Internal diffusion resistance

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

If only internal diffusion resistances are significant, the boundary conditions are... 

The essence of monolithic catalysts is the very thin layers, in which internal diffusion resistance is small. As such, monolithic catalysts create a possibility to control the selectivity of many complex reactions. Pressure drop in straight, narrow channels through which reactants move in the laminar regime is smaller by two or three orders of magnitude than in conventional fixed-bed reactors. Provided that feed distribution is optimal, flow conditions are practically the same across a monolith due to the very high reproducibility of size and surface characteristics of individual monolith passages. This reduces the probability of occurrence of hot spots resulting from maldistributions characteristic of randomly packed catalyst beds. 

Because internal diffusion resistance is also significant, not all of the interior surface of the pellet is accessible to the concentration at the external surface of the pellet, C j. We have already learned that the effectiveness factor is a measure of this surface accessibility [see Equation (12-38)] ... 

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),... 

Finally, the first-order kinetics allowed a direct display of the relative importance of the diffusion resistances on the global rate. The quantities 0.92 and 6.15 in the denominator of the previous equation measure the resistances of external diffusion and internal diffusion plus reaction. The value of rj divides the latter into internal-diffusion resistance and the resistance of the intrinsic reaction on the interior catalyst site . 

If, however, the alumina particles are represented by the pseudohomoge-neous model, assuming slab geometry, neglecting the axial dispersion and the film mass transfer resistance (it has been demonstrated (ref. 2) that the effects of these two variables are negligible in comparison with the effect of the internal diffusion resistance), the theoretical transfer function for the study zone is given by ... 

The effective diffusion coefficient depends on the particle porosity, the pore diameter, the tortuosity, and the nature of the diffusing species. For gas-filled pores, the above factors can be allowed for to make a reasonable estimate of the effective diffusivity in the gas phase. However, diffusion of adsorbed molecules along the pore walls, called surface diffusion, often contributes much more to the total flux than diffusion in the gas phase. This is particularly evident in the adsorption of water vapor on silica gel and the adsorption of hydrocarbon vapors on carbon, where the measured values of correspond to internal and external coefficients of comparable magnitude or even to external film control, For adsorption of solutes from aqueous solutions, surface migration is much less important, and the internal diffusion resistance generally dominates the transfer process. 

The predicted breakthrough curve is shown as a solid line in Fig. 25,10. The slope increases with time, and c/cg becomes 1.0 at N(x — 1) = 1,0. In practice, the breakthrough curves are usually S shaped, because the internal diffusion resistance is not negligible, and it increases somewhat when the solid becomes nearly saturated. 

The actual length of the adsorption zone may be much greater than the estimated value of L,ni of 11.1 cm because of internal diffusion resistance, which is hard to predict. However, even if the length of unused bed is 25 cm, the time to breakthrough would be about 10 years. 

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. 

Using artificial neural networks (ANN) the reaction system, including intrinsic reaction kinetics but also internal mass transfer resistances, is considered as a black-box and only input-output signals are analysed. With this approach the conversion rate of the i-th reactant into the j-th species can be expressed in a general form as a complex function, being a mathematical superposition of all above mentioned functional dependencies. This function includes also a contribution of the internal diffusion resistances. So each of the rate equations of Eq. 5 can be described with the following function based on the vanables which uniquely define the state of the system ... 

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. 

The film mass transfer resistance is taken as negligible compared to the internal diffusion resistance. 

Increasing values of Bi correspond to increasing internal diffusion resistance compared to the external mass transfer. For Bi — oo, the external mass transfer is fast and the reactant concentration on the outer catalyst surface is equal to the bulk-phase concentration the reaction is only influenced by internal transport processes. The influence of internal and external mass transfers on the overall efficiency (Eq. (11.2)) is shown in Figure 11.4 as an example. The catalyst efficiency decreases strongly at small mass Biot numbers. This is because of the reduced... 

