Mass transfer equation diffusion, reaction

Diffusion-reactions with charged substrate. When the rate of the enzyme reaction is no longer negligible compared to that of the diffusion, the diffusion or mass-transfer equation must take account of the charge effects produced by a polyelectrolytic carrier. 

The basic theory of mass transfer to a RHSE is similar to that of a RDE. In laminar flow, the limiting current densities on both electrodes are proportional to the square-root of rotational speed they differ only in the numerical values of a proportional constant in the mass transfer equations. Thus, the methods of application of a RHSE for electrochemical studies are identical to those of the RDE. The basic procedure involves a potential sweep measurement to determine a series of current density vs. electrode potential curves at various rotational speeds. The portion of the curves in the limiting current regime where the current is independent of the potential, may be used to determine the diffusivity or concentration of a diffusing ion in the electrolyte. The current-potential curves below the limiting current potentials are used for evaluating kinetic information of the electrode reaction. 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

The latter strongly depends on the specific reaction mechanism, the stoichiometry, and the presence or absence of parallel reaction schemes (69). The rate expressions for Rt usually represent nonlinear dependences on the mixture composition and temperature. Specifically for the coupled reaction-mass transfer problems, such as Eqs. (A10), it is always essential as to whether or not the reaction rate is comparable to that of diffusion (68,77). Equations (A10) should be completed by the boundary conditions relevant to the film model. These conditions specify the values of the mixture composition at both film boundaries. For example, for the liquid phase ... 

As mentioned, from the reaction kinetics viewpoint the behavior of zeolite catalysts shows large variability. In addition, the apparent kinetics can be affected by pore diffusion. The compilation of literature revealed some kinetic equations, but their applicability in a realistic design was questionable. In this section we illustrate an approach that combines purely chemical reaction data with the evaluation of mass-transfer resistances. The source of kinetic data is a paper published by Corma et al. [7] dealing with MCM-22 and beta-zeolites. The alkylation takes place in a down-flow liquid-phase microreactor charged with catalyst diluted with carborundum. The particles are small (0.25-0.40 mm) and as a result there are no diffusion and mass-transfer limitations. 

When a first order irreversible chemical reaction (e.g. oxygen absorption and oxidation) takes place simultaneously with diffusion in food for example, then one obtains the following expression from the general mass transfer equation (7-10) ... 

The mass transfer equation is written in terms of the usual assumptions. However, it must be considered that because the concentration of the more abundant species in the flowing gas mixture (air), as well as its temperature, are constant, all the physical properties may be considered constant. The only species that changes its concentration along the reactor in measurable values is PCE. Therefore, the radial diffusion can be calculated as that of PCE in a more concentrated component, the air. This will be the governing mass transfer mechanism of PCE from the bulk of the gas stream to the catalytic boundaries and of the reaction products in the opposite direction. Since the concentrations of nitrogen and oxygen are in large excess they will not be subjected to mass transfer limitations. The reaction is assumed to occur at the catalytic wall with no contributions from the bulk of the system. Then the mass balance at any point of the reactor is... 

We now discuss some of the main features of LLPTC models developed for reaction under neutral conditions. Evans and Palmer (1981) were among the first to consider the effect of diffusion and mass transfer inPTC. They considered PTC in liquid-liquid systems by considering two well-mixed bulk phases of uniform composition separated by a uniform stagnant mass-transfer layer at the interface, and set up equations for bulk phase species balance and mass conservation equations for simultaneous diffusion and reaction in the film. Dynamics of the interaction between reaction and diffusion were studied under these assumptions for two special cases (a) reaction which is pseudo-first-order in the quaternary ion-pair (b) mass-transfer controlled instantaneous reaction. 

The equations governing the voltammetric method (e.g., assuming only species O is present initially) include the same ones as used previously, namely the mass-transfer equations [such as (5.4.2)] and the initial and semi-infinite conditions (5.4.3) and (5.4.4). However, the flux condition at the electrode surface is different, because the net reaction involves the electrolysis of diffusing O as well as O adsorbed on the electrode, to produce R that diffuses away and R that remains adsorbed. The general flux equation is then... 

The following discussion represents a detailed description of the mass balance for any species in a reactive mixture. In general, there are four mass transfer rate processes that must be considered accumulation, convection, diffusion, and sources or sinks due to chemical reactions. The units of each term in the integral form of the mass transfer equation are moles of component i per time. In differential form, the units of each term are moles of component i per volnme per time. This is achieved when the mass balance is divided by the finite control volume, which shrinks to a point within the region of interest in the limit when aU dimensions of the control volume become infinitesimally small. In this development, the size of the control volume V (t) is time dependent because, at each point on the surface of this volume element, the control volnme moves with velocity surface, which could be different from the local fluid velocity of component i, V,. Since there are several choices for this control volume within the region of interest, it is appropriate to consider an arbitrary volume element with the characteristics described above. For specific problems, it is advantageous to use a control volume that matches the symmetry of the macroscopic boundaries. This is illustrated in subsequent chapters for catalysts with rectangular, cylindrical, and spherical symmetry. 

DIMENSIONLESS FORM OF THE GENERALIZED MASS TRANSFER EQUATION WITH UNSTEADY-STATE CONVECTION, DIFFUSION, AND CHEMICAL REACTION... 

The product of the Reynolds and Schmidt numbers, which counts as one dimensionless number, is equivalent to the Peclet number for mass transfer, PeMx- The Peclet number represents the ratio of the convective mass transfer rate process to the diffusion rate process of component, and it appears on the left-hand side of the dimensionless mass transfer equation for component i. The remaining r dimensionless transport numbers can be treated simultaneously because they represent ratios of scaling factors for the reactant-product conversion rate due to the jth independent chemical reaction relative to the rate of diffusion of component I. Hence,... 

