Diffusional resistances, addition

Equation (63) shows that, unlike diffusion in series, the total diffusional resistance for diffusion in parallel is no longer additive. For diffusion in series, the total diffusional resistance is higher than any individual resistance, whereas for diffusion in parallel, the total diffusional resistance is between the minimum and maximum individual resistances. 

The design of fixed-bed ion exchangers shares a common theory with fixed-bed adsorbers, which are discussed in Chapter 17. In addition, Thomas(14) has developed a theory of fixed-bed ion exchange based on equation 18.21. It assumed that diffusional resistances are negligible. Though this is now known to be unlikely, the general form of the solutions proposed by Thomas may be used for film- and pellet-diffusion control. 

When a molecule passes across an interface without chemical reaction, it encounters a total resistance R which is the sum of three separate diffusional resistances. These originate in phase 1, in the interfacial region (perhaps lOA thick) and in phase 2 (see Fig. 1). This additivity of resistances is expressed by ... 

One now has both kinetic and diffusional resistances that are additive. However, ion adsorption processes in soils or clay minerals are assumed to occur in a series rather than in a parallel-reaction mode. As noted earlier, the diffusion steps include both FD and PD processes. One can therefore assign two resistances to the diffusion resistance,... 

The stratum corneum has been estimated to contribute 1000 times the diffusional resistance to chemical penetration as the layers beneath it, except for extremely lipid-soluble compounds with tissue/water partition coefficients greater than 400. As in most other epithelial tissues, the two other layers of the skin (dermis and subcutaneous tissue) offer little resistance to penetration. Once a substance has penetrated the outer epithelium, these tissues are rapidly traversed. This may not be true for highly lipid-soluble compounds, because the dermis may function as an additional aqueous barrier preventing a chemical that has penetrated the epidermis from being absorbed into the blood. 

The second term in Equations (1) and (2) accounts for diffu-sional transfer across the bubble boundary. (A factor e /(1+e p is sometimes (e.g. 49) included in the bracket of Eq. 2 o account for the dense phase diffusional resistance.) There is some question (30) of the extent to which there is interference between the bulk flow and diffusion terms. Nevertheless, most experimental evidence suggests that the two terms are additive and that the diffusional term is described by the penetration theory. With these changes, and including a small enhancement factor for bubble interaction. Sit and Grace (35) have recommended the following equations as being in best accord with existing experimental data ... 

Rate of Growth Crystal growth is a layer-by-layer process, and since growth can occur only at the face of the crystal, material must be transported to that face from the bulk of the solution. Diffusional resistance to the movement of molecules (or ions) to the growing crystal face, as well as the resistance to integration of those molecules into the face, must be considered. As discussed earlier, different faces can have different rates of growth, and these can be selectively altered by the addition or elimination of impurities. 

Under linear-equilibrium conditions, the combined effect of two diffusional resistances in series is treated by simple addition of the resistances. In the irreversible case, either the external or the internal resistance alone controls, with the latter over-taking the former part-way along the breakthrough curve (Section III, B, 2d). However, when the resistances are in relatively equal balance, with 3NP > Nf > 0.3NP, it is possible throughout the range of r values to calculate the breakthrough behavior by assuming an equivalent reaction-kinetic resistance. 

In addition to the above, the model for the catalyst bed module should include equations for the pressure drop across the bed. Although these equations are reasonably simple and their solution as a part of the model equation is a straightforward exercise, they are of crucial importance for without appreciation of the pressure drop consideration fast increases in reactor productivity can be theoretically estimated using the model with very fine particles. Obviously, this is not practically possible because of the excessive pressure drop associated with fine particles. In fact, it is the excessive pressure drop associated with small catalyst pellets that necessitates the use of relatively large catalyst particles in fixed beds. The use of these relatively large particles in turn, is the reason behind the existence of diffusional resistances and thus all the complexities associated with reliable modelling of industrial fixed bed catalytic reactors. 

The three-halves power of dimensionless temperature in the expression for eA( ) is based on the temperature dependence of gas-phase ordinary molecular diffusion coefficients when the catalytic pores are larger than 1 p.m. In this pore-size regime, Knudsen diffusional resistance is negligible. The temperature dependence of the collision integral for ordinary molecular diffusion, illustrated in Bird et al. (2002, pp. 526, 866), has not been included in ea) ). The thermal energy balance given by equation (27-28), which includes conduction and interdiffu-sional fluxes, is written in dimensionless form with the aid of one additional parameter,... 

The effect of the particle density on the catalyst settling and its influence on the reactor performance is shown in Fig. 5 in a labscale size column, where, in addition to the computed catalyst concentration profiles, the corresponding profiles of the conversion and the intraparticle effectiveness factor are plotted. Lower particle density causes a more uniform catalyst distribution and decreases intraparticle diffusional resistances. 

A potential problem associated with adding a support material on the membrane surface is the resistance to substrate diffusion caused by the additional layer and the biofilm growing inside it. Further research is necessary to elucidate whether the advantages of retaining a larger amount of biomass outweigh the disadvantages with increased diffusional resistance for specific applications. 

As noted in Section 1.5, many commercial adsorbents consist of smaU microporous crystals formed into a macroporous pellet Such adsorbents offer two distinct diffusional resistances to mass transfer the micrq>ore resistance of the adsorbent crystals or microparticles and the ihacropore diffdkional resistance of the pellet. When adsorption occurs from a binary (or multicomponent) fluid mixture, there may be an additional resistance to mass transfer associated with transport through the laminar fluid boundary layer surrounding the particle (see Section 6.7). The general situation is as sketched in Figure... 

Wagner begins with the assumption that in certain cases the solubilities and mobilities of A ions in BX and of ions in AY are so low, and consequently the product layers are formed so slowly, that another reaction mechanism may predominate once nucleation has occurred. This reaction mechanism is shown in Fig. 6-11 (b). Essentially, a closed circular flow of cations occurs in the product phases such that the cations diffuse only in their own respective compounds, A quantitative approximation to the reaction rate can be made as follows. Since three phases are in simultaneous contact, it is sufficient to specify only one additional variable in addition to P and T in order to uniquely determine the problem. If the partial pressure Px2 is chosen as this variable, then it is a simple matter to calculate the activity gradient of A in AY if the free energies of formation of the individual compounds are known. This is essentially given by the standard free energy A of the reaction AX 4- BY == AY 4- BX. Then, if the diffusional resistances in AY and BX are known, it is possible to calculate the rate of the displacement reaction for this limiting case as well. 

For the permeable adlayer model, adsorption may occur on the entire platinum surface up to the saturation limits that hold for the carbon-free surface. The slightly reduced uptakes of hydrogen and carbon monoxide on the carbon-covered surface are indeed consistent with the carbon-free limits. In addition, the permeable adlayer model can explain the 100 mV increase in overpotential as an effect due either to steric hindrance of the reaction by the adlayer, or diffusional resistance of H2O (the reactant) or CO2 (the product) through the adlayer. Thus, we attribute the increase in... 

On examining Eq. (4.3.78) it is seen that the time required to attain a certain conversion for a nonporous system is the sum of two terms the time to reach the same conversion under chemical reaction control and that under pure diffusion control. This is an important result that is generally true for systems consisting of first-order rate processes in series. It is to be noted, moreover, that even in the case of a porous system where diffusion and reaction occur in parallel, this additivity holds, if approximately, as seen in Eq. (4.3.75). The first term on the r.h.s. corresponds to t in the absence of diffusional resistance and the second term to t under diffusion control. 

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