Limiting density

Flare Gas - Molecular weight, lower explosive limit, density at flare tip, fraction F of heat release radiated by the flame. 

For general rules, a first-order statistical approximation for limiting densities Pi t —> oo) can be obtained by a method akin to the mean-field theory in statistical mechanics (more sophisticated approaches will be introduced in chapter 4). 

Although differences between mean-field estimates and exact limiting densities are to be expected because of the presence of correlations in actual evolutions, the mean-field approach nonetheless typically comes to within about 10 — 20% of the exact values for all elementary complex rules. 

We recall that the MFT assumes that does not induce any correlation between separated sites if all of the sites are mutually uncorrelated at i = 0, MFT assumes that they remain uncorrelated at all later times t > 0. A virtue of this approach is that it permits an easy derivation of the limiting value density, pt->oo- Because the underlying assumption is generally not valid, however, we should hardly be surprised to learn that the limiting densities obtained for most of the interesting (i.e, nonlinear) rules differ significantly from those obtained by Monte Carlo simulations of those same rules. 

The variation from Avogadro s theorem may be expressed in terms of the density per unit pressure. Let m be the weight in grams of v litres of a gas under a pressure p atm. at 0° C. The density is m/v and the density per unit pressure mjpr. When p = 1 the expression represents the absolute normal density, and, when p is very small, the absolute limiting density. At intermediate pressures the densities per unit pressure are ... 

Since, however, the ratio of the absolute limiting densities is the ratio of the masses of equal volumes of the two gases when the latter are in the ideal limiting state, it follows from Avogadro s theorem that this is also equal to the ratio of the molecular weights Mi mi (lkfl oh M2 m2 Ow 0)1... 

This limiting density is all the less the smaller is the value of the pressure, and tends to the limiting value D/2 as P tends to zero. 

Similarly, for T —> oo, it can be shown that the curves have contact of infinite order with their asymptotes, i.e., the horizontals through the limiting densities. 

The limiting density, P, may be defined as the lowest p at which. deton velocity will reach its maximum value, called limiting velocity, De. For HE s like TNT, PETN, RDX,... 

In detonation of weak explosives like mixtures contg large amounts of nonexplosive ingredients (such as AN, K chlorate, carbonaceous material, etc), there also are critical and limiting densities, diameters and detonation velocities, but curves of their relationships have different shapes than those for HE s... 

Experimental Procedures. For determination of critical and limiting densities and diameters, the method described in Ref 8, pp 277-79 Ref 9, pp 196-97 can be used. As the method is practically identical with that used for determination of "Critical Length. of Propagation of Detonation given in Section 1, it is not necessary to repeat its description... 

The first determinations of the at. wt. of chlorine were obtained by the analysis of the chlorates later determinations have been directed towards the quantitative synthesis of hydrogen chloride, or to the use of physical methods—mol. vol., limiting densities, and critical data. General calculations have been made by F. W. Clarke, B. Brauner, and others.1... 

In order to achieve the maximum velocity of detonation for a homogeneous explosive, it is necessary to consolidate the explosive composition to its maximum density. For a crystalline explosive the density of compaction will depend upon the consolidation technique (i.e. pressing, casting, extrusion, etc.). The limiting density will be the density of the explosive crystal. The velocity of detonation can be calculated from the density of the explosive composition using Equation 3.5,... 

In most cases of loading ammunition, the higher the absolute density of an explosive, the higher is its loading density (See also Limiting Density and Maximum Loading Density)... 

Limiting Density (Densite7 Limite, in Fr). Dautriche et al (Addnl Ref C) observed that for certain expl mixts, such as Cheddites (See Vol 2, pp 057 059), Explosif P (KC103 90 paraffin 10%), Explosif S(NaClC>3 89 paraffin 11%), Minelites (See Ref 9, p 361), etc, the detonation rate increases with increasing loading density to a certain maximum value, which is characteristic for each expl mixt. This value is called by Davis (Ref 9, p 361) the optimum density. Further increase of d causes the diminution of the deton rate, as well as of the sensitivity to initiation and a point could finally be reached at which no detonation is produced. The density at which the expl ceases to detonate was named in France densite limite... 

A table in Ref 9, PP 362-63, gives optimum and limiting densities for several expl mixts. 

Pepin Leh2lleur (Ref 5, p 54), gives maximum loading densities for the following expls TNT 1.58, PA 1.69, Blasting Gelatin (92/8) 1.63> PETN 1.62 RDX 1.70 (See also Optimum Density under Limiting Density)... 

K Optimum Loading Density. See under Limiting Density... 

Initial supercritical fluid work was on C02 (21-23) and indicated weak interactions between the fluid and solute. Additional work has appeared on Xe, SF6, C2H, and NH3 (24,25). For all fluids, spectral shifts were observed with fluid density. Yonker, Smith and co-workers (24-26,28) compared their results to the McRae continuum model for dipolar solvation (56,57), which is based on Onsanger reaction field theory (58). Over a limited density range, there was agreement between the experimental data and the model (24-26,28), but conditions existed where the predicted linear relationship was not followed (28). At low fluid densities, this deviation was attributed (qualitatively) to fluid clustering around the solute (28). 

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