Simulations of mass transfer limited reaction in a moving droplet to study transport limited characteristics

Transport limited heterogeneous reactions with asymmetric transport rates in the non-reacting phase can exhibit an interesting switch in the concentrations of the reactants in the reacting phase from one limiting reactant to the other. This switch, called “cross-over” [Mchedlov-Petrossyan P.O., Khomenko G., Zimmerman W.B., 2003a. Nearly irreversible, fast heterogeneous reactions in premixed flow. Chemical Engineering Science 58, 3005-3023; Mchedlov-Petrossyan P.O., Zimmerman W.B., Khomenko G.A., 2003b. Fast binary reactions in a heterogeneous catalytic batch reactor. Chemical Engineering Science 58, 2691-2703], relates to the optimum design of the tubular reactor as all the reactants in the reacting phase are completely consumed at cross-over. The cross-over phenomenon, which has been studied by a number of researchers using phenomenological modelling, is investigated here by developing a distributed model using level-set simulations, in order to explore the possibility of the existence of cross-over in the frame of reference of a moving droplet. Cross-over occurs for a droplet moving due to buoyancy with asymmetric transfer rates of the reactants in the non-reacting phase and an instantaneous reaction occurring inside the droplet (reacting phase). The cross-over length obtained using the level-set simulation is found to be within 0.7-8% of that obtained using the phenomenological model. Computational experiments are performed by varying the ratios of the initial concentrations of the reactants and the transfer rates of the reactants, in order to obtain the parametric region for the existence of cross-over which is also compared with the theoretical prediction.     

The fast reaction asymptote of the coalescence-redispersion model

The coalescence-redispersion (CRD) model is examined in the fast chemistry limit for the single bimolecular reaction A+B M R and the series parallel reactions A+B M R;B+R M S occurring in a plug flow reactor with unmixed feed streams. For the single bimolecular reaction it is shown that in this limit the CRD model is asymptotically equivalent to the 3E fast closure as t M 0. For the series parallel reactions the CRD model predictions tend to be bracketed by the 3E slow and fast closure formulations. Representative results are presented for the variance decay of the individual reactants, and it is seen that even in the fast reaction limit where mixing is controlling, the shape of the feed stream PDFs has negligible influence on the progress of the reaction.     

THE FAST REACTION ASYMPTOTE OF THE COALESCENCE-REDISPERSION MODEL

The coalescence-redispersion (CRD) model is examined in the fast chemistry limit for the single bimolecular reaction A+B →R and the series parallel reactions A+B →R;B+R →S occurring in a plug flow reactor with unmixed feed streams. For the single bimolecular reaction it is shown that in this limit the CRD model is asymptotically equivalent to the 3E fast closure as t →0. For the series parallel reactions the CRD model predictions tend to be bracketed by the 3E slow and fast closure formulations. Representative results are presented for the variance decay of the individual reactants, and it is seen that even in the fast reaction limit where mixing is controlling, the shape of the feed stream PDFs has negligible influence on the progress of the reaction.     

Molecular weight determination of high polymers by means of vapor pressure osmometry and the solute dependence of the constant of calibration

Data are presented to show that the calibration constant of the vapor pressure osmometer is in fact not a constant, but rather depends on the nature of the solute. It is shown that the assumption of the constancy of the calibration constant is particularly severe when comparing low molecular weight standards and polymeric materials. A model is presented which visualizes either condensation or evaporation taking place at the solution drop surface, depending on the relative magnitudes of the concentration and Thermistor self-heating. In terms of this model, the solute dependence of the calibration constant is attributed to the formation of a diffusion-controlled surface concentration of the solution drop that differs from the concentration of the drop as a whole. Experimental evidence consistent with a diffusion-controlled surface layer is given. A method based on this model is given for operating the instrument so that the solute dependence of the calibration constant disappears. When the instrument is run in this manner, M _n determinations from vapor pressure and membrane osmometry are in significantly better agreement than when the instrument is operated as recommended by the manufacturer.     

