Critical saturation ratio

Several modifications of this nucleation equation have been suggested by Zeldovitch [24] that allow Figure 7.5 [25] to be drawn, which shows the nucleation rate as a fimction of the saturation ratio and the number of atoms in the critical embiyo (see Section 6.3 for details). The nucleation rate is veiy low for small values of the saturation ratio. But, at a critical saturation ratio, the nucleation rate increases drastically and saturates at a maximum rate corresponding to the condensation rate, times the molecular density, PJk T. 

Figure 10.1 The droplet current (nudeation rale) for supersaturated water vapor at T = 300 K calculated from (10.16). The critical saturation ratio, corresponding to / s cm"" sec , is about 3.1.

Figure lOJ Comparison of theoretical and experimental critical saturation ratios for toluene. The dashed line is the theoretical prediction (10.16) with / = 1 cm sec, and the solid line is the experimental result, the envelope to the numbered individual chamber stale curves. (After Katz et al,. 1975 data for many other n-alkyl benzenes are given in this reference.)... 

Determine the size of the smallest stable drop at the critical saturation ratio for toluene at 300 K. Of how many molecules are these drops composed ... 

Two effects are observed in homogeneous nucleation experiments for all substances. First, the nucleation rate is always a steep function of saturation ratio S. The second feature common to all systems is that the critical saturation ratio Sc decreases as T increases, and J increases as T increases at constant S. Also, critical nuclei become smaller as 5 increases and as T increases. 

Figure 17.7 presents the Kohler curves for (NH4)2S04 for complete dissociation and no dissociation. Note that dissociation lowers the critical saturation ratio of the particle (the particle is activated more easily) and increases the drop critical diameter (the particle absorbs more water). 

The traditional method of studying gas-liquid nucleation involves the use of a cloud chamber. In such a chamber the saturation ratio 5 is changed until, at a given temperature, droplet formation is observable. Because once clusters reach the critical size for nucleation, subsequent droplet growth is rapid, the rate of formation of macroscopically observable droplets is assumed to be that of formation of critical nuclei. In such a device it is difficult to measure the actual rate of nucleation because the nucleation rate changes so rapidly with S. J is very small for S values below a critical saturation ratio 5, and very large for S > 5,. Thus what is actually measured is the value of 5, defined rather arbitrarily by the point where the rate of appearance of droplets is 1 cm s . ... 

The self-similar solution of an unsteady rarefaction wave in a gas-vapour mixture with condensation is investigated. If the onset of condensation occurs at the saturation point, the rarefaction wave is divided into two zones, separated by a uniform region. If condensation is delayed until a fixed critical saturation ratio Xc > 1 is reached, a condensation discontinuity of the expansion type is part of the solution. Numerical simulation, using a simple relaxation model, indicates that time has to proceed over more then two decades of characteristic times of condensation before the self-similar solution can be recognized. Experimental results on heterogeneous nucleation and condensation caused by an unsteady rarefaction wave in a mixture of water vapour, nitrogen gas and chromium-K)xide nuclei are presented. The results are fairly well described by the numerical rdaxation model. No plateau formation could be observed. 

A numerical evaluation of the condensation discontinuity is performed for several values of the critical saturation ratio x - State 1 is related isentropically to a fixed reference state 0. In Fig. 1. the Rankine-Hugoniot curves and the Ma2 Ma relations are shown for a mixture of water vapour and nitrogen gas. Only those parts of the curves are shown that correspond to entropy increase, to real massflux and to positive droplet mass fraction downstream the discontinuity. The Chapman-Jouguet points, defined by Ma2 = 1, separate the curves in four different regions ... 

If condensation starts at a fixed critical saturation ratio change of the... 

A typical experiment is shown in Fig. 4. Pressure, temperature, vapour mass fraction and saturation ratio are compared with numerical calculation. The characteristic time r, required for numerical evaluation, is obtained by a fit of the experimental vapour mass fraction signal, resulting in r = 5 ms. The chosen values of critical saturation ratio and piston velocity are rather arbitrary. The experimental saturation ratio is calculated from pressure, temperature and vapour mass fraction. When no liquid mass can be detected, the vapour mass fraction is set to the initial value. Experiment and numerical simulation agree fairly well. The first part of the expansion of the gas-vapour mixture is isentropic and accounts for an increase of the saturation ratio. Condensation on the heterogeneous nuclei starts at a value of the measured saturation ratio of about three. After the onset of condensation a rise in temperature is observed due to the release of latent heat. The saturation ratio tends to unity as time increases. The plateau formed in the numerical solution is not observed in the experimental signal. Obviously, the experimental condition is far from self-similarity, and the expansion process is still in its early stage, where relaxation is dominant. The simple numerical model does not describe accurately the... 

