“FORWARD MIXING” MODEL: APPLICATION TO PULSED PLATE EXTRACTION COLUMN OPERATION IN THE EMULSION REGION

The “Forward Mixing” model has been applied to data obtained from a 22 cm diameter pulsed plate extraction column. Measurements of drop size distributions, dispersed phase hold-up and concentration profiles for two systems (toluene-acetone-water and n-butanol-succinic acid-water) of quite different properties were made with the column operating in the emulsion region. Generated drop size distribution function parameters, size-dependent slip velocities and mass transfer coefficients, and continuous phase axial dispersion coefficients were accurate in predicting dispersed phase hold-up and extraction efficiencies (or the related plug flow number of transfer units). These parameters were correlated with phase superficial velocities and pulse velocities. The influence of continuous phase axial dispersion was much greater than the influence of drop size variation, and was not accurately predicted by most previous tracer-based correlations. An inlet dispersed phase distributor was beneficial to the performance with the high interfacial tension system.     

Gas phase flow in bubble columns: A convective phenomenon

The axial dispersion model has been commonly used to describe gas phase flow in bubble columns. Scatter in dispersion coefficients reported to date may be a result of the misuse of the axial dispersion model when a convective model would be more appropriate. Using simple tests with radioactive tracer response curve moments, convective and dispersive behaviours are differentiated. A convective model is presented. The model fits both tracer response curves and mean gas velocity well in both the bubbly and churn turbulent flow regimes, and may be used as a technique to calculate bubble rise velocity distributions.     

用单纯形法同时确定轴向混合系数与传质系数

在内径5cm的梯形波空气脉冲柱内,测定了30%TBP(煤油)-Th(NO_3)_4-HNO_3(H_2O)体系在两种板段结构条件下钍的浓度剖面.用扩散模型描述萃取过程,用单纯形法直接由浓度剖面同时确定轴向混合系数、传质系数与真实传质单元高度.由此获得的计算的浓度剖面与实验测定的浓度剖面比较符合.结果表明:梯形波脉冲柱的轴向混合系数较小,用于补偿轴向混合的柱高约占表观传质单元高度的32—44%;用单纯形法寻优,对原始微分方程采用差分近似得两组线性代数方程组,对每组线性代数方程组用追赶法直接解,两组方程之间用迭代法,程序简单,收敛较快.在PDP11/23小型计算机上,约1分钟即算得一组结果.     

On the equivalence of hermite and power moments in the construction of dispersion approximations

The m + 1 dispersion coefficients, which define the mth dispersion approximation to the transverse average of the solution of the convective diffusion equation, can be gotten on exacting equality of the first m + 1 moments of the solution and its approximation. We establish the equivalence of the Hermite and power moments in the construction of dispersion approximations.     

Exact Analysis of Field-Flow Fractionation

Abstract

A rigorous convective diffusion theory is formulated for the predictive modeling of field-flow fractionation (FFF) columns used for the separation of colloidal mixtures. The theory is developed for simulating the behavior of a colloid introduced into fluid in time-dependent flow in a parallel plate channel across which a transverse field is applied. The methodology of generalized dispersion theory is used to solve the model equations. The theoretical results show that the cross-sectional average concentration of the colloid satisfies a dispersion equation with time-dependent coefficients. The results of this work, in principle, are valid for all values of time since the introduction of the colloid. It is shown that these results asymptotically approach those of the nonequilibrium theory formulated by Giddings for large values of time.

Illustrative numerical results are obtained for the case of steady laminar flow and a uniform initial distribution. The behavior of the coefficients in the dispersion equation is explained on physical grounds. Of particular interest is the fact that at large values of the transverse Peclet number P, Taylor dispersion in the FFF column is very small. Under these conditions, axial molecular diffusion as well as Taylor dispersion in the connecting tubing could make a substantial contribution to the axial dispersion observed in practical FFF columns.

The theoretical predictions are compared with the experimental data of Caldwell et al. and Kesner et al. on electrical FFF columns. The comparisons indicate that the theory has potential in predicting the performance of such systems.     

