Experimental internal diffusion

Adsorption equilibrium of CPA and 2,4-D onto GAC could be represented by Sips equation. Adsorption equilibrium capacity increased with decreasing pH of the solution. The internal diffusion coefficients were determined by comparing the experimental concentration curves with those predicted from the surface diffusion model (SDM) and pore diffusion model (PDM). The breakthrough curve for packed bed is steeper than that for the fluidized bed and the breakthrough curves obtained from semi-fluidized beds lie between those obtained from the packed and fluidized beds. Desorption rate of 2,4-D was about 90 % using distilled water. 

It ought to be verified, however, in all cases, that the experimental Q-9 curve truly represents the distribution of surface sites with respect to a given adsorbate under specified conditions. The definition of differential heats of adsorption [Eq. (39) 3 includes, in particular, the condition that the surface area of the adsorbent A remain unchanged during the experiment. The whole expanse of the catalyst surface must therefore be accessible to the gas molecules during the adsorption of all successive doses. The adsorption of the gas should not be limited by diffusion, either within the adsorbent layer (external diffusion) or in the pores (internal diffusion). Diffusion, in either case, restricts the accessibility to the adsorbent surface. 

Other results also confirm the important role of internal diffusion. Experimental activation energies (67—75 kJ mol"1) of the sucrose inversion catalysed by ion exchangers [506—509] were considerably lower than those of a homogeneously catalysed reaction (105—121 kJ mol"1) [505, 506,508] and were close to the arithmetic average of the activation energy for the chemical reaction and for the diffusion in pores. The dependence of the rate coefficient on the concentration in the resin of functional groups in the H+-form was found to be of an order lower than unity. A theoretical analysis based on the Wheeler—Thiele model for a reaction coupled with intraparticle diffusion in a spherical bead revealed [510,511] that the dependence of the experimental rate coefficient on acid group concentration should be close to those found experimentally (orders, 0.65 and 0.53 for neutralisation with Na+ and K+ ions respectively [511] or 0.5 with Na+ ions [510]). 

A controls the overall rate of conversion, equation 3.81 could be used directly as the rate equation for design purposes. If, however, external mass transfer were important the partial pressures in equation 3.81 would be values at the interface and an equation (such as equation 3.66) for each component would be required to express interfacial partial pressures in terms of bulk partial pressures. If internal diffusion were also important, the overall rate equation would be multiplied by an effectiveness factor either estimated experimentally, or alternatively obtained by theoretical considerations similar to those discussed earlier. 

Based on the analysis of the obtained experimental regularities of the electrochemical process, it may be fundamentally concluded that the process at electrode displays a selfoscillation mechanism. Obviously, self-oscillations happen due to internal diffusion of the surface components of the mimetic electrode, which is not affected by solution mixing intensity. Mixing causes a strong influence on the external diffusion and OH- anion drainage from the interface layer. 

In this case the experimental determination of the rate and the knowledge of the reaction order is sufficient to assess the importance of internal diffusion limitations. Note that only n > -1 is meaningful. 

An experimental test to verify the absence of significant concentration gradients inside the catalyst pellet is based on the inverse proportional relation between the effectiveness factor and the pellet diameter for strong internal diffusion limitations. Hence, a measured rate which is independent of the pellet size indicates that internal diffusion limitations can be neglected. Care should be taken to avoid artifacts. External heat transfer effects also depend on pellet size and for exothermic reactions might compensate the internal diffusion limitations. If the catalyst pellet consists of a support with an non-uniformly distributed active phase, crushing and sieving to obtain smaller pellets is hazardous. 

A qualitative evidence of the above are the data reported in [52]. It has been established that there is a correlation between the calculated rate of internal diffusion foam collapse and the experimentally determined rate. To obtain a stable foam from poor surfactants (alcohols, acids, etc.) under these conditions is hardly possible because of either insufficient dynamic elasticity of foam films or the lack of equilibrium elasticity (for films from insoluble surfactants). Furthermore, the n barrier for films from acid or alcohol solutions is low and the typical capillary pressures for a real foam are sufficient to induce disturbance of the film equilibrium and, respectively, foam collapse. 

