Non-isothermal effectiveness factors

The isothermal and non-isothermal effectiveness factors for the single unimolecular irreversible reaction. Equilibrium adsorption-desorption model with linear isotherm For simplicity of presentation and without any loss of generality we consider here the case where the bulk temperature and concentration are taken as the reference temperature and concentration. In this case the boundary conditions (5.127) become at fu= 1.0... 

Non-isothermal ejfectiveness factor The non-isothermal effectiveness factor can be obtained numerically only by integrating the two points boundary value differential equations using different numerical techniques, the most efficient of these techniques is the orthogonal collocation method. 

For a spherical particle the non-isothermal effectiveness factor t] is defined as ... 

FIGURE 5.49 Intraphase non-isothermal effectiveness factor versus 70 for linear kinetics y = 20, Sh, Nu = oo. 

Fig. 15. Non-isothermal effectiveness factor for spherical particle as function of the Thiele modulus, . Adapted from Weisz and Hicks (1962) and Trambouze et al. (1988).

Garside, J. and Tavare, N.S. (1981) Non-isothermal effectiveness factors for crystal growth. Chemical Engineering Science, 36, 863-866. 

Tavera EM. Analytical expression for the non-isothermal effectiveness factor The nth-order reaction in a slab geometry. Chemical Engineering Science 2005 60 907-916. 

Hoyos B, Cadavid JG, Rangel H. Formulation and numerical calculation of non-isothermal effectiveness factor for finite cylindrical catalysts with bidimensional diffusion. Lat. Am. Appl. Res. 2004 34 17. 

The last factor reflects the role of the non-isothermal effect on the kinetics of the process. The complete kinetic equation for non-isothermal polymerization can be written as... 

In practice there are a number of other factors to be taken into account. For example, the above analysis assumes that this plastic is Newtonian, ie that it has a constant viscosity, r). In reality the plastic melt is non-Newtonian so that the viscosity will change with the different shear rates in each of the three runner sections analysed. In addition, the melt flow into the mould will not be isothermal - the plastic melt immediately in contact with the mould will solidify. This will continuously reduce the effective runner cross-section for the melt coming along behind. The effects of non-Newtonian and non-isothermal behaviour are dealt with in Chapter 5. 

Fig. 3. Effectiveness factor as a function of Thiele modulus for a non-isothermal catalyst pellet.

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)]... 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

The experimentally-determined effectiveness factor is determined as the ratio of the experimental macro reaction rate to the intrinsic reaction rate under the same interface (bulk) composition and temperature. Based on the experimental conditions of the macrokinetics, the predicted effectiveness factors of the methanation reaction and the WGSR are obtained by solving the above non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components. Table 1 shows the calculated effectiveness factors and the experimental values. By... 

We have used CO oxidation on Pt to illustrate the evolution of models applied to interpret critical effects in catalytic oxidation reactions. All the above models use concepts concerning the complex detailed mechanism. But, as has been shown previously, critical. effects in oxidation reactions were studied as early as the 1930s. For their interpretation primary attention is paid to the interaction of kinetic dependences with the heat-and-mass transfer law [146], It is likely that in these cases there is still more variety in dynamic behaviour than when we deal with purely kinetic factors. A theory for the non-isothermal continuous stirred tank reactor for first-order reactions was suggested in refs. 152-155. The dynamics of CO oxidation in non-isothermal, in particular adiabatic, reactors has been studied [77-80, 155]. A sufficiently complex dynamic behaviour is also observed in isothermal reactors for CO oxidation by taking into account the diffusion both in pores [71, 147-149] and on the surfaces of catalyst [201, 202]. The simplest model accounting for the combination of kinetic and transport processes is an isothermal continuously stirred tank reactor (CSTR). It was Matsuura and Kato [157] who first showed that if the kinetic curve has a maximum peak (this curve is also obtained for CO oxidation [158]), then the isothermal CSTR can have several steady states (see also ref. 203). Recently several authors [3, 76, 118, 156, 159, 160] have applied CSTR models corresponding to the detailed mechanism of catalytic reactions. 

