Mass diffusion process evaluation

Table 1 Summary of Experimental Methods for Evaluating Diffusion Coefficients and Investigating Mass Transport Processes of Pharmaceutical Interest... 

The following, well-acceptable assumptions are applied in the presented models of automobile exhaust gas converters Ideal gas behavior and constant pressure are considered (system open to ambient atmosphere, very low pressure drop). Relatively low concentration of key reactants enables to approximate diffusion processes by the Fick s law and to assume negligible change in the number of moles caused by the reactions. Axial dispersion and heat conduction effects in the flowing gas can be neglected due to short residence times ( 0.1 s). The description of heat and mass transfer between bulk of flowing gas and catalytic washcoat is approximated by distributed transfer coefficients, calculated from suitable correlations (cf. Section III.C). All physical properties of gas (cp, p, p, X, Z>k) and solid phase heat capacity are evaluated in dependence on temperature. Effective heat conductivity, density and heat capacity are used for the entire solid phase, which consists of catalytic washcoat layer and monolith substrate (wall). 

Nonequilibrium sorption due to mass-transfer limitations (including slow external or internal diffusion) and sorption to two different sorbents have been incorporated into a single ADE to evaluate the conditions under which mass-transfer processes may be important [206]. Simulations with this model, using mass-transfer parameters estimated from empirical correlations, reveal nonequilibrium conditions (i.e., mass-transfer limitations) when groundwater velocities increase (such as those that might occur in a funnel-and-gate system). 

In conjunction with heat conduction we will also investigate mass diffusion. As a result of the analogy between these two molecular transport processes many results from heat conduction can be applied to mass diffusion. In particular the mathematical methods for the evaluation of concentration fields agree to a large extent with the solution methods for heat conduction problems. 

When Eq. (4) is coupled with controlling equations of mass and momentum for gas phase and solid phase, the detailed flow-reaction—diffusion process in a MTO fluidized bed reactor can be simulated (Zhao et al., 2013). In practical applications, simplified models for two-phase hydrodynamics are also proposed (Abba et al., 2003 Bos et al., 1995 Zhang et al., 2012), in which the detailed flow patterns cannot be calculated but it is very efficient in the overall reactor performance evaluation. 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

The former usually involves process temperature or isolation. Sohds surface characteristics are important in that they control the extent to which an operation is diffusion-limited, i.e., diffusion into and out of the pores of a given sohds particle, not through the voids among separate particles. The size of the solids parti(des, the surface-to-mass ratio, is also important in the evaluation of surface characteristics and the diffusion problem. 

Here, issues in relation to the trickle flow regime—isothermal operation and plug flow for the gas phase—will be dealt with. Also, it is assumed that the flowing liquid completely covers the outer surface particles (/w = 1 or aLS = au) so that the reaction can take place solely by the mass transfer of the reactant through the liquid-particle interface. Generally, the assumption of isothermal conditions and complete liquid coverage in trickle-bed processes is fully justified with the exception of very low liquid rates. Capillary forces normally draw the liquid into the pores of the particles. Therefore, the use of liquid-phase diffusivities is adequate in the evaluation of intraparticle mass transfer effects (effectiveness factors) (Smith, 1981). 

It is necessary to compare atomic diffusion and mass loss velocities to other possible transport velocities in order to evaluate the relative importance of each. Each process is briefly presented in this section. 

Further, mixing/turbulent diffusivity derived by considering that the mixing process in a liquid is caused by the random movement of inner substances based on the turbulent flow can also be used as an index for the evaluation of the local mixing rate. The mass balance in the flow field is written as... 

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