Diffusion and External Mass-Transfer Resistance

Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, Diffusion-Reaction Interactions in Catalyst Pellets, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor r, can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction  

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Here, as in Section 8.5.4, we treat the isothermal case for ijo, and relate tj0 to 17. may then be interpreted as the ratio of the (observed) rate of reaction with pore diffusion and external mass transfer resistance to the rate with neither of these present. 

Acrivos, A., On the combined effect of longitudinal diffusion and external mass transfer resistance in fixed bed operations. Chem. Eng. Sci. 13, 1 (1960). 

Solutions to these equations presented by Wise et al. (195, 233) for conditions where pore diffusion and external mass transfer resistance can be neglected are... 

Intraparticle Diffusion and External Mass-Transfer Resistance. 7-22... 

The curves calculated in this way at constant for different combinations of Pe and S [i.e., for different combinations of axial dispersion and mass transfer resistance but the same linear combination 1/Pe + 5(1 + 5/0/15] show close agreement with each other and with the curve calculated from the simple linearized rate model using an overall lumped coefficient to account for the combined effects of axial dispersion, diffusion, and external mass transfer resistance. 

A summary of some of the available solutions for irreversible systems is given in Table 8.2. All solutions assume plug flow. The limiting cases of solid diffusion control and pore diffusion control were solved by Cooper - and by Cooper and Libermann while the solution for the case of combined pore diffusion and external mass transfer resistance was obtained by Weber and Chakravorti using the method of Cooper and Libermann. Weber and... 

Internal and external mass transfer resistances are important factors affecting the catalyst performance. These are determined mainly by the properties of the fluids in the reaction system, the gas-liquid contact area, which is very high for monolith reactors, and the diffusion lengths, which are short in monoliths. The monolith reactor is expected to provide apparent reaction rates near those of intrinsic kinetics due to its simplicity and the absence of diffusional limitations. The high mass transfer rates obtained in the monolith reactors result in higher catalyst utilization and possibly improved selectivity. 

In another example, Beveridge studied the oxidation of zinc sulfide spheres and reported that the global rate was, in turn, controlled by the surface chemical reaction at low temperatures, diffusion through the zinc oxide product layer at intermediate temperatures, and external mass transfer resistance at higher temperatures. 

The fact that the Biot number of mass (a measure of the ratio of internal diffusion to external mass transfer resistance) is much larger than unity, implies that the major resistance lies in the internal diffusion process. A simple analysis can be made to assess the relative importance of internal and external mass transport processes now that the pellet can be considered isothermal. For an isothermal pellet, Eq. 4.32 can be written as ... 

A kinetics or reaction model must take into account the various individual processes involved in the overall process. We picture the reaction itself taking place on solid B surface somewhere within the particle, but to arrive at the surface, reactant A must make its way from the bulk-gas phase to the interior of the particle. This suggests the possibility of gas-phase resistances similar to those in a catalyst particle (Figure 8.9) external mass-transfer resistance in the vicinity of the exterior surface of the particle, and interior diffusion resistance through pores of both product formed and unreacted reactant. The situation is illustrated in Figure 9.1 for an isothermal spherical particle of radius A at a particular instant of time, in terms of the general case and two extreme cases. These extreme cases form the bases for relatively simple models, with corresponding concentration profiles for A and B. 

The results reviewed above suggest that gas-phase diffusion can contribute significantly to polarization as O2 concentrations as high as a few percent and are not necessarily identifiable as a separate feature in the impedance. Workers studying the P02 -dependence of the electrode kinetics are therefore urged to eliminate as much external mass-transfer resistance in their experiments as possible and verify experimentally (using variations in balance gas or total pressure) that gas-phase effects are not obscuring their results. 

We have presented a general reaction-diffusion model for porous catalyst particles in stirred semibatch reactors applied to three-phase processes. The model was solved numerically for small and large catalyst particles to elucidate the role of internal and external mass transfer limitations. The case studies (citral and sugar hydrogenation) revealed that both internal and external resistances can considerably affect the rate and selectivity of the process. In order to obtain the best possible performance of industrial reactors, it is necessary to use this kind of simulation approach, which helps to optimize the process parameters, such as temperature, hydrogen pressure, catalyst particle size and the stirring conditions. 

There are three distinct mass-transfer resistances (1) the external resistance of the fluid film surrounding the pellet, (2) the diffusional resistance of the macropores of the pellet, and (3) the diffusional resistance of the zeolite crystals. The external mass-transfer resistance may be estimated from well-established correlations (4, 5) and is generally negligible for molecular sieve adsorbers so that, under practical operating conditions, the rate of mass transfer is controlled by either macropore diffusion or zeolitic diffusion. In the present analysis we consider only systems in which one or other of these resistances is dominant. If both resistances are of comparable importance the analysis becomes more difficult. 

