External and internal diffusion

In many situations of practical interest, an appreciable drop in concentration arises between a fluid phase and the external surface of the catalyst because of diffusional resistance. In the steady state, the rate of diffusion to the external surface equals the rate of input to the pore mouth, rd = kga(Cg-Cs) = D(dC/dr)r=RfC=Cs (7.37) 

When the rate of reaction is known as a function of the concentration, rc = f(C) (7.38) 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration, 

For a second order reaction in a slab or a sphere, analytical solutions proceed in terms of elliptic functions beyond the solution of P7.03.ll, although a numerical solution throughout may be preferable. Such a numerical procedure is adopted in P7.03.19 for a second order reversible reaction. 

In those cases where the internal effectiveness is known in equation form, the steady state rate relation may be written 

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P7.03.19. REVERSIBLE REACTION WITH EXTERNAL AND INTERNAL DIFFUSION. 

Considerable progress has been made within the last decade in elucidating the effects of the microenvironment (such as electric charge, dielectric constant and lipophilic or hydrophilic nature) and of external and internal diffusion on the kinetics of immobilized enzymes (7). Taking these factors into consideration, quantitative expressions have been derived for the kinetic behavior of relatively simple enzyme systems. In all of these derivations the immobilized enzymes were treated as simple heterogeneous catalysts. 

In addition, the reduction of NOj is a very fast reaction and is controlled by external and internal diffusion [27, 30]. In contrast, the oxidation of SO2 is very slow and is controlled by the chemical kinetics [31]. Accordingly, the SCR activity is increased by increasing the catalyst external surface area (i.e. the cell density) to favor gas-solid mass transfer while the activity in the oxidation of SO2 is reduced by decreasing the volume of the catalyst (i.e. the wall thickness) this does not affect negatively the activity in NO removal because significant ammonia concentrations are confined near the external geometric surface of the catalyst. 

When external and internal diffusion resistances affect simultaneously the rate of the biocatalytic reaction, the relative contributions of each effect must be estimated... 

Previously to any kinetic experiment, we have studied the influence of external and internal diffusion by changing the stirring speed and the particle size respectively. It has been found that for more than 1000 r.p.m. and particle size < 0.25 mm the reaction is not controlled by either external or internal diffusion. 

Horvath, C. and Engasser, J-M. Biotechnoi. Bioeng. 16 (1974) 909. External and internal diffusion in heterogeneous enzyme systems. 

Basic assumptions 1 to 3 were put forward as a result of analysis, firstly, of the reaction-diffusion mechanism described in detail by V.I. Arkha-rov1,46,47 and, secondly, of a linear-parabolic equation derived for the first time by U.R. Evans from somewhat different considerations. It should be noted that similar assumptions were used earlier by B.Ya. Pines,9,131 who in deriving differential forms of kinetic equations summed up the duration of external and internal diffusion. 

Thus at a temperature of 600 °C, the reaction in the catalyst layer of particle diameter 0.029 cm occurs in the kinetic region. In the layer of conunercial catalyst of diameter 0.6 cm, external and internal diffusions considerably affect the rate of oxidation. 

Generally, the overall kinetics is primarily governed by the external and internal diffusion (also called the two-step mass transport mechanism Steps 2 and 3 above). The intraparticle diffusion model described below can therefore be used for calculation of ion uptake by IX resins. When the mass transfer is due only to the diffusion of adsorbate molecules through the pore liquid, a pore diffusion model is often used. On the other hand, in the case where the intraparticle mass transfer is contributed by the diffusion... 

An isomerization reaction has the simple form A —> B. Assuming that operating conditions and the condition of the catalyst are such that the external- and internal-diffusion steps have negligible concentration gradients, propose rate equations for the following cases ... 

The selectivity at a position in a fluid-solid catalytic reactor is equal to the ratio of the global rates at that point. The combined effect of both external and internal diffusion resistance can be displayed easily for a set of parallel reactions. We shall do this first and then consider how internal resistance influences the selectivity for other reaction sequences. 

This might be the dehydrogenation of a mixed feed of propane and n-butane, where the desired catalyst is selective for the K-butane dehydrogenation. Suppose that the temperature is constant and that both external and internal diffusion resistances affect the rate. At steady state, the rate (for the pellet, expressed per unit mass of catalyst) may be written in terms of either Eq. (10-1) or Eq. (11-44),... 

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. 

A temperature gradient would also be expected. For an isothermal case, with rj set equal to 1, multiple steady-state solutions may be found (see Figure 10), and the concentration gradient is very significant at temperatures above 427°C (800°F). The non-isothermal catalytic effectiveness factors for positive order kinetics under external and internal diffusion effects were studied by Carberry and Kulkarni (8) they also considered negative order kinetics. 

Diffusion of reactants to the external surface is the first step in a solid-catalyzed reaction, and this is followed by simultaneous diffusion and reaction in the pores, as discussed in Chapter 4. In developing the solutions for pore diffusion plus reaction, the surface concentrations of reactants and products are assumed to be known, and in many cases these concentrations are essentially the same as in the bulk fluid. However, for fast reactions, the concentration driving force for external mass transfer may become an appreciable fraction of the bulk concentration, and both external and internal diffusion must be allowed for. There may also be temperature differences to consider these will be discussed later. Typical concentration profiles near and in a catalyst particle are depicted in Figure 5.6. As a simplification, a linear concentration gradient is shown in the boundary layer, though the actual concentration profile is generally curved. 

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

In addition, measurements of the intrinsic reaction rate (free of external and internal diffusion limitations) were achieved by the strict control of the thickness of SIM shell on SIM/alumina beads. The Hnearity observed between the variation of the MOF layer thickness and the conversion observed for the Knoevenagel condensation demonstrated that the reaction takes place inside the whole MOF layer through the porosity and not just at the external surface. 

However, in other cases internal diffusion limitation can be significant even with very thin washcoat thicknesses [127], when temperature is high (> 700°C). This refers, for example, to catalytic combustions, which are extremely fast. Hayes et al. [135] evaluated the extent of intraphase and interphase resistances to the catalytic conversion of low concentrations of carbon monoxide in air in a tube wall reactor (coated with a platinum-alumina deposit). Above 610 K there was strong evidence of both intraphase and interphase resistances to catalytic conversion. In Sections 8.3.2, 8.3.3, and 8.3.4, we provide a systematic analysis for prediction of the extension of external and internal diffusion limitations. 

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