Global effectiveness factor

The analysis will be done in three steps. In the fist step, differential equations will be developed by combining enzyme kinetics and mass transfer to obtain the substrate (and product) profile within the biocatalyst particle in the second step, local effectiveness factor profiles will be obtained from the previous step in the third step a global effectiveness factor will be obtained by adequately averaging that distribution. This global effectiveness factor describes the behavior of the biocatalyst particle as a whole and will be obtained in terms of measured and calculated parameters, being a useful way of incorporating IDR into enzyme reactor design and performance evaluation, as considered in section 5.3. 

The final step is the determination of a global effectiveness factor from the profile of local effectiveness factors that adequately describe the behavior of the biocatalyst particle (membrane in this case) as a whole. Since the distribution of r values is... 

Fig. 4.13 Global effectiveness factor (mean integral value) of a membrane immobilized enzyme with Michaelis-Menten kinetics as a function of bulk substrate concentration and Thiele modulus...

Substrate (and product) profiles are obtained from the numerical resolution of the above differential equations (system of differential equations in the case of product inhibition). The corresponding local effectiveness factors (ratio of effective and intrinsic reaction rates) are then calculated and the global effectiveness factor determined from their profiles, as in the case of simple Michaels-Menten kinetics. Results are represented in three-dimensional plots in Figs. 4.15 to 4.18 respectively. 

Mass transfer limitations can be relevant in heterogeneous biocatalysis. If the enzyme is immobilized in the surface or inside a solid matrix, external (EDR) or internal (IDR) diffusional restrictions may be significant and have to be considered for proper bioreactor design. As shown in Fig. 3.1, this effect can be conveniently incorporated into the model that describes enzyme reactor operation in terms of the effectiveness factor, defined as the ratio between the effective (or observed) and inherent (in the absence of diffusional restrictions) reaction rates. Expressions for the effectiveness factor (rj), in the case of EDR, and the global effectiveness factor (t ) for different particle geometries, in the case of IDR, were developed in sections 4.4.1 and 4.4.2 (see Eqs. 4.39-4.42,4.53,4.54,4.71 and 4.72). Such functions can be generically written as ... 

DRX - x-ray diffraction TPR - Temperature programmed reaction MEV - Scanning electronic microscopy TEM - Transmission Electronic microscopy T>n - Thiele modulus rj - Effectiveness factor - Global effectiveness factor E f = Height bed at minimum fluidization mf = void fraction at minimum fluidization Ps = density of solid p = density of gas fh - bubble fraction fe - emulsion fraction of gas ki - interface coefficient (mfl nr bed x h) u - mean velocity collision velocity Ub - superficial velocity (m/s)... 

Equation 3.39 expresses an averaged reaction rate evaluated with the actual concentration and temperature profiles that are observed inside the pellet, in the presence of reaction superposed with the eventual presence of significant limitations from diffusion through the catalyst pores. If transport to the catalyst surface is also limited, conditions on the interface may differ significantly from those in the bulk fluid. In this case, it maybe more useful to refer the observed average rate of reactant consumption to the conditions in the interparticular bulk fluid, defining a global effectiveness factor fj, which is related to (3.39) by... 

The effectiveness factor (referred to surface conditions //) and the global effectiveness factor (referred to bulk conditions if) can be described by approximations in limiting cases (see Chapter 3 for examples) and rational approximations can be derived. 

The global design equation used for Steps 1 and 9 is modified to include the effectiveness factor ... 

Equation (10.29) is the appropriate reaction rate to use in global models such as Equation (10.1). The reaction rate would be —ka if there were no mass transfer resistance. The effectiveness factor rj accounts for pore diffusion and him resistance so that the effective rate is — 7ka. 