An enzyme is immobilized by copolymerization technique. The diameter of the spherical particle is 2 mm and the number density of the particles in a substrate solution is 10,000/L. Initial concentration of substrate is 0.1 mole/L. A substrate catalyzed by the enzyme can be adequately represented by the first-order reaction with k0 = 0.002 mol/Ls. It has been found that both external and internal mass-transfer resistance are significant for this immobilized enzyme. The mass-transfer coefficient at the stagnant film around the particle is about 0.02 cm/s and the diffusivity of the substrate in the particle is 5 x 10-6 cm2/s. 

In general case, as was mentioned, the diffusion coefficient and/or convective velocity can depend on the space coordinate, thus D=D(y), u(y), [or on the concentration, D = D(c) or both of them, D = D(c, y)]. In the boundary conditions the external mass-transfer resistance is also taken into account. 

It can be anticipated that prediction of diffiisivities should be best with larger particles since the diffusion path is physically longer. A few simulation runs with different values of diffusivity, solubility and external mass transfer coefficient, show that the most sensitive parameter (or resistance) are intraparticle diffusivity and solubility while the effect of external mass transfer coefficient is small. 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

Here, the parameter F = Uo]dJ2De( — t) considers the effect of intraparticle diffusion, Pe = V dJlEzi. takes into account the effect of axial dispersion, S = 3(1 — e)Kt/U0L considers the effect of total external mass-transfer resistance, and A0 = /j (l — )k dp/2UoL considers the effect of surface reaction on the conversion. In these reactions L/0l, s the superficial liquid velocity, dp is the particle... 

For the given rate expression, equations (7-124) to (7-127) can be numerically integrated, e.g., in Fig. 7-13 for reaction control and Fig. 7-14 for intraparticle diffusion control, both with negligible external mass-transfer resistance x is the fractional conversion. 

A plot of H/(2mq) versus 1/Ug is a straight line with a slope equal to Di and an ordinate equal to 3f + S )/Sq. The coefficient of external mass transfer is estimated using one of the several correlations available for it (see Chapter 5, subsection 5.2.5, correlation of Wilson and Geankoplis [62], Kataoka et al. [87], or the penetration theory [88]). Correcting for the contribution due to the external mass transfer resistance gives the last term in the plate height equation, 5, hence the intraparticle diffusion coefficient, Dg. 

Eq. (11) shows the relationship between the three reaction resistances. The first term represents the external mass transfer resistance, the second the resistance associated with diffusion through the product layer, and the third the chemical reaction resistance at the reactant-product interface. 

Costa and Smith " studied the hydrofiuorination of nonporous uranium dioxide pellets under conditions where external mass transfer resistance was negligible. The global rate was initially controlled by the surface chemical reaction resistance, but switched to product layer diffusion as the reaction progressed and the product layer thickness increased. 

The study of a particular adsorption process requires the knowledge of equilibrium data and adsorption kinetics [4]. Equilibrium data are obtained firom adsorption isotherms and are used to evaluate the capacity of activated carbons to adsorb a particular molecule. They constitute the first experimental information that is generally used as a tool to discriminate among different activated carbons and thereby choose the most appropriate one for a particular application. Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum [3]. Therefore, all equations derived firom monolayer gas adsorption remain vafid. Some of these equations, such as the Langmuir and Dubinin—Astakhov, are widely used to determine the adsorption capacity of activated carbons. Batch equilibrium tests are often complemented by kinetics studies, to determine the external mass transfer resistance and the effective diffusion coefficient, and by dynamic column studies. These column studies are used to determine system size requirements, contact time, and carbon usage rates. These parameters can be obtained from the breakthrough curves. In this chapter, I shall deal mainly with equilibrium data in the adsorption of organic solutes. 

These conclusions differ somewhat from those of Pirkle and Siegell in their analysis of adsorption chromatography in a crossflow magnetically fluidized bed (14). They found the dominant effects to be the width of the feed band and the external mass transfer resistance. It is not surprising that the effect of internal diffusion would be more important in size exclusion chromatography with macromolecular solutes. 

As shown earlier, the mass transfer coefficient is inversely proportional to the channel diameter and increases with the molecular diffusion coefficient. Accordingly, the influence of mass transfer on the observed reaction rate can be estimated by varying the diameter [84], This method was applied in experiments by Fichtner et al. [91], who varied the hydraulic channel diameter between 125 and 85 pm. The authors investigated the partial oxidation of methane to synthesis gas, a fast reaction, in a MSR made of rhodium. Another method to vary the external mass transfer resistance consists of varying the molecular diffusion coefficient. The... 

The reactant molecules diffuse from the bulk of the fl iiid phase to the surface of the catalyst pellets. This process is usually described by mass transfer rate over a hypothetical external mass transfer resistance. The mass transfer coefficient is usually calculated using J-factor correlations (e.g. Hill, 1977). This step is dependent upon the properties of the gas mixture and the flow conditions around the catalyst pellets as well as the size and shape of the catalyst pellets. This step is usually referred to as external mass transfer of reactant molecules. 

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