Intraparticle transport

Intraparticle Transport Meclianisms Intraparticle transport may be hmited by pore dijfusion, solid dijfusion, reaction kinetics at phase boundaries, or two or more of these mechanisms together. 

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

Rapid Adsorption-Desorption Cycles For rapid cycles with particle diffusion controlling, when the cycle time is much smaller than the time constant for intraparticle transport, the LDF approximation becomes inaccurate. The generalized expression... 

This discussion also applies to the original variable Y s, which represents the ensemble-average temperature of particles located at a particular point at a given time. Basically, we know the total enthalpy of each particle, but we do not know how it is distributed inside any given particle. Since the reaction rate can be very sensitive to the local temperature, we will need a SGS model to describe the coupling between intraparticle transport processes and chemical reactions. 

Penetration of the reactants into the pores of the catalyst particle (intraparticle transport)... 

For subsequent tests of intraparticle transport resistance, the catalyst was dry sieved into seven fractions. The mean particle size of these seven fractions were 18, 29, 35,44, 57, 81, and 93 pm. The particle size distributions of these seven fractions are shown in Figure 1, the platinum loading and dispersion are depicted in Figure 2. It is clear from Figure 2 that the platinum loading is a strong function of the catalyst particle size i.e. 5.42% for the 18 pm fraction and 3.25% for... 

The viscosity of most gases at atmospheric pressure is of the order of 10"7 Ns/m2, so for pores of about 1 /mi radius DP is approximately 10"5 m2/s. Molecular diffusion coefficients are of similar magnitude so that in small pores forced flow will compete with molecular diffusion. For fast reactions accompanied by an increase in the number of moles an excess pressure is developed in the interior recesses of the porous particle which results in the forced flow of excess product and reactant molecules to the particle exterior. Conversely, for pores greater than about 100/im radius, DP is as high as 10"3 m2/s and the coefficient of diffusion which will determine the rate of intraparticle transport will be the coefficient of molecular diffusion. 

Based on the studies on the KD306-type sulfur-resisting methanation catalyst, the non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key-components have been established, which were solved using an orthogonal collocation method. The simulation values of the effectiveness factors for the methanation reaction ch4 and the shift reaction fco2are in fair agreement with the experimental values, which indicates that both models are able to predict intraparticle transport and reaction processes within catalyst pellets. 

Fig. 4.19. Schematic representation of intraparticle flow. In pressure driven flow there is no flow through the particle (A) in electrically driven flow there is intraparticle transport. In (A) transport of solute into the pores is accomplished solely by diffusion, whereas in (B) the EOF enhances transport through the pores. Reprinted from ref. [125] with permission. Copyright Elsevier 1999.

For simultaneous treatment of interphase and intraparticle transport resistances, the boundary conditions at the external pellet surface are now given by eqs 15 and 16, whereas eqs 11 and 12 still hold for the case of negligible interphase transport resistance. The bound-... 

TaUe 1. Dimensionless representation of the stationary mass and enthalpy balance equations for combined interphase and intraparticle transport and reaction (single, nth order, irreversible reactions). 

Table 2. Experimental diagnostic criteria for the absence of intraparticle transport effects in simple, irreversible reactions (power law kinetics only).

At this point, it should be mentioned that there may be some doubt about how successful the nonisothermal criteria are at involving observable quantities only. This concerns the fact that in the nonisothermal case one has to specify the Arrhenius number which contains the true activation energy of the catalyzed reaction. The above statement would obviously define the true activation energy as a directly observable quantity in the nonisothermal criteria. However, this would presumably be an experimental value derived from studies in which intraparticle transport effects were absent, which is precisely what one is attempting to define [12]. 

Table 4 summarizes a number of well-known theoretical diagnostic criteria for the estimation of intraparticle transport effects on the observable reaction rate. Tabic 5 gives a survey of the respective criteria for interphase transport effects. It is quite obvious that these are more difficult to use than the simple experimental criteria given in Tables 2 and 3. In general, the intrinsic rate expression has to be specified and, additionally, either the first derivative of the intrinsic rate with respect to concentration (and temperature) at surface... 

Table 4. Theoretical diagnostic criteria for the absence or intraparticle transport effects (simple reactions with arbitrary kinetics).

To access the potential influence of spillover on catalysis and interfacial transport, more qualitative studies are required. Further, it is, for instance, necessary to isolate the individual steps in the phenomena and account for the reaction kinetics of the process. As an example, what is the difference between inter- and intraparticle transport on the support ... 

The intraparticle transport effects, both isothermal and nonisothermal, have been analyzed for a multitude of kinetic rate equations and particle geometries. It has been shown that the concentration gradients within the porous particle are usually much more serious than the temperature gradients. Hudgins [17] points out that intraparticle heat effects may not always be negligible in hydrogen-rich reaction systems. The classical experimental test to check for internal resistances in a porous particle is to measure the dependence of the reaction rate on the particle size. Intraparticle effects are absent if no dependence exists. In most cases a porous particle can be considered isothermal, but the absence of internal concentration gradients has to be proven experimentally or by calculation (Chapter 6). 

Representative rate data for 2,4,5-T and parathion for the experiments on adsorption of pesticides on active carbon are presented in Figures 1 and 2. The (C0 — C)/m values in these plots represent the amount of solute, both in micromoles and milligrams, removed from solution per gram of carbon. Good linearization of the data is observed for the experiments, in accord with expected behavior for intraparticle-transport rate control. Similar linearization was obtained also for data for the other pesticides. The linear traces facilitate comparison of relative rates of adsorption of pesticides, and such comparison is made in column 1 of Table III, using the square f the slope pf each plot as the relative rate constant for the experiment. 

For uptake of solute from solution by porous solids the rate will be endothermic rather than exothermic if intraparticle transport is the rate-limiting mechanism. Because diffusion is an endothermic process while adsorption is exothermic, rate of uptake of solute by porous solids will often increase with increasing temperature while for the same system the equilibrium position of adsorption or adsorption capacity will decrease with increasing temperature. 

Comparison of the values of E for the various pesticides and the neutral and anionic species of the simple nitrophenol indicates that a much higher activation energy is associated with adsorption of neutral molecules (parathion and the neutral nitrophenol) than with adsorption of anions (2,4-D, DNOSBP, and the anionic nitrophenol). This observation suggests the possibility of two different rate-limiting steps in the intraparticle transport mechanism. Current studies are being directed toward more detailed exploration of the observed thermokinetic phenomena. 

For the particle sizes used in industrial reactors (> 1.5 mm), intraparticle transport of the reactants and ammonia to and from the active inner catalyst surface may be slower than the intrinsic reaction rate and therefore cannot be neglected. The overall reaction can in this way be considerably limited by ammonia diffusion through the pores within the catalysts [211]. The ratio of the actual reaction rate to the intrinsic reaction rate (absence of mass transport restriction) has been termed as pore effectiveness factor E. This is often used as a correction factor for the rate equation constants in the engineering design of ammonia converters. 

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