Scaling considerations

Another approach, neglecting the details of the chemical structure and concentrating on the universal elements of chain relaxation, is based on dynamic scaling considerations [4, 11], In particular in polymer solutions, this approach offers an elegant tool to specify the general trends of polymer dynamics, although it suffers from the lack of a molecular interpretation. 

The reason why simple scaling considerations do not lead to a unique result for the exponent a is due to the fact that the entanglement problem as a geometrical phenomenon contains two independent lengths the lateral distance between the chains s = (L/V) 1/2 and the step width of the random walk... 

Based on the analogy between polymer solutions and magnetic systems [4,101], static scaling considerations were also applied to develop a phase diagram, where the reduced temperature x = (T — 0)/0 (0 0-temperature) and the monomer concentration c enter as variables [102,103]. This phase diagram covers 0- and good solvent conditions for dilute and semi-dilute solutions. The latter will be treated in detail below. 

Patience, G. S., Chaouki, J., Berruti, F., and Wong, R., Scaling Considerations for Circulating Fluidized Bed Risers, Powder Technol., 72, 31 (1992)... 

This is an important scaling consideration as the spray momentum is a controlling factor in the droplet trajectories. 

But the entire conception here is that of equilibrium solvation of the transition state by the Debye ionic atmosphere, and closer inspection [51] indicates that this assumption can hardly be justified indeed, time scale considerations reveal that it will nearly always be violated. The characteristic time for the system to cross the reaction barrier is cot, 0.1 ps say. On the other hand, the time required for equilibration of the atmosphere is something like the time for an ion to diffuse over the atmosphere dimension, the Debye length K- this time is = 1 ns for a salt concentration C= 0.1M and only drops to lOps for C 1M. Thus the ionic atmosphere is perforce out of equilibrium during the barrier passage, and in analogy with ionic transport problems, there should be an ionic atmosphere friction operative on the reaction coordinate which can influence the reaction rate. 

A chemical reaction can be viewed as occurring via the formation of an excited state that can be any one of the degrees of freedom of the collection of N atoms. That is, translational, rotational, vibrational, and electronic excitation can lead to a chemical reaction. We often do not need to consider explicitly the quantized nature of rotational and vibrational energies in practical applications because of time scale considerations. For example, when a chemical reaction proceeds via a vibrationally excited state, in which the average lifetime typically is about 3 x 10" where T is in Kelvins... 

Walter JK, Werz W, Berthold W. Process scale considerations in evaluation studies and scale-up. Dev Biol Stand 1996 88 99-108. 

A more subtle problem occurs for quantities involving several characteristic length scales, Consider for instance the density correlation function in the limit of large momenta (qi ff)2 > 1 where 1/q defines a length scale of interest, which is much smaller than Rg. In the excluded volume limit simple scaling considerations (cf. Sect. 9.1, Eq. (9.20)) suggest... 

All work reviewed so far in this subsection concerns thin films with neutral surfaces, but we feel that the general scaling description Eqs. (129)-(133), Fig. 23) should also apply to thin films with symmetric surfaces that both favor the same component (say B, cf. Fig. 5) relative to the other. The additional feature, not present in Fig. 22, then is a shift of the critical volume fraction critCD) with thickness. Scaling considerations [216,224,225] predict for this shift... 

The preceding results are applicable provided that there are no unbridged segments along the matrix crack. Unbridged regions concentrate the stress in the adjacent fibers and weaken the composite.13,75 Simple linear scaling considerations indicate that the diminished UTS depends on a nondimensional flaw index (Table 1.2a),... 

We stress that the present approach is essentially self-contained. In addition to the self-consistent derivation of the average chain configuration at equilibrium, with the only exception of the internal viscosity, chain dynamics will be derived from the equilibrium results under classical assumptions (e.g., the preaveraged approximation, the linear Langevin equation). In particular, no a priori use will generally be made of scaling considerations. 

Although our numerical results (Fig. 1) show that the HF approximated solution Eq. (10) is rather crude, it nonetheless provides a qualitative guidance to the physical behavior. Simple scaling considerations lead us to the conclusion that ctHF depends on two parameters the dimensionless time variable 7hf t and K [Eq. (9)]. Eq. (10) predicts that at t —> 0 the decay is more rapid than exponential, so the survival probability Iff) = o (t) 2 behaves in the HF approximation as PhfV) 1 — 1 Kc ( 1 I 2) ft/7t, becoming exponential at larger times, PhfV) exp(— 7hf t). The deviation from exponential decay is appreciable only for K < 1. There are two ways to attain K < 1. One is to take a small difference between the impurity atom velocity Vi and the critical... 

To determine precipitation temperature on a commercial scale, consideration is also given to such factors as production time, material handling, ease of subsequent processes, and production efficiency. Many catalyst manufacturing processes choose a precipitation temperature between 70°C and 80°C. 

In this chapter, an example of the synthesis of an a-amino acid, N-Boc-(S)-3-fluorophenylalanine (Section 2.2.1) is provided, highlighting the process requirements for the 50-200 kg scale. Consideration is then given to more general and practical details necessary to provide an economic process, with reference to other examples. 

A summary of cardiovascular allometric relationships is given in Tables 7.4.3 and 7.4.4. For further scaling considerations for the cardiovascular system, consult Dawson (1991). 

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