Interpretation of mass transfer measurements in bubble columns considering dispersion of both phases

Mass transfer measurements in two bubble columns with an inner diameter of 100 resp. 140 mm with the systems air/water/carbon dioxide and nitrogen/n-propanol/carbon dioxide have been evaluated with the axial dispersion model. The dispersion coefficients of both phases have been determined in separate investigations. As the results revealed a strong influence of the liquid viscosity, additional dispersion coefficient measurements have been carried out with the system air/glycol. It could be shown that the liquid phase dispersion coefficient decreases with increasing viscosity while the gas phase dispersion coefficient increases with increasing liquid viscosity. Both coefficients are strongly dependent on the gas throughput and the column diameter. Using these coefficients, the mass transfer coefficients have been calculated by fitting the calculated concentration profile to the measured values and by splitting the volumetric mass transfer coefficient with the experimental value of the interfacial are a. The results agree best with a correlation of Calderbank and Moo-Young.     

Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions

One-dimensional models for porous catalysts and laminar flow tubular reactors with first-order reactions at the walls are developed. The method which is used replaces the transport equations containing the local concentrations with equations, similar to those used in the dispersion theory, for the concentrations averaged over the cross section. The main assumption is the approximation of the coefficients in the dispersion equations by those valid for a pulse of concentration introduced at time zero at the mouth of the pore or at the inlet of the tube.The effectiveness factor depends not only on the Thiele modulus but also on the length-to-radius ratio of the catalyst pore. The one-dimensional model for the catalytic tubular reactor provides a good approximation to the area-mean concentration for β (k_sR/D) up to 1·0. The comparison is made with an exact orthogonal expansion solution developed also in the paper. For larger values of β, say β ≈ 100, the discrepancies between the exact solution and the one-dimensional model may be as high as 50 per cent. Therefore, a generalized dispersion solution is presented to obtain more accurate predictions for β > 1·0.     

Numerical simulation of interphase mass transfer with the level set approach

A level set approach is applied for simulating the interphase mass transfer of single drops in immiscible liquid with resistance in both phases. The control volume formulation with the SIMPLEC (semi-implicit method for pressure-linked equations consistent) algorithm incorporated is used to solve the governing equations of incompressible two-phase flow with deformable free interface on a staggered Eulerian grid. The solution of convective diffusion equation for interphase mass transfer is decoupled with the momentum equations. Different spatial discretization schemes including the fifth-order WENO (weighted essentially nonoscillatory), second-order ENO (essentially nonoscillatory) and power-law schemes, are tested for the solution of mass transfer to or from single drops. The conjugate cases with different equilibrium distribution coefficients are simulated successfully with the transformation of concentrations, molecular diffusivities, mass transfer time and velocities. The predicted drop concentration, overall mass transfer coefficient and flow structure are compared with the reported experimental data of a typical extraction system, i.e., n-butanol-succinic acid-water, and good agreement is observed.     

Simulation of true moving bed adsorptive reactor: Detailed particle model and linear driving force approximations

A new true moving bed (TMB) adsorptive reactor model with a detailed particle approach is presented introducing the formulation of the mass balance for the solid phase in counter-current moving systems. The system studied here is the enzymatic inversion of sucrose into fructose and glucose and subsequent separation of glucose/fructose; the reaction occurs both in the outer fluid phase and inside particles. Model equations include film mass transfer, intra-particle diffusion resistance, axial dispersion for the outer fluid phase, plug flow of the solid phase and linear adsorption equilibrium of glucose/fructose. This new model is compared with previous LDF-type approximations for reactive systems and applied to pure separative TMB process. The numerical solution of model equations is obtained for transient and steady state with commercial and public domain packages (gPROMS and COLNEW). The influence of the particle size and reaction rate constant is analyzed in the (γ₂×γ₃) reactive/separation region.     