The question remains as to when the various diffusion effects really influence the conversion rate in fluid-solid reactions. Many criteria have been developed in the past for the determination of the absence of diffusion resistance. In using the many criteria no more information is required than the diffusion coefficient DA for fluid phase diffusion and for internal diffusion in a porous pellet, the heat of reaction and the physical properties of the gas and the solid or catalyst, together with an experimental value of the observed global reaction rate (R ) per unit volume or weight of solid or catalyst. For the time being the following criteria are recommended. Note that intraparticle criteria are discussed in much greater detail in Chapter 6. 

Madon and Boudart propose a simple experimental criterion for the absence of artifacts in the measurement of rates of heterogeneous catalytic reactions [R. J. Madon and M. Boudart, Ind. Eng. Chem. Fundam., 21 (1982) 438]. The experiment involves making rate measurements on catalysts in which the concentration of active material has been purposely changed. In the absence of artifacts from transport limitations, the reaction rate is directly proportional to the concentration of active material. In other words, the intrinsic turnover frequency should be independent of the concentration of active material in a catalyst. One way of varying the concentration of active material in a catalyst pellet is to mix inert particles together with active catalyst particles and then pelletize the mixture. Of course, the diffusional characteristics of the inert particles must be the same as the catalyst particles, and the initial particles in the mixture must be much smaller than the final pellet size. If the diluted catalyst pellets contain 50 percent inert powder, then the observed reaction rate should be 50 percent of the rate observed over the undiluted pellets. An intriguing aspect of this experiment is that measurement of the number of active catalytic sites is not involved with this test. However, care should be exercised when the dilution method is used with catalysts having a bimodal pore size distribution. Internal diffusion in the micropores may be important for both the diluted and undiluted catalysts. 

The effective diffusion coefficients were calculated from the experimentally observed data (time, amount of cation exchanged, temperature), using Paterson s solution of Fick s second law, or published approximate solutions (8, 16). Taking into consideration particle shape and particle size distribution, the differential coefficients of internal diffusion in ion exchange can be ascertained by a method previously described (9). 

Thus, in spite of a satisfactory agreement of Equation 17 with experimental data for systems with weakly convex adsorption isotherms, the internal diffusion coefficients, evaluated for these cases according to Equation 17, are in fact below their actual value. 

At the other end of the liquid range, in the supercooled region, experimental internal energy is quantitatively reproduced by SPC/E at 255 K, while the increase of both dielectric constant and heat capacity is only qualitatively accounted for [163]. The agreement with experiment worsen for thermal expansion coefficient, a, see Table 4. Diffusion is well reproduced at 300 K, but its slowing down at 255 K is underestimated by SPC/E. 

In view of the calculated values of < )c and ( )ni (Table 2) the limitation to internal oxygen diffusion under the experimental conditions is even lower in the zeolite erystal than in the catalyst particle. The Thiele modulus in zeolite crystals for a severe deactivation state corresponding to activity = 0.20 [7] is < )c = 3.09 10 3 (Sample 5), for an initial coke content of 1.9 wt% (approximately 40 wt% of the coke content needed for blockage of the internal zeolite channels). This low value of Thiele modulus is evidence that oxygen-coke contact is not limited by internal diffusion in the deactivated eatalyst. 

The previous equations describing the adsorption/desorption behavior of gases lead to models describing sink effects in indoor environments. In addition, transport of molecules within the sink material can have a major impact on desorption rates thus, models accounting for internal diffusion have been developed for indoor sinks. Models based on fundamental theories are preferred over empirical approaches, but some studies rely on experimental data to fit empirical models [21-23]. 

