Similarity theory

Similarity theory is used to study a physical phenomenon at a reduced scale compared to the teal scale (there are also cases where one wishes to enlarge the scale, although this is rather infrequent). The classical example in fluid mechanics is the use of wind turmel models to study an aircraft prototype before considering its real-size constraction. The same process is followed in chemical engineering when building a pilot experiment. 

To represent a given phenomenon at a differerrt scale, it should be ensured that the physics of the phenomenon is preserved through the change of scale. Depending on the case, the equations of continuum mechanics, Navier-Stokes equations, the transport equations for thermal energy, etc. will provide the appropriate framework to identify, through dimensional analysis, the dimensionless numbers characterizing 

As a continuation of the previous section, let us take the example of a flow governed by Navier-Stokes equations. In the steady-state case, the values of the Euler number, the Reynolds number, and the Froude number should be preserved through the change of scale. Denoting by U, L, P, p, and as the characteristic quantities of the real flow and im, Pvol, Pm, and Pm as the characteristic quantities of the model flow, the following relations should be verified  

The first two relations (equality of Reynolds and Froude numbers) show that it is not possible to change the geometrical scale without changing the fluid (density and viscosity), if the first two conditions are to be fulfilled. If they are, the third condition will also be fulfilled in the flow, insofar as the full-scale and model-scale pressnre bonndary conditions are adapted to verify the latter condition. 

An application example faU velocity of a spherical particle in a viscous fluid at rest 

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The preceding treatment relates primarily to flocculation rates, while the irreversible aging of emulsions involves the coalescence of droplets, the prelude to which is the thinning of the liquid film separating the droplets. Similar theories were developed by Spielman [54] and by Honig and co-workers [55], which added hydrodynamic considerations to basic DLVO theory. A successful experimental test of these equations was made by Bernstein and co-workers [56] (see also Ref. 57). Coalescence leads eventually to separation of bulk oil phase, and a practical measure of emulsion stability is the rate of increase of the volume of this phase, V, as a function of time. A useful equation is... 

Shortly thereafter but independently of Kekule Archibald S Couper a Scot working m the laboratory of Charles Adolphe Wurtz at the Ecole de Medicine m Pans and Alexan der Butlerov a Russian chemist at the University of Kazan proposed similar theories... 

Hofmann (H8) has proposed, on the basis of similarity theory, a correlation for the liquid Peclet number of the following form ... 

On the basis of similarity theory Hofmann (H8) derived the following relationship describing axial dispersion in the liquid phase ... 

The cornerstone of LES methodology is the self-similarity theory of Kolmogorov stating that even though the large structures of a turbulenf flow depend on fhe boundary and initial conditions, the finer scales have a universal... 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

Predictions of high explosive detonation based on the new approach yield excellent results. A similar theory for ionic species model43 compares very well with MD simulations. Nevertheless, high explosive chemical equilibrium calculations that include ionization are beyond the current abilities of the Cheetah code, because of the presence of multiple minima in the free energy surface. Such calculations will require additional algorithmic developments. In addition, the possibility of partial ionization, suggested by first principles simulations of water discussed below, also needs to be added to the Cheetah code framework. 

Patrick Bultinck, Xavier Girones and Ramon Carbo-Dorca, Molecular Quantum Similarity Theory and Applications. 

When a similar theory (which appears objectionable to the present reviewer also on other grounds) was applied to the formation of ice in water droplets160), the critical nucleus < was > assumed to be a hexagonal prism of height equal to the short diameter . No capillary pressure acts across plane faces of a prism. Nevertheless the author found a value (for the 7s] of water - ice) near 20 erg/cmz for drops of about 0.002 cm in diameter at —37 °C. 

Monin-Obukhov similarity theory, as expressed by (7.10), can be used in (9.2) to give... 

Monin-Obukhov similarity theory can be used to prescribe the form of Ka in the surface layer. can be expressed as... 

Myrup and Ranzieri (1976) developed an approach based on similarity theory and a set of empirical formulas. For unstable conditions (z/L < -5) their profile is specified by... 

Under neutral conditions the atmospheric lapse rate is adiabatic. Close to the ground the vertical eddy diffusivity profile can be based on Monin-Obukhov similarity theory, in which case = 1 and = ku,z- With this formulation, increases without limit—clearly a physically unrealistic situation. Myrup and Ranzieri (1976) proposed a set of empirical roll off functions for altitudes above the surface layer ... 

In the surface layer, similarity theory can be used to give an expression for eddy diffusivity under stable conditions ... 

In fact, Cuy s idea was not completely original. Many years before, Fliirscheim [4] and Fry [5], had postulated similar theories of alternating polarities, and the idea was soon extended by Hanke and Koessler [6], Kermack and Robinson [7] and Stieglitz [8] in order to predict the site of reactivity in both aliphatic and aromatic systems. However, as has been stated by A.E. Remick [9], "it would profit us but little to pursue further the similarities and differences of these theories of alternating polarity. Suffice it to say that they were eventually shown to be wrong [10] [11] at least in regard to saturated molecules". In spite of this, it is worthwhile referring here to the work of Lapworth. 

The basic terms vp and dp and the liquid properties are made non-dimensional in the equations according to the similarity theory ... 

