Mass transfer coefficient diffusion

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. 

The general approach for modelling catalyst deactivation is schematically organised in Figure 2. The central part are the mass balances of reactants, intermediates, and metal deposits. In these mass balances, coefficients are present to describe reaction kinetics (reaction rate constant), mass transfer (diffusion coefficient), and catalyst porous texture (accessible porosity and effective transport properties). The mass balances together with the initial and boundary conditions define the catalyst deactivation model. The boundary conditions are determined by the axial position in the reactor. Simulations result in metal deposition profiles in catalyst pellets and catalyst life-time predictions. 

Thickness of mass transfer film D = Mass transfer diffusion coefficient N = Mass transfer diffusion... 

Note that the above formulation includes allowance for the fractional phase holdup volumes, hL and ho, the phase flow rates, L and G, the diffusion coefficients Dl and Dq, and the overall mass transfer capacity coefficient Klx a, all to vary with position along the extractor. 

Coefficient of heat transfer Diffusion coefficient Flux of a quantity x Heat flow rate Kinematic viscosity Mass flow rate Mass-transfer coefficient Thermal conductivity Thermal diffusion coefficient Thermal diffusivity Viscosity Volume flow rate... 

In the frontal analysis experiment described in Section 5.3.2, the transport model of chromatography was used to fit the experimental data [40]. Neglecting axial and eddy diffusion, band broadening was accounted for by one single mass transfer rate coefficient. The mass transfer rate coefficients estimated were small and strongly dependent on the temperature and solute concentration, particularly the rate coefficients corresponding to the imprinted L-enantiomer (Fig. 5.12). Above a concentration of ca. 0.1 g/L the mass transfer rate coefficients of the two enantiomers are similar. 

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. 

Compared to our discussion of the general rate model in Chapter 6 for linear chromatography, the only important changes are the need to use nonlinear competitive isotherms and to consider the possible dependence of diffusion and the mass transfer rate coefficients on the local concentrations of the feed components. These changes increase dramatically the mathematical complexity of the problem and only numerical solutions are possible. 

The mass transfer film coefficients (kx and l y) are more directly related to physical properties, such as diffusivities and hydrodynamic conditions, than the overall mass transfer coefficients (Kx and Ky and are, therefore, easier to predict and correlate. On the other hand, since the interfacial compositions are difficult to measure or predict, it is more convenient to use overall mass transfer coefficients for process design and analysis. The relationship between the two sets of mass transfer coef-hcients may be derived by equating different groups in Equations 15.11 and 15.12 as follows ... 

In commercial absorption equipment, both the liquid and vapor are usually in turbulent flow and the effective stagnant film thickness AZ is not known. The common practice, therefore, is to rewrite (16-2) in terms of an empirical mass transfer (film) coefficient that replaces both the diffusivity and film thickness. Thus, by definition. 

It appears possible to make the following two important generalizations concerning the relative rates of mass transfer to the catalyst pellet and diffusion into the pellet (a) Mass transfer to the external catalyst surface is always faster than diffusion into the internal catalyst surface. This is because turbulence in the fluid stream enhances the effective diffusion coeflacient in the flowing fluid to much larger values than those possible inside a catalyst pellet. Even in the absence of turbulence, the presence of small pores in catalysts depresses the diffusion coefficient to (Knudsen) values lower than the bulk values in the flowing stream, (b) Hence, whenever mass transfer to the external catalyst surface is influencing reaction rate, then the internal surface area can be only partly available to the reaction. We thus get the elementary theorem that whenever a catalyst is sufficiently active so that the reaction rate is influenced by mass transfer (diffusion) to the catalyst pellet, then the internal surface area of that catalyst can be only partially available to the reaction. 

For both intrinsic and mass transfer diffusion, it is useful to analyze the surface diffusion coefficient terms of enthalpies and entropies [91Loml] ... 

The mass transfer film coefficients kx and ky) are more directly related to physical properties, such as diffusivities and hydrodynamic conditions, than the overall mass transfer coefficients Kx and Ky)... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

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. 

Neither the penetration nor the surface renewal theory can be used to predict mass transfer coefficients directiy because T and s are not normally known. Each suggests, however, that mass transfer coefficients should vary as the square root of the molecular diffusivity, as opposed to the first power suggested by the film theory. 

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

Cg = the concentration of the saturated solution in contact with the particles, D = a diffusion coefficient (approximated by the Hquid-phase diffusivity), M = the mass of solute transferred in time t, and S = the effective thickness of the liquid film surrounding the particles. For a batch process where the total volume H of solution is assumed to remain constant, dM = V dc and... 

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