Growth of a Droplet Population

FIGURE 15.15 Simulated evolution of temperature, liquid water content, supersaturation, and particle diameters during the lifetime of a cloud (Pandis et al., 1990). The sections denote 7 sizes of initial particles. 

Let us describe the above situation quantitatively, by deriving the equation for the rate of change of the supersaturation The water vapor mixing ratio is related to the water vapor partial pressure Pu, by (15.46), while by definition 

Differentiating this expression with respect to time and rearranging the terms we obtain 

The change of the air pressure with time can be calculated assuming that the environment is in hydrostatic equilibrium so that 

The maximum supersaturation reached inside a cloud/fog is an important parameter. Particles with critical supersaturations lower than this value will become activated and become cloud droplets. The rest remain close to equilibrium but never grow enough to be considered droplets and are called interstitial aerosol. The aerosol population inside a cloud is therefore separated into two groups, interstitial aerosols that contain significant amounts of water but are not activated (their sizes are usually smaller than 2 (im) and cloud droplets, with size increases corresponding to mass changes of three orders of magnitude. 

The growth of an aerosol population to cloud droplets can be investigated using the growth equation derived in the previous section. In general, one would need to integrate simultaneously the differential equations derived in Section 17.3.4 for the air parcel 

FIGURE 17.14 Diffusional growth of individual drops with different dry masses as a function of time. The drops are initially at equilibrium at 80% RH. 

because r 1 decreases rapidly with size. Therefore, as more and more particles get activated, the rate of transport of water from the vapor to the particulate phase increases, while the rate of supersaturation increase due to the cooling remains approximately constant. The result is that the supersaturation increase slows down and after 6 min reaches a maximum value of 0.1%. 

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Nucleation scavenging of aerosols in clouds refers to activation and subsequent growth of a fraction of the aerosol population to cloud droplets. This process is described by (17.70) and has been discussed in Section 17.5. 

A population balance is a balance on a defined set of countable or identifiable entities in a given system as a result of all phenomena which add or remove entities from the set. If the set in question is the number of droplets between diameter D and (D + dD) in a vessel, the set may receive droplets by flow into the vessel, by coalescence or by growth from smaller droplets. The set may also lose droplets by outflow from the vessel, by coalescence or by growth out of the set s size range. 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

The comparison of this equation with Eq. (62) shows that the contribution of the omitted terms in the derivation of Eq. (62) can be accounted for by the introduction of the factor F known as the Zeldovich nonequilibrium factor. The factor is always less than unity, having an order of magnitude of 10 and accounts for the depletion of the cluster population due to the process of growth.It seems that the Zeldovich factor depends on the geometrical form of the cluster (see Refs. 72 and 78) and for a liquid droplet has the value r = (AGc/37rfc7TV ) /l... 

Surface active organic compounds, often present in atmospheric wet aerosols (i.e., clouds and fogs), alter the surface tension of the tiny liquid droplets. A large decrease of the surface tension value may change the processes of droplet nucleation and growth. As a consequence, the changes in droplet population significantly affect the cloud albedo as well as the formation of atmospheric precipitation. 

Figure 6.4. A schematic model for the nucleation and growth of latex particles in the acrylamide microemulsion polymerization stabilized by sodium bis(2-ethylhexyl)sulfosuccinate. (I) The initial condition of the polymerization system consists of a very large population of the acrylamide/ water-swollen micelles ( 6nm in diameter) dispersed in the continuous oily phase. Nucleation of particle nuclei occurs when free radicals are absorbed by the microemulsion droplets. (II) Growth of latex particles are achieved by (a) collision and then coalescence between two particles and (b) diffusion of monomer molecules from the microemulsion droplets through the continuous oily phase and then into the particles, (c) The polymerization system comprises water-swollen polyacrylamide particles ( 40nm in diameter) and acrylamide/water-swollen micelles ( 3nm in diameter) dispersed in the continuous oily phase at the end of polymerization [81].

In two other studies the predicted evolution of a bimodal particle size distribution was examined. Kabalnov e( al. examined hexane-in-water emulsions stabilized by varying levels of hexadecane (A 02 = 0.1, 0.01, 0.001). For the higher mole fractions, the emulsion growth rates were reliably predicted by eqn. (9.11). Also, these emulsions had a physical appearance identical to that of an emulsion containing only hexadecane, i.e. they did not cream. The = 0.001 sample behaved quite differently, however. Upon storage, this emulsion quickly separated into two layers a sedimented layer with a droplet size of ca. 5 pm and a dispersed population of submicron droplets (i.e. a bimodal distribution). The 5 pm droplets were equivalent to the droplet size observed for a fresh perfiuorohexane emulsion prepared under similar conditions. In light of the fact that the stability criterion was not met for this mole fraction of hexadecane, the observed bimodal distribution of droplets is predictable. A bimodal size distribution was also observed by the SdFFF technique by Weers and Arlauskas in PFOB-in-water emulsions stabilized by low levels of PFDB. 

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