Diffusion criteria

The main difference between the cases discussed in this section on selectivity and the previously discussed case of a simple polystep reaction resides in the fact that here the single component which generates the species that becomes the intermediate in the polyfunctional composite can itself generate a distinct product species with appreciable yield. Since the coupling between the Y- and the X-system occurs in any event through mass-transport of intermediates between X-sites and T-sites, the diffusion criteria already discussed must apply or the kinetic schemes which accomplish interception or selectivity control will not be physically and effectively accomplished. The criterion, of formula (15) should be satisfied. ... 

The relative insensitivity of this type of diffusion criterion to particle shape and to assumption of exact kinetics, has been discussed in connection with the macroscopic reactant diffusion problem on catalyst granules (7). The condition (15) is a general order-of-magnitude criterion defining the physical conditions of intimacy between the component systems for no mass-transport inhibition. It defines a requirement for realizing the formal kinetics of polystep reactions. 

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

Since the diffusion coefficient is constant for a given material, Eq. (2.63) shows that the time required for a displacement increases with the square of the distance traveled. This can be understood by thinking that the displacement criterion would be met by finding the diffused particle anywhere on the surface of a sphere of radius x after time t if it started at the origin. The surface area of a sphere is proportional to the square of its radius. 

For many laboratoiy studies, a suitable reactor is a cell with independent agitation of each phase and an undisturbed interface of known area, like the item shown in Fig. 23-29d, Whether a rate process is controlled by a mass-transfer rate or a chemical reaction rate sometimes can be identified by simple parameters. When agitation is sufficient to produce a homogeneous dispersion and the rate varies with further increases of agitation, mass-transfer rates are likely to be significant. The effect of change in temperature is a major criterion-, a rise of 10°C (18°F) normally raises the rate of a chemical reaction by a factor of 2 to 3, but the mass-transfer rate by much less. There may be instances, however, where the combined effect on chemical equilibrium, diffusivity, viscosity, and surface tension also may give a comparable enhancement. 

An example where one metal melts before the densihcation process, is the formation of bronze from a 90 10 weight percentage mixture of copper and tin. The tin melts at a temperature of 505 K, and the liquid immediately wets the copper particles, leaving voids in the compact. The tin then diffuses into the copper particles, leaving further voids due to dre Kirkendall effect. The compact is therefore seen to swell before the hnal sintering temperature of 1080 K is reached. After a period of homogenization dictated by tire criterion above, the alloy shrinks on cooling to leave a net dilatation on alloy formation of about 1%. 

This could occur if the separation ratio of another solute pair, although larger, was very close to that of the critical pair but contained solutes, for example, of widely different molecular weight (and, consequently, very different diffusivities). Fortunately, the possibility of this situation arising is remote in practice, and will not be considered in this discussion. It follows that the efficiency required to separate the critical pair, numerically defined, is the first performance criterion. 

An injection moulding is in the form of a flat sheet 100 mm square and 4 nun thick. The melt temperature is 230°C, the mould temperature is 30°C and the plastic may be ejected from the mould at a centre-line temperature of %°C. If the runner design criterion is that it should be ejectable at the same instant as the moulding, eshmate the required runner diameter. The thermal diffusivity of the melt is 1 x 10 m /s. 

Among the commonly used criteria is the air distribution performance index (ADPI), defined as the percentage of ItKadons where a combination of air temperature and air velocity meets comfort requirements. This criterion is based on experimental results of air diffuser performance for specifically tested room configurations. Data on the ADPI are available only for sedentary activity. 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

The occurrence of the mesophase in the fiber is confirmed by x-ray diffraction examination. The occurrence of three equatorial reflections 010, 110, and 100, the absence of layer and meridional reflections, and the manifestation of the intensity maximum of diffusively scattered radiation at 20 = 19 in the fiber diffraction pattern are the criterion for the presence of the mesophase. The... 

Provided the mole fraction of A does not fall below N, then the oxide AO will be formed exclusively. The important criterion is the ratio of the oxidation parabolic rate constant to that of the diffusion coefficient of For A1 in Fe, the parabolic rate constant is very low, whilst the diffusion coefficient is relatively high, whereas the diffusion coefficient of Cr is much lower. Hence, the bulk alloy composition of A1 in iron required for the exclusive formation of AI2O3 at any given temperature is lower than the Cr concentration required for the exclusive formation of CrjOj. 

