Similarity solutions necessary

Example. Add a solution of 0 5 ml. of benzyl alcohol in 5 ml. of petroleum (b.p. 100-120 ) to a similar solution of 0 5 ml. of phenylisocyanate, and boil the mixture gently under reflux for 20 minutes. Filter hot if necessary from any insoluble diphenylurea, and cool. Filter off the crystalline urethane, and recrystallise from the petroleum colourless crystals, m.p. 76 . 

It is only po.ssible to obtain similar solutions in situations where the governing equations (Eqs. (12.40) to (12.44)) are identical in the full scale and in the model. This tequirement will be met in situations where the same dimensionless numbers are used in the full scale and in the model and when the constants P(i, p, fj.Q,.. . have only a small variation within the applied temperature and velocity level. A practical problem when water is used as fluid in the model is the variation of p, which is very different in air and in water see Fig. 12.27. Therefore, it is necessary to restrict the temperature differences used in model experiments based on water. 

Pulvermacher and Ruckenstein (P10) have recently reexamined Kapur s similarity solution to the nonrandom coalescence equation (Eq. 74), and have established a necessary condition for the existence of a solution to Eq. (79). From the slope of the plots in Fig. 20 they have determined that the exponents x and y in Eq. (83) are related by... 

In 1933 and 1934, Guth and Mark (37, 38) and Kuhn (39) independently studied the problem and arrived at similar solutions. These investigations furnished the background necessary to more clearly define the relationship of molecular weight and high viscosity. 

This example illustrates that it is necessary to exercise great care and thoroughness in finding self-similar solutions. 

A Some ambiguity must be attached to this statement in view of the conclusions reached by Luzzati el al. (1961) on the basis of small angle X-ray scattering studies from dilute solutions of polybenzyl-L-glutamate in dimethylformamide, pyridine, and TO-cresol. These authors conclude that the conformation of the polypeptide in these solutions is that of the 3io-helix (Donohue, 1953) and not that of the a-helix. The value of 6o = —630° in Eq. (13) was obtained from optical rotatory dispersion measurements with similar solutions, and therefore if the conclusions of Luzzati et al. are correct, this value characterizes the 3io-helix. It would then be necessary to determine whether the 6o value for the a-helix is significantly different. 

Necessary conditions for the existence of a self-similar solution are that (1) the governing PDE must reduce to an ODE for F as a function ot// alone, and (2) the original boundary and initial conditions must reduce to a number of equivalent conditions for F that are consistent with the order of the ODE. Of course, a proof of sufficient conditions for existence of a selfsimilar solution would require a proof of existence of a solution to the ODE and boundary conditions that are derived for F. In general, however, the problems of interest will be nonlinear, and we shall be content to derive a self-consistent set of equations and boundary conditions and attempt to solve this latter problem numerically rather than seeking a rigorous existence proof. Let us see how the systematic solution scheme based on the general form (3-135) works for the Rayleigh problem. 

The necessary condition for the existence of a similarity solution is that the coefficients in the equation for /[obtained by substituting (6-50) and (6-51) into (6-49)] are either functions of rj or constants but do not depend on either t or r independently. Combining the t dependence from the two sides of (6-49), we see that this leads to the condition... 

Thus a necessary condition for the existence of similarity solutions for a, p 0 is that ue = xm. The sufficient condition is that ue = xm and a solution /(>]) exists that satisfies (10-95) in the form... 

Hence a necessary condition for existence of a similarity solution is that g2 satisfy (11-80), i.e.,... 

The reactivity of the normal alkyl halides varies with the halogen and with the alkyl group the more reactive are those with the smaller number of carbon atoms. The iodides are the most reactive, the chlorides the most inert. Methyl iodide is much used in organic syntheses, as it reacts readily with many substances and serves as a means of introducing the methyl group into such compounds. An alcoholic solution of methyl iodide reacts with a similar solution of silver nitrate in the cold, and silver iodide is formed. With ethyl iodide the reaction proceeds much more slowly, and with the higher alkyl iodides the application of heat is necessary to bring about reaction. 

