Pseudokinetics

The characteristics of a linear growth pattern as the consequence of an enzyme system with constant activity have been thoroughly investigated and modeled by Knorre et al. (1978a,b). It is thought that linearity is caused by constant activity of enzymes in systems where substrate concentration is limited. Linearities always indicate the presence of some limitations even their exact nature cannot be readily determined. Beyond a lack in nutrients, linear growth phases can also be the result of transport limitations. Thus, from a systematic point of view, these data repesent pseudokinetics. 

Pseudokinetic parameters result from a kinetic study whenever the model (either knowingly or unknowingly) is simpler than the real situation. Generally, falsification can be the result of the undetected influence of other reactions or of transport phenomena (macrokinetics). 

Pseudokinetic phenomena become evident only when process kinetic analysis is carried out with mathematical models. Most bioprocesses are basically heterogeneous systems. Generally, pseudohomogeneous rates measured in L phase analyses are used, because they are thought to reflect directly the intrinsic reaction rate of metabolism in the solid phase (biomass). Even under steady-state conditions, however, this assumption is not necessarily valid. 

Many factors are known to influence the shape of Monod-type biokinetics (A. Moser, 1981 Fiechter, 1982) most of them alter K. Apparent values can be the result of  

One might also think that the value depends on the size of the cells (transport limitation by Dg), and that only in some ideal cases is the same as that appropriate to the enzyme or mitochondria (Hartmeier, Bronn, and Dellweg, 1971 Kessick, 1974). 

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To obtain the kinetic order n, it has been worked at constant pressure, 12.5 MPa, and it has been defined a pseudokinetic rate constant K. 

The linear equations (4) and (5) represent the pressure dependence of the rate pseudokinetics constants at 375°C for V and Ni, respectively. 

The pseudokinetic rate constants and the intrinsic rate constants of the Ni removal reactions at 375, 400 and 415°C are indicated in Table 3. The activation energies for Ni and V removal reactions were obtained from the semilogarithmic representation of In K versus 1/T by applying the Arrhenius equation. Figure 3. The values of m, n and Ea are indicated in Table 4. 

The overall termination pseudokinetic rate parameter may be defined as... 

Similarly, all other rate parameters may be defined using the pseudokinetic approach. For example, the overall chain transfer to CTA (Chain Transfer Agent) pseudokinetic rate parameter is calculated as... 

Pseudo-Homopolymer Approach or Pseudokinetic Rate Constants Method (PKRCM)... 

The second approach implies that the MWD equations for a multicomponent monomer system are treated as those of a homopolymer. These ideas appeared in the literature under slightly different names but around the same period, when more researchers started studying copolymerization systems (e.g., Ballard et al. [20]). Hamielec s group formalized this approach under the name of pseudokinetic rate constants method (PKRCM) and illustrated its use for linear, branched, and crosslinked copolymerization systems [21], as well as for batch, semibatch, and continuous reactors [22], Ray s group also made use of what they referred to as apparent rate constants [23]. The group of Morbidelli used a similar idea that they termed as pseudohomopolymer approach [24]. 

Note that by defining the terms in parenthesis as pseudokinetic or apparent rate constants (Table 12.4) Equations 12.42 and 12.43 are identical to those describing the MWD of the live polymer in a homopolymerizafion. 

Mass balance for polymer radicals TABLE 12.5 Pseudokinetic Rate Constants (Pseudo-Homopolymer Approach) ... 

Figure 12.4 shows a schematic representation of the system studied by Herndndez-Ortiz et al. [47, 48]. The polymerization scheme for that case is listed in Table 12.6. On the basis of that polymerization scheme, the kinetic equations summarized in Table 12.7 and Equation 12.46 were derived. The pseudokinetic rate constants used are summarized in Table 12.8. The reader is referred to the paper by Hernandez-Ortiz et al. [47] for more details. What is interesting here is to point out, as listed in Table 12.9, that simpler cases can be adequately described by simplifying that compiex model ... 

Required initial conditions [M]q, [I]q, [S]q Fixed initial conditions /20 = 0, [NO ]o = 0 Comments pseudokinetic rate constants method not required. 

Comments pseudokinetic rate constants method required. If two monomers are employed, M = 2,... 

Logically, the accuracy of parameter estimation using this simulation method depends on the quality of the model function chosen. An inadequate model results in pseudokinetic parameters. The fitting of simulation to experimental data is carried out with the aid of statistical criteria such as the F-test, minimizing deviations erf ... 

Similarly results have been reported by Toda (1975) examining the dependence of pseudokinetic parameters r ax.app s.app on fluid flow rate. The following equation was found to be valid for describing the dependence of on /cl2 ... 

The complete computer simulation of the BRE is given by dotted lines for various film thicknesses d, and the area of experimental verification is indicated with solid lines. Obviously, high S concentration cannot be attained with thick films. As can be seen from Fig. 5.69, the appearance of a simple first-order reaction rate in cases of biofilm processing can often be understood as an observed case of pseudokinetics. 

Finally, a case of pseudokinetics appears in case of global measurement techniques, as in biological waste water treatment. 

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