Kinetic modeling equations

Table 2. Final forms of kinetic modeling equations... 

The equation derived by Troelstra and Kruyt is only valid for coagulating dispersions of colloids smaller than a certain maximum diameter given by the Rayleigh condition, d 0.10 A0. Equation 4 applies in cases where particles are transported solely by Brownian motion. Furthermore, the kinetic model (Equations 2 and 3) has been derived under the assumption that the collision efficiency factor does not change with time. In the case of some partially destabilized dispersions one observes a decrease in the collision efficiency factor with time which presumably results from the increase of a certain energy barrier as the size of the agglomerates becomes larger. 

Andrew s kinetic model (Equation 21) (Andrews, 1968), for example, relates the specific growth rate (px) with the substrate concentration S, applying three parameters (p max, ks and k s). 

Table 3 Kinetic model equations for free radical polymerization...

RTD experiments showed that the fixed-bed almost behaves like a plug-flow reactor and the infrared cell like a continuous stirred tank reactor. This fixed-bed is described by the tanks-in-series model, using 9 tanks for the catalyst compartment. The two kinetic models (Equations 1-6) are able to describe the stop-effect experiments at 180 and 200°C, and the considerations made in this work are valid for both temperatures. However, for the sake of clarity, only model discrimination at 180°C will be presented here. In the experimental conditions used here, both models can be simplified the first adsorption step is considered as irreversible, and instantaneous equilibrium is assumed for the second one. With these hypothesis the total number of kinetic parameters is reduced from five (ki, Li, k2, k.2 and ks) to three (ki, K2 and ks), and the models can be expressed as follows ... 

The denaturation kinetics of proteins during convective drying can generally be expressed by the first-order reaction kinetics model (Equation 49.5) (Skelte and Anthony, 1996 Oldfield et al., 1998 Haque et al., 2013b) ... 

Cross-linked PVA membranes were also used for the PV separation of the liquid mixtures of water-acetic acid and water-acetic acid-n-butanol-BuAc (Liu et al. 2001). The permeation fluxes of water and acetic acid as a function of the composition were studied. The esterification of acetic acid with -butanol catalyzed by Zr(S04) 4H2O was carried out at a temperature range of 60°C-90°C. It was noticed that mostly water, less acetic acid, much less -butanol, and actually no BuAc permeated through the membrane during PV separation of the quaternary mixture of water-acetic acid-n-butanol-BuAc. From the results obtained from this study, Liu et al. (2001) developed a kinetic model equation for the esterification then, it was taken as a model reaction to study the coupling of PV with esterification. 

Another S-utilization mechanism has been proposed consisting of a consequence of two phenomena (adsorption and penetration of substrate molecules on specific sites followed by transportation through the cell membrane). In this approach, the following generalized kinetic model equation was derived (Borzani and Hiss, 1979 Hiss and Borzani, 1983) ... 

Kinetic Model Equations for Inhibition by Substrates and Products... 

In most cases of inhibition, the formal kinetic model equations are, like Monod s relationship, derived from theories of the inhibition of single enzymes. The equations are, however, only hypotheses they may be replaced by any other adequate model. 

The predictions of different kinetic model equations for pellet growth are compared in Fig. 5.76 (van Suijdam et al, 1982). The main objection against the models presented is that they are autonomous, that is, the biomass rate equation is not related to the concentration of the limiting substrate, and that mass transfer limitations are not explicitly considered in these macrokinetics. To introduce these phenomena, a combination of mass transfer limitation effects and biokinetics must be used. Such an integrated model for pellet growth was developed (Metz, 1975) and extended (van Suijdam et al., 1982), based on balance equations for X, S, and O2 in liquid and solid phase. 

Equation (154) represents a solution to our kinetic model equation when the explicit expressions given in this section are used to avaluate the thermodynamic properties and transport coefficients. The solution is known to be valid at long wavelengths, or more precisely when Tk c. At the opposite extreme of very short wavelengths all the collision effects become unimportant compared to free molecular flow. Then all kinetic model solutions will tend to the distribution... 

Table 8. Ratio of the First-Order Adsorption Rate Constants h = fenpoyAiOH-DDPO. Estimated by Fitting the Kinetic Model Equation to XPS Measurements, Contact-Angle Data, and ToF-SIMS Data...

Table 9.1 shows some drying kinetic model equations used to fit microwave drying data. 

Most assessments of the activity, selectivity and deactivation have been based on the comparison of the light-off curves under the same reaction conditions. Several efforts have been made in order to obtain kinetic model equations for catalytic... 

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