THE EXACT PENETRATION MODEL OF DIFFUSION IN MULTICOMPONENT IDEAL GAS MIXTURES. ANALYTICAL AND NUMERICAL SOLUTIONS

An exact analytical solution is derived for the penetration model of diffusion in multicomponent ideal gas mixtures at constant pressure and temperature. It takes the form of a matrizant solution to the continuity and Maxwell-Stefan equations transformed by introduction of a similarity variable, and includes as special cases the corresponding binary and linearized theory solutions

Direct numerical implementation of the analytical solution is computationally inefficient, but an alternative finite-difference algorithm is developed in which the transformed equations are solved by Euler's method with a simple shooting technique. Sample calculations are reported for two ternary diffusion problems

It is concluded on the basis of the theoretical and numerical results that the linearized theory predictions should provide an excellent approximation to the exact solution of the penetration model.     

THE EXACT PENETRATION MODEL OF DIFFUSION IN MULTICOMPONENT IDEAL GAS MIXTURES. ANALYTICAL AND NUMERICAL SOLUTIONS

An exact analytical solution is derived for the penetration model of diffusion in multicomponent ideal gas mixtures at constant pressure and temperature. It takes the form of a matrizant solution to the continuity and Maxwell-Stefan equations transformed by introduction of a similarity variable, and includes as special cases the corresponding binary and linearized theory solutions

Direct numerical implementation of the analytical solution is computationally inefficient, but an alternative finite-difference algorithm is developed in which the transformed equations are solved by Euler's method with a simple shooting technique. Sample calculations are reported for two ternary diffusion problems

It is concluded on the basis of the theoretical and numerical results that the linearized theory predictions should provide an excellent approximation to the exact solution of the penetration model.     

Stefan-Maxwell mass transport

Stefan-Maxwell multicomponent transport equations have a lot of appeal because of their symmetry and because the diffusion coefficients are independent of composition for ideal gases. A simple numerical solution method, which can easily be extended to more components, is illustrated for the popular chemical-engineering film, penetration, and boundary-layer models with a particular example of evaporation of acetone and methanol through air. Extension of each model to other geometries and chemical systems is discussed. In the boundary-layer model there is no need to assume that both total concentration and density are constant.     

Modelling of microporous diffusion of n-paraffins in zeolite 5A

Sorptive liquid-phase diffusion of two n-paraffins, C₁₀H₂₂ and C₁₁H₂₄, dissolved in isooctane, onto micropore of 5A zeolite was studied to assess multicomponent diffusion and competitive effects. Diffusion coefficients for adsorbing components are determined from experimental batch reactor data. The experimental data indicate that diffusion through the microporous zeolite crystals is the primary diffusional resistance. A mathematical model of the rate of adsorption of a solute from a liquid by micropore adsorbent in a batch system was developed. The equation describing the mass transport by diffusion in a micropore adsorbent has been solved in order to obtain theoretical uptake curves for systems when the adsorption equilibrium isotherm is the favourable and nonlinear one. A computer simulation of the microporous diffusion is performed by use of the ISIM-Interactive Simulation Language. The effect of main term and cross-term coefficients of micropore diffusion for the system considered is investigated.     

Drying of Wheat Based on the Non-Equilibrium Thermodynamics: A Numerical Study

A new mathematical model to describe simultaneous heat and mass (liquid and vapor) transfer and shrinkage during drying of capillary-porous bodies with particular reference to prolate spheroid solid is presented. As an application, the methodology was used to predict drying of soft red winter wheat (Arthur). The mathematical model was based on the nonequilibrium thermodynamics considering variable transport coefficients and convective boundary conditions at the surface of the solid. All the partial differential equations presented in the model have been written in prolate spheroidal coordinates and solved numerically by a finite-volume method using implicit fully formulation. Results of the drying and heating kinetics and moisture content and temperature distributions in a wheat kernel during drying process are presented and analyzed. The methodology allows verification of the heat, liquid, and vapor fluxes, taking into account the thermal and hydrical gradients inside the grain.     

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.     

Features of Multicomponent Mass Transfer in Gas Mixtures Containing Hydrocarbon Components

The features characterizing the transition from a diffusion process to the concentration gravitational convection in multicomponent gas mixtures containing hydrocarbon components are analyzed. It is demonstrated that this transition is possible due to the difference in interdiffusion coefficients of components, initial concentration of components, and the pressure of the experiment. A mathematical model for describing the transition from the diffusion process to the concentration gravitational convection in multicomponent gas mixtures is considered. The key feature indicating the transfer from the diffusion regime to the convective one is the curvature of an isoline of the heavy component of the gas mixture.     

