Type Empirical Kinetics

Monod-Type Empirical Kinetics Many bioreactions show increased biomass growth rate with increasing substrate concentration at low substrate concentration for the limiting substrate, but no effect of substrate concentration at high concentrations. This behavior can be represented by the Monod equation (7-92). Additional variations on the Monod equation are briefly illustrated below. For two essential substrates the Monod equation can be modified as 

This type of rate expression is often used in models for water treatment, and many environmental factors can be included (the effect of, e.g., phosphate, ammonia, volatile fatty acids, etc.). The correlation between parameters in such complicated models is, however, severe, and very often a simple Monod model (7-92) with only one limiting substrate is sufficient. 

Here the reactants (substrates) are glucose (CH20), 02, NH3, and a sulfur-providing nutrient S3, and the products are biomass X, C02, metabolic product P and H20. 

The products of bioreactions can be reduced or oxidized, and all feasible pathways have to be redox neutral. There are several cofactors that transfer redox power in a pathway or between pathways, each equivalent to the reducing power of a molecule of H2, e.g., nicotinamide adenine dinucleotide (NADH), and these have to be included in the stoichiometric balances as H equivalents through redox balancing. For instance, for the reaction of glucose to glycerol (CHs/30), j NADH equivalent is consumed  

The stoichiometry in the biochemical literature often does not show H20 produced by the reaction however, for complete elemental balance, water has to be included, and this is easily done once an 02 requirement has been determined based on a redox balance. Likewise for simplicity, the other form of the cofactor [e.g., the oxidized form of the cofactor NADH in Eq. (7-148)] is usually left out. In 

O2 is typically a substrate that in high concentrations leads to substrate inhibition, but a high concentration of the carbon source can also be inhibiting (e.g., in bioremediation of toxic waste a high concentration of the organic substrate can well lead to severe inhibition or death of the microorganism). 

Here the typical example is the inhibitor effect of ethanol on yeast growth. Considerable efforts are made by the biocompanies to develop yeast strains that are tolerant to high ethanol concentrations since this will give considerable savings in, e.g., production of biofuel by fermentation. 

The various component reaction rates for a single reaction can be related to the growth rate by using the stoichiometric (yield) coefficients, e.g., from Eq. (7-147)  

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When a simple, fast and robust model with global kinetics is the aim, the reaction kinetics able to predict correctly the rate of CO, H2 and hydrocarbons oxidation under most conditions met in the DOC consist of semi-empirical, pseudo-steady state kinetic expressions based on Langmuir-Hinshelwood surface reaction mechanism (cf., e.g., Froment and Bischoff, 1990). Such rate laws were proposed for CO and C3H6 oxidation in Pt/y-Al203 catalytic mufflers in the presence of NO already by Voltz et al. (1973) and since then this type of kinetics has been successfully employed in many models of oxidation and three-way catalytic monolith converters... 

Since the end of the 90 s, our group has been developing a non-empirical kinetic model, named KINOXAM, for the lifetime prediction of polymers and polymer matrix composites in their use conditions. The model is totally open. It is composed of a core, common to all types of polymers, derived from the now well-known closed-loop mechanistic scheme (/). Around this core, various optional layers can be added according to the complexity of oxidation mechanisms and the relationships between the structural changes taking place at the molecular scale and the resulting ones at larger scales (the macromolecular and macroscopic scales). 

The data in Table II pertaining to pyrolysis conditions shows that all four feedstocks were pyrolyzed under substantially similar conditions, namely steam-to-hydrocarbon weight ratios of 0.9 0.1, residence times of 0.3 sec, reactor exit pressures of 2.0 bar absolute, and reactor exit temperatures of 835°C. Care also was taken to maintain identical axial temperature profiles in the reactor for each of these runs. No unambiguous measure of substrate conversion during pyrolysis is possible for distillate feedstocks of the type used in the present experiments in terms of the empirical kinetic severity function of Zdonik et al. (5), all of the present experiments were conducted at a severity of about 2. 

