Mass transfer diffusion equation

Four strategies are generally employed to demonstrate mass transfer limitation in aquatic systems. Most commonly, measured uptake rates are simply compared with calculated maximal mass transfer rates (equation (17)) (e.g. [48,49]). Uptake rates can also be compared under different flow conditions (e.g. [52,55,56,84]), or by varying the biomass under identical flow conditions (e.g. [85]). Finally, several recent, innovative experiments have demonstrated diffusion boundary layers using microsensors [50,51]. Of the documented examples of diffusion limitation, three major cases have been identified ... 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

The mass transfer model. In our previous work [6] the mass transfer model equations and their mathematical treatment have been described extensively. The relevant differential equations, describing the process of liquid-phase diffusion and simultaneous reactions of the species according to the penetration theory, are summarized in table 1. Recently we derived from this penetration theory description a film model version, which is incorporated in the evaluation of the experimental results. Details on the film model version are given elsewhere [5]. 

Equilibrium partitioning and mass transfer relationships that control the fate of HOPs in CRM and in different phases in the environment were presented in this chapter. Partitioning relationships were derived from thermodynamic principles for air, liquid, and solid phases, and they were used to determine the driving force for mass transfer. Diffusion coefficients were examined and those in water were much greater than those in air. Mass transfer relationships were developed for both transport within phases, and transport between phases. Several analytical solutions for mass transfer were examined and applied to relevant problems using calculated diffusion coefficients or mass transfer rate constants obtained from the literature. The equations and approaches used in this chapter can be used to evaluate partitioning and transport of HOP in CRM and the environment. 

Assuming a constant surface area, dissolution at a solution-solid interface (Case I) results in linear kinetics in which the rate of mass transfer is constant with time (equation 1). Analytical solutions to the diffusion equation result in parabolic rates of mass transfer (, 16) (equation 2). This result is obtained whether the boundary conditions are defined so diffusion occurs across a progressively thickening, leached layer within the silicate phase (Case II), or across a growing precipitate layer forming on the silicate surface (Case III). Another case of linear kinetics (equation 1) may occur when the rate of formation of a metastable product or leached layer at the fresh silicate surface becomes equal to the rate at which this layer is destroyed at the aqueous... 

The diffusion coefficient D(y) is a fimction of temperature, and it varies with position near the electrode according to the local temperature variation. However, as the thermal layer thickness is about five times larger than the diffusion layer thickness, the dif ion coefficient has in fact a variation that can be assumed to be negligible within the mass-transfer diffusion layer corresponding to the integration domain of equation (14.51). Thus, in the following development, D(y) = D, and dD/dy = 0. 

In problems where the flux ratios are known (e.g., condensation and heterogeneous reacting systems where the reaction rate is controlled by diffusion) the mole fractions at the interface are not known in advance and it is necessary to solve the mass transfer rate equations simultaneously with additional equations (these may be phase equilibrium and/or reaction rate equations). For these cases it is possible to embed Algorithms 8.1 or 8.2 within another iterative procedure that solves the additional equations (as was done in Example 8.3.2). However, we suggest that a better procedure is to solve the mass transfer rate equations simultaneously with the additional equations using Newton s method. This approach will be developed below for cases where the mole fractions at both ends of the film are known. Later we will extend the method to allow straightforward solution of more complicated problems (see Examples 9.4.1, 11.5.2, 11.5.3, and others). 

The distance d may be equated approximately with the thickness of the stationary liquid film, df, so that slow mass transfer is equated with high liquid loadings. It is also desirable to choose liquids within which the solute molecule has a large diffusion coefficient, Ds. 

The results obtained in equations (8-136) to (8-142) assume constant B, i.e., the reaction is pseudo-first-order in A. Another limiting case that yields to analytical solution is that in which the rate of reaction is very rapid and the reaction occurs wholly within the film. Here we consider the reaction A -I- P to occur very rapidly compared to mass-transfer/diffusion rates. The profiles look as in Figure 7.17b, and the overall flux and enhancement factor are given by... 

Time-Averaged Properties. The unsteady-state macroscopic mass balance for mobile component A is applied to the quiescent liquid, where the rate of interphase mass transfer via equation (11-205) is interpreted as an input term due to diffusion across the gas-liquid interface. There are no output terms, sources, sinks, or contributions from convective mass transfer in the macroscopic mass balance. Hence, the accumulation rate process is balanced by the rate of interphase mass transfer across time-varying surface S t), where both terms have dimensions of moles per time ... 

Lf equated to 3x10 m /s (fiom the average Lf value from mass transfer diffusion on metals). 

In this section, surface boundary motion is considered as an essential feature for solving such an equation. Thus, diffusion accompanied by an electrode thickness increment due to diffusion of metal cations (solute) and immobile species is considered to be a particular diffusion problem that resembles mass transfer diffusion toward the faces of cathode sheets in an electrowinning EW) cell. 

The formulae presented above are based on ideas put forward by Damkholer.From analysis of the competition between reaction,heat and mass transfer (diffusion) inside catalysts he obtained an equation which relates temperature and concentration at a point within the pellet,i.e.,... 

Keywords Computational mass transfer Reynolds averaging Closure of time-averaged mass transfer equation Two-equation model Turbulent mass transfer diffusivity Reynolds mass flux model... 

Theoretically, the unknown diffusivity can be obtained directly by the closure of the mass transfer differential equation by a proper method in order to solve at once all unknown parameters in the equation. In the following sections, the... 

Abstract In this chapter, an exothermic catalytic reaction process is simulated by using computational mass transfer (CMT) models as presented in Chap. 3. The difference between the simulation in this chapter from those in Chaps. 4,5, and 6 is that chemical reaction is involved. The source term in the species conservation equation represents not only the mass transferred from one phase to the other, but also the mass created or depleted by a chemical reaction. Thus, the application of the CMT model is extended to simulating the chemical reactor. The simulation is carried out on a wall-cooled catalytic reactor for the synthesis of vinyl acetate from acetic acid and acetylene by using both c — Sc model and Reynolds mass flux model. The simulated axial concentration and temperature distributions are in agreement with the experimental measurement. As the distribution of lx shows dissimilarity with Dj and the Sci or Pri are thus varying throughout the reactor. The anisotropic axial and radial turbulent mass transfer diffusivities are predicted where the wavy shape of axial diffusivity D, along the radial direction indicates the important influence of catalysis porosity distribution on the performance of a reactor. 

The difiusion equation taking into account the rate of chemical reaction can be written based on each of the mass transfer diffusion models. 

The values of some of the parameters in these equations, such as the diffusion coefficient D and the characteristic length parameter d, will depend on specific models and definitions (see below). Using the definitions of the Sherwood number, Sh = kd/D, the ratio of total and molecular mass transfer (with k the mass transfer coefficient), and the Schmidt number. Sc = r]/pD the ratio of momentum and molecular mass transfer, the equation can be written as ... 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

Equations 11 and 12 caimot be used to predict the mass transfer coefficients directly, because is usually not known. The theory, however, predicts a linear dependence of the mass transfer coefficient on diffusivity. 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

The equations of combiaed diffusion and reaction, and their solutions, are analogous to those for gas absorption (qv) (47). It has been shown how the concentration profiles and rate-controlling steps change as the rate constant iacreases (48). When the reaction is very slow and the B-rich phase is essentially saturated with C, the mass-transfer rate is governed by the kinetics within the bulk of the B-rich phase. This is defined as regime 1. 

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