Pore diffusion determination

There are data showing that at the same contact time, but different linear velocities, there is no difference in the performance of a carbon system. It is obvious then that the effect of linear velocity on the diffusion through the film around the particle and the ratio of the magnitude of the film diffusion to the pore diffusion are the factors that determine the effects, if any, that occur. Therefore, the linear velocity cannot be ignored completely when evaluating a system. Systems at the higher linear velocity (LV) treat more liquid per volume of carbon at low-concentration levels and the mass-transfer zone (MTZ) is shorter. 

Schiesser and Lapidus (S3), in later studies, measured the liquid residencetime distribution for a column of 4-in. diameter and 4-ft height packed with spherical particles of varying porosity and nominal diameters of in. and in. The liquid medium was water, and as tracers sodium chloride or methyl orange were employed. The specific purposes of this study were to determine radial variations in liquid flow rate and to demonstrate how pore diffusivity and pore structure may be estimated and characterized on the basis of tracer experiments. Significant radial variations in flow rate were observed methods are discussed for separating the hydrodynamic and diffusional contributions to the residence-time curves. 

Models of chemical reactions of trace pollutants in groundwater must be based on experimental analysis of the kinetics of possible pollutant interactions with earth materials, much the same as smog chamber studies considered atmospheric photochemistry. Fundamental research could determine the surface chemistry of soil components and processes such as adsorption and desorption, pore diffusion, and biodegradation of contaminants. Hydrodynamic pollutant transport models should be upgraded to take into account chemical reactions at surfaces. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

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. 

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

The occurrence of pore diffusion can usually be determined by simply grinding the catalyst into smaller and smaller particles. If the rate per gram of catalyst increases as the particles become smaller and smaller, then pore diffusion is likely to be occurring. This effect is due to the fact that the pore lengths are decreased by the catalyst particles being ground into smaller and smaller pieces. Eventually, the pores become short enough that the reactants can readily diffuse in and out of them faster than the chemical reaction occurs on the surface. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

The presence (or absence) of pore-diffusion resistance in catalyst particles can be readily determined by evaluation of the Thiele modulus and subsequently the effectiveness factor, if the intrinsic kinetics of the surface reaction are known. When the intrinsic rate law is not known completely, so that the Thiele modulus cannot be calculated, there are two methods available. One method is based upon measurement of the rate for differing particle sizes and does not require any knowledge of the kinetics. The other method requires only a single measurement of rate for a particle size of interest, but requires knowledge of the order of reaction. We describe these in turn. 

The first-order decomposition of A is run in an experimental mixed flow reactor. Find the role played by pore diffusion in these runs in effect determine whether the runs were made under diffusion-free, strong resistance, or intermediate conditions. 

The first term represents the reaction rate on the external area (no pore diffusion), while the second term represents the reaction rate occurring inside the pores. The terms in the denominators represent chemical and diffusion processes it is the relative size of these terms which determine whether or not diffusion is affecting the rate of the chemical reaction. The following table gives estimates of the size of these terms as well as the name of the process which they represent. 

The magnitude of the diffusion coefficients given in Table I can be compared with a value of 3.3 X 10 5 cm.2/sec. determined experimentally by Stokes (26) for HCl in bulk solution at infinite dilution. The pore diffusion coefficients listed in Table I for HCl vary by a factor of (2 - 4) X 10"2 from that given by Stokes. McNeill and Weiss (15) have indicated that active carbon can be considered as a weak-base anion-exchange sorbent. According to Helfferich (13), diffusion coefficients in such resins can be several orders of magnitude less than the corresponding bulk solution coefficients. The Cl" ion probably limits the rate of diffusion, since its mobility in aqueous solution is much less than that of the H30+ ion. Further evidence to support this conclusion has been obtained in the present work from determinations of pore diffusion... 

Decomposition reactions are solid-gas reactions which do not involve diffusional transport through the solid. Their reaction rates are determined by surface kinetics and possibly pore diffusion. The assumption of local equilibrium is not valid. The course of an isothermal decomposition is schematically illustrated in Figure 15-15. There is often an induction period followed by a rapid increase in relative yield until, after the inflection point, the reaction eventually ceases (the yield will not always be 100%). Since atomic transport in crystals is normally not involved in these decomposition reactions, we shall restrict ourselves to a few comments only. 