Axial dispersion in packed beds: The effect of particle size non-uniformities

The effect of particle size non-uniformities on axial dispersion coefficients during laminar liquid flow through packed beds has been studied. The investigations were carried out for binary mixtures of particles with diameters d₁ = 0.169 mm and d₁ = 0.360 mm.A generalized function to determine the increase of the axial dispersion coefficients in non-uniform beds relative to those obtained in uniform beds has been proposed.     

Numerical investigation of the evaporation of two-component droplets

A numerical model for the complete thermo-fluid-dynamic and phase-change transport processes of two-component hydrocarbon liquid droplets consisting of n-heptane, n-decane and mixture of the two in various compositions is presented and validated against experimental data. The Navier-Stokes equations are solved numerically together with the VOF methodology for tracking the droplet interface, using an adaptive local grid refinement technique. The energy and concentration equations inside the liquid and the gaseous phases for both liquid species and their vapor components are additionally solved, coupled together with a model predicting the local vaporization rate at the cells forming the interface between the liquid and the surrounding gas. The model is validated against experimental data available for droplets suspended on a small diameter pipe in a hot air environment under convective flow conditions; these refer to droplet’s surface temperature and size regression with time. An extended investigation of the flow field is presented along with the temperature and concentration fields. The equilibrium position of droplets is estimated together with the deformation process of the droplet. Finally, extensive parametric studies are presented revealing the nature of multi-component droplet evaporation on the details of the flow, the temperature and concentration fields.     

Solid-liquid mass transfer in a gas-liquid-solid fluidized bed

Solid-liquid mass transfer in co-current two- and three-phase fluidized beds of water, air and benzoic acid pellets is studied. An axial dispersion model is used to describe the liquid flow when evaluating the solid-liquid mass transfer. The axial concentration profile of benzoic acid in the liquid is compared to that obtained experimentally and is found to be accurate. Three-phase fluidized bed solid-liquid mass-transfer coefficients are higher than the corresponding two-phase bed coefficients. The mass-transfer coefficient increases with increasing gas rate and is independent of liquid rate over the entire range studied. The mass-transfer coefficient also appears to be dependent on particle size, but only at high gas rates. At low or zero gas rates, k is nearly independent of particle size. A generalized correlation is developed which accurately and conveniently predicts the mass transfer in both two- and three-phase fluidized beds. Comparison to the solid-liquid mass-transfer characteristics of slurry bubble columns is also performed.     

High flux,single solute mass transfer in spherical rigid drops

High flux (i.e. high solute concentrations) mass transfer in spherical, rigid drops has been studied for the case of a single transferring solute. Model equations have been solved for the cases of (a) infinite and (b) finite continuous phase mass-transfer coefficients—or constant, and variable, interface compositions.The partial differential equations were solved to illustrate the influences of high solute concentrations and the continuous phase mass-transfer coefficients. In fact, the model diffusion equation resulting from a differential solute balance remains unchanged from the well-known low flux equation. It is only at the boundary, where the convective flux contribution must be taken into account, as it increases the total flux into (or out of) the drop as solute concentrations increase. Extraction efficiencies were also calculated. These proved relatively insensitive to the concentration level because the solute convective flux contribution simply increases the drop size. The diffusive flux into the drop interior changes little with the concentration level, so resulting in the small differences in the extraction efficiencies.     

Modelling the liquid-phase adsorption in packed beds at low Reynolds numbers: An improved hydrodynamic model

