Pseudo-kinetic constants

Van Deemter (1980) compared the kinetic constant fcc calculated from A and E obtained in the laboratory with the pseudo-kinetic constants k c from several commercial and pilot plants, and found that the values for k c are always lower than those corresponding to kc, as shown in Fig. 13. It appears, therefore, that the process of coke burning is principally controlled by gas interchange and mass transfer in the bubbling or turbulent bed. Fast... 

The models considered earlier were developed for homopolymerization of olefins with single- and multiple-site catalysts. As has aheady been seen, several industrial polyolefins are, however, copolymers of ethylene, propylene and higher a-olefins. Because, for copolymerization, the kinetic rate constants depend on monomer and chain end type (in the terminal model), modeling these systems may seem daunting at first sight, but it will now be shown that, using the concept of pseudo-kinetic constants, the same equations derived for homopolymerization can be applied for copolymerization as well. 

The important property of Equation 2.97 is that it is analogous to Equation 2.3 with the pseudo-kinetic constants replacing the actual polymerization kinetic constants. The same result would be obtained if it were decided to develop equations for any other species in the reactor. As a consequence, all the equations derived above for the homopolymerization model are applicable to copolymerization as well, provided that the polymerization kinetic constants are replaced with pseudo-kinetic constants. Equations in Table 2.10 can be used either for homo- and copolymerization This elegant approach can considerably reduce the time and effort spent on developing models for copolymerization. Even though it was demonstrated for binary copolymers, this approach is equally valid for higher copolymers. Table 2.12 summarizes the pseudo-kinetic constants associated with the equations shown in Table 2.10. 

To calculate the pseudo-kinetic constants, the values of Pa and at each polymerization time need to be known. If the reactor is being operated at steady-state. Pa and fA do not change and neither do the pseudo -kinetic constants. The value of is calculated by realizing... 

Table 2.12 Pseudo-kinetic constants for binary copolymerization...

Copolymers made with multiple-site catalysts can be modeled with the same equations derived for homopolymers produced with multiple-site catalysts in Table 2.10. The only modification required is the use of the pseudo-kinetic constants shown in Table 2.12 instead of the actual kinetic constants in the equations presented in Table 2.10. 

Remember, however, that for non-steady-state operation, the pseudo-kinetic constants will vary as a function of /a and Pa and must, therefore, be updated throughout the polymerization until a steady-state is reached. 

In this section, we will first illustrate how to use the method of moments for homopolymerization and then show how these equations can be easily adapted to copolymerization using the method of pseudo-kinetic constants. 

Equation (94) can be expressed in the more compact form of Eq. (95), where the pseudo-kinetic constants are defined by Eqs. (96)-(102). 

Notice that Eqs. (56) and (95) are equivalent, with the only difference that Eq. (95) uses pseudo-kinetic constants in place of the actual kinetic constants found in Eq. (56). The beauty of this modeling approach is that it is not necessary to develop new equations for copolymerization (binary or higher) the equations developed for homopolymerization, including the moment equations, are equally applicable to copolymerization, provided that pseudo-kinetic constants are used to replace the actual polymerization kinetic constants. 

To calculate the pseudo-kinetic constants one must know the values of /a and < a at each polymerization time. Values for /a are easily calculated from the molar balance for the monomers, Eq. (103). 

For those pesticides that are cometabolized, ie, not utilized as a growth substrate, the assumption of first-order kinetics is appropriate. The more accurate kinetic expression is actually pseudo-first-order kinetics, where the rate is dependent on both the pesticide concentration and the numbers of pesticide-degrading microorganisms. However, because of the difficulties in enumerating pesticide-transforming microorganisms, first-order rate constants, or half-hves, are typically reported. Based on kinetic constants, it is possible to rank the relative persistence of pesticides. Pesticides with half-hves of <10 days are considered to be relatively nonpersistent pesticides with half-hves of >100 days are considered to be relatively persistent. 

Experiments with [ArH] [Hg] followed pseudo-first-order kinetics. Plots of 1 /k versus 1/tArH] were linear. Interpret this information in terms of a reaction scheme, and show how the intercept and slope of the plot are related to the kinetic constants. 