In Fig. 6.2a AGj is plotted as a function of j for a few values of the saturation ratio a/ao = S. Obviously the activation energy AG decreases with increasing saturation ratio, as does the size of the critical nucleus, r or rj. 

As before, the saturation ratio S can also be expressed as a/ao, where a and a0 are the actual and equilibrium activities, respectively, of the solutes that characterize the solubility. Once nuclei of critical size Xj+1 (in Eq. 6.1) have been formed, crystallization is spontaneous. 

Figure 14.3 Saturation ratio for water as a function of critical particle diameter, single ion, atmospheric pressure, T = 273°C.

Similar to the case for homogeneous nucleation, Eq. 14.12 can be differentiated, set equal to zero, and used to determine an expression for the saturation ratio at critical drop diameter ... 

Figure 14.7 Plot of saturation ratio or supersaturation as a function of critical particle diameter for soluble nuclei of 10-1B and 10-16 g.

FIGURE 6.4 Critical nuclei size (i.e., AG = AG ) as a function of saturation ratio. For a given value of S, all r s r will grow and all r < r will dissolve. Redrawn with permission from Dirksen and Ring [4a], = 2j8 yV/( i gD. Reprinted from [4a],... 

Figure 6.4 shows the critical nuclei size as a function of saturation ratio S. The standard critical size nuclei is given by... 

FIGURE 6.6 Generalized nucleation rate diagram that describes the homogeneous nucleation rate as a function of the saturation ratio. The number of ions in a critical nucleus, n. is given by equation (6.21) and A = 4j8 -y V / 27/ [feBTln(10)P. Experimental nucleation rates, , are from a BaS04 precipitation reaction. Redrawn, with permission... 

The critical radius, r, is a radius above which an embryo will grow spontaneously. A plot of the critical radius as a function of the saturation ratio, S, is given in Figure 7.4. As the partial pressure ratio increases the critical size decreases for each value of the surface energy. 

With the clean system, no aerosol forms unless the expansion exceeds a limit cone-sponding to a saturation ratio of about four. At this critical value, a shower of drops forms and fails. The number of drops in the shower remains about the same no matter how often the expansion process is repeated, indicating that these condensation nuclei arc regenerated. 

Further experiments show a second crilical expansion ratio corresponding to a satu ration ratio of about eight. At higher saturation ratios, dense clouds of fine drops form, the number increasing with the supersaturation. The number of drops produced between the two critical values of the saturation ratio i.s small compared with the number produced above the second limit. 

For each critical chamber state, the distribution of the saturation ratio and temperature can be calculated as shown in Fig. 10.2. The set of curves for the critical chamber states based on measurements with toluene is shown in Fig. 10.3. The experimental saturation ratio passes through a maximum with respect to temperature in the chamber. Condensation occurs not at the peak supersaturation but at a value on the high-temperature side because the critical supersaturation decreases with increasing temperature. Hence the family of... 

Here d d ) /dt represents the change in the critical particle size with time due to the variation of the saturation ratio with time. 

As already noted in the introductory sections, AIDA ice nucleation studies primarily focus on a precise determination of the critical ice saturation ratio for relevant aerosol types like H2SO4/H2O solution droplets, mineral dust and soot particles, as well as internally mixed aerosol particles. The ice saturation ratio Sice(T)... 

Ice particle measurements in the expansion experiment with 40% OC soot aerosol markedly differ from the 16% OC sample. Note that the optical particle spectrometer hardly detects any ice particles. Additionally, extinction signatures of ice are barely visible in the infrared spectra and diere is only a weak intensity increase of the back-scattered laser light in course of the expansion. The number concentration of ice crystals is less than 10 cm, thus < 1% of the seed aerosol particles act as deposition ice nuclei. In contrast to the 16% OC experiment, no precise critical ice saturation ratio can be specified for the 40% OC soot sample. RHi continues to increase to 190% because very little water vapour is lost on the small surface area of the scarce ice crystals. In summary, die comparison of the two expansion experiments provides first evidence that a higher fraction of organic carbon notably suppresses the ice nucleation potential of flame soot particles. 

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