Hougen-Watson kinetic equations

Consider the Hougen-Watson kinetics of Section 7.4.4. Using Equation 7.58, verify the foEowing limits... 

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 ... 

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. 

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 Hougen-Watson rate equations go further than the mass action kinetic equations in that they account exphcitly for the interaction of the reacting species with the catalyst sites, but as to the mechanism they don t go very far beyond what is expressed by the stoichiometric equation, hi Sections 2.4.2 and 2.4.3 on the other hand, the reaction was decomposed in elementary steps. This is now illustrated by means of an example. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

Of particular value in kinetic studies are residual plots using the linearized form of the Hougen-Watson equation. For the model of Eq. (18), for example, we obtain... 

From a consideration of either Eqs. (113) or (114) (K3), it is evident that a saddle point is predicted from the fitted rate equation. This could eliminate from consideration any kinetic models not capable of exhibiting such a saddle point, such as the generalized power function model of Eq. (1) and the several Hougen-Watson models so denoted in Table XVI. 

In general, the use of Langmuir-Hinshelwood-Hougen-Watson (LHHW)-type of rate equation for representing the hydrogenation kinetics of industrial feedstocks is complicated, and there are too many coefficients that are difficult to determine. Therefore, simple power law models have been used by most researchers to fit kinetic data and to obtain kinetic parameters. 

Further investigations showed significant disadvantages of the above assumptions. Nevertheless, Hinshelwood, Schwab, Hougen, Watson and others derived equations which adequately described a particular kinetic experiment within a certain range of parameters. 

Steps 3-5 are strictly chemical and consecutive to each other Hougen-Watson-Langmuir-Hinshelwood rate equations describing the rate of the purely chemical phenomenon consisting of steps 3-5 have been derived in Chapter 3 on the kinetics of catalysed reactions. In the transport-limited situation the supply of reactant and/or the removal of reaction product will not be sufficiently fast to keep pace with the potential intrinsic rate, and the concentrations of A and B inside the pores will be different from the corresponding concentrations in the bulk of the fluid phase. 

A higher form of interpretation of the effect of solvents on the rate of heterogeneously catalyzed reactions was represented by the Langmuir-Hinshelwood kinetics (7), in the form published by Hougen and Watson (2), where the effect of the solvent on the reaction course was characterized by the adsorption term in the kinetic equation. In catalytic hydrogenations in the liquid state kinetic equations of the Hougen-Watson type very frequently degrade to equations of pseudo-zero order with respect to the concentration of the substrate (the catalyst surface is saturated with the substrate), so that such an interpretation is not possible. At the same time, of course, also in these cases the solvent may considerably affect the reaction. As is shown below, this influence is very adequately described by relations of the LFER type. 

Rate expressions of the form of Equation 5.153 are known as Hougen Watson or Langmuir-Hinshelwood kinetics [17, This form of kinetic expression is often used to describe the species production rates for heterogeneously catalyzed reactions. We complete the section on the kinetics of elementary surface reactions by returning to the methane synthesis reaction listed in Section 5.2. The development proceeds exactly as outlined in Section 5.2. But now it is necessary to add a site-balance expression (Equation 5,129) in Step 3. 

In addition to calculating kinetic and equilibrium constants in the Hougen-Watson model via numerical values for ao, a, and U2, the success of this procedure suggests that the mechanism proposed and the choice of a rate-limiting step are reasonable, based on actual experimental data. For completeness, simultaneous solution of the three equations in steps 4 to 6 yields the following results for the kinetic and adsorption/desorption equilibrium constants in terms of the parameters in the polynomial model ... 

PSEUDO-FIRST-ORDER KINETIC RATE EXPRESSIONS THAT CAN REPLACE HOUGEN-WATSON MODELS AND GENERATE LINEARIZED ORDINARY DIFFERENTIAL EQUATIONS FOR THE MASS BALANCE... 

The correction factor Nsur ce in the E vs. Aa correlation for complex kinetics is given by the inverse of the dimensionless rate law evaluated at the external surface of the catalyst, where the dimensionless molar density of reactant A is unity, by definition. Hence, the correction factor surface for the Hougen-Watson model described by equations (19-1) and (19-8) is ... 

TABLE 19-1 Numerical Solution of the Mass Transfer Equation for One-Dimensional Diffusion and Hougen-Watson Chemical Kinetics with Dissociative Adsorption of Reactant A2 in Porous Catalysts with Rectangular Symmetry"... 

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