Similarity solutions

Let be a new variable, which is a combination of r and f as in the previous case of capillary spreading h(t,r) should depend on this new single variable. Such solution is referred to as a similarity solution. 

Let = rf(t), where/(O is a new unknown function. Substituting the latter definitions into Equation 3.28, we find that 

This equation must be independent of t this gives two conditions  

Similarity solution (3.30) should be substituted in Equation 3.27. Calculations similar to the previous case results in an identical expression for the time derivative. Using the definition 

Equation 3.31 should not depend on time but on variable only hence. 

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If an ethanolic solution of picric acid is similarly added to one of aniline, no precipitation occurs, owing to the high solubility of aniline picrate in ethanol. If, however, a cold aqueous solution of aniline hydrochloride is added to a similar solution of sodium picrate and the mixture shaken, yellow crystals of aniline picrate, m.p. 165 , soon separate. 

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 . 

Nitration at a rate independent of the concentration of the compound being nitrated had previously been observed in reactions in organic solvents ( 3.2.1). Such kinetics would be observed if the bulk reactivity of the aromatic towards the nitrating species exceeded that of water, and the measured rate would then be the rate of production of the nitrating species. The identification of the slow reaction with the formation of the nitronium ion followed from the fact that the initial rate under zeroth-order conditions was the same, to within experimental error, as the rate of 0-exchange in a similar solution. It was inferred that the exchange of oxygen occurred via heterolysis to the nitronium ion, and that it was the rate of this heterolysis which limited the rates of nitration of reactive aromatic compounds. 

The observation of nitration in nitromethane fully dependent on the first power of the concentration of aromatic was made later. The rate of reaction of /)-dichlorobenzene ([aromatic] = 0-2 mol [HNO3] = 8-5 mol 1 ) obeyed such a law. The fact that in a similar solution 1,2,4-trichlorobenzene underwent reaction according to the same kinetic law, but about ten times slower, shows that under first-order conditions the rate of reaction depends on the reactivity of the compound. 

Evidence from the viscosities, densities, refractive indices and measurements of the vapour pressure of these mixtures also supports the above conclusions. Acetyl nitrate has been prepared from a mixture of acetic anhydride and dinitrogen pentoxide, and characterised, showing that the equilibria discussed do lead to the formation of that compound. The initial reaction between nitric acid and acetic anhydride is rapid at room temperature nitric acid (0-05 mol 1 ) is reported to be converted into acetyl nitrate with a half-life of about i minute. This observation is consistent with the results of some preparative experiments, in which it was found that nitric acid could be precipitated quantitatively with urea from solutions of it in acetic anhydride at —10 °C, whereas similar solutions prepared at room temperature and cooled rapidly to — 10 °C yielded only a part of their nitric acid ( 5.3.2). The following equilibrium has been investigated in detail ... 

Certain features of the addition of acetyl nitrate to olefins in acetic anhydride may be relevant to the mechanism of aromatic nitration by this reagent. The rapid reaction results in predominantly cw-addition to yield a mixture of the y -nitro-acetate and y5-nitro-nitrate. The reaction was facilitated by the addition of sulphuric acid, in which case the 3rield of / -nitro-nitrate was reduced, whereas the addition of sodium nitrate favoured the formation of this compound over that of the acetate. As already mentioned ( 5.3. i), a solution of nitric acid (c. i 6 mol 1 ) in acetic anhydride prepared at — 10 °C would yield 95-97 % of the nitric acid by precipitation with urea, whereas from a similar solution prepared at 20-25 °C and cooled rapidly to —10 °C only 30% of the acid could be recovered. The difference between these values was attributed to the formation of acetyl nitrate. A solution prepared at room... 