The main problem related to SLPC is the loss of solvent because of evaporation in a continuously operated catalytic reactors. This problem can be overcome by using ionic liquids as solvent [17-20]. Ionic liquids are molten salts and their partial pressure is low under conditions commonly used for hydroformylation and hydrogenation reactions. As generally observed for SLPC, the catalytic activity and product selectivity depends on the liquid loading and the nature of the porous support [21]. A detailed discussion can be found in [22]. In order to diminish internal diffusion resistances within the supported liquids by using microstruc-tured supports with high porosity like foams or fibrous materials, are proposed for SLPC [23]. 

Figure 5.75 shows the bulk reaction rate of the biofilm k in dimensionless form as a function of the bulk concentration A with different biofilm characteristic values of A/B (Harremoes, 1978). Even though the depth of penetration may be small compared with the entire film thickness, the effect of internal diffusion resistances is to mask the true zero-order kinetics (see Equ. 5.111), yielding an apparent reaction order of one-half. 

The entrapment process involves physically entrapping the enzyme with insoluble polymers of synthetic or natural origin. It is a type of gel entrapment in which the lipase is brought into a monomeric solution, which upon polymerization leads to its entrapment. By using this method, the lipase is kept free in solution however, it is also restricted by the polymer (Murty et al., 2002). The advantage of such a technique is that it is simple, the enzyme does not interact with the polymer, denaturation is avoided, and it has a high initial yield. However, the mass transfer limitation, in the form of internal diffusion resistance, is a problem, as the substrate/product diffusion rate across the membrane is a limiting factor (Villeneuve et al., 2000). 

To increase the amount of immobilized enzyme per unit weight of the support, immobilization is carried out in porous supports with large internal surface areas. However, the substrate has to diffuse through the internal pores of the particle in order to reach the active site. This results in additional diffusion resistance. The substrate concentration profile in this case is shown in Figure 4.4. Internal diffusion resistance depends on particle size and shape, external substrate concentration, and effective diffusion within the particle, D. The latter depends on the molecular diffusion of the substrate in the support matrix, and on the porousness (ep and tortuosity (t) of the pores in the particle (Park et al., 2006 Young and Al-Duri, 1996). The effective diffusion can be determined from different mathematical expressions as shown in (Equations 4.50 to 4.52) (Pilkington et al, 1998) ... 

Fluid flow rate is also considered in extraction optimization. It is usually used to determine whether the extraction is solubility or internal diffusion controlled. Typically, solubility controlled extractions show a direct correlation to the flow rate, whereas internal diffusion controlled extractions show this much less. In internal diffusion-controlled processes, the extraction yield can be increased by using smaller particles, as the specific area increases and the internal diffusion resistance lessens, due to a shorter diffusion path (Snyder et al., 1984). However, this is not always the case, as smaller particles may cause channeling (Eggers, 1996). 

In many SCR applications, the thickness and effective diffusivity of the active layer may not allow the simplifying assumption of negligible internal diffusion resistance. In these cases, a more detailed approach which models mass transfer both in the gas phase and in the washcoat/active volume pores is needed. 

Based on the equations derived in Sections 4.5.3 and 4.5.4, we now define an overall effectiveness factor tjoveraih vvhich includes external and internal diffusion resistances. Here we only consider irreversible and reversible first-order isothermal reactions for more complex cases see Baerns et al. (2006) or Westerterp, van Swaaij, and Beenackers (1998). 

Obviously, system pressure has serious influence on internal diffusion resistance. Firstly, molecular diffusion coefficient decreases proportionally with the increase of pressure. Secondly, high pressure leads to the enhancement of productivity of catalysts which would lead to the increase of internal diffusion resistance. 

If internal diffusion resistance in the catalyst pores affects the reaction rates, the observed reaction rate, R, is usually lower than the reaction rate obtained with the surface concentrations, R. The ratio between these rates is called the effectiveness factor ... 

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