A high Damkohler number means that the global rate is controlled by mass transfer phenomena. So, the process rate can be rewritten in terms of the Damkohler number and the external effectiveness factor for each reaction order can be deduced, as shown in Table 5.5. In Figure 5.3, the external effectiveness factor versus the Damkohler number is depicted for various reaction orders. It is clear that the higher the reaction order, the more obvious the external mass transfer limitation. For Damkohler numbers higher than 0.10, external mass transfer phenomena control the global rate. In the case of n = 1, the external effec-... 

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

It is possible to combine the resistances of internal and external mass transfer through an overall effectiveness factor, for isothermal particles and first-order reaction. Two approaches can be applied. The general idea is that the catalyst can be divided into two parts its exterior surface and its interior surface. Therefore, the global reaction rates used here are per unit surface area of catalyst. 

First-order reactions without internal mass transfer limitations A number of reactions carried out at high temperatures are potentially mass-transfer limited. The surface reaction is so fast that the global rate is limited by the transfer of the reactants from the bulk to the exterior surface of the catalyst. Moreover, the reactants do not have the chance to travel within catalyst particles due to the use of nonporous catalysts or veiy fast reaction on the exterior surface of catalyst pellets. Consider a first-order reaction A - B or a general reaction of the form a A - bB - products, which is of first order with respect to A. For the following analysis, a zero expansion factor and an effectiveness factor equal to 1 are considered. 

In Eqn. 5.3-1, rj is the effectiveness factor of the catalyst with respect to the dissolved gaseous reactant and the temperature of the outer surface. The rate of reaction within the catalyst pores is comprised in rj. R is the reaction rate expressed in moles of gaseous reactant, A, per unit of bubble-free liquid, per unit of time. Reaction is irreversible. In equation (1) it has not been assumed that the gas is pure gas A, its concentration in the bubbles being Cg. Also, Henry s law for the gas is assumed and written as in Eqn. 5,3-4. Using Henry s law, Eqn. 5.3-4, the intermediate concentrations (Cs, CL) can be eliminated using the above system of equations. This provides an expression of the global rate in terms of an apparent constant, ko, that contains the various kinetic and mass transfer steps. Therefore, the observed rate can be written as ... 

Apart from oxidation of the lubricant and the metal surfaces, there can be complex tribo-chemical reactions. Chemical reactions at the surfaces can be stimulated by different factors. One factor is heating due to friction. This can either be a global effect (elevated mean temperature of surfaces and lubricant) or a localized phenomenon. Especially in situations where mixed or boundary lubrication exists, the direct contact of surface asperities can lead to high flash temperatures. At these hot spots temperatures in excess of 1000°C promote chemical reactions and surface melting. Other factors promoting chemical reactions are ... 

In contrast, EET has been historically modelled in terms of two main schemes the Forster transfer [15], a resonant dipole-dipole interaction, and the Dexter transfer [16], based on wavefunction overlap. The effects of the environment where early recognized by Forster in its unified theory of EET, where the Coulomb interaction between donor and acceptor transition dipoles is screened by the presence of the environment (represented as a dielectric) through a screening factor l/n2, where n is the solvent refractive index. This description is clearly an approximation of the global effects induced by a polarizable environment on EET. In fact, the presence of a dielectric environment not only screens the Coulomb interactions as formulated by Forster but also affects all the electronic properties of the interacting donor and acceptor [17],... 

As indicated In Figure 1 (m), drug molecules can exit the villus venules. The global effect of this exchange is a decreased absorption capacity of the blood circulation. When the concentration of absorbed material is high, as is the case in food absorption, the efficiency of the blood supply is decreased by a factor of up... 

Figure 22 illustrates how relatively simple global quality factors can be used as filters in the search for optimum solutions in the parameter space that defines multiple-pulse sequences. Suppose for typical coupling constants = 10 Hz a multiple-pulse sequence with a constant rf amplitude = 10 kHz is desired that effects efficient Hartmann-Hahn transfer in the offset range of +4 kHz. Here, the simple two-dimensional parameter... 

Further generalization of the Thiele modulus and effectiveness factor for a general global reaction and various shapes is... 

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