First-order nitrations. The kinetics of nitrations in solutions of acetyl nitrate in acetic anhydride were first investigated by Wibaut. He obtained evidence for a second-order rate law, but this was subsequently disproved. A more detailed study was made using benzene, toluene, chloro- and bromo-benzene. The rate of nitration of benzene was found to be of the first order in the concentration of aromatic and third order in the concentration of acetyl nitrate the latter conclusion disagrees with later work (see below). Nitration in solutions containing similar concentrations of acetyl nitrate in acetic acid was too slow to measure, but was accelerated slightly by the addition of more acetic anhydride. Similar solutions in carbon tetrachloride nitrated benzene too quickly, and the concentration of acetyl nitrate had to be reduced from 0-7 to o-i mol 1 to permit the observation of a rate similar to that which the more concentrated solution yields in acetic anhydride. 

Minimizing Chemical Interferences The quantitative analysis of some elements is complicated by chemical interferences occurring during atomization. The two most common chemical interferences are the formation of nonvolatile compounds containing the analyte and ionization of the analyte. One example of a chemical interference due to the formation of nonvolatile compounds is observed when P04 or AP+ is added to solutions of Ca +. In one study, for example, adding 100 ppm AP+ to a solution of 5 ppm Ca + decreased the calcium ion s absorbance from 0.50 to 0.14, whereas adding 500 ppm POp to a similar solution of Ca + decreased the absorbance from 0.50 to 0.38. These interferences were attributed to the formation of refractory particles of Ca3(P04)2 and an Al-Ca-O oxide. 

The second type of solution polymerization concept uses mixtures of supercritical ethylene and molten PE as the medium for ethylene polymerization. Some reactors previously used for free-radical ethylene polymerization in supercritical ethylene at high pressure (see Olefin POLYMERS,LOW DENSITY polyethylene) were converted for the catalytic synthesis of LLDPE. Both stirred and tubular autoclaves operating at 30—200 MPa (4,500—30,000 psig) and 170—350°C can also be used for this purpose. Residence times in these reactors are short, from 1 to 5 minutes. Three types of catalysts are used in these processes. The first type includes pseudo-homogeneous Ziegler catalysts. In this case, all catalyst components are introduced into a reactor as hquids or solutions but form soHd catalysts when combined in the reactor. Examples of such catalysts include titanium tetrachloride as well as its mixtures with vanadium oxytrichloride and a trialkyl aluminum compound (53,54). The second type of catalysts are soHd Ziegler catalysts (55). Both of these catalysts produce compositionaHy nonuniform LLDPE resins. Exxon Chemical Company uses a third type of catalysts, metallocene catalysts, in a similar solution process to produce uniformly branched ethylene copolymers with 1-butene and 1-hexene called Exact resins (56). 

OC-Hydroxycarboxylic Acid Complexes. Water-soluble titanium lactate complexes can be prepared by reactions of an aqueous solution of a titanium salt, such as TiCl, titanyl sulfate, or titanyl nitrate, with calcium, strontium, or barium lactate. The insoluble metal sulfate is filtered off and the filtrate neutralized using an alkaline metal hydroxide or carbonate, ammonium hydroxide, amine, or alkanolamine (78,79). Similar solutions of titanium lactate, malate, tartrate, and citrate can be produced by hydrolyzation of titanium salts, such as TiCl, in strongly (>pH 10) alkaline water isolation of the... 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Removal of phenols from similar solutions by washing with aqueous alkali. 

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. 

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions ... 

FIgura 4.1. Overpressure as a function of flame speed for three geometries. The relationships are based on calculations by use of a self-similar solution (Kuhl et al. 1973). 

The similarity solution for a flow field in front of a steady piston is a special case from a much larger class of similarity solutions in which certain well-defined variations in piston speed are allowed (Guirguis et al. 1983). The similarity postulate for variable piston speed solutions, however, sets stringent conditions for the gas-dynamic state of the ambient medium. These conditions are unrealistic within the scope of these guidelines, so discussion is confined to constant-velocity solutions. 

Figure 4.14. Flow-field parameter distributions in front of an expanding piston. Soiution by matched asymptotic expansions by Guirao et al. compared to exact similarity solutions for various piston Mach numbers.

