Hougen-Watson kinetic models

1 Langmuir-Hinshelwood Mechanisms for Surface-Catalyzed Reactions 

This mechanism, which was developed in 1940, employs Langmuir isotherms to describe adsorption/desorption equilibria of all reactants and products. Chemical reaction on the catalytic surface is the rate-limiting step, which governs the overall rate of reaction. Each component adsorbs without preference on one active site. The five-step sequence of elementary steps is 

The fractional surface coverage by each component follows the Langmuir isotherm for single-site adsorption. Hence, 

The dependence of JElsurf Rx on (0y) reveals that the rate-limiting step in the mechanism reqnires two active sites on the catalyst. In general, if n active sites are required for the slowest step in the mechanism, then the reaction rate depends on the nth power of the vacant-site fraction. The combination of eqniUbrium constants in the backward rate of eqnation (14-61) is simplified by using the adsorption isotherm (i.e., i = KiPi y) to re-evaluate X eq.surf.Rx- Hence, 

Term A is a product of kinetic and adsorption/desorption equilibrium constants. The kinetic contribution is given by the forward rate constant of the slowest step. In the example above, equilibrium constants are included only for those reactants that adsorb on the catalytic surface. Term B is written in terms of partial pressures and represents the forward rate minus the backward rate. All reactant partial pressures appear in the forward rate, and all product partial pressures appear in the backward rate, regardless of whether or not each gas adsorbs. The equilibrium constant in the backward rate is based on gas-phase partial pressures. Term C represents the vacant-site fraction on the catalytic smface and includes a contribution from each component that adsorbs. The exponent of this adsorption term in the denominator of the rate law corresponds to the number of active sites that are required in the rate-limiting step. 

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Examples of Hougen-Watson kinetic models, which are also called Langmuir-Hinshelwood models, can be derived for a great variety of assumed surface mechanisms. See Butt and Perry s Handbook (see Suggestions for Further reading in Chapter 5) for collections of the many possible models. The models usually have numerators that are the same as would be expected for a homogeneous reaction. The denominators reveal the heterogeneous nature of the reactions. They come in almost endless varieties, but all reflect competition for the catalytic sites by the adsorbable species. 

Figure 3. Effectiveness factor versus Thiele modulus at various times of poisoning (Hougen-Watson kinetic model for benzene hydrogenation).

The data fit obtained when benzene hydrogenation is represented by Hougen-Watson kinetic model is poor the results are complicated by changes in equilibrium adsorption constants with extent of poisoning. 

Skrzypek el al. mode (19H5) Skrzypek el al. (1985) developed this model based on the Langmuir-Hinshelwood-Hougen-Watson kinetic model to explain the non-monotonic behaviour observed by Calder-bank (1974). They suggested that the reaction rate behaviour can be related to the Langmuir-Hinshelwood kinetic model for bimolecular reactions, where the surface reaction between o-Xylene and oxygen chemisorbed on the active centers is the rate determining step. The rate of appearance of various components can be written as ... 

The Hougen-Watson kinetic model that is consistent with the Langmuir-Rideal mechanism can be obtained from the rate law in equations (14-63) and (14-64) via the following modification of each generic term ... 

HOUGEN-WATSON KINETIC MODELS The adsorption isotherm for each species is... 

The Hougen-Watson kinetic model is expressed in terms of the partial pressnre of H2. If H2 dissociation is an important step that cannot be neglected in the mechanism, then the hydrogen dissociation equilibrium... 

The Hougen-Watson kinetic model is constructed from the second step in the mechanism ... 

The feed stream is stoichiometric in terms of the two reactants. Diatomic A2 undergoes dissociative adsorption. Components B, C, and D experience single-site adsorption, and triple-site chemical reaction on the catalytic surface is the rate-controlling feature of the overall irreversible process. This Langmuir-Hinshelwood mechanism produces the following Hougen-Watson kinetic model for the rate of reaction with units of moles per area per time ... 

Two-dimensional diffusion occurs axially and radially in cylindrically shaped porous catalysts when the length-to-diameter ratio is 2. Reactant A is consumed on the interior catalytic surface by a Langmuir-Hinshelwood mechanism that is described by a Hougen-Watson kinetic model, similar to the one illustrated by equation (15-26). This rate law is linearized via equation (15-30) and the corresponding simulationpresented in Figure 15-1. Describe the nature of the differential equation (i.e., the mass transfer model) that must be solved to calculate the reactant molar density profile inside the catalyst. 

The most important characteristic of this problem is that the Hougen-Watson kinetic model contains molar densities of more than one reactive species. A similar problem arises if 5 mPappl Hw = 2CaCb because it is necessary to relate the molar densities of reactants A and B via stoichiometry and the mass balance with diffusion and chemical reaction. When adsorption terms appear in the denominator of the rate law, one must use stoichiometry and the mass balance to relate molar densities of reactants and products to the molar density of key reactant A. The actual form of the Hougen-Watson model depends on details of the Langmuir-Hinshelwood-type mechanism and the rate-limiting step. For example, consider the following mechanism ... 

Since 1 a is only a function of spatial coordinate r, the partial derivative in (19-38) is replaced by a total derivative, and the dimensionless concentration gradient evaluated at the external surface (i.e., ] = 1) is a constant that can be removed from the surface integral in the numerator of the effectiveness factor. In terms of the Hougen-Watson kinetic model and the dimensional scaling factor for chemical reaction that agree with the Langmuir-Hinshelwood mechanism described at the beginning of this chapter ... 

Consider the Hougen-Watson kinetic model for the production of methanol from CO and H2, given by equation (22-38). Do not linearize the rate expression. Write the rate law in dimensionless form if the chemical reaction is essentially irreversible (i.e., A eq, 00). 

The final expression for the Hougen-Watson kinetic model is... 

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