Dynamic similarity, principle

This section describes how this set of equations can be solved analytically by the introduction of various simplifications. First, gas dynamics is linearized, thus permitting an acoustic approach. Next, a class of solutions based on the similarity principle is presented. The simplest and most tractable results are obtained from the most extensive simplifications. 

The principle of dynamical similarity expresses the fact that tivo pumps geometrically similar to each other will... 

Dynamic similarity requires geometric and kinematic similarity in addition lo force ratios at corresponding points being equal, involving properties of gravitation, surface tension, viscosity and inertia [8, 21]. With proper and careful application of this principle scale-up from test model lo large scale systems is often feasible and quite successful. Tables 5-... 

Irrespective of the approach taken to scale-up, the scaling of unit operations and manufacturing processes requires a thorough appreciation of the principles of similarity. Process similarity is achieved between two processes when they accomplish the same process objectives by the same mechanisms and produce the same product to the required specifications. Johnstone and Thring (56) stress the importance of four types of similarity in effective process translation (1) geometric similarity (2) mechanical (static, kinematic, and dynamic) similarity (3) thermal similarity and (4)... 

As a conclusion, there is no one scale-up rule that applies to many different kinds of mixing operations. Theoretically we can scale up based on geometrical and dynamic similarities, but it has been shown that it is possible for only a few limited cases. However, some principles for the scale-up are as follows (Oldshue, 1985) ... 

Even in the first publications concerning the copolymerization theory [11, 12] their authors noticed a certain similarity between the processes of copolymerization and distillation of binary liquid mixtures since both of them are described by the same Lord Rayleigh s equations. The origin of the term azeotropic copolymerization comes just from this similarity, when the copolymer composition coincides with monomer feed composition and does not drift with conversion. Many years later the formal similarity in the mathematical description of copolymerization and distillation processes was used again in [13], the authors of which, for the first time, classified the processes of terpolymerization from the viewpoint of their dynamics. The principles on which such a classification for any monomer number m is based are presented in Sect. 5, where there is also demonstrated how these principles can be used for the copolymerization when m = 3 and m = 4. 

Figure 2-15. Photographs of the relaxation of a pair of initially deformed viscous drops back to a sphere under the action of surface tension. The characteristic time scale for this surface-tension-driven flow is tc = fiRi 1 + X)/y. The properties of the drop on the left-hand side are X = 0.19, /id = 5.5 Pa s, ji = 29.3 Pa s, y = 4.4 mN/rn, R = 187 /an, and this gives tc = 1.48 s. For the drop on the right-hand side, X = 6.8, lid = 199 Pa s, //. = 29.3 Pa s, y = 4.96 mN/m, R = 217 /an, and tc = 9.99 s. The photos were taken at the times shown in the figure. When compared with the characteristic time scales these correspond to exactly equal dimensionless times (/ = t/tc) (a) t = 0.0, (b) t = 0.36, (c) t = 0.9, (d) t = 1.85, (e) t = 6.5. It will be noted that the drop shapes are virtually identical when compared at the same characteristic times. This is a first illustration of the principle of dynamic similarity, which will be discussed at length in subsequent chapters.

The fact that only two dimensionless combinations of the dimensional parameters appear in (7-2) and (7-3) is one demonstration of the so-called principle of dynamic similarity. This principle is nothing more than the observation that the form of the velocity and pressure fields for a typical flow problem will depend on the dimensionless parameters (7-4) and (7-5) rather than on any of the dimensional parameters alone. It may not be obvious without examining the details of specific flow problems that additional dimensionless parameters will not appear in the boundary conditions. However, with the exception of problems that involve multiple scales, as previously described, no additional parameters will appear for flows that involve only solid boundaries. 

Other groups also work on similar principles for the characterisation of zeolite membranes by the so-called Dynamic Desorption Porosimetry or by dynamic desorption of adsorbing species [130],... 

The dynamic resolution strategy has been applied to related s) tems with success, for example the amide (3.42), a-chloro-substituted compound (3.43), and the a-amido p ketophosphonate (3.44). In each case a similar principle operates, where the starting material is rapidly racemising, and the catalyst selects only one enantiomer in the reduction process. 

There are myriad correlations between dimensionless groups. We will examine a few more examples in this chapter. Again, it is not necessary to memorize each correlation. Rather, it is important to know how to apply the principles of dynamic similarity to create new design tools. 

Before embarking on a discussion of the results of these studies let us add one historical note. The difficulty with swinging the polymer tails in a conformational transition has been recognized for many years. A means of circumventing was proposed by Schatzki. Verdier and Stockmayer had earlier invoked a similar principle but used it only to produce Rouse modes. We know now that slow Rouse modes are insensitive to the details of the faster time-scale dynamics. The proposed motions are completely local, and involve going from one equilibrium rotational isomeric state to another by moving only a finite, small number of atoms. Mechanisms of this class have come to be known as crankshaft motions (a term applicable in the strictest sense only to the Schatzki proposal). Because of the limited amount of motion and the simplicity of the dynamics these models are easy to understand, analyze, and simulate. This probably contributes to the continued attention devoted to them. The crankshaft idea has helped to focus attention on the necessity to localize the motion associated with conformational transitions, but complete localization is too restrictive. There are theoretical objections that can be raised to the crankshaft mechanism, but the bottom line is that no signs of it are found in our simulations. 

The term viscoelasticity combines viscous and elastic stress-strain flow characteristics. If materials behavior is dominated by viscous flow it is generally referred to as a fluid, whereas if the elastic properties dominate the mechanical properties of a material it is considered to be solid. Most adhesives are applied in a liquid or pasty condition to allow wetting and promote spreading and then are required to phase change into a solid. In the liquid state, rheology provides the methods to differentiate between elastic and viscous flow characteristics while, for example, dynamic mechanical analysis of cured adhesive polymers uses similar principles to access elastic and viscous parameters of the stress-strain response. 

The dynamic picture of a vapor at a pressure near is then somewhat as follows. If P is less than P , then AG for a cluster increases steadily with size, and although in principle all sizes would exist, all but the smallest would be very rare, and their numbers would be subject to random fluctuations. Similarly, there will be fluctuations in the number of embryonic nuclei of size less than rc, in the case of P greater than P . Once a nucleus reaches the critical dimension, however, a favorable fluctuation will cause it to grow indefinitely. The experimental maximum supersaturation pressure is such that a large traffic of nuclei moving past the critical size develops with the result that a fog of liquid droplets is produced. 

The axial filter (Oak Ridge National Laboratory) (30) is remarkably similar to the dynamic filter in that both the rotating filter element and the outer shell are also cylindrical. An ultrafiltration module based on the same principle has also been described (31). Unlike the disk-type European dynamic filters described above, the cylindrical element models are not so suitable for scale-up because they utilize the space inside the pressure vessel poorly. 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

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