The solid lines in Figure 4.5 represent extrapolations of experimental data to full-scale vessel bursts on the basis of dimensional arguments. Attendant overpressures were computed by the similarity solution for the gas dynamics generated by steady flames according to Kuhl et al. (1973). Overpressure effects in the environment were determined assuming acoustic decay. The dimensional arguments used to scale up the turbulent flame speed, based on an expression by Damkohler (1940), are, however, questionable. 

Clarke and Andrew have developed a similar solution further activated by addition of chloride ions which deposits more substantial films shown to contain hexavalent chromium. The films give appreciable protection against salt spray and tarnishing by sulphur dioxide. 

Variations in solution composition throughout a test should be monitored and, if appropriate, corrected. Variations may occur as a result of reactions of one or more of the constituents of the solution with the test specimen, the atmosphere or the test vessel. Thus, it is important that the composition of the testing solution is what it is supposed to be. Carefully made-up solutions of pure chemicals may not act in the same way as nominally similar solutions encountered in practice, which may, and usually do, contain other compounds or impurities that may have major effects on corrosion. This applies particularly to artificial sea-water, which is usually less corrosive than natural sea-water. This subject is discussed in detail in a Special Technical Publication of ASTM, and tests with natural, transported and artificial sea-water have been described . Suspected impurities may be added to the pure solutions in appropriate concentrations or, better still, the testing solutions may be taken directly from plant processes whenever this is practical. 

Consider next a similar solution of PbCl2. The radius ascribed to the ion Pb++ is only a little larger than that of Ca++. The question arises whether the form of the function p(r) for contact between Pb++ and Cl- will be similar to that for contact between Ca++ and Cl- in the same solvent. It has been mentioned in Sec. 20 that in aqueous solution PbCh has an abnormally low conductivity. This has been ascribed to the association of Pb++ and Cl- ions to form molecular ions of the form (PbCl)+. We are led to inquire whether we can draw a clear distinction between such a molecular ion, on the one hand, and a pair of ions that happen to remain in contact for a time r. 

Procedure. Prepare a solution of benzoic acid in distilled water by dissolving 0.100 g in a 100 mL graduated flask and making up to the mark (solution A i). Prepare similar solutions of the other two acids giving solutions A2 and A3. Now take 10.0 mL of solution A t and dilute to the mark in a 100 mL graduated flask with distilled water giving a solution containing 0.01 mg of benzoic acid mL-1. Prepare similar solutions B2 and B3 from solutions A2 and A3. 

If possible, relate the problem to another situation and look for a similar solution. This... 

Values for c in each method are obtained by solving the equation for various values of each c and choosing the solution of lowest energy. In practice, both methods give similar solutions for molecules that contain only localized electrons, and these are in agreement with the Lewis structures long familiar to the organic chemist. Delocalized systems are considered in Chapter 2. 

The cold jet theory was applied in the region between the nozzle and the edge of lifted flame, which, for the axisymmetric jets, has the following similarity solutions [10,11] ... 

Similarity solutions of the velocity profile functions for the Von Karman problem. (From Von Karman, Th, Z., Angew. Math. Mech., 1,231,1921.)... 

The gas motion near a disk spinning in an unconfined space in the absence of buoyancy, can be described in terms of a similar solution. Of course, the disk in a real reactor is confined, and since the disk is heated buoyancy can play a large role. However, it is possible to operate the reactor in ways that minimize the effects of buoyancy and confinement. In these regimes the species and temperature gradients normal to the surface are the same everywhere on the disk. From a physical point of view, this property leads to uniform deposition - an important objective in CVD reactors. From a mathematical point of view, this property leads to the similarity transformation that reduces a complex three-dimensional swirling flow to a relatively simple two-point boundary value problem. Once in boundary-value problem form, the computational models can readily incorporate complex chemical kinetics and molecular transport models. 

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