Pseudo Kinetic Rate Constant Method for Copolymers with Long Branches... 

In this paper, the pseudo-kinetic rate constant method in which the kinetic treatment of a multicomponent polymerization reduces to that of a hcmopolymerization is extensively applied for the statistical copolymerization of vinyl/divinyl monomers and applications to the pre- and post-gelation periods are illustrated. 

The pseudo-kinetic rate constant method for multicomponent polymerization has been applied in some copolymerization studies (3-5), and its derivation and specific approximations have been made clear (6,7). The pseudo-kinetic rate constants basically... 

Symbols used are defined at the end of this paper. The definitions of other pseudo-kinetic rate constants can be found in earlier papers (6,7). 

Necessary conditions for the validity of the pseudo-kinetic rate constants are ... 

Applying the pseudo-kinetic rate constants, the explicit formulation of the kinetics of a multicomponent polymerization reduces to that of a homopolymerization. 

Application of the Method of Mcanents. In order to apply the method of moments (6,7), the pseudo-kinetic rate constant for the crosslinking reaction should be defined as follows. 

Liquid phase oxidation reaction of acetaldehyde with Mn acetate catalyst can be considered as pseudo first order irreversible reaction with respect to oxygen, and the reaction occurred in liquid film. The value of kinetic constant as follow k/ = 6.64.10 exp(-12709/RT), k2 = 244.17 exp(-1.8/RT) and Lj = 3.11.10 exp(-13639/RT) m. kmor. s. The conversion can be increased by increasing gas flow rate and temperature, however the effect of impeller rotation on the conversion is not significant. The highest conversion 32.5% was obtained at the rotation speed of 900 rpm, temperature 55 C, and gas flow rate 10" m. s. The selectivity of acetic acid was affected by impeller rotation speed, gas flow rate and temperature. The highest selectivity of acetic acid was 70.5% at 500 rpm rotation speed, temperature of 55 C... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

The kinetic constants, measured by monitoring the disappearance of the BrN3 under the experimental conditions, are pseudo order constants, kpseud0.2(obsj> = k2(obslinearized form, eq (13), can be written. 

Table 23.1 HMF formation kinetics in isothermal heating as a function of treatment temperature, first order reaction pseudo rate constant and regression coefficients...

Based on this equation, when the pseudo-first-order kinetic constant ( ga) was estimated at 150 Lg of (TSS)J, the half-life of E2 was established to be 0.2 h, with nearly all of the E2 being converted to El. El was removed more slowly at a half-life of 1.5 h and a kinetic constant of approximately 20 L g of (TSS)J, and EE2 was not significantly degraded under those same conditions. By comparison, in similar experiments conducted by Layton et al. (2000) at higher temperamres (30°C), at least 40% of the EE2 was mineralized in activated sludge within 24 h. 

Comparison of Pseudo First-Order Kinetic Constant... 

The kinetic parameters chosen for comparison are rate constants and t1/2. Radiation influences and the effect of reactor design are usually identical when these kinetic data are compared between the various AOPs tested. The values for pseudo first-order kinetics and half-lives for various processes are given in Table 14.3. In most cases, the values of f3/4 are equal to two times those of t1/2 therefore, the reactions obey a first-order kinetics. Figure 14.5. shows that Fenton s reagent has the largest rate constant, e.g., approximately 40 times higher than UV alone, followed by UV/F C and Os in terms of the pseudo first-order kinetic constants. Clearly, UV alone has the lowest kinetic rate constant of 0.528 hr1. 

The pseudo first-order kinetic constants (kt) of p-hydroxybenzoic acid are given in Table 14.6. UV/HA has a higher rate constant than that of UV/ 03 because the former requires only 0.5 peroxide molecule and 0.5 photon, whereas the latter process requires 1.5 ozone molecules and 0.5 photon. This is shown in Equation (14.8) and Equation (14.9) ... 

The single pulse voltammograms of a pseudo-first-order catalytic process are easily characterized by the increase of the limiting current with the time or the chemical kinetic constants, whereas its half-wave potential remains unchanged. 

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