Relations, which are also sets, play an important role in set theory and in the similarity theory, but due to space limitations are not formally considered in this work. 

People have been thinking about tiny objects for a long time. The ancient Greek philosopher Democritus (ca. 460-370 b.c.e.) believed that properties of matter depended on the shapes of small, indivisible bits of matter called atoms. Although this idea failed to catch on at the time— no one could see these atoms because they were so small—in 1803, the British chemist John Dalton (1766-1844) proposed a similar theory. Dalton s theory was an important advance and helped scientists understand chemical reactions—for example, the reaction of two atoms of hydrogen (H) and one atom of oxygen (O) to form H O—but atoms themselves remained cloaked in mystery. 

Cook also stated that the geometrical model theory was developed without his knowing that a similar theory was formulated, as early as 1928 by Roth 8c Wohler and described in Roth s thesis (Ref 1). 

A similar theory was published by Frey-Wyssling [37] who suggested a scheme composed of two projections outlined in Fig. 79, where the crystalline phases of the cellulose fibre are marked as dotted line rectangles. 

Problems of how chemical reactions (including catalytic reactions and combustion) run under real conditions naturally led Ya.B. to hydrodynamics, heat transfer and problems of turbulence. Another significant factor was his contact with the prominent scientist D. A. Frank-Kamenetskii, who joined the Institute of Chemical Physics in 1935 with broad interests in these areas and in similarity theory. 

Along with the methods of similarity theory, Ya.B. extensively used and enriched the important concept of self-similarity. Ya.B. discovered the property of self-similarity in many problems which he studied, beginning with his hydrodynamic papers in 1937 and his first papers on nitrogen oxidation (25, 26). Let us mention his joint work with A. S. Kompaneets [7] on selfsimilar solutions of nonlinear thermal conduction problems. A remarkable property of strong thermal waves before whose front the thermal conduction is zero was discovered here for the first time their finite propagation velocity. Independently, but somewhat later, similar results were obtained by G. I. Barenblatt in another physical problem, the filtration of gas and underground water. But these were classical self-similarities the exponents in the self-similar variables were obtained in these problems from dimensional analysis and the conservation laws. 

In Part II (Sections 7-10) are described experiments on the decomposition of nitric oxide which was added to the explosive mixture in advance. These experiments establish a proportionality between the rate of decomposition and the square of the concentration of nitric oxide and give the heats of activation for the formation and decomposition of nitric oxide. The similarity theory and the exact mathematical theory of a reversible bimolecular... 

However, even after such simplifications the number of factors determining the yield of nitric oxide still remains large. The aim of the similarity theory consists in deducing the relationships between the quantities of interest in a simple form convenient for experimental investigation. The first step consists in computing the nitric oxide content [NO] at the theoretical temperature. This quantity which is expressed in terms of 0 2, N2, B, E, Tm provides a natural measurement unit for the yield of nitric oxide NO in the explosion products. We can thus define the dimensionless yield as the ratio NO/[NO],... 

Formulas (8.6) and (8.7) are an example of the power of the similarity theory which allows for deriving a by no means evident relation between the yield of nitric oxide and the reaction rate or the time r from considerations of the temperature dependence of the yield of nitric oxide. 

On the whole the similarity theory reduces the problem of the amount of nitric oxide formed in explosions to one series of experiments, which in principle is sufficient for determining the characteristic curve in dimensionless coordinates, NO/[NO] as a function of fcTO[NO]r. Figure 14 represents such a curve obtained from experiments with hydrogen mixtures with equal oxygen and nitrogen content in the explosion products at p0 = 200 mm and a volume of the vessel equal to 3 liters the quantity fcm[NO]r is plotted in the logarithmic scale. In plotting the curve we made use of the expression... 

Given the condition NO/[NO] = 0 at t/r = 0 we must find the limit of NO/[NO] at t/r —> oo. It follows from the form of the equation that at given fi and /2 the quantity NO/[NO] depends only on the product fcm[NO]r. This statement coincides with the content of the similarity theory introduced in the preceding section on dimensional considerations. We see that the validity of the theory depends on the existence of the functions fx and /2, which must be the same for different explosions. But the rate coefficient and the equilibrium quantity depend on the temperature. Hence the form and the very existence of the functions f1 and /2 depend on the law of cooling. On the other hand, it is evident that the law of cooling must be formulated in such a manner that under the given conditions the cooling rate will depend only on the temperature of the gas, but not on the heat of activation of the reaction or the equilibrium quantity of nitric oxide. It can be shown that both conditions are satisfied only by the law ... 

As long as the oxygen concentration remains constant throughout the cooling of the explosion products, the previously developed similarity theory and the mathematical theory of a reversible bimolecular reaction in an explosion remain valid in their entirety. 

The similarity theory and an exact mathematical theory of the course of the reversible reaction during cooling of the explosion products are set forth. A universal relationship between the ratio of the yield to the equilibrium... 

Anbar and Hart (2) have demonstrated an interesting empirical correlation between the absorption maxima of the solvated electron in various solvents and those of the iodide ion in these same solvents. The halide ion spectra have also been discussed (23) in terms of a similar theory. 

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