Note that the kernel of coagulation (13.27) does not obey the necessary conditions of existence of a self-similar solution. Therefore we shall consider some results of the numerical solution and obtain an approximate solution of equation (13.33) by the method of moments. 

A more complex kind of partition is that of the distribution of small organic molecules between proteins and an aqueous phase. A number of such studies 41) have been correlated by an equation similar to Eq. (16) with the octanol-water partition coefficient and K2 the binding constant of the solute with protein or the ratio of the percent bound solute to the percent free solute. In other studies (47), the binding constant is expressed as 1/C, where C is the molar concentration of the solute necessary to produce a 1 1 (or higher) complex of protein and the solute molecule. The way that K2 is defined greatly affects the intercept and makes it difficult to compare with the data of ordinary solvent-... 

As seen from Eq. 30, by measuring the total current during a steady electroosmotic flow, the bulk conductivity, the applied electrical field, and the channel dimensions, the surface conductivity can be determined. Generally, this is a relatively simple method with a reasonable accuracy. It should be noted that for the measurement of the specific surface conductivity, the replacement between two similar solutions is not necessary. 

If it is necessary to extract the rare metal from an acid or alkaline leach liquor, or similar solution, a resin should be selected which is known to be stable in the leaching reagent. For extraction of the rare metal ion itself, the resin would normally be of the cation-exchange type, unless the rare metal ion could be converted to an anionic complex. However, the possibility of designing a process in which the impurities are absorbed by the resin and the rare metal remains unextracted, should not be neglected. An example of this type has been developed by Ayres,i > in which iron, titanium, lanthanum and beryllium impurities are extracted from a zirconium nitrate solution by operation at a pH where the zirconium is converted to a non-ionic hydrated oxide sol. 

What will be of interest to us in this chapter is whether a similarity solution is feasible for a population balance equation. In other words, can the population balance equation admit a self-similar solution Feasibility is of course necessary (but far from sufficient ) for the existence of a self-similar... 

An elegant example of a similar approach was employed toward the total synthesis of epothilone A (52) by Nicolaou et al., utihzing the solid support for the synthesis and elaboration of the natural product core scaffold 50 and cleaving it from the resin via RCM to generate the macrolactone 51 (Scheme 6.13). In case of similar solution-phase reactions (without the solid support), high-dilution conditions were necessary to prevent side reactions and obtain the RCM product in high yield. 

Separation of molecular species i from species j in an open separator having bulk flow requires a knowledge of how C, and Cj are distributed along the separator length. This knowledge is acquired from a solution of equation (6.2.5g) for each of species i and j. Similarly, solution of equation (6.2.5m) for species i and j will provide the profiles of pi and pj In the separator. Tables 6.2.1 and 6.2.2 illustrate these equations in terms of molar fluxes and mass fluxes, respectively. It is useful, however, to consider such equations in terms of various constituent terms of a flux expression. In Section 6.2.1.1 certain special expressions will be used for the diffusive term in multiphase systems it is necessary to provide a limited fundamental background here to approaches and treatments that will be routinely employed in that section and in Chapters 7 and 8. 

Hydrolysis of Potassium Ethyl Sulphate. Dissolve about i g. of the crystals in about 4 ml. of cold distilled water, and divide the solution into two portions, a) To one portion, add barium chloride solution. If pure potassium ethyl sulphate were used, no precipitate should now form, as barium ethyl sulphate is soluble in water. Actually however, almost all samples of potassium ethyl sulphate contain traces of potassium hydrogen sulphate formed by slight hydrolysis of the ethyl compound during the evaporation of its solution, and barium chloride almost invariably gives a faint precipitate of barium sulphate. b) To the second portion, add 2-3 drops of concentrated hydrochloric acid, and boil the mixture gently for about one minute. Cool, add distilled water if necessary until the solution has its former volume, and then add barium chloride as before. A markedly heavier precipitate of barium sulphate separates. The hydrolysis of the potassium ethyl sulphate is hastened considerably by the presence of the free acid Caustic alkalis have a similar, but not quite so rapid an effect. 

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