Reaction rates, intraparticle

This involves knowledge of chemistry, by the factors distinguishing the micro-kinetics of chemical reactions and macro-kinetics used to describe the physical transport phenomena. The complexity of the chemical system and insufficient knowledge of the details requires that reactions are lumped, and kinetics expressed with the aid of empirical rate constants. Physical effects in chemical reactors are difficult to eliminate from the chemical rate processes. Non-uniformities in the velocity, and temperature profiles, with interphase, intraparticle heat, and mass transfer tend to distort the kinetic data. These make the analyses and scale-up of a reactor more difficult. Reaction rate data obtained from laboratory studies without a proper account of the physical effects can produce erroneous rate expressions. Here, chemical reactor flow models using matliematical expressions show how physical... 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

In this case the reaction rate will depend not only on the system temperature and pressure but also on the properties of the catalyst. It should be noted that the reaction rate term must include the effects of external and intraparticle heat and mass transfer limitations on the rate. Chapter 12 treats these subjects and indicates how equation 8.2.12 can be used in the analysis of packed bed reactors. 

The term in brackets is a dimensionless group that plays a key role in determining the limitations that intraparticle diffusion places on observed reaction rates and the effectiveness with which the catalyst surface area is utilized. We define the Thiele modulus hT as... 

If there were no limitations placed on the reaction rate by intraparticle diffusion (i.e., if the reactant concentration were C0 throughout the pore), the reaction rate would be given by... 

At steady state, the rates of each of the individual steps will be the same, and this equality is used to develop an expression for the global reaction rate in terms of bulk-fluid properties. Actually, we have already employed a relation of this sort in the development of equation 12.4.28 where we examined the influence of external mass transfer limitations on observed reaction rates. Generally, we must worry not only about concentration differences between the bulk fluid and the external surface of the catalyst, but also about temperature differences between these points and intraparticle gradients in temperature and composition. 

The most difficult problem to solve in the design of a Fischer-Tropsch reactor is its very high exothermicity combined with a high sensitivity of product selectivity to temperature. On an industrial scale, multitubular and bubble column reactors have been widely accepted for this highly exothermic reaction.6 In case of a fixed bed reactor, it is desirable that the catalyst particles are in the millimeter size range to avoid excessive pressure drops. During Fischer-Tropsch synthesis the catalyst pores are filled with liquid FT products (mainly waxes) that may result in a fundamental decrease of the reaction rate caused by pore diffusion processes. Post et al. showed that for catalyst particle diameters in excess of only about 1 mm, the catalyst activity is seriously limited by intraparticle diffusion in both iron and cobalt catalysts.1... 

This discussion also applies to the original variable Y s, which represents the ensemble-average temperature of particles located at a particular point at a given time. Basically, we know the total enthalpy of each particle, but we do not know how it is distributed inside any given particle. Since the reaction rate can be very sensitive to the local temperature, we will need a SGS model to describe the coupling between intraparticle transport processes and chemical reactions. 

To evaluate the effect of intraparticle resistance on the overall reaction rate, an approach based on the introduction of effectiveness factor q is usually proposed ... 

The porous structure of either a catalyst or a solid reactant may have a considerable influence on the measured reaction rate, especially if a large proportion of the available surface area is only accessible through narrow pores. The problem of chemical reaction within porous solids was first considered quantitatively by Thiele [1] who developed mathematical models describing chemical reaction and intraparticle diffusion. Wheeler [2] later extended Thiele s work and identified model parameters which could be measured experimentally and used to predict reaction rates in... 

If intraparticle diffusion controls the overall reaction rate, the Thiele modulus will be large (0 > 2) and then the effectiveness factor 77 is approximately 0. From eqn. (10) defining the Thiele modulus, it follows that, for a given reaction, the effectiveness factor will be... 

The usual experimental criterion for diffusion control involves an evaluation of the rate of reaction as a function of particle size. At a sufficiently small particle size, the measured rate of reaction will become independent of particle size. The reaction rate can then be safely assumed to be independent of intraparticle mass transfer effects. At the other extreme, if the observed rate is inversely proportional to particle size, the reaction is strongly influenced by intraparticle diffusion. For a reaction whose rate is inhibited by the presence of products, there is an attendant danger of misinterpreting experimental results obtained for different particle sizes when a differential reactor is used, because, under these conditions, the effectiveness factor is sensitive to changes in the partial pressure of product. 