Example 10-1 Experimental, global rates are given in Table 10-2 for two levels of conversion of SOj to SO3. Evaluate the concentration difference for SO2 between bulk gas and pellet surface and comment on the significance of external diffusion. Neglect possible temperature differences. The reactor consists of a fixed bed of x -in. cylindrical pellets through which the gases passed at a superficial mass velocity of 147 lb/(hr)(ft ) and at a pressure of 790 mm Hg. The temperature of the catalyst pellets was 480°C, and the bulk mixture contained 6.42 mole % SOj and 93.58 mole % air. To simplify the calculations compute physical properties on the basis of the reaction mixture being air. The external area of the catalyst pellets is 5.12 ft /lb material. The platinum covers only the external surface and a very small section of the pores of the alumina carrier, so that internal diffusion need not be considered. 

For the reaction under consideration, values of the reaction rate as a function of concentration and catalyst decay time may be obtained by numerical differentiation of the experimental data obtained by Prasad and Doraiswamy (1974) at various space velocities and catalyst decay times in a fixed-bed reactor. The bed length was maintained small enough to give isothermal conditions to within 2°C. It was also ensured that the feed velocity was high enough, and the particle size small enough, to eliminate external and internal diffusion effects, respectively. The kinetic parameters, including decay time, will vary with the type of silica gel used. The data obtained for the silica gel used are summarized in Table CS3.1. 

An experimental approach to verify the impact of internal diffusion is to perform experiments with catalyst of different particle sizes. The experimental results on the impact of particle size in cinnamaldehyde hydrogenation are presented in Table 9.11, demonstrating that a decrease in mean particle size increases both the catalyst activity and selectivity and that the smallest catalyst fraction (< 45 pm) can be considered to be sufficient to safely eliminate the influence of internal diffusion, as confirmed by the calculations. 

Most of these long-term emissions are due to internal diffusion within the building material itself. In such cases, with solid and dried materials, SER, is not affected to any great extent by the air velocity over the material surface. An overview of parameters affecting SER of VOCs from building products is given by Wolkoff (1995) and experimentally evaluated (Wolkoff, 1996b). 

From the data in Table 1 it may be concluded that, under the experimental conditions applied, both the hydrogenation of benzene and of toluene are externally diffusionally controlled and that at the same time we are dealing with internal diffusion of hydrogen and of the substrates, viz. in the pores of the catalyst. 

In our laboratories extensive studies on the catalytic hydrogenation of aromatic nitrocompounds, as an example of the catalytic three-phase reactions, have been carried out in reactors of different types - e g. see [8-10]. In all cited cases the time consumed for kinetic investigations had a very significant contribution to the total experimental effort [11-13]. Particularly for the hydrogenation over palladium on alumina catalyst, the experimental investigations leading to the detection and quantitative description of internal diffusion resistances in catalyst pellets have taken a lot of time. 

The application of ANN for a representation of reaction kinetics can be a very promising method to solve modelling problems. Besides intrinsic kinetics also internal diffusion resistances can be included into the neural network based model. This approach significantly reduces the time required for experimental studies. Despite that neural networks do not help to understand and develop a real reaction mechanism, they make the prediction of the reactor behaviour possible. This approach can be essential in the case of complex or uncertain kinetics - e.g. for polymerization reactions. In this study the neural network approach has been tested for a batch reactor. A trained network can be successfully implemented into any type reactor model. 

In practice, lower values than those calculated according to Eq. (30) are found for intermediate Reynolds numbers, only approaching them in the presence of oscillation. A compilation of experimental R factors is available (Cl). The experimental effective diffusivity in gas absorption with internal circulation has been plotted against the axial velocity representing the flow adjacent to the interface (G7), and shows a direct relation between circulation and internal resistance. Similar results are reported (C6) for other gas-liquid and liquid-liquid systems. 

Example 3.6.d-l Experimental Differentiation Between External and Internal Diffusion Control... 

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