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22.3 Multicomponent Particle Theory

A number of industrially important processes, such as distillation, absorption and extraction, bring into contact two phases which are not at equilibrium. The rate at which a specie is transferred from one phase to the other depends on the departure of the system from equilibrium. The quantitative treatment of these rate processes requires knowledge of the equilibrium states of the system. Apart from these cases, vapor-liquid equilibrium (VLE) relationships in multicomponent systems are needed for the solution of many other classes of engineering problems, such as the computation of evaporation rates in spray combustion applications.

In FLUENT the rate of vaporization of a single component droplet is computed from Equation  22.9-20, where $C_{i,s}$ is the equilibrium concentration of the droplet species in the gas phase, and is computed in Equation  22.9-21 as:

 C_{i,s} = p_{sat}/RT_p (22.3-1)

where $T_p$ is the droplet temperature, and $p_{sat}$ is the saturation pressure of the droplet species at $T_p$.

For the general case where N components are evaporating from a droplet (distillation), the evaporation rate of each species is again given by Equation  22.9-20; however, $p_{sat}$ in Equation  22.3-1 must be replaced by $p_{i}$, the partial pressure of species $i$, to calculate the concentration of $i$ at the droplet surface.

The partial pressure of species $i$ can be obtained from the general expression for vapor liquid equilibrium [ 343],

 \hat{\phi_{i}} y_{i} p = \gamma_{i} x_{i} {\phi_{sat,i}} {p_{sat,i}} exp\left[\frac{{V_{i}}^{L} (p - {p_{sat,i}})}{RT}\right] (22.3-2)

which is obtained by equating the fugacity of the liquid and vapor mixtures. Here, $\hat{\phi_{i}}$, is the fugacity coefficient for species $i$ in the mixture, and accounts for nonideality in the gas; ${\phi_{sat,i}}$ is the fugacity coefficient for pure $i$ at the saturation pressure; $\gamma_{i}$ is the activity coefficient for species $i$ in the mixture, and accounts for nonideality in the liquid phase; $p$ is the absolute pressure; $T$ is the temperature; $R$ is the universal gas constant; ${V_{i}}^{L}$ is the molar volume of the liquid; ${p_{sat,i}}$ is the saturation pressure of species $i$ ; and $x_{i}$ and $y_{i}$ are the equilibrium compositions of species $i$ in the liquid and gas phases, respectively. The exponential term is the Poynting correction factor and accounts for compressibility effects within the liquid. Except at high pressures, the Poynting factor is usually negligible.

Under low pressure conditions where the gas phase may be assumed to be ideal, $\hat{\phi_{i}} \approx 1$ and ${\phi_{sat,i}} \approx 1$. Furthermore, if the liquid is also assumed to be ideal, $\gamma_{i}\approx 1$ and Equation  22.3-2 reduces to Raoult's law,

 y_i p = x_i {p_{sat,i}} (22.3-3)

Raoult's law is the default vapor-liquid equilibrium expression used in the FLUENT multicomponent droplet model. However, there is a UDF hook available for user-defined vapor-liquid equilibrium models.

While Raoult's law represents the simplest form of the VLE equation, keep in mind that it is of limited use, as the assumptions made for its derivation are usually unrealistic. The most critical assumption is that the liquid phase is an ideal solution. This is not likely to be valid, unless the system is made up of species of similar molecular sizes and chemical nature, such as in the case of benzene and toluene, or n-heptane and n-hexane.

When Raoult's law is applicable, the vaporization rate of each species from a multicomponent droplet can be computed from Equation  22.9-20, with the equilibrium concentration of species i in the gas phase $C_{i,s}$ computed as:

 C_{i,s} = x_i p_{sat,i} /R T_p (22.3-4)

where $T_p$ is the droplet temperature, $x_i$ is the mole fraction of species i in the droplet, and $p_{sat,i}$ is the saturation pressure of species i at $T_p$.

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