Mass transfer analysis basic equation

Obviously, there is a significant error possible ( 18%) for both Eq. fl5-44al and fl5-44bl from just the multiplier for ripple formation. This type of error is not unusual for semi-empirical and enpirical mass-transfer correlations. These equations are semi-empirical because the basic form is from theoretical analysis, but the coefficient is from experimental data. 

Eq.(79) is the second Pick s low. Its structure is the same as that of the differential equation of the convective h t transfer (in case of a st y state process Eq. (56)). This gives the possibility, as shown later, to calculate the heat transfer processes by means of experimental data or equations for mass transfer. The basic methods for these calculations are the similarily theory and the dimensional analysis. That is why before considering the theory of mass transfor processes, we present these important methods largely used in chemical engineering and in particular In the area of packed columns. 

Hixson and Knox employed a more fundamental theoretical approach in their analysis of the effect of agitation on growth rate of single crystals (H7). They started with the basic equation for mass transfer of solute across a plane parallel to an interface and at a fixed distance from it ... 

Fundamentals The basic reaction and transport steps in trickle bed reactors are similar to those in slurry reactors. The main differences are the correlations used to determine the mass transfer coefficients. In addition, if there is more than one component in the gas phase (e.g., liquid has a high vapor pressure or one of the entering gases is inert), there is one additional transport step in the gas phase. Figure 12-17shows the various transport steps in trickle bed reactors. Following our analysis for slurry reactors we develop the equations for the rate of transport of each step. The steps involving reactant A in the gas phase are... 

Finally, we present an interpretation of our observations in terms of diffusion paths. Basically, the diffusion equations are solved for the case of two semi-infinite phases brought into contact under conditions where there is no convection and no interfacial resistance to mass transfer. Other simplifying assumptions such as uniform density and diffusion coefficients in each phase are usually made to simplify the mathematics. The analysis shows that the set of compositions in the system is independent of time although the location of a particular composition is time-dependent. The composition set can be plotted on the equilibrium phase diagram, thus showing the existence of intermediate phases and, as explained below, providing a method for predicting the occurrence of spontaneous emulsification. 

The same approaches that were successful in linear chromatography—the use of either one of several possible liunped kinetic models or of the general rate model — have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotiienn provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. 

Fairly rigorous formulas for the interfacial heat and mass transfer terms are defined in sect 3.3 for the different averaging methods commonly applied in chemical reactor analysis. However, since the modeling concepts are mathematically similar for the different averages, we choose to examine these constitutive equations in the framework of the volume averaging method described in sect 3.4.1. This modeling framework is used extensively in chemical reactor analysis because the basic model derivation is intuitive and relatively easy to understand. 

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 ... 

The basic equations covering combined mass- and heat-transfer phenomena have been covered in the literature. The analysis combines the sensible and latent heat transfer into an overall process based on enthalpy potential as the driving force. 

The basics of charge transfer may also be presented in the form of two analogies. One involves using equations that describe the collision mechanics between particles and the wall, as presented by Timoshenko (1951) and developed by Soo (1967). This is quite similar to the basic heat-transfer analysis. The second approach is to use the penetration theory as given by Higbie (1935) and Danckwerts (1951) for heat, mass, and momentum transfer for the analysis of charge transfer. 

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