Enhancement factors for gas absorption with reversible reaction

An approximate analytical expression is derived for the enhancement factor for absorption with a reversible chemical reaction which is second-order in each direction, with corresponding stoichiometry and equal diffusivities of reactants, using the Danckwerts model of mass transfer. The result agrees with simpler limiting expressions for the enhancement factors for irreversible reactions, reactions slow enough to give negligible depletion of reactants, and instantaneous reactions. The result agrees satisfactorily with the available corresponding data for numerical solutions of the differential equations for the Higbie model, with the exception of one case, where the deviation is probably due to an error in the numerical solution, or its graphical representation. No numerical solutions are available for comparison when concentrations of solute and reaction products in the bulk of the liquid are appreciable. The method seems applicable also to more complex kinetics.     

Experimental study of mass transfer limited reaction—Part II: Existence of cross-over phenomenon

In Part II, we extend the inverse methodology which is discussed in Part I to various experiments performed in a tubular reactor to infer mass transfer coefficients. Mass transfer coefficients, inferred from the concentrations of the reactants which are extracted from the absorbance of the reaction mixture using multicomponent spectrum analysis, are then used to solve the convection-diffusion-reaction equations for the concentrations of the reactants in the bulk phase and in the dispersed phase, to explore the possibility of cross-over for a mass transfer limited reaction. Experiments are performed incorporating the asymmetry in the transport rates of the premixed reactants, which is the potential reason for the existence of cross-over. The different mass transfer coefficients of the two premixed reactants indeed indicate a switch in the concentration of the reactants in the dispersed phase, which is termed as “cross-over”. The experimental results are further analysed by validating the theoretical criterion proposed by Mchedlov-Petrossyan et al. (2003, Chem. Eng. Sci. 58, 3005-3023 & 2691-2703) to obtain the parametric space for the existence of cross-over, in order to optimize the length of the tubular reactor.     

A novel reverse flow reactor coupling endothermic and exothermic reactions. Part I: comparison of reactor configurations for irreversible endothermic reactions

A new reactor concept is studied for highly endothermic heterogeneously catalysed gas phase reactions at high temperatures with rapid but reversible catalyst deactivation. The reactor concept aims to achieve an indirect coupling of energy necessary for endothermic reactions and energy released by exothermic reactions, without mixing of the endothermic and exothermic reactants, in closed-loop reverse flow operation. Periodic gas flow reversal incorporates regenerative heat exchange inside the reactor. The reactor concept is studied for the coupling between the non-oxidative propane dehydrogenation and methane combustion over a monolithic catalyst.Two different reactor configurations are considered: the sequential reactor configuration, where the endothermic and exothermic reactants are fed sequentially to the same catalyst bed acting as an energy repository and the simultaneous reactor configuration, where the endothermic and exothermic reactants are fed continuously to two different compartments directly exchanging energy. The dynamic reactor behaviour is studied by detailed simulation for both reactor configurations. Energy constraints, relating the endothermic and exothermic operating conditions, to achieve a cyclic steady state are discussed. Furthermore, it is indicated how the operating conditions should be matched in order to control the maximum temperature. Also, it is shown that for a single first order exothermic reaction the maximum dimensionless temperature in reverse flow reactors depends on a single dimensionless number. Finally, both reactor configurations are compared based on their operating conditions. It is shown that only in the sequential reactor configuration the endothermic inlet concentration can be optimised independently of the gas velocities at high throughput and maximum reaction coupling energy efficiency, by the choice of a proper switching scheme with inherently zero differential creep velocity and using the ratio of the cycle times.In this first part, both the propane dehydrogenation and the methane combustion have been considered as first order irreversible reactions. However, the propane dehydrogenation is an equilibrium reaction and the low exit temperatures resulting from the reverse flow concept entail considerable propane conversion losses. How this ‘back-conversion’ can be counteracted is discussed in part II Chemical Engineering Science, 57, (2002), 855-872.     

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.     