A second important type of kinetic study relates to the way in which rates depend on temperature. The most satisfactory way of dealing with this problem is to investigate how rate constants—or in the case of complex rate equations, the constants appearing in the empirical rate equations—depend upon temperature. Such studies have been of great importance in chemical kinetics, because the temperature dependence leads to a theoretical interpretation of reaction rates that is of very great significance. The temperature dependence is related to molecular properties of the reaction system,... 

Following failure mode identification (le, the material suspected of significant aging, the stress implicated in its aging, and the appropriate damage parameter, which quantifies the expected failure), Phase II kinetic analysis for the predictions of lifetimes is conducted. The usual kind of kinetic analysis is one that uses Arrhenius principles, but other types of kinetics may be necessary depending on the empirical form of the data. 

The following figures include selected data to illustrate the types of kinetic curves that can be generated using this approach and which may subsequently be subjected to empirical or compartmental modeling. 

The simplest empirical kinetic law of the type (62) for this reaction can be written as follows ... 

Assuming that oxygen supply is sufficient to avoid local oxygen limitations, the kinetic model required for the simulation includes only the material balance equation for the substrate. As suggested in earfier simulations based on recirculation models (micro-macromixer) by Bajpai and Reuss [60], the uptake kinetics are only considered in the vicinity of the so-called critical sugar concentration. Thus, a rather simple unstructured empirical model is chosen for the purpose of this study. It involves a Monod type of kinetics for substrate uptake... 

Incidentally, this type of kinetic equation also acts as an empirical model of order (m, n) for a homogeneous noncatalytic reaction. Taking the natural logarithm of both sides of Eq. (9.35) gives... 

The way by which all the factors involved influence the course of a reaction varies from case to case, and prediction is largely empirical. For catalytic processes, the actual species acting as catalyst is often unknown because coordination number, type of ligands, stereochemistry of the complex, and formal charge are difficult to establish in the reaction medium. Often many species are present, and the most active may be the one having the lowest coordination number and being present in a concentration so low that it cannot be detected spectroscopically. Only kinetic studies can provide evidence for such species. 

Co2(CO)q system, reveals that the reactions proceed through mononuclear transition states and intermediates, many of which have established precedents. The major pathway requires neither radical intermediates nor free formaldehyde. The observed rate laws, product distributions, kinetic isotope effects, solvent effects, and thermochemical parameters are accounted for by the proposed mechanistic scheme. Significant support of the proposed scheme at every crucial step is provided by a new type of semi-empirical molecular-orbital calculation which is parameterized via known bond-dissociation energies. The results may serve as a starting point for more detailed calculations. Generalization to other transition-metal catalyzed systems is not yet possible. 

Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications (refer to the examples in Chapters 11 through 16). These models are conceptually attractive because a general model for any system size can be developed even before the system is constructed. A detailed exposition of fundamental mathematical models in chemical engineering is beyond our scope here, although we present numerous examples of physiochemical models throughout the book, especially in Chapters 11 to 16. Empirical models, on the other hand, are attractive when a physical model cannot be developed due to limited time or resources. Input-output data are necessary in order to fit unknown coefficients in either type of the model. 

The electrostatic precipitator in Example 2.2 is typical of industrial processes the operation of most process equipment is so complicated that application of fundamental physical laws may not produce a suitable model. For example, thermodynamic or chemical kinetics data may be required in such a model but may not be available. On the other hand, although the development of black box models may require less effort and the resulting models may be simpler in form, empirical models are usually only relevant for restricted ranges of operation and scale-up. Thus, a model such as ESP model 1 might need to be completely reformulated for a different size range of particulate matter or for a different type of coal. You might have to use a series of black box models to achieve suitable accuracy for different operating conditions. 