A hydrodynamic model including only one parameter (λ_O) for the prediction of both axial dispersion and external mass transfer in fixed-bed adsorbers at low Reynolds numbers (creeping flow regime) has been developed. The theoretical analysis is based on the application of the (two-dimensional) uniform dispersion model originally proposed by Bischoff and Levenspiel [1962a. Fluid dispersion—generalization and comparison of mathematical models—I. Generalization of models. Chemical Engineering Science 17, 245-255] to the representative capillary of a tube bundle model for describing the flow and mixing behaviour in packed beds. The combination of this model with the relationship between longitudinal and radial dispersion leads to the definition of the sole hydrodynamic parameter λ_O (one-parameter hydrodynamic model). Furthermore, the detailed investigation reveals that the one-parameter concept may be utilized for the application of the (one-dimensional) axial dispersed plug flow model as well. The functional dependence of the parameter λ_O on the flow conditions is elaborated from axial dispersion measurements. Both the new (one-parameter) hydrodynamic model and the classical model including axial dispersion and external mass transfer coefficients (two-parameter model) are utilized to simulate the breakthrough curves for the adsorption of naphthalene onto silica gel. This simulation study reveals that only the one-parameter hydrodynamic model is able to predict the adsorber dynamics over a large range of flow rates.     

一个气体在液体中溶解度的关联式

基于两个假设,本文提出了一个气体在液体中的溶解度随温度、压力变化的关联式。用该式关联H_2-H_2O、N_2-H_2O、He-H_2O、C_2H_5-H_2O体系以及外推至低压,都得到满意的结果。     

A study on flow characteristics in a rotating screen blade extraction column

Studies have been carried out on the axial dispersion in the continuous phase and the hold-up of dispersed phase in a rotating screen-blade extraction column, by employing an axial dispersion model. Experiments on both single and two phase operations have been conducted with the plate spacing, the mesh size, the impeller speed (RPM), and the superficial velocities as the system parameters. By regression analysis of experimental data, empirical equations correlating the dispersion coefficients and the fractional hold-up of dispersed phase with the system parameters were obtained.     

Computational techniques for liquid-liquid extraction column model parameter estimation,using the forward mixing model

Here is presented the first step toward the practical application of a model of liquid-liquid extraction column performance which includes the influence of drop size distribution, or of ‘forward mixing’. The theory, previously developed and described, has been used successfully to obtain model parameter values from experimental extraction data, including drop size distributions and solute concentration profiles. The presence of a significant settling zone height complicates the theory and poses difficulties. These were overcome by the reduction of the settling zone height to an insignificant level. Values of the continuous phase mass transfer, and axial dispersion, coefficients for an assumed (Handlos-Baron) drop-side model are reported. The overall mass transfer coefficients are confirmed to increase with drop size.     

Mass transfer with axial diffusion and chemical reaction

Mass transfer with axial diffusion and chemical reaction is studied for three different cases. The first one is plug flow mass transfer in semi-infinite region. The second one is laminar flow mass transfer in a semi-infinite region. The third one is laminar flow convective diffusion in the region of x ≤ 0 while mass transfer with chemical reaction in the region of x ≥ 0. The first problem is solved in terms of Bessel functions while the second in terms of confluent hypergeometric functions. In order to solve the third problem, orthonormal functions are constructed by linear combination of the eigenfunctions of the two semi-infinite regions respectively. The series expansion coefficients of the solutions are determined by solving a set of forty simultaneous equations obtained through matching boundary conditions at x = 0.Solutions are generally obtained for low Peclet number (Pe = 1, 5, 10) and high reaction rate parameters. Effects of axial diffusion and chemical reaction on the concentration field are discussed.     

Modeling of Size Exclusion Parametric Pumping

Abstract

Sephadex G and Biogel P, well-known gels in size exclusion chromatography, show a large change in their elution behaviors as temperature changes. These phenomena were exploited to separate the binary model solutes Blue Dextran 2000 and nickel nitrate. A series of batch size exclusion parametric pumping experiments in the direct thermal mode was carried out previously to separate binary mixtures. In this paper the experimental separations in batch size exclusion parametric pumping are compared with the predictions of both a local equilibrium model and a dispersion model. The dispersion model, which includes an axial dispersion term, gives a better fit of the experimental data. Experimentally obtained adsorption isotherms are used in the dispersion model to predict the separation performance of nickel nitrate in the Sephadex gel column. The resulting simultaneous partial differential equations are solved by quasilinearization of the equations followed by numerical integration. For both models, all parameters were estimated independently.