Intrinsic Rate Expressions from Equality of Rates

1 Intrinsic Rate Expressions from Equality of Rates 

Suppose a gradientless reactor is used to obtain intrinsic rate data for a catalytic reaction. Gas-phase concentrations are measured, and the data are fit to a rate expression using the methods of Chapter 7. The rate expression can be arbitrary  

As discussed in Chapter 7, this form can provide a good fit of the data if the reaction is not too close to equilibrium. However, most reaction engineers prefer a mechanistically based rate expression. This section describes how to obtain plausible functional forms for based on simple models of the surface reactions and on the observation that aU the rates in Steps 2 through 8 must be equal at steady state. Thus, the rate of transfer across the film resistance equals the rate of diffusion into a pore equals the rate of adsorption equals the rate of reaction equals the rate of desorption, and so on. This rate is the pseudohomo-geneous rate shown in Steps 1 and 9. 

Example 10.1 Consider the heterogeneously catalyzed reaction A — P. Derive a plausible form for the intrinsic kinetics. The goal is to determine a form for the reaction rate that depends only on gas-phase concentrations. 

Solution Under the assumption of intrinsic kinetics, all mass transfer steps are eliminated, and the reaction rate is determined by Steps 4-6. The simplest possible version of Steps 4-6 treats them all as elementary, irreversible reactions  

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This result is experimentally indistinguishable from the general form, Equation (10.12), derived in Example 10.1 using the equality of rates method. Thus, assuming a particular step to be rate-controlling may not lead to any simplification of the intrinsic rate expression. Furthermore, when a simplified form such as Equation (10.15) is experimentally determined, it does not necessarily justify the assumptions used to derive the simplified form. Other models may lead to the same form. 

The apparent reaction rate r at the level of one pore results from the exchange of mass between the liquid flow and the porous structure of the catalyst particle as depicted in the close-up of figure 8. In the absence of external mass transfer limitations, ra equals the product of the intrinsic reaction rate r by the particle effectiveness factor rip,the variables being expressed in terms of the liquid bulk concentrations ... 

Separation from mixtures is achieved because the membrane transports one component more readily than the others, even if the driving forces are equal. The effectiveness of pervaporation is measured by two parameters, namely flux, which determines the rate of permeation and selectivity, which measures the separation efficiency of the membrane (controlled by the intrinsic properties of the polymer used to construct it). The coupling of fluxes affecting the permeability of a mixture component can be divided into two parts, namely a thermodynamic part expressed as solubility, and a kinetic part expressed as diffusivity. In the thermodynamic part, the concentration change of one component in the membrane due to the presence of another is caused by mutual interactions between the permeates in the membrane in addition to interactions between the individual components and the membrane material. On the other hand, kinetic coupling arises from the dependence of the concentration on the diffusion coefficients of the permeates in the polymers [155]. 

Generally, as the Thiele modulus of main reaction is smaller than unity, the order of deactivation is equal to unity and, therefore, the apparent deactivation rate constant can he extracted from the slope of -In Xs(t) versus t Krishnaswamy and Kittrell (ref. 10) have shown that the relationship between the intrinsic deactivation rate constant k and the apparent deactivation rate constant kda can be expressed as... 

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