Diffusion Transport Vector for Multicomponent Gas Separation in Ultracentrifuge

Abstract

Simplified diffusion transport vector expressions for multicomponent gas mixture mass transfer analysis have been derived. Approximate relations for the general diffusion coefficients for multicomponent gas separation in an ultracentrifuge for both isotope and nonisotope mixtures are developed. Taking into account that diffusion coefficients matrices are diagonally dominant, a simple relationship for diffusion transport vectors for the case of isotope separation is derived. It is shown that the relative inaccuracies in separative power and separation factors calculation are less than 1-2%. Analogous relationships for diffusion coefficients for the separation of a nonisotope mixture containing small admixtures in the main gas are suggested. These relationships can be used when the total mole fraction of the admixtures is less than 5%.     

Modeling of Vacuum Desorption in Freeze-Drying Process

A mathematical model of multicomponent vacuum desorption, which occurs in vacuum freeze-drying process, was developed. In freeze-drying porous biomaterials and pharmaceuticals are considered and the vacuum freeze-drying process, especially the moisture desorption in its final stage, is investigated. In this article, the drying with conductive heating and constant contact surface temperature was considered. Pressure drop is taken into account in the model formulation but was neglected in process simulation because of thin material layers undergoing freeze-drying. Model equations were solved by numerical method of lines. Moisture content and temperature distributions within the drying material were predicted from the model as a function of drying time.     

Mechanism of bubble formation in the drying of polymer films

A common problem in making thin polymer films by solution processing is the undesirable formation of bubbles during the drying process. These bubbles appear well below the boiling point of the solvent. Experience indicates, however, that the degassing of the polymer solutions reduces bubble formation, indicating a relationship with the presence of air. This work is based on a hypothesis that if the solubility of air in the polymer solution increases with solvent concentration, then the solution can become supersaturated with air as the concentration of the solvent is reduced during the drying process. To test this hypothesis the system poly(vinyl acetate)‐toluene‐nitrogen was chosen. Previously published data on the solubility and diffusion of nitrogen in the polymer‐solvent system were used. Different diffusion models based on the friction coefficients and free‐volume model were then used to correlate the diffusivity data so that the diffusion behavior of the ternary system can be predicted over a broad range of conditions. Finally, the thermodynamic and diffusivity correlations were incorporated into a multicomponent drying model which included main and cross‐diffusion terms to predict saturation behavior in the polymer solution during the drying process. The model without the cross‐diffusion terms represents the ideal system in which the diffusion of one component does not affect the diffusion of others. The drying model did not predict supersaturation of nitrogen when cross‐diffusion terms were neglected. Supersaturation of nitrogen was predicted, however, when the cross‐diffusion terms are included. Therefore, the cross‐diffusion terms in the mass transfer model are essential for the development of nitrogen supersaturation. Also different diffusion models based on the friction coefficients led to qualitatively similar predictions for the supersaturation of nitrogen. The simulation's results supported our experimental observations regarding bubble formation. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Effects of non-ideal adsorption equilibria of gas mixtures on column dynamics

A theoretical study has been made for simulating the dynamic behavior of non-ideal gas mixtures in an isothermal fixed-bed adsorber. A mathematical model was developed which takes into account the non-ideality of adsorbable species on the adsorbed phase under equilibrium. The model is based on both the real adsorbed solution theory (RAST), which incorporates the activity coefficients in the multicomponent isotherm equations to account for the deviations from ideality, and the linear driving force (LDF) model for representing diffusion resistance inside the adsorbent particles. To describe the effect of non-ideal adsorption equilibrium of gas mixtures on the breakthrough curves, we considered several model mixtures of binary and ternary components which exhibit non-ideal behavior with azeotropic crossovers in the composition domains at equilibrium. Sample calculations of a fixed-bed adsorption were done with various inlet gas compositions of binary and ternary mixtures, respectively, at a fixed total concentration. From the calculation results, it was shown that the order of breakthrough curves could be changed at a certain value of inlet gas composition ratio. This result implies that the dynamic behaviors of fixed-bed adsorption are greatly influenced by multicomponent equilibrium models. Furthermore, the reversal phenomenon of breakthrough curves could not be simulated by the ideal adsorbed solution theory (IAST).     

Drying of Lentil Including Shrinkage: A Numerical Simulation

This article presents a theoretical study about drying of lentil including shrinkage. The two-dimensional unsteady-state diffusion modeling written in the oblate spheroidal coordinates system considers the volume variation effect, convective boundary condition at the surface of the solid, and variable thermo-physical properties. The governing equation was discretized using the finite-volume method and the linear equations system was solved by Gauss-Siedel iterative method. To validate the model, numerical results of the average moisture content were compared with experimental data from eight experiments and a good agreement was obtained. The diffusion coefficients for all drying experiments are determined using the least square error technique.     

ANALYSIS OF HEAT AND MASS TRANSFER DURING DRYING OF PAPER/BOARD

A mathematical model has been successfully developed to study the heat and mass transfer process during paper drying. This model takes into account the consective transfer of vapor and liquid apart from the known transport mechanisms of capillary flow of liquid, diffusion, vaporization-condensation, and heat conduction. The partial differential equations describing temperature, saturation and pressure change within the web during drying with associated boimdary conditions and initial conditions were solved using finite difference method. The model predictions show that during the drying process the web can be conveniently divided into three different zones, namely dry zone, wet zone and an intermediate zone. The movement of liquid and vapor in opposite directions in the intermediate zone is similar to the action of a heat pipe. Also, as drying proceeds the location of the intermediate zone and hence the heat pipe advances progressively through the thickness of the web.     