The expression for the effectiveness factor q in the case of zero-order kinetics, described by the Michaelis-Menten equation (Eq. 8) at high substrate concentration, can also be analytically solved. Two solutions were combined by Kobayashi et al. to give an approximate empirical expression for the effectiveness factor q [9]. A more detailed discussion on the effects of internal and external mass transfer resistance on the enzyme kinetics of a Michaelis-Menten type can be found elsewhere [10,11]. 

Many studies investigating one or more of these potential rate-determining steps have been carried out over the years. These studies have shown that the rate of reaction depends upon many factors such as temperature [15, 27-29], pellet size [27-29], crystallinity [28], additive types and concentrations [30], process gas type and quantity [31, 32], molecular weight [22, 31] and end group concentrations [16, 33] - all of which will be addressed individually later in this section. Various models have also been proposed involving kinetics [33] and/or by-product diffusion [11, 16, 21, 27-29, 34, 35] through to empirical Equations [15]. The variety of models used and the wide range of kinetic and physical data published demonstrate the complexity of the mechanisms involved. 

The hyperbolic model types have very commonly been used in the analysis of kinetic data, as discussed in Section I. Such applications are sometimes justified on the theoretical bases already alluded to, or simply because models of the form of Eq. (2) empirically describe the existing reaction-rate data. Considerably more complex models are quite possible under the Hougen-Watson formalism, however. For example, Rogers, Lih, and Hougen (Rl) have proposed the competitive-noncompetitive model... 

The complex nature of heterogeneous catalytic reactions, which consist of a sequence of at least three steps (adsorption, surface reaction and desorption), the possible intervention of transport processes and the uncertainty about the actual state of the surface makes every attempt to obtain a complete formal kinetic description without simplifying assumptions futile. In this situation, some authors prefer fully empirical equations of the type... 

For a formal kinetic description of vapour phase esterifications on inorganic catalysts (Table 21), Langmuir—Hinshelwood-type rate equations were applied in the majority of cases [405—408,410—412,414,415]. In some work, purely empirical equations [413] or second-order power law-type equations [401,409] were used. In the latter cases, the authors found that transport phenomena were important either pore diffusion [401] or diffusion of reactants through the gaseous film, as well as through the condensed liquid on the surface [409], were rate-controlling. 

As appears from the examination of the equations (giving the best fit to the rate data) in Table 21, no relation between the form of the kinetic equation and the type of catalyst can be found. It seems likely that the equations are really semi-empirical expressions and it is risky to draw any conclusion about the actual reaction mechanism from the kinetic model. In spite of the formalism of the reported studies, two observations should be mentioned. Maatman et al. [410] calculated from the rate coefficients for the esterification of acetic acid with 1-propanol on silica gel, the site density of the catalyst using a method reported previously [418]. They found a relatively high site density, which justifies the identification of active sites of silica gel with the surface silanol groups made by Fricke and Alpeter [411]. The same authors [411] also estimated the values of the standard enthalpy and entropy changes on adsorption of propanol from kinetic data from the relatively low values they presume that propanol is weakly adsorbed on the surface, retaining much of the character of the liquid alcohol. 

The section on crystallization comprises zeolite synthesis, kinetics and mechanism of formation, stability relationships, recrystallization processes as well as the genesis of natural zeolites. Recent advances in this field have been surveyed, and some new perspectives have been outlined in the review by E. M. Flanigen. Most of the studies in this field are still empirical because of the complexity of the systems involved. Considerable progress has been made, however, towards a better understanding of the processes and mechanisms governing zeolite crystallization. It is not unreasonable to expect that conditions for synthesizing new zeolite structure types can eventually be predicted. 

In the literature, one can find other empirical or semi-empirical equations representing the kinetics of powder reactions. One can certainly take into account grain size distribution, contact probability, deviations from the spherical shape, etc. in a better way than Carter has done. Even more important are parameters such as evaporation rate, gas transport, surface diffusion, and interface transport in this context. As long as these parameters are neglected in quantitative work, the kinetic equations are inadequate. Nevertheless, considering its technological relevance, a particular type of powder reaction will be discussed in the next section. 

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