Analytic Solution of Diffusion‐Reaction in Spherical Porous Catalyst

The model for coupled diffusion and nt^h‐order reaction in a spherical catalyst pellet, which is a strong nonlinear differential equation with boundary values, is solved using the Adomian decomposition method. The explicit solutions for concentration profiles and effectiveness factors, for the model equation, are obtained and compared with the solutions obtained by a finite difference numerical method. For reaction orders less than 1.0, the Adomian polynomial series solution with 4 terms gives comparable results to those obtained with a finite difference numerical procedure. For reaction orders greater than 1.0, series solutions with more than 6 terms, are required for good accuracy.     

Unsteady mass transfer around axisymmetric drops of revolution in variable diffusion coefficient liquids

Unsteady mass transfer in the continuous phase around axisymmetric drops of revolution at high Peclet numbers has been theoretically studied. The liquid is a binary system, having a variable diffusion coefficient, which depends on the solute concentration. The solution to the problem was obtained by extending the theory of Favelukis and Mudunuri, developed for a constant diffusion coefficient liquid. The procedure consists of transforming the differential mass balance, for a binary system, into a partial differential equation which has an analytical solution, and an ordinary differential equation that needs to be solved numerically. Solutions to a large number of problems can be immediately obtained with the only requirements being the shape of the drop, the tangential velocity at the surface of the drop and an expression for the variable diffusion coefficient liquid. An approximate analytical solution is also suggested which is in excellent agreement with the numerical results.     

伴有二级不可逆反应的非牛顿幂律流体薄膜流中化学吸收的研究

本文对伴有二级不可逆反应的非牛顿幂律流体降膜流中的吸收过程进行了研究,提出扩散反应方程和它的解,并得到实验验证.     

Mass transfer of sodium chloride in simulated crystallizing pond during the rainfall period

The mass transfer equation describing the transport phenomena of sodium chloride is derived for the simulated crystallizing pond during the rainfall period. According to the theoretical and experimental results, the dissolution rate of NaCl is shown to be controlled by the combination of salt transport including the diffusion and convection of solute in solution and surface dissolution reaction including the desorption and surface diffusion of solute. It is found that both the dissolution and convective mass transfer coefficients are proportional to the total rainfall quantity, and inversely proportional to the brine depth. The variation of dissolution mass transfer coefficient to the total rainfall quantity and brine depth is correlated from the experimental results. Correlation of the convective mass transfer coefficient is also obtained by using the numerical solution of the mass transfer equation.     

海藻酸纤维对水溶液中Fe~(3+)的吸附动力学研究

以海藻酸纤维作为吸附材料研究了纤维对水溶液中Fe3+的吸附性能,用准一级动力学方程、准二级动力学方程、Elovich方程、粒子扩散方程、双常数方程和指数函数方程数学模型对不同Fe3+浓度、吸附时间的吸附曲线进行分析,研究了其吸附的动力学机理。结果表明:在研究的浓度和条件范围内,准二级动力学模型、Elovich方程和双常数方程的拟合度较好,模拟值与实验值吻合较好,说明海藻酸纤维吸附Fe3+是复杂的非均相扩散的化学吸附过程。粒子扩散方程表明颗粒内扩散不是控制海藻酸纤维吸附Fe3+过程的唯一步骤,而是由膜扩散和颗粒内扩散联合控制。     

Fick's First Law Correction by an exact solution of the boltzmann equation

It is shown that there exists a dominant solution of the Boltzmann equation for an artificial model gas in terms of a relation for the diffusion flux differing from the regular formulation of Fick's first law. Its mathematical structure corresponds to the Cattaneo-formulation based on heuristics of the Fourier heat flux vector. This dominant solution allows a realistic estimation of a diffusion relaxation time and serves as the basis for valuable information concerning two actual problems of hypersonic gas dynamics: first, a useful relation for the calculation of collision integrals for diffusion coefficients, especially at high temperatures; second, a new type of a chemical reaction equation, which is particularly relevant in very fast relaxation processes encountered in the chemistry of fuels.     

Diffusion in porous media of a random structure

The present paper deals with diffusion in a porous solid, which is considered as a discrete random medium composed of random structural elements (pores) chaotically connected with each other. Each structural element is characterized by the passage time distribution function calculated with taking into account the correlations of flows passing through the neighboring pores. The analysis of a nonstationary diffusion process, after transition to the limit of times much exceeding the mean passage time of a single pore, enables to obtain general expressions connecting the effective diffusivity in a porous medium with the statistical parameters of structural elements. At the same time the values are calculated which characterize the deviation of the concentration distribution in a porous medium from the gaussian one obtained as a solution of the quasi-homogeneous diffusion equation. To make the general expressions more concrete it is necessary to introduce a model of structural elements (pores) which determines the passage time distribution. The models of straight and corrugated pores considered yield expressions for the effective diffusion coefficient in the Fick and Knudsen regions. A similar analysis of the nonstationary diffusion accompanied by the first order reaction allows to get the expression for the effective rate constant of the catalytic reaction in a porous medium.     