Modeling of Vacuum Desorption in Freeze-Drying Process

Abstract

A mathematical model of multicomponent vacuum desorption, which occurs in vacuum freeze-drying process, was developed. In freeze-drying porous biomaterials and pharmaceuticals are considered and the vacuum freeze-drying process, especially the moisture desorption in its final stage, is investigated. In this article, the drying with conductive heating and constant contact surface temperature was considered. Pressure drop is taken into account in the model formulation but was neglected in process simulation because of thin material layers undergoing freeze-drying. Model equations were solved by numerical method of lines. Moisture content and temperature distributions within the drying material were predicted from the model as a function of drying time.     

多孔催化剂效率因子的多组分扩散模型 (Ⅰ)多组分扩散模型及数值计算方法

本文按多孔催化剂内多组分分子扩散及努森扩散联合过程的特性,导出了描述等温球形催化剂内单一反应的反应物和产物浓度分布的二阶非线性常微分方程组.微分方程中各个反应组分的浓度梯度强烈相互耦合.在计算机上用打靶法求解此常微分方程组提出的两点边值问题,即可求得各个反应组分的浓度分布和催化剂效率因子的数值解.由于反应组分的有效扩散系数间的差异,催化剂内反应组分的组成变化之间的关系不再符合反应的化学计量关系.当内扩散过程的阻滞作用严重时,催化剂颗粒内会出现“死区”,死区内反应组分的浓度趋近于平衡浓度,实际上不再继续进行反应.存在死区时,使用此数学模型及解题方法可求出死区的半径及此情况下催化剂的效率因子.工业上常用的圆柱状催化剂按照相等比外表面积的球体处理.     

ANALYSIS OF HEAT AND MASS TRANSFER DURING DRYING OF PAPER/BOARD

ABSTRACT

A mathematical model has been successfully developed to study the heat and mass transfer process during paper drying. This model takes into account the consective transfer of vapor and liquid apart from the known transport mechanisms of capillary flow of liquid, diffusion, vaporization-condensation, and heat conduction. The partial differential equations describing temperature, saturation and pressure change within the web during drying with associated boimdary conditions and initial conditions were solved using finite difference method. The model predictions show that during the drying process the web can be conveniently divided into three different zones, namely dry zone, wet zone and an intermediate zone. The movement of liquid and vapor in opposite directions in the intermediate zone is similar to the action of a heat pipe. Also, as drying proceeds the location of the intermediate zone and hence the heat pipe advances progressively through the thickness of the web.     

ISOTHERMAL AND NON-ISOTHERMAL MULTICOMPONENT ADSORPTION KINETICS

The model equations in the relaxation form for the multicomponent kinetics of isothermal and non-isothermal adsorption, taking into account all major distinctive features of the interphase heat and mass exchange inside porous grains and at their surface (see points 1 to 4 below) for P (“pore”) and S (“solid”) models of mass transfer within porous grains of the adsorbent, have been obtained.

First for isothermal and non-isothermal kinetics in the mixed kinetics region of mass and heat exchange in the absence natural mutual diffusion and natural thermal-diffusion the essential influence effective mutual diffusion and effective thermal-diffusion is shown.

Recommendations on the use of model equations of adsorption kinetics for describing isothermal and non-isothermal adsorption dynamics of multicomponent mixtures in the inner-diffusion and mixed (outer- and inner-diffusion) kinetic region of heat and mass exchange are made.     

Drying of Lentil Including Shrinkage: A Numerical Simulation

Abstract

This article presents a theoretical study about drying of lentil including shrinkage. The two-dimensional unsteady-state diffusion modeling written in the oblate spheroidal coordinates system considers the volume variation effect, convective boundary condition at the surface of the solid, and variable thermo-physical properties. The governing equation was discretized using the finite-volume method and the linear equations system was solved by Gauss-Siedel iterative method. To validate the model, numerical results of the average moisture content were compared with experimental data from eight experiments and a good agreement was obtained. The diffusion coefficients for all drying experiments are determined using the least square error technique.     

Generalized linear driving force approximation for adsorption of multicomponent mixtures

This work deals with the development of efficient numerical tools for the solution of diffusion dominated parabolic partial differential equations. This study finds its application in the modeling of the intraparticle mass balance necessary for describing dynamic adsorption processes.The orthogonal collocation method is proposed as the basis for developing generalized linear driving force approximations for adsorption and desorption of multicomponent mixtures in a single particle, independently of the mass transport model adopted. Based on this approach, it was possible to derive some approximations previously obtained from the analytical series of the homogeneous diffusion equation.Orthogonal collocation is also compared to other numerical methods found in the literature, using both the homogeneous diffusion and the dusty-gas mass transport models. The results show that orthogonal collocation is the more consistent approach.