Global rate expression

The global reaction rate depends on three factors (i) chemical kinetics (the intrinsic reaction rate), (ii) the rates that chemical species are transported (transport hmitations), and (hi) the rnterfacial surface per imit volume. Therefore, even when a kinetic-transport model is carefully constructed (using the concepts described above), it is necessary to determine the interfacial surface per unit volume. The interfacial surface depends on the way the two phases are contacted (droplet, bubble, or particle size) and the holdup of each phase in the reactor. All those factors depend on the flow patterns (hydrodynamics) in the reactor, and those are not known a priori. Estimating the global rate expression is one of the most challenging tasks in chemical reaction engineering. 

5 Species Balance Equation and Reactor Design Equation 

The genesis of the reactor design equations is the conservation of mass. Since reactor operations involve changes in species compositions, the mass balance is written for individual species, and it is expressed in terms of moles rather than mass. Species balances and the reactor design equations are discussed in detail in Chapter 4. To obtain a complete description of the reactor operation, it is necessary to know the local reaction rates at all points inside the reactor. This is a formidable task that rarely can be carried out. Instead, the reactor operation is described by idealized models that approximate the actual operation. Chapters 5-9 cover the applications of reactor design equations to several ideal reactor conflgurations that are commonly used. 

For flow reactors, the plug-flow and the CSTR models represent two limiting cases. The former represents continuous reactor without any mixing, where the reactant concentrations decrease along the reactor. The latter represents a reactor with complete mixing where the outlet reactant concentration exists throughout 

When the behavior of a reactor is not adequately described by one of the idealized models, a more reflned model is constructed. In such models the reactor is divided into sections, each is assumed to have its own species concentrations and temperature, wifli material and heat interchanged between them [6, 7, 10, 11, 43]. The volume of each zone and flie interchanges are parameters determined from the reactor operating data. The advantage of such refined models is that fliey provide a more detailed representation of flie reactor, based on actual operating data. However, their application is limited to existing reactors. 

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The model considers the noble-metal catalyzed oxidation reactions of CO, two hydrocarbons of differing reactivities and H2, and the reaction kinetics was described by the global rate expressions of the dual-site Langmuir-Hinshelwood type [2]. 

For isothermal systems, it is occasionally possible to eliminate the external surface concentrations between equations 12.6.1 and 12.6.2 and arrive at a global rate expression involving only bulk fluid compositions (e.g., equation 12.4.28 was derived in this manlier). In general, however, closed form solutions cannot be achieved and an iterative trial and error procedure must be employed to determine thq global rate. One possible approach is summarized below. 

Pseudo homogeneous models require global rate expressions of the type discussed in Section... 

For our present purposes, the global rate expression may be presumed to be identical with that of equation A. 

A global rate expression for CO methanation over a nickel catalyst is given by Lee (1973) and Vatcha (1976). They report that a Langmuir-Hinshelwood rate law of the form... 

The global rate expression and the external effectiveness factor for an isothermal catalytic reaction... 

Here the units of concentration are mol/m3 s. According to this global rate expression, oxidation of CO is first order in the CO concentration, half order in [02] and 0.25 order in [H20]. The notion of reaction order is described in Section 9.3. Notice that the reaction order is not related to the stoichiometry of the reaction this is typical of rate expressions for global reactions. Notice further that the complex dependence of the concentrations of 02 and H20 confirms that this is not a simple reaction. 

Assume that we wish to design a high-pressure combustion chamber where complete oxidation of CO to C02 is an important design consideration. For this purpose we extrapolate our global rate expression for CO oxidation to higher pressure. The right-hand side of Eq. 13.6 can be rewritten in terms of mole fractions and the total molar concentration [M],... 

In our design considerations we have extrapolated the global rate expression for CO oxidation outside the conditions for which it was derived, and this extrapolation leads to erronous results. Experimental results on oxidation of CO in a flow reactor at varying pressure are shown in Fig. 13.3. The results clearly show that in the medium temperature range around 1000 K, an increased pressure acts to lower, not increase, the rate of CO oxidation. To secure adequate oxidation of CO, we would probably need to increase the postflame residence time in a high-pressure reactor compared to an atmospheric pressure reactor. 

Evaluate the validity of the global rate expression for thermal NO formation... 

To date, numerous model compounds simulating the pollutants in common waste streams have been studied under laboratory-scale conditions by many researchers to determine their reactivities and to understand the reaction mechanisms under supercritical water oxidation conditions. Among them, hydrogen, carbon monoxide, methanol, methylene chloride, phenol, and chlorophenol have been extensively studied, including global rate expressions with reaction orders and activation energies [58-70] (SF Rice, personal communication, 1998). 

Validation of the Global Rates Expressions. In order to validate the global rate expressions employed in the model, temperature and concentration profiles determined by probing the flames on a flat flame burner were studied. Attention was concentrated on Flames B and C. The experimental profiles were smoothed, and the stable species net reaction rates were determined using the laminar flat-flame equation described in detail by Fristrom and Westenberg (3) and summarized in Reference (8). 

Initially, an attempt was made to develop original global rate expressions for Reactions 14 to 16 from these rate data. It soon became clear, however, that the number of experimental points was too few to allow the attainment of this goal. Moreover, since a ternary system was being analyzed, the concentration profiles had an intricate form which made numerical differentiation to retrieve the rates somewhat inaccurate. It was therefore decided to use these rate data to check the overall rate expressions derived by other authors and used in the present model. 

For low conversions (less than 20%) of organic, the assumption of a differential reactor (dc/ct Ac/At=([C]in-[C]out)/ is a good one and rates are calculated on this basis. For higher conversions (greater than 20%), the lobal stoichiometric oxidation equation is numerically integrated and the inetic parameters are obtained by regression to conversion data. Rates are then calculated from the global rate expression. 

Table 8 Global Rate Expressions for Oxidation of Select Model Organic Compounds in Supercritical Water ... 

If a given step is much slower than the others (i.e., rate determining), that appropriate term predominates in the global rate constant. If there is more than one slow step, or if the surface reaction is not first order, then a global rate expression is too complicated and cannot be solved analytically. 

The rates at which chemical transformations take place are in some circumstances strongly influenced by mass and heat transfer processes (see Sections 12.3 to 12.5). In the design of heterogeneous catalytic reactors, it is essential to utilize a rate expression that takes into account the influence of physical transport processes on the rate at which reactants are converted to products. Smith (94) has popularized the use of the term global reaction rate to characterize the overall rate of transformation of reactants to products in the presence of heat and mass transfer limitations. We shall find this term convenient for use throughout the remainder of the chapter. Global rate expressions then include both external heat and mass transfer effects on the reaction rate and the efficiency with which the internal... 

This chapter deals with the microkinetics of gas-solid catalytic reaction systems. An applied approach is adopted in the discussion, which starts with the formulation of intrinsic rate equations that account for chemical processes of adsorption and surfece reaction on solid catalysts and then proceeds with the construction of global rate expressions that include the individual and simultaneous effects of physical external and internal mass and heat transport phenomena occurring at the particle scale. 

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