Limiting Cases of the Equations of the Wave Model of Longitudinal Dispersion Accompanied by a Chemical Reaction

The relationship between the wave (relaxation) and diffusion models of longitudinal dispersion accompanied by a chemical reaction in an apparatus is analyzed. Of greatest interest is the limiting form of the wave-model equations for a case where the relaxation time is relatively short as compared to the time of mass (heat) transfer through the apparatus. The limiting equation of the wave model corresponds to the equation of the diffusion model if the dispersion coefficient depends on the concentration of the desired substance in a certain way. Comparison is performed with other models and the exact (numerical) solution to the problem of the dispersion of the substance throughout a round pipe in which there is a Poiseuille flow accompanied by a first-order chemical reaction.     

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.     

二维颗粒反应的分数维模型

本文将分数维理论引入颗粒与流体的非均相反应系统,提出不规则颗粒二维反应的分数维模型.在假稳态条件下,对不规则颗粒的核收缩反应方程求得解析解.结果表明,当反应受膜扩散控制、灰层扩散控制和化学反应控制时,颗粒的分数维越高,反应总速率越快;而在膜扩散控制阶段,分数维对相对反应时间没有影响.     

聚乙烯醇水溶液中H_2O_2的催化分解反应动力学研究

研究了在聚乙烯醇 ( PVA)水溶液中 H2 O2 的催化分解反应动力学 ,讨论了 PVA浓度、反应温度、搅拌速率等对 H2 O2 分解反应的影响。结果表明由于小分子在 PVA水溶液中的扩散影响 ,致使 H2 O2 的分解反应动力学行为明显偏离经典的 H2 O2 分解反应动力学曲线 ,其偏离程度依赖于扩散的影响     

THE LAMINAR FLOW TUBULAR REACTOR WITH HOMOGENEOUS AND HETEROGENEOUS REACTIONS I. Integral Equations for Diverse Reaction Rate Regimes

Design equations for non-isothermal Laminar Flow Tubular Reactors (LFTRs) with homogeneous and heterogeneous - at the reactor wall - reactions with arbitrary kinetic equations have been satisfactorily treated transforming the original P.D.E. problem into a system of integral equations. The kernels of the integral operators are related to an eigenvalue problem which does not depend on the kinetic equations; this avoids repetitive computational effort in the treating of different reaction kinetics.

To render a more efficient numerical treatment and according to the governing reaction rate regime, modified expressions of the general solution were obtained was follows: (i) a solution with kernels depending only on the diffusion and convective times was obtained for a low reaction regime; (ii) another solution with kernels including the reaction lime, besides the diffusion and convective ones, was necessary for a fast reaction regime and (iii) the local quasi-steady-state approximation was obtained as limiting case of solution (ii) for a instantaneous reaction regime.     

THE LAMINAR FLOW TUBULAR REACTOR WITH HOMOGENEOUS AND HETEROGENEOUS REACTIONS I. Integral Equations for Diverse Reaction Rate Regimes

Design equations for non-isothermal Laminar Flow Tubular Reactors (LFTRs) with homogeneous and heterogeneous - at the reactor wall - reactions with arbitrary kinetic equations have been satisfactorily treated transforming the original P.D.E. problem into a system of integral equations. The kernels of the integral operators are related to an eigenvalue problem which does not depend on the kinetic equations; this avoids repetitive computational effort in the treating of different reaction kinetics.

To render a more efficient numerical treatment and according to the governing reaction rate regime, modified expressions of the general solution were obtained was follows: (i) a solution with kernels depending only on the diffusion and convective times was obtained for a low reaction regime; (ii) another solution with kernels including the reaction lime, besides the diffusion and convective ones, was necessary for a fast reaction regime and (iii) the local quasi-steady-state approximation was obtained as limiting case of solution (ii) for a instantaneous reaction regime.     

Self-diffusion in Solutions of Carboxylated Acrylic Polymers as Studied by Pulsed Field Gradient NMR. 1. Solvent Diffusion Studies

The diffusion constants of solvent in solutions of carboxylated acrylic random copolymers in isopropanol (IPA) have been investigated using the PFG NMR technique. The copolymers are composed of butyl methacrylate and methacrylic acid with a range of molar ratios (BMA/MAA 100/0 through to 60/40 in 10 mol% steps). It was found that the echo attenuation function depended on the diffusion time, and experiments with short and long diffusion times have been analysed separately. Two questions are addressed: the first concerns the influence of concentration on the polymer aggregation in solution, and the second concerns the influence of the BMA/MAA molar ratio on the diffusion constants of the solvent. The data obtained have been discussed using established physical models for diffusion. Using the model of Wang gave the relative mass of bound solvent per mass of polymer which increased with the mol% of MAA in the copolymer.     