Theoretical Aspects Of The Kinetics Of Competitive Ozone Reactions In Water

Competition of direct and decomposition reactions that ozone can undergo in water when a given compound is present is studied according to concepts of absorption theories. The importance of different parameters, such as pH, rate constants, mass transfer coefficients and concentration of reactants, is considered to clarify which reaction prevails. Parameters like Hatta numbers (film theory) or diffusion and reaction times (surface renewal theories) are key factors to check the competition of these reactions.     

Kinetic and Adsorption Effects in the Polarographic Reduction of Chlorate Catalyzed by Molybdenum-Tungsten Mixtures

The polarographic reduction of chlorate catalyzed by molybdenum-tungsten mixtures is investigated. The reduction currents are limited by the rate of the chemical reactions in the bulk of the solution. The reactants are adsorbed on the mercury drop. Their rate of adsorption is fast compared to the rate of the chemical reactions.     

Effect of Viscosity on Membrane Fluxes in Cross-Flow Ultrafiltration

Abstract

For practical applications of ultrafiltration (UF), an estimation of membrane fluxes under various operational conditions is very important. This study analyzed concentration polarization (CP) as a coupled transport problem with concentration-dependent solute viscosity. Besides the effects of variable viscosity, the model includes the effects of solute osmotic pressure, solute rejection at the membrane surface, and the axial pressure drop. This provides a fundamental understanding of the effects of various operating parameters on concentration polarization and transmembrane flux. A finite-difference solution of the transport equations is presented to model the concentration polarization in a thin-channel UF system. Simulation results for ultrafiltration of Dextran T-70 show that concentration-dependent solute viscosity adversely affects the transmembrane flux and needs to be carefully considered in modeling concentration polarization in membrane filtration.     

New method for the determination of precipitation kinetics using a laminar jet reactor

In this paper a new experimental method for determining the kinetics of fast precipitation reactions is introduced. Use is made of a laminar jet reactor, which is also frequently applied to determine the kinetics of homogeneous gas-liquid reactions. The liquid containing one or more of the precipitating reactants passes a gas-filled reactor as a stagnant jet in which no mixing occurs. The remaining reactant needed for precipitation is supplied in gaseous form and causes the precipitation reaction to occur while it is diffusing into the jet. Hydrodynamics as well as transport phenomena are precisely known for this system, whereas agglomeration can be minimized by adjustment of the concentration of the solute supplied by the gas. The kinetics of the different crystallization steps can be determined by analyzing the size distribution of the produced particles. This new method is experimentally demonstrated for the precipitation of CuS using H₂S gas. The obtained data were successfully used to simulate a packed bed absorber in which H₂S is absorbed by a CuSO₄ solution.     

TAYLOR DISPERSION OF MULTICOMPONENT SOLUTES

Coupled transport of multicomponent solutes in globally continuous systems is considered in the framework of the Generalized Taylor dispersion theory. Coupling between transports of n different species at the local (or micro-) scale, is considered to result from first-order irreversible surface reactions occurring on the local space boundaries, or from the off-diagonal terms of the solute diffusivity matrices.

General expressions are obtained for the global effective (long-time) solute dispersion matrix cofficients: mean global scalar reactivity, velocity vector and dispersivity dyadic.

The effect of surface chemical reactions is to partition the matter between different solute constituents. This is manifested in a coupling of the global transport coefficients, which may be mathematically removed by a linear (canonic) transformation applied to the effective global transport equation. This type of coupling does not exist for inert solutes.

The second type of the global coupling is represented by the off-diagonal terms of the global velocity and dispersivity matrices. It exists for both reactive and inert solutes. This coupling stems from the convective dispersion process (dependence or the global velocity vector on the local space coordinate). Is shown to be irremovable from the global transport equation by any linear transformation via the solute partition matrix. In the canonic form of the global equation the irremovable coupling is manifested by the traceless parts of the global solute velocity matrix and the global solute dispersivity.