For unsteady-state diffusion into a quiescent medium with no chemical reaction, the mass transfer Peclet number does not appear in the dimensionless mass transfer equation for species i because it is not appropriate to make variable time t dimensionless via division by L/ v) if there is no bulk fluid flow (i.e., (d) = 0). In this case, the first term on each side of equation (10-11) survives, which corresponds to the unsteady-state diffusion equation. However, the characteristic time for diffusion of species i over a length scale L, given by L /50i,mix. replaces L/ v) to make variable time t dimensionless. Now, the accumulation and diffusional rate processes scale as CAo i.mix/A, with dimensions of moles per volume per time. Since both surviving mass transfer rate processes exhibit the same dimensional scaling factor, there are no dimensionless numbers in the mass transfer equation which describes unsteady-state diffusion for species i in nonreactive systems. 

Consider several overlapping subsets of the dimensionless mass transfer equation from Section 10-2 which correspond to various combinations of convection, diffusion, and chemical reaction that may or may not exhibit transient behavior. 

If one constructs the appropriate dimensionless equation that governs the molar density profile fi for component i, then xj/i depends on all the dimensionless independent variables and parameters in the governing equation and its supporting boundary conditions. Geometry also plays a role in the final expression for in each case via the coordinate system that best exploits the summetry of the macroscopic boundaries, but this effect is not as important as the dependence of on the dimensionless numbers in the mass transfer equation and its boundary conditions. For example, if convection, diffusion, and chemical reaction are important rate processes that must be considered, then the governing equation for transient analysis... 

What important dimensionless number(s) appear in the dimensionless partial differential mass transfer equation for laminar flow through a blood capillary when the important rate processes are axial convection, radial diffusion, and nth-order irreversible chemical reaction ... 

In spherical coordinates, the dimensional mass transfer equation with radial diffusion and first-order irreversible chemical reaction exhibits an analytical solution for the molar density profile of reactant A. If the kinetics are not zeroth-order or first-order, then the methodology exists to find the best pseudo-first-order rate constant to match the actual rate law and obtain an approximate analytical solution. The concentration profile of reactant A in the liquid phase must satisfy... 

Now that one has obtained the basic information for the molar density of reactant A within the liquid-phase mass transfer boundary layer, it is necessary to calculate the molar flux of species A normal to the gas-liquid interface at r = l bubbie, and define the mass transfer coefficient via this flux. Since convective mass transfer normal to the interface was not included in the mass transfer equation with liquid-phase chemical reaction, it is not necessary to consider the convective mechanism at this stage of the development. Pick s first law of diffusion is sufficient to calculate the flux of A in the r direction at r = /fbubbie- Hence,... 

Most important, heterogeneous surface-catalyzed chemical reaction rates are written in pseudo-homogeneous (i.e., volumetric) form and they are included in the mass transfer equation instead of the boundary conditions. Details of the porosity and tortuosity of a catalytic pellet are included in the effective diffusion coefficient used to calculate the intrapellet Damkohler number. The parameters (i.e., internal surface area per unit mass of catalyst) and Papp (i.e., apparent pellet density, which includes the internal void volume), whose product has units of inverse length, allow one to express the kinetic rate laws in pseudo-volumetric form, as required by the mass transfer equation. Hence, the mass balance for homogeneous diffusion and multiple pseudo-volumetric chemical reactions in one catalytic pellet is... 

Solve the dimensionless mass transfer equation (i.e., the mass balance for reactant A) with homogeneous one-dimensional diffusion and zeroth-order irreversible chemical reaction to obtain an expression for 4molar density of reactant A. 

Numerical Integration of the Mass Transfer Equation with Diffusion and Reaction... 

Hence, at the center of spherical catalytic pellets, the first term on the right side of the mass transfer equation with diffusion and chemical reaction depends on the intrapellet Damkohler number and adopts a value between zero and — when the... 

The dimensionless scaling factor in the mass transfer equation for reactant A with diffusion and chemical reaction is written with subscript j for the jth chemical reaction in a multiple reaction sequence. Hence, A corresponds to the Damkohler number for reaction j. The only distinguishing factor between all of these Damkohler numbers for multiple reactions is that the nth-order kinetic rate constant in the 7th reaction (i.e., kj) changes from one reaction to another. The characteristic length, the molar density of key-limiting reactant A on the external surface of the catalyst, and the effective diffusion coefficient of reactant A are the same in all the Damkohler numbers that appear in the dimensionless mass balance for reactant A. In other words. 

Catalysts with Cylindrical Symmetry. This analysis is based on the mass transfer equation with diffusion and chemical reaction. Basic information has been obtained for the dimensionless molar density profile of reactant A. For zeroth-order kinetics, the molar density is equated to zero at the critical value of the dimensionless radial coordinate, criticai = /(A). The relation between the critical value of the dimensionless radial coordinate and the intrapeUet Damkohler number is obtained by solving the following nonlinear algebraic equation ... 

Consider one-dimensional diffusion and zeroth-order chemical reaction in a flat-slab porous wafer-type catalyst. The conditions are approximately isothermal and the inirapellet Damkohler number of reactant A is Aa. intrapellet = VS. The mass transfer equation is solved numerically, not analytically. 

Answer The dimensionless mass transfer equation in rectangular coordinates with one-dimensional diffusion and nth-order chemical reaction represents the starting point for a generic solution to part (a). The dimensionless molar density of reactant A must satisfy... 

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