General modulus

Fig. 19. Generalized modulus—temperature curves for polymeric materials showing the high modulus glassy state, glass-transition regions for cured and uncured polymers, plateau regions for cross-linked polymers, and the dropoff in modulus for a linear polymer.

Generalizati on. It has been observed that all plots of effectiveness against Thiele modulus are similar and that a single plot can represent the nine main cases fairly adequately by defining a generalized modulus as... 

Fig. 2.4 Classical representation of General Modulus-Temperature plots showing the different steps p...

The reformer tube operation was simulated on the basis of a set of continuity-, energy- and momentum equations using one and two dimensional heterogeneous models. Intraparticle gradients in the rings were accounted for by the use of the generalized modulus concept. 

So far the generalized modulus of Bischoff has only been applied for a single reaction. Since two reactions are occurring simultaneously, each with their effectiveness factor, (18) can only be conveniently applied when a simple relationship exists between dCj and dC2. The experimental program led to a fairly constant ratio of CO/COj over a rather broad range of total conversion, confirming previous observations by Akers and Camp (18). Therefore, the ratio of dCj/dC2 was approximated by (cf-cf T/ (cl-C ), which is the more accurate the more equilibrium is approached. 

The effectiveness factors based upon the generalized modulus concept were calculated in each axial increment used in the numerical integration of (19/ 20 and 21). The saving in computer time, as compared with solving the differential equations for mass transfer inside the particle in each increment, is enormous, in particular since the particle shape is an additional complication. The simulation results based upon the complete model (equations 19-27) cure represented in Figures 5 and 6 for typical operating conditions. 

Equation 56 can be used only for spherical catalyst pellets and first order, irreversible reactions. However, for convenience, and in analogy to the Thiele modulus, a generalized modulus ij/pn can be defined as well which applies to arbitrary pellet shape and arbitrary reaction order. This is defined as... 

Voss, D. T. (1988). Generalized modulus-ratio tests for analysis of factorial designs with zero degrees of freedom for error. Communications in Statistics Theory and Methods, 17, 3345-3359. 

For a cylinder and other geometries, a general modulus for a first-order reaction is... 

Generalizations for the reaction kinetics have also been made. Petersen [8] has shown that for a sphere a generalized modulus can be postulated for nth-order kinetics ... 

Using this generalized modulus the effectiveness factor in the low ij region (or high 0, region) can be calculated from... 

The generalized modulus defined in Equation (6.3.51) has been normalized so that the effectiveness factor is approximately 1/< q large values of Q, as illustrated in Figure 6.3.9. 

In ITT, the slow stress fluctuations in g(f,y) are approximated by following the slow structural rearrangements, encoded in the transient density correlators. The generalized modulus becomes, using the approximation (10a) and the vertex (lib) ... 

The parameter pco characterizes a short-time, high frequency viscosity and models viscous processes which require no structural relaxation, like in the general case (15). Together with F, it is tire only model parameter affected by Hl. Steady state shear stress under constant shearing, and viscosity then follow via integrating up the generalized modulus ... 

Figure 17.3. Generalized Modulus-Temperature curves for engineering polymer blends for automotive applications combinations of high amorphous polymers with lower Tg crystalline polymers. Reproduced with permission from L. M. Robeson, in Contemporary Topics in Polymer Science, Vol. 6, Multiphase Macromolecular Systems , B. M. Culbertson, Ed., Plenum Press, New York, 1989.

We may take equation (vi) as defining a generalized modulus, compatible with any kinetics. 

Use of this generalized modulus brings effectiveness factor curves that are nearly superimposed upon each other for th-order power law kinetics. The results of this are shown in Figure 7.5 for slab geometry. 

Show me that the generalized modulus reduces to the form of equation (7-5) for a first-order irreversible reaction with constant D ff. Further, what about equation (7-12) ... 

Intraparticle diffusion can have a significant effect on the kinetic behavior of enzymes immobilized on solid carriers or entrapped in gels. In their basic analysis of this problem. Moo-Young and Kobayashi (1972) derived a general modulus and effectiveness factor. The results also predicted possible multiple steady-states as well as unstable situations for certain systems. While these results are very interesting it should be remembered that they are primarily mathematical and await extensive experimental support data. 

If a solid substrate is sufficiently porous, the enzyme can diffuse into it and degradation can proceed inside the material. The water-soluble substrate fragments must also diffuse out of the solid matrix through these same pores into the bulk solution where they may be subject to further enzymatic attack. The reaction may concurrently proceed at the exterior of the matrix surface and for substrates of low porosity this is where much of the degradation takes place. As before, utilization of the effectiveness factor and general modulus is convenient in solving the relevant differential equations. 

Aris [79] noted this, and from similar results for cylinders and other geometries found that a general modulus for all shapes could be defined with — tj curves practically superimposed ... 

Certain reaction rate forms can lead to unusual bdiavior, that is not u ll represented by the general modulus approach, over the entire range of modulus values. For isothermal systems, these are assocuOed with rate equations that can exhibit empirical or approximate negative order bdiavior. For example, Satterfield, Roberts, and Hartman [87,88] have shown that rate equations erf the form... 

For an nth order irreversible reaction, the generalized modulus becomes... 

The results of Eqs. b and d were then introduced into the general modulus, Eq. 3.6.b-8,... 

We have now seen how possible pore diffusion problems can be evaluated compute the generalized modulus and use Figs. 3.6.a-3 or 3.6.b-l to see if the value of t) is less than unity. In the design situation this procedure can be used since is presumably known, but when determining kinetic constants from laboratory or pilot plant data this can t be done since k is udiat is being sought. Thus, criteria for the importance of pore diffusion, independent of k are also useful. There are two main types that are generally used. 

Using the generalized modulus, the criterion Eq. 3.6.C-5 was extended by Petersen [85] and by Bischoif [102] to the case where the reaction rate may be... 

Bischoff [102] showed that the generalized modulus concept of Eq. 3.6.b-8 can be extended by substituting the right hand side of Eq. 3.7.a-14 or Eq. 3.7.a-14a as the rate form. This then asymptotically unified all the isothermal and endothermic curves, and the suitable portions of the exothermic curves, but still would not permit prediction of the maxima or stability aspects. 

The only unknown parameter left in Eq. (o) is the tortuosity factor, r. This factor was determined from a comparison between the experimental rate at zero coke content, measured in a differential reactor and the surface fluxes. The latter were calculated using Pick s law and for a given t from the concentration profiles obtained by numerical integration of the system (Eqs. (g), (h), (i), and (j)). A value of T = 5 led to the best fit of all six experiments. This is the generally accepted value for the tortuosity factor in a catalyst of the type used in this work. It was also possible to calculate an effectiveness factor from these results. A value of 0.20 was obtained for a particle radius of 2.3 mm at 550°C. The Bischoff general modulus approach, presented in Chapter 3, leads to a value ofO.28. Finally, the heat transfer coefficients were calculated from the correlation of Handley and Heggs, mentioned in Chapter 3. 

Lee and Reilly (1981) defined a more rigorous form of the Thiele modulus based on the generalized modulus of Bischolf and Aris (see Chapter 7) which is particularly useful in analyzing the role of diffusion in deactivation. Their analysis shows that, in reactant-independent deactivation, the presence of a strong dilfusional limitation lowers the rate of deactivation to half the diffusion-free value. Thus, surprisingly, diffusion seems to have a favorable effect on the performance of a deactivating immobilized enzyme catalyst. 

---