Design equations with effectiveness factors

A 2 factorial design with two factors requires four runs, or sets of experimental conditions, for which the uncoded levels, coded levels, and responses are shown in Table 14.4. The terms Po> Po> Pfc> and Pafc in equation 14.4 account for, respectively, the mean effect (which is the average response), first-order effects due to factors A and B, and the interaction between the two factors. Estimates for these parameters are given by the following equations... 

Designers of most structures specify material stresses and strains well within the pro-portional/elastic limit. Where required (with no or limited experience on a particular type product materialwise and/or process-wise) this practice builds in a margin of safety to accommodate the effects of improper material processing conditions and/or unforeseen loads and environmental factors. This practice also allows the designer to use design equations based on the assumptions of small deformation and purely elastic material behavior. Other properties derived from stress-strain data that are used include modulus of elasticity and tensile strength. 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Equations (14-37) and (14-42) represent two different ways of obtaining an effective factor, and a value of Ae obtained by taking the reciprocal of S, from Eq. (14-42) will not check exactly with a value of A, derived by substituting At = 1/Si and A2 = 1/S2 into Eq. (14-37). Regardless of this fact, the equations generally give reasonable results for approximate design calculations. 

Since most practical problems do not have analytical expressions for the effectiveness factor, the use of Equations (10.2.6) and (10.2.7) are more generally applicable. If Equations (10.2.6) and (10.2.7) are to be solved numerous times (e.g., when performing a reactor design/optimization), then an alternative approach may be applicable. The catalyst particle problem can first be solved for a variety of Cab, Tg, (p, Bi , Bi, that may be expected to be realized in the reactor design. Second, the values can be fit to a function with Cab, Bi ... 

The objective of Chaps. 10 and 11 is to combine intrinsic rate equations with intrapellet and fluid-to-pellet transport rates in order to obtain global rate equations useful for design. It is at this point that models of porous catalyst pellets and effectiveness factors are introduced. Slurry reactors offer an excellent example of the interrelation between chemical and physical processes, and such systems are used to illustrate the formulation of global rates of reaction. 

Application of the theory in this case is quite simple, in fact, since we have been careful to define quantities such as the effectiveness factor or the enhancement factor in terms of the rate under observable conditions. Consider as an example a PFR catalytic reactor model in which we wish to include possible diffusion limitations on the catalyst activity. This becomes, de facto, a two phase model since the inclusion of an effectiveness factor means that we are considering the catalyst as a separate phase. We start with the design equation written in the following form... 

When external or internal mass transfer resistances are negligible, ijg= 1 or i]i= 1, respectively. If intrinsic kinetic parameters (determined while using free enzymes or cells, with no mass transfer limitations) are known, the total effectiveness factor can thus be used together with the reactor design equations as... 

The reaction penetration depths. Id or la, are highly insightful parameters to evaluate catalyst layer designs in view of transport limitations, uniformity of reaction rate distributions, and the corresponding effectiveness factor of Pt utilization, as discussed in the sections Catalyst Layer Designs in Chapter 1 and Nonuniform Reaction Rate Distributions Effectiveness Factor in Chapter 3. Albeit, these parameters are not measurable. The differential resistances, Rd or Ra, can be determined experimentally either as the slope of the polarization curve or from electrochemical impedance spectra (Nyquist plots) as the low-frequency intercept of the CCL semicircle with the real axis. The expressions in Equation 4.33 thus relate the reaction penetration depths to parameters that can be measured. 

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