In preparative selective chromatography, the formation of broad zones of the substances is determined by the formation of sharp boundaries of each zone. The formation of these sharp boundaries of substance zones in column sorption processes for systems in which the interphase transfer is limited by substance diffusion in sorbent grains [104, 122, 123] is determined by the dimensionless criterion X ... 

It is therefore important to examine under what conditions the above criterion is met (i.e. fast ion backspillover relative to its desorption or consumption) for otherwise the promotional process will be internally diffusion limited not due to slow diffusion of the reactants but due to slow diffusion (backspillover) of the promoting species. 

Checking the absence of internal mass transfer limitations is a more difficult task. A procedure that can be applied in the case of catalyst electrode films is the measurement of the open circuit potential of the catalyst relative to a reference electrode under fixed gas phase atmosphere (e.g. oxygen in helium) and for different thickness of the catalyst film. Changing of the catalyst potential above a certain thickness of the catalyst film implies the onset of the appearance of internal mass transfer limitations. Such checking procedures applied in previous electrochemical promotion studies allow one to safely assume that porous catalyst films (porosity above 20-30%) with thickness not exceeding 10pm are not expected to exhibit internal mass transfer limitations. The absence of internal mass transfer limitations can also be checked by application of the Weisz-Prater criterion (see, for example ref. 33), provided that one has reliable values for the diffusion coefficient within the catalyst film. 

The Merrill and Hamrin criterion was derived for a first-order reaction. It should apply reasonably well to other simple reactions, but reactions exist that are quite sensitive to diffusion. Examples include the decomposition of free-radical initiators where a few initial events can cause a large number of propagation reactions, and coupling or cross-linking reactions where a few events can have a large effect on product properties. 

Molecules must come into contact for a reaction to occur, and the mechanism for the contact is molecular motion. This is also the mechanism for diffusion. Diffusion is inherently important whenever reactions occur, but there are some reactor design problems where diffusion need not be explicitly considered, e.g., tubular reactors that satisfy the Merrill and Hamrin criterion. Equation (8.3). For other reactors, a detailed accounting for molecular diffusion may be critical to the design. 

To ensure an apples-to-apples comparison, reduce kt until aoutlatn matches the value of 0.44321 achieved in the tube. This is found to occur at A t = 0.9311. Diffusion is now added until ,/ ,>, = 0.43849 as in the case of a circular tube with F/7 = 0.003. This is found to occur at about SiaVY = 0.008. Thus, the flat-plate counterpart to the Merrill and Hamrin criterion is... 

Flow in a Tube. Laminar flow with a flat velocity profile and slip at the walls can occur when a viscous fluid is strongly heated at the walls or is highly non-Newtonian. It is sometimes called toothpaste flow. If you have ever used Stripe toothpaste, you will recognize that toothpaste flow is quite different than piston flow. Although Vflr) = u and z(7) = 1, there is little or no mixing in the radial direction, and what mixing there is occurs by diffusion. In this situation, the centerline is the critical location with respect to stability, and the stability criterion is... 

The echoplanar imaging thrombolysis evaluation trial (EPITHET) is the first large study designed specifically to assess whether the existence of a diffusion-perfusion mismatch should be an eligibility criterion for thrombolysis. 

Table 1 summarizes several of the experimental methods discussed in this chapter. A need exists for new or revised methods for transport experimentation, particularly for therapeutic proteins or peptides in polymeric systems. An important criterion for the new or revised methods includes in situ sampling using micro techniques which simultaneously sample, separate, and analyze the sample. For example, capillary zone electrophoresis provides a micro technique with high separation resolution and the potential to measure the mobilities and diffusion coefficients of the diffusant in the presence of a polymer. Combining the separation and analytical components adds considerable power and versatility to the method. In addition, up-to-date separation instrumentation is computer-driven, so that methods development is optimized, data are acquired according to a predetermined program, and data analysis is facilitated. 

Criteria 1-3 are the cardinal characteristics of Fickian diffusion and disregard the functional form of D(ci). Violation of any of these is indicative of non-Fickian mechanisms. Criterion 4 can serve as a check if the D(ci) dependence is known. As mentioned, it is crucial that the sorption curve fully adhere to Fickian characteristics for a valid determination of D from the experimental data. At temperatures well above the glass transition temperature, 7 , Fickian behavior is normally observed. However, caution should be exercised when the experimental temperature is either below or slightly above 7 , where anomalous diffusion behavior often occurs. 

Weisz-Prater criterion uses measured values of the rate of reaction to determine if internal diffusion is limiting the reaction. 

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

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