Transverse lengths

The transverse length across the domains, the average distance of straight lines drawn across a domain, are given by [39,40]... 

In Figure 10, the transverse length of the polystyrene domains Increase steadily during the polymerization of monomer II, showing a more rapid increase later in the polymerization. Specific Interfaclal surface area (Figure 11) does not Increase monotonously with PS content. Rather, it shows a maximum at near the midrange of PS content depending on the synthetic detail. 

The rapid Increase of transverse length and the presence of a maximum in the specific surface area suggests a macroscopic development of dual phase continuity. A similar trend was obtained in another study of IPN s with SANS [13]. 

Figure 10. Increase of the polystyrene transverse length with polystyrene content [A ].

Here, p is the local (transverse) Peclet number, which is the ratio of transverse diffusion time to the convection time. Per is the radial Peclet number (ratio of transverse diffusion time to a convection time based on pipe radius). We assume that p <4 1 while Per is of order unity. (Remark The parameter Pe /p — ux)L/Dm is also known as the axial Peclet number. Also note that for any finite Per or tube diameter, the axial Peclet number tends to infinity as p tends to zero.) When such scale separation exists, we can average the governing equation over the transverse length scale using the L-S technique and obtain averaged model in terms of axial length and time scales. 

It is well known that a less-dense fluid beneath a more-dense fluid is buoyantly unstable [181]. Therefore, flames propagating upward may be expected to experience an instability that mathematically is a consequence of the term F in equations (89) and (90). If the entire flame is viewed as a discontinuity moving at velocity Vq with respect to the fresh gas and with respect to the burnt mixture, then through dimensional reasoning, a characteristic transverse length of the instability in the presence of a gravitational acceleration g may be guessed to be of order VqV Iq and a characteristic... 

Thus, the experiment appears indecisive. The most likely answer is that both kinetic mechanisms are active through the larger portion of the polymerization, and/or the theory illustrated above in equations (5) and (6) does not accurately portray the physical situation. For example, even though the data in Figure 8 and 9 represent conversions up to about 40% polystyrene, the transverse lengths increase as shown in Figure 6. 

Figure 6. Increase in the polystyrene transverse lengths with polystyrene content. (Reproduced from ref. 18. Copyri t 1988 American Chemical Society.)...

A major simplification in the modeling of monolithic channels is to decouple fluid mechanics from the problem by assuming a well-defined velocity profile. In this respect, the laminar flow assumption is likely to be valid since the timescale for viscous diffusion over a transverse length a is much smaller than the one for convection through a channel with length L ... 

In this expression, the relationship between channel volume, wall surface area (or hydraufic diameter <4), the characteristic transverse length a, and the shape factor S is expressed as follows ... 

Figure 8.8 Calculation approaches for the effectiveness hictor in thin catalytic coatings. The distinction between the conventional method and the one proposed in Lopes et aL [94] is illustrated. The Thiele modulus for a first order reaction is given by = K- kfD. The ratio between the catalyst and channel transverse length scales is given hy e = t la and o is the catalyst shape factor.

The Debye et al. theory can also be interpreted in terms of the transverse lengths across the domains Li and L2 of phases 1 and 2,... 

In this section, we discuss what happens if the Debye length X]) becomes comparable with the transverse length scale a of the channel and the Debye layer from various parts of the wall overlaps at the center of the channel. Eor a standard electrolyte with concentration about... 

If one could assume that the cavity angles were all about the same, then the size of each jump could be used to estimate the relative transverse length of each cavity. Cavity width could also be estimated by the scanning length during the gradual drop. These equivalences are Illustrated in Figure 14. 

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