Intraparticle diffusion external mass-transfer resistance

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... 

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 ... 

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. 

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

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. 

Finally, Rosen [J.B. Rosen, J. Chem. Phys., 20, 387 (1952) Ind. Eng. Chem., 46, 1590 (1954)] presented the solution for linear equilibrium and intraparticle diffusion-controlled adsorption (no external mass-transfer resistance). In this case... 

Solutions are provided for external mass-transfer control, intraparticle diffusion control, and mixed resistances for the case of constant Vj and F, out = 0- The results are in terms of the fractional... 

Solutions are provided for external mass-transfer control, intraparticle diffusion control, and mixed resistances for the case of constant Vf and F0 in = FVi out = 0. The results are in terms of the fractional approach to equilibrium F = (ht — hf)/(nT — nf), where hf and are the initial and ultimate solute concentrations in the adsorbent. The solution concentration is related to the amount adsorbed by the material balance - (hi - nf )M,Ay. 

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. 

Three main types of mass transfer resistances are recognized film diffusion (which occnrs at the external surface of the adsorbent), intraparticle diffusion (which occnrs within the pores or amorphous structure of the adsorbent), and adsorption/desorption kinetics (which occnrs at the internal surface of the adsorbent). 

In a catalytic reactor, the fluid flows through the catalyst particles and may face a resistance caused by the concentration gradient between the bulk fluid and the external particle surface. This resistance (interparticle or external mass transfer limitation) must be added to intraparticle diffusion limitation. 

On the other hand, customized models for catalytic reactors include all of the main processes taking place inside the catalytic reactors. The most important of these processes for catalytic reactors are those associated with the catalyst pellets, namely intrinsic kinetics (which includes chemisorption and surface reaction), intraparticle diffusion of mass and heat, external mass and heat transfer resistances between the catalyst surface and the bulk of the fluid, as well as all the heat production and heat consumption accompanying the catalytic reaction. 

The mass transfer resistance term is the sum of two coefficients such as the external or film mass transfer coefficient, hjum, and the transparticle mass transfer coefficient, htransparticie-Tho formcr accounts for the difficulties encountered by analyte molecules to penetrate into the network of mesopores inside the particles, and the latter, for the time that it takes for them to diffuse across this network, once they have entered into it. The transparticle mass transfer resistance for shell particles was derived by Kaczmarski and Guiochon (68). According to this theory the intraparticle diffusivity depends on the ratio (p = R/Rg) of the diameter of the solid core to that of the particle in a core-shell particle (Figure 5.6). As this ratio increases, the mass transfer kinetics becomes faster through the shell particles. 

Since Sh > 2 and < D /r, the minimum value of (Bi) , is given by t/3c ("-3.0). Thu, even under these rather extreme assumptions the internal concentratiort gradient is appreciably greater than the external gradient. Any additional resistance to mass transfer from either Knudsen diffusion or intracrystalline diffusion will decrease further, so one may conclude that under most practically realisable conditions the intraparticle resistance is more important than film resistance in determining the mass transfer rate. 

For heat transfer the relative importance of the internal and external resistances is reversed. For a gaseous system X,/Xy--10 -10 so it is evident from Eqs. 7.20 and 7.23 that at any reasonable Reynolds number (Bi) < 1.0, indicating that the external temperature gradient is much greater than the temperature gradient within the particle. The model of an isothermal particle in which all resistance to mass transfer is due to intraparticle diffusion while resistance tp heat transfer is confined to the external film thus emerges as a realistic representation for most conditions of practical importance. The validity of this approximation has been verified experimentally for a sin e isolated adsorbent particle. ... 

The external fluid film resistance (the corresponding mass-transfer coefficient ki from equations (3.4.32a,b)) is in series with the intraparticle transport resistance. The flux of a species through a porous/mesoporous/microporous adsorbent particle consists, in general, of simultaneous contributions from the four transport mechanisms described earlier for gas transport in Section 3.1.3.2 (for molecular diffusion, where (Dak/T>ab) 2> 1) ... 

Possibilities for a single resistance include a linear rate expression with a lumped parameter mass transfer coefficient based either on the external fluid film or on a hypothetical solid film, depending on which film is controlling the rate of uptake of adsorbate. A quadratic driving force expression, again with a lumped parameter mass transfer coefficient, may be used instead. Alternatively, intraparticle diffusion, if the dominant form of mass transfer, may be described by the general diffusion equation (Pick s second law) with its appropriate boundary conditions, as described in Chapter 4. 

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