If, however, both reactions were influenced by intraparticle diffusion effects, the rate of reaction of a particular component would be given by the product of the intrinsic reaction rate, fecg, and the effectiveness factor, Tj. Substituting eqn. (6) for the effectiveness factor gives (for a first-order isothermal reaction) the overall rate as 0tanh< >. As is often the case, the molecular weights of the diffusing reactants are similar and can be... 

Research with pilot scale units has shown that the major resistances to mass transfer of reactant to catalyst are within the liquid film surrounding the wetted catalyst particles and also intraparticle diffusion. A description of these resistances is afforded by Fig. 14. Equating the rate of mass transfer across the liquid film to the reaction rate, first order in hydrogen concentration... 

Fig. 2. Reactant concentration as a function of distance from the center of a catalyst particle for fast mixing (A) and slow mixing (B). In both figures, (I) represents a reaction rate limited by intrinsic reactivity at the active site, (2) represents a reaction rate limited by mass transfer, and (3) represents a reaction rate limited by a combination of intraparticle diffusion and intrinsic reactivity. (Reprinted with permission from Ref.73). Copyright 1981 American Chemical Society)...

At one extreme diffusivity may be so low that chemical reaction takes place only at suface active sites. In that case p is equal to the fraction of active sites on the surface of the catalyst. Such a polymer-supported phase transfer catalyst would have extremely low activity. At the other extreme when diffusion is much faster than chemical reaction p = 1. In that case the observed reaction rate equals the intrinsic reaction rate. Between the extremes a combination of intraparticle diffusion rates and intrinsic rates controls the observed reaction rates as shown in Fig. 2, which profiles the reactant concentration as a function of distance from the center of a spherical catalyst particle located at the right axis, When both diffusion and intrinsic reactivity control overall reaction rates, there is a gradient of reactant concentration from CAS at the surface, to a lower concentration at the center of the particle. The reactant is consumed as it diffuses into the particle. With diffusional limitations the active sites nearest the surface have the highest turnover numbers. The overall process of simultaneous diffusion and chemical reaction in a spherical particle has been described mathematically for the cases of ion exchange catalysis,63 65) and catalysis by enzymes immobilized in gels 66-67). Many experimental parameters influence the balance between intraparticle diffusional and intrinsic reactivity control of reaction rates with polymer-supported phase transfer catalysts, as shown in Fig. 1. 

Such a high reaction rate strongly suggested the potential for mass transport problems. Indeed, a turn-over frequency on the order of 1 s 1 is considered appropriate for the purpose of mechanistic studies normally conducted in the gas phase (ref. 5). At higher rates, various complications including intraparticle diffusion problems, often arise. The situation is even more severe in the liquid phase where the bulk diffusivity of species is considerably reduced. A... 

When intraparticle diffusion occurs, the kinetic behaviour of the system is different from that which prevails when chemical reaction is rate determining. For conditions of diffusion control 0 will be large, and then the effectiveness factor tj( 1/ tanh 0, from equation 3.15) becomes. From equation 3.19, it is seen therefore that rj is proportional to k Ul. The chemical reaction rate on the other hand is directly proportional to k so that, from equation 3.8 at the beginning of this section, the overall reaction rate is proportional to k,n. Since the specific rate constant is directly proportional to e"E/RT, where E is the activation energy for the chemical reaction in the absence of diffusion effects, we are led to the important result that for a diffusion limited reaction the rate is proportional to e E/2RT. Hence the apparent activation energy ED, measured when reaction occurs in the diffusion controlled region, is only half the true value ... 

Studies with porous catalyst particles conducted during the late 1930s established that, for very rapid reactions, the activity of a catalyst per unit volume declined with increasing particle size. Mathematical analysis of this problem revealed the cause to be insufficient intraparticle mass transfer. The engineering implications of the interaction between diffusional mass transport and reaction rate were pointed out concurrently by Damkohler [4], Zeldovich [5], and Thiele [6]. Thiele, in particular, demonstrated that the fractional reduction in catalyst particle activity due to intraparticle mass transfer, r, is a function of a dimensionless parameter, 0, now known as the Thiele parameter. 

Figure 5. Effectiveness factor rj as a function of the Thiele modulus for different pellet shapes. Influence of intraparticle diffusion on the effective reaction rate (isothermal, first order, irreversible reaction).

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