If tj - 1 the reaction is not, or not significantly, influenced by pore diffusion. If tj pore diffusion is the sole dominating rate-limiting step. For the determination of Tj, the combined diffusion and reaction equation has to be solved. With a sequential model of the two rate phenomena, diffusion and reaction, and with the assumption of spherical geometry and validity of the Michaelis-Menten equation for the en2yme kinetics, r = kcat[E] [S]/(JCM + [S]), Eq. (5.58) results. 

A model Is presented for char gasification with simultaneous capture of sulfur In the ash minerals as CaS. This model encompasses the physicochemical rate processes In the boundary layer, In the porous char, and around the mineral matter. A description of the widening of the pores and the eventual collapse of the char structure Is Included. The modeling equations are solved analytically for two limiting cases. The results demonstrate that pore diffusion effects make It possible to capture sulfur as CaS In the pores of the char even when CaS formation Is not feasible at bulk gas conditions. The model predictions show good agreement with experimentally determined sulfur capture levels and reaction times necessary to complete gasification. 

In many laboratory experiments with powder catalysts, with a relatively small particle size, the pore diffusion is usually dominant. The presence or absence of this effect can be determined with the Thiele modulus, cf. Equation (20). Generally, experiments are done with the same amount of catalyst, but different particle size. If r is independent of the particle size, then mass transport limitations are excluded, indicated by being smaller than 1 ... 

Based on the study of Sugano et al. (2000) and our predictive VolSurf model for this series, it can be concluded that factors like size and shape previously reported to affect paracellular permeability are indeed important to explain the local structure-permeability relationship of this chemotype. Usually, permeability via paracellular aqueous pore diffusion depends on the size of the solute and its diffusion coefficient in water. Another important factor is lipophilicity. Between intestinal absorption and both volume and lipophilicity, a negative correlation was reported for this series of thrombin inhibitors. In addition, hydrogen bonding properties and dipolarity are factors that determine... 

The pore diffusivity used in this analysis was determined by the Renkin equation4, the axial dispersion coefficient calculated by assuming a constant Peclet number of 0.2, and the mass transfer coefficient from the bulk to the particle surface calculated by the correlation of Wakao and Kaguei. The product of the heat capacity and density of the solid phase was taken to be the same as that used by Raghavan and Ruthven17. The density of the fluid phase was assumed to be that of pure C02 and was calculated from data provided by the Dionix Corporation in their AI-450 SFC software. Constant pressure heat capacities for the mobile phase were also assumed to be that of pure C02 and were taken from Brunner3. 

For the same catalytically active material but with different catalyst carriers, different reaction rates and rate equations can be expected. Consider the hydrogenation of 2,4 DNT as discussed in Section 9.2 for 5% Pd on an active carbon catalyst with an average particle size of 30 (im [3]. These experiments were later repeated but with a Pd on an alumina catalyst [5]. This catalyst consisted of 4 x 4mm pellets, crushed to sizes of lower than 40/um in order to avoid pore diffusion limitations. In Figure 2.9 the measured conversion rates are given as a function of the averaged catalyst particle diameter, showing that above a diameter of 80/im the rate measured diminishes. For small particles they determined the rate equations under conditions where there were no pore diffusion lim-... 

Arve and Liapis [34] suggest estimating the parameters characterizing the intraparticle diffusion and the adsorption-desorption step mechanisms of affinity chromatography from the experimental data obtained in a batch system. The numerical simulations of the chromatographic process will use the values of the parameters of the adsorption isotherm and those of the effective pore diffusion as determined from stirred tank experiments together with the film mass transfer coefficients calculated from chemical engineering expressions found in the literature. 

Thus the surface temperature is fixed at 65.6°C with pore diffusion the rate determining step. At all other surface temperatures, pore diffusion gives the largest t value for the mass transfer steps, and boimdary layer heat transfer gives the largest t value for the heat transfer steps. 

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