Analytical Solution of Mass Transfer Equation with Interfacial Resistance in Food Drying

A simplification of boundary conditions of the mass transfer equation in food drying was obtained. An average distribution constant between food and air humidity was introduced. With this simplification and analytical solution for the mass transter equation with interfacial resistance was; found. and a medihcd Bi number is proposed: Thc analytical solution for infinite flat slab was comparcd to the numcrical solution without the simplification. The analytical solution showed a good approximation to thc numerical solution. The equation oblained from thc analytical solution predicted ha1 for modified Bi numbers greater han 30. the drying curve bccamcs indcpcndcnt o f thc boundary conditions, which means that drying is diffusion controlled. Experimental drying curves of carrots were obtained at modified Bi numbers between 10 and 60. The crperimenwl drying cumes did not show statistical diffcrencc for modified Bi numbcrs grcater than 40, which approaches to the same limit predicted by the analytical solution.     

离子交换分离磷矿脱镁废水中镁离子的动力学

用动边界模型积分方程描述了001′7树脂脱除磷矿脱镁废水中的Mg2+的有限浴交换动力学，得到了动力学总方程. Mg2+的交换过程属颗粒扩散控制，表观活化能为24.11 kJ/mol，[Mg2+]的表观反应级数为1.0.     

Study of impedance model and water transport behavior of modified solvent-free epoxy anti-corrosion coating by EIS II

This work studied the evolution of an impedance model of 60 μm modified solvent-free epoxy anti-corrosion coating on a Q235 steel surface in 3.5% NaCl solution using electrochemical impedance spectroscopy. The electrochemical process of the system was divided into five stages. During the early stage of immersion, water absorption occurred mainly at the coating, coating resistance decreased, and coating impedance deviated from purely capacitive characteristics. The corrosion reaction started after water permeated into the metal/coating interface. During the mid-immersion stage, the electrochemical reaction at the coating/metal interface was controlled by semi-infinite diffusion of corrosion products due to the barrier effect of the coating. The types of corrosion product diffusion gradually became finite layer diffusion and barrier layer diffusion with the clogging of coating pores. Logarithm of coating capacitance and the square root of time showed a linear relationship in the early immersion stage, which was a typical characteristic of Fick’s diffusion. Afterwards, increase in the coating capacitance slowed down, and a non-Fickian diffusion process occurred, which presented the two-stage absorption characteristics. Water diffusion coefficient in coating was calculated to be 2.95 × 10^?10 cm²/s, while volume fraction and total water absorption at saturation of coating were 2.3% and 185 μg, respectively, indicating good water resistance and protective properties of the coating. The water kinetic equation in the coating which can reflect the whole Fick diffusion process of water and containing time and location variables was obtained via the Fick diffusion equation and three-dimensional images of water distribution in the coating were drawn by transforming the dynamic equation into programs.     

Diffusion of Degradation Products in Ceramic Moldings during Pyrolysis: Effect of Geometry

A general equation applicable to a sphere, an infinite plate, and an infinite cylinder is used to calculate the transient diffusion of degradation product in a ceramic molding during pyrolysis. The solution includes the concentration-dependent diffusion coefficient for diffusion of monomer in its parent polymer which forms the continuous phase in the ceramic in the as-molded state. In the early stages of pyrolysis, before continuous porosity has developed, the model searches for the critical heating rate above which defects are produced. Reasonable agreement is obtained with experimentally measured critical heating rates for 3-mm-thick plates.     

Solubility and diffusion coefficient of antioxidants in polyethylene

As an aid in assessing the ability of antioxidant additives to persist in polymers and thus remain effective in protecting against oxidation, the solubility and diffusion coefficient of two antioxidants in branched polyethylene have been determined in this work. A method was developed for this purpose by which the diffusion coefficient and solubility could be determined simultaneously. The method consists of analyzing the concentration profile across a stack of polyethylene sheets through which the antioxidant was allowed to diffuse. The concentration of antioxidants in polyethylene was determined by a thermogravimetric technique which relies directly on the ability of the additives to suppress oxidation reaction. The diffusion coefficients determined showed excellent agreement with values in the literature which were obtained by a radiotracer method. The solubility of the antioxidants in three normal hydrocarbon solvents of varying molecular sizes was also determined by a conventional technique at various temperatures and found to correlate well with their solubility in polyethylene determined by the diffusion method. In particular, the dependence of the solubility on the size of solvent molecules and on temperature agrees well with an equation derived on the basis of the regular solution theory of liquid mixtures.