The effects of antioxidants on OT of SME by non-isothermal (conventional) DSC, static mode P-DSC, and dynamic mode P-DSC were investigated by Dunn (2006a), which is summarized in Table 1.15. Results from all three methods consistently showed that treating SME with antioxidants TBHQ and a-tocopherol increased OT with respect to untreated SME. Statistical comparison of P-DSC results with those from isothermal analysis of OSI at 60°C was facilitated by calculation of the corresponding response factors (defined ratios of OT of the sample to that of methyl oleate, and of OSI of the sample to that of methyl oleate). Data for the sample and reference material (methyl oleate) were measured under the same experimental conditions. Results showed the highest degree of correlation (P = 0.79) between dynamic-mode P-DSC and isothermal OSI analyses. 

Operative. For the non isothermal case, effectiveness factors greater than unity are possible. Weisz and Hicks have considered this problem in some detail and constructed a number of graphs for various heats of reaction and activation energies. When a reaction is limited by pore diffusion, the reaction rate is proportional to yjky. If the temperature effects can be expressed as a simple Arrhenius relationship = A txp —E/RT), then the measured activation energy E will be about half the true activation energy. Very low values of the activation energy, i.e, 1-2 kcal. mole are only observed when mass transfer to the external catalyst surface is limiting the rate. 

The preceding discussion of effectiveness factors is valid only for isothermal conditions. When a reaction is exothermic and non isothermal, the effectiveness factor can be significantly greater than 1 as shown in Figure 12-7. Values of T greater than 1 occur because the externa surface temperature of the pellet is le.ss than the temperature inside the pellet where the exothermic reaction is taking place. Therefore, the rate of reaction inside the pellet is greater than the rate at the surface. Thus, because the effect vene.s,s factor is the... 

Although for simple cases of isothermal non-porous pellets with linear kinetics the concept of the effectiveness factor is not practically important it is quite useful to present it since the dehnition and the principles involved are the same as for the more complex cases which will be discussed later. 

For the simple linear kinetics of the isothermal non-porous catalyst pellet described by equations (5.3, 5.4), the effectiveness factor is simply given by ... 

Another interesting phenomenon can emerge under non-isothermal conditions for strongly exothermic reactions there will be multiple solutions to the coupled system of energy and mass balances even for the simplest first-order reaction. Such steady-state multiplicity results in the existance of several possible solutions for the steady state overall effectiveness factor, usually up to three with the middle point usually unstable. One should, however, note that the phenomenon is, in practice, rather rarely encountered, as can be understood from a comparison of real parameter values (Table 9.2). 

A temperature gradient would also be expected. For an isothermal case, with rj set equal to 1, multiple steady-state solutions may be found (see Figure 10), and the concentration gradient is very significant at temperatures above 427°C (800°F). The non-isothermal catalytic effectiveness factors for positive order kinetics under external and internal diffusion effects were studied by Carberry and Kulkarni (8) they also considered negative order kinetics. 

Figure 7.19 Effectiveness factor versus normalized Thiele modulus for a first-order reaction in a non isothermal spherical pellet.

Figure I .9.a-I Effectiveness factor diagram for non-isothermal situations (after McGreavy and Cresswell [108], from Froment [9]). 

Cassiere, G. and Carberry, J.J. (1973) Interphase catalytic effectiveness factor activity, yield and non-isothermality. Chem. Eng. Educ., 7 (1), 22—26. 

For Pshooting method is equal to 1) such interval of the values of Thiele modulus exists in which the effectiveness factor T] = ro /rs exceeds unity. Consequently, the presence of an internal resistance to mass transport may lead to serious increase in the overall rate of the isothermal and non-isothermal, heterogeneous autocatalytic reactions compared to the values obtained for the vanishing or very large resistance. 

For non-isothermal kinetics, we adapt the result derived by Tavera [100] originally for Dirichlet conditions, and as expected b will be a function of kinetic descriptors (e.g., order of reaction w), nonisothermal parameters (y and P), and surface conditions (c j, and T, y). The dependence on the latter quantities (generally unknown) leads to an implicit calculation of the effectiveness factor. Substituting Equation 3.42 into 3.54a,... 

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