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.     

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%.     

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.     

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.     

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.     

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.     

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.     

Convective drying of osmotically dehydrated apples described by three-dimensional numerical solution of the diffusion equation with analysis of water effective diffusivity spatial distribution

Abstract

This study presents two liquid diffusion models to represent the convective drying of apple, osmotically dehydrated in sucrose solution, cut into parallelepiped-shaped pieces. Model 1 considered water diffusivity and the volume of the slices with constant values. Model 2 considered water effective diffusivity and the dimensions of the slices as variable. The numerical solution of the three-dimensional diffusion equation in Cartesian coordinates was obtained through the finite volume method, with a fully implicit formulation and boundary condition of the third kind. Process parameters were estimated by an optimizer using experimental data. A spatial distribution analysis was carried out for water effective diffusivity and moisture content in the apple slices. The results showed that the concentration of the osmotic solution used in the pretreatment influenced the drying process and that the mathematical model that considered a variable diffusivity and shrinkage was more suitable to describe the experimental data.     

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.     

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.     

Mathematical Modeling of Drying of Liquid/Solid Slurries in Steady State One-Dimensional Flow

ABSTRACT

A mathematical model of simultaneous mass, heat and momentum transfer for two-phase flow of a gas and a solid/liquid slurry was developed. The model was applied to calculation of the drying process of coal-water slurry droplets in a gas medium in a steady one-dimensional flow. The model was based on the well-known two-stage drying process for slurry droplets. After the first period of drying, in which the evaporation rate is controlled by the gas phase resistance, the evaporating liquid diffuses through the porous shell (crust) and then, by convection, into the gas medium. Inside the dry external crust of the drop, a wet central core forms, which shrinks as evaporation proceeds. The temperature of the slurry droplet rises. The process ends when the temperature of the dry outer crust reaches the coal ignition temperature in the case of combustion or when the moisture of the particle reaches the final required moisture. The developed model was based on one-dimensional balance equations of mass, energy and momentum for the liquid/solid and gas phases. The system of governing equations was represented by first-order differential equations and solved simultaneously. The numerical solution of the governing equations was obtained using Gear's method. The model permitted calculation     

AN ANALYTICAL SOLUTION OF THE GRANULAR PRODUCT IN DEEP-BED FALLING RATE DRYING PROCESS

Abstract

According to analysis of drying characteristics of granular product, the paper describes analytical solutions on a single grain with two falling rale drying stages during corresponding period of deep-bed drying process. Changes of drying and physical parameters and their correlation were discussed. A mathematical model was developed and solved numerically. The application of this analytical solution is also given.     

A Sharp Drying Front Model

Abstract

The drying of materials is often described by nonlinear diffusion equations. Up to now the only way to solve these equations is by numerical simulations. Recently an analytic solution has been proposed for the drying problem. Based on this solution a sharp drying front model is presented. Measured moisture profiles during drying and the drying curve of gypsum are compared with approximate models.     

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.     

DRYING OF POLYMER SOLUTION

ABSTRACT

Drying process of polymer solution,which i s carryed out industrially by three shapes of the drying material,is investigated comprehensively. The three shapes are slab as various polymer films by film casting dryer, cylinder as various synthetic. fibers by solution spinning and sphere (drop) as particle or povder of various food materials and polymers by spray drying. These are analysed basically in common manner by silnul tanews solution of diffusion equations and heat balance equations with activity and concentration dependent diffusion coefficient of polymer-solvent system. The conceptions concerning the     

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     

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.