of parallel combination of the electrochemical capacitances of the phases and charge-transfer resistances between the phases. The origin of the circuit elements is analysed using the principle of detailed balance and the master equation, as a generalization of different systems, such as intercalation batteries, electrochromics and solar cells based on nanostructured semiconductors, and systems with traps. The Butler-Volmer type equation for interfacial charge-transfer at the solid-solid interface is derived. The model is extended with the relaxation of charge in one of the phases providing an inductive behaviour at high frequency. Experimental results of intercalation of Li into nanostructured TiO₂, in the configuration TiO₂/room temperature ionic liquid/PEDOT show the inductive behaviour.