LHHW models

Most standard chemical engineering tests on kinetics [see those of Car-berry (50), Smith (57), Froment and Bischoff (19), and Hill (52)], omitting such considerations, proceed directly to comprehensive treatment of the subject of parameter estimation in heterogeneous catalysis in terms of rate equations based on LHHW models for simple overall reactions, as discussed earlier. The data used consist of overall reaction velocities obtained under varying conditions of temperature, pressure, and concentrations of reacting species. There seems to be no presentation of a systematic method for initial consideration of the possible mechanisms to be modeled. Details of the methodology for discrimination and parameter estimation among models chosen have been discussed by Bart (55) from a mathematical standpoint. 

In general, a reaction kinetics following a LHHW model is suitable, but the identification of parameters remains demanding. For some catalysts power-law models may be appropriate, for others not. For example, reaction orders identical with stoichiometric coefficients were suitable for Pd/Al203 doped with different metals. On the contrary, for Pd/MgO reaction orders with respect to phenol ranging from -0.5 to 0.5 were observed [17]. However, the bibliographic search was not able to find a quantitative kinetic model for Pd-type catalysts suitable for reactor design. 

In this project, we make use of platinum-type catalyst on silica gel. Although this is less selective than more modem palladium-based catalysts, kinetic data are available in the literature as an LHHW model [2], better suited for flexible reactor design. The reaction rate equations are ... 

So, power rate expressions may be considered as special cases of the LHHW models, but valid over a limited range of conditions. 

In heterogeneous catalysis these models are generally referred to as the Langmuir-Hinshelwood-Hougen-Watson (LHHW) models. The term Michaelis-Menten kinetics is often used in homogeneous catalysis, enzyme reactions and reactions of microbial systems. 

The models described above are termed Langmuir-Hinshelwood-Hougen-Watson (LHHW) models, named after the scientists that contributed a lot to the development of these engineering models. The characteristics of these models are that adsorption follows the Langmuir isotherm, and that reaction takes place between adsorbed species. Sometimes, one distinguishes Eley Rideal models, whereby a molecule reacts directly from the gas phase with a surface complex ... 

In homogeneous catalysis, enzyme reactions, and reactions of microbial systems the same types of equations are used as in the LHHW models. In the latter disciplines, however, they are often referred to as Michaelis-Menten kinetics. 

As shown in Section 11.1.2.2, hyperbolic (LHHW) models are usually the most appropriate for reactions occnrring on solid catalysts. These are nonlinear models whose parameters can be determined using statistical methods. Several examples are available. 

When this procedure is applied to LHHW models, caution should be exercised in claiming a model to be the best. The need for this increases with the number of unknown parameters, since several models may then show almost equal convergence. In general, it is advisable to confine the methods to models with no more than three or four unknown parameters. Furthermore, it should be ensured that all data points are of equal precision. 

Langmuir-Hinshelwood-Hougen-Watson (LHHW) Models... 

In the interest of generality, we consider hypothetical reactions and derive rate equations for a few typical LHHW models (Hougen and Watson, 1947 Yang and Hougen, 1950 Satterfield, 1980, Butt, 1980 Doraiswamy and Sharma, 1984 Boudart and Djega-Mariadassou, 1984). As the Langmuir isotherm is the basis of all LHHW models, we begin by a simple derivation of this isotherm. 

Focusing on the case where surface reaction is controlling, the basic procedure in developing a LHHW model is to write the rate equation in terms of the surface coverage of reactant A rather than its concentration [/ ]. Sometimes, as in reactions requiring a second (vacant) site for adsorbing a product (e.g.,... 

All LHHW models can be consolidated into a single general form... 

In some bimolecular reactions like disproportionation of propylene to butylene and ethylene and hydrogenation of ethylene to ethane, a modified form of LHHW models has to be used. Here, the reaction is assumed to occur by a molecule of one of the reactants (say A) striking an obsorbed molecule of B (or another A). Thus the rate equation would be (or aPb if reacts... 

Table 7.1 LHHW models for a few selected reaction types"...

A simple way to check roughly for the consistency of the selected LHHW model is to apply a test proposed by Everett (1950) and Boudart et al. (1967). This is not a powerful test. 

The general conclusion from these studies is that the methods suggested earlier in this chapter for obtaining the best LHHW model for a given reaction using isothermal data are quite valid. However, in a strict sense, they cannot be used to describe transient phenomena (Simon and Vortmeyer, 1978 Graham and Lynch, 1987), or extrapolated to nonisothermal reactors without testing them in laboratory reactors with different values of L operated under nonisothermal conditions. 

All sites are equally active with equal heats of adsorption this is not true because there is usually a distribution of activity on the surface that is ignored in the LHHW models. 

An interesting feature of LHHW kinetics is worth noting. Many reactions on surfaces known to be nonideal surprisingly follow the ideal LHHW models, a situation that can only be described as the placebo effect or the paradox of heterogeneous kinetics (Boudart et al., 1967 Boudart, 1986). In the same vein but with less justification, it has also been argued for more than four decades—... 

In the developments already presented, the rate equations used were simple power law expressions. But, as discussed previously, catalytic rate equations are much more complex and often require the use of LHHW models. Many attempts have been made to incorporate these models in the analysis (e.g., Chu and Hougen, 1962 Krasuk and Smith, 1965 Roberts and Satterfield, 1965, 1966 Hutchings and Carberry, 1966 Schneider and Mitschka, I966a,b Kao and Satterfield, 1968 Rajadhyaksha et al., 1976 see in particular Aris, 1975 Luss, 1977). Clearly, graphical representation becomes cumbersome when a large number of adsorbed species is involved. However, the problem is quite tractable where only one species is adsorbed. 

It was pointed out earlier that regime 3 is controlling (for simple power law kinetics) if /M 1. It can similarly be shown (Chaudhari, 1984) that regime 3 is controlling for LHHW models if... 

One of the earliest LHHW models to be developed for a slurry reaction was for the hydrogenation of glucose to sorbitol, considered in Example 17.1. Models for several other three-phase catalytic reactions have since been developed, many of which are listed in Table 17.3. 

An effectiveness factor e can be defined and incorporated in any LHHW model as, for example, in a typical single-site surface-reaction model, giving... 

Table 17.3 Examples of LHHW models for gas-liquid-solid catalytic reactions in organic synthesis (from Mills and Chaudhary, 1997)... 

In Ali Nakhaei Pour and coworker s (Ali et al., 2013) work, the chain-growth probabilities of the FTS reaction are developed by LHHW modeling in gradientless reactor over Fe/Cu/La/Si catalyst, which can simultaneously provide a chain-growth probabilities ai and a. To calculate and 012, they give a method as shown in Eqs. (32)—(38), where and x are the reaction rate and molar fraction of each species, A and B are two important parameters which are related to the system. 

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