The solution scheme is illustrated by calculating the mean global diffusivity of a solute consisting of two components, transport of which is coupled at the microscale via the molecular diffusivity matrix. At the macroscale the coupling is shown to be represented by negative off-diagonal terms of the global diffusivity matrix,     

Vapor pressure osmometry: A model accounting for the solute dependence of the calibration constant

A model describing the operation of the vapor pressure osmometer has been developed to account for the solute dependence of the calibration constant. This model envisions the existence of a diffusion-controlled surface concentration that differs from the concentration of the drop as a whole. Three limiting cases approximated for the surface concentration depend on the relative magnitudes of thermistor self-heating and solution concentration. Quantitative predictions of the dependence of d(ΔR)/d(V²) (the variation of thermistor resistance difference with thermistor self-heating) on both solute molecular weight and solution concentration are in good agreement with experimental data.     

Model development for multicomponent mass transfer with rapid chemical reactions in a small drop

Multicomponent mass transfer accompanied by instantaneous chemical reactions in a small drop has been modeled and simulated for the case where two different solutes diffuse from a continuous phase into the drop and react rapidly with a third reactant in the drop. The computational results obtained by Galerkin’s finite element method are reported in terms of concentration profiles, the locations of reaction front, the cumulative mass flux, and the enhancement factor. The effects of physical parameters, such as diffusivities of the solutes and the reactant, the interfacial concentration of solutes, and the relative amount of the reactant, on the calculated quantities are discussed.     

Characterization and modelling of diffusion and reaction of low molecular weight reactants in molten polymer

A model reactive system was defined for studying experimentally and by simulation the competition between reaction and diffusion of two low molecular weight reactants, 2,3-epoxypropyl-phenylether (EPPE) and dipentylamine (DPA). Both reactants are miscible in a high-viscous molten polymer, poly(ethylene-co-vinyl acetate) (EVA). The comparison of the experimental rates of reaction for initially homogeneous samples and bi-layer unpremixed samples proved that the reaction was diffusion controlled. A kinetic model of the epoxy-amine reaction was coupled to mutual diffusion coefficients of reacting species in a transport model and the simulations were compared with experimental results. The diffusion/reaction process was finally related to typical mixing conditions encountered in reactive polymer processes. For the model reactive system, the simulations have established that actual mixing conditions with shear rate values encountered in polymer processing machines, were able to homogenize the system in less than 10 s. In other words, the reaction should no longer be controlled by molecular diffusion as soon as a relatively low intensity mixing is applied (shear rate > 10 s⁻¹).     

DIFFUSION LIMITATIONS IN ENZYME MIMICING POLYMER MEDIATED REACTIONS

Diffusional effects in reactions involving macromolecular substrates and reactants in heterogeneous media lead to unique features. We report investigations on the hydrolysis of 2-hydroxy ethyl methacrylate copolymers catalyzed by imidazole and poly (W-vinyl imidazole) selected as an enzyme mimicing polymer. This process has been mathematically modelled, and the predictions compared with experimental findings. The model framework is general for diffusion-reaction problems of this class. The work has also pragmatic relevance in the design of drug delivery systems, since the reaction takes place in pH ranges of physiological significance.     

A Study on Concentration Polarization in Ultrafiltration

Abstract

A finite-difference solution of coupled transport equations for momentum and solute continuity is presented to model the concentration polarization in a tubular ultrafiltration (UF) system. The model includes the effects of solute osmotic pressure and solute rejection at the membrane surface, axial pressure drop and resistance of the gel layer. This provides a fundamental understanding of the dynamics of various operating parameters on concentration polarization and transmembrane flux. Simulation results are presented for a wide range of operating variables to show their effects on local variation of solute concentration and transmembrane flux. The numerical results were also compared with previously published experimental data, which shows that a concentration polarization model based on constant membrane permeability (usually obtained from pure water flux data) grossly overestimates the flux behavior. If the effect of gel polarization is included, the model can predict the actual permeate flux very closely. Thus, in modeling ultrafiltration, one needs to be careful in using the appropriate membrane permeability terms. The commonly used intrinsic membrane permeability which is usually a constant, may not describe the true flux behavior in ultrafiltration. Actually the nature of the feed, solute-surface interaction and gel layer formation control the effective permeability, which varies axially along the membrane length.     

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.