### 6.3.3. Liquid Evaporation Model

The liquid evaporation model is a model for particles with heat and mass transfer. The model uses two mass transfer correlations depending on whether the droplet is above or below the boiling point.

The boiling point is determined through an Antoine equation (vapor pressure equation that describes the relation between vapor pressure and temperature for pure components, see Antoine Equation in the CFX-Solver Modeling Guide) that is given by

| (6–46)

where , and are user-supplied coefficients. The particle is boiling if the vapor pressure, , is greater than the ambient gas pressure, .

When the particle is above the boiling point, the mass transfer is determined by:

| (6–47)

where is the latent heat of evaporation of the particle component and where and are the convective and radiative heat transfers, respectively.

When the particle is below the boiling point, the mass transfer is given by:

| (6–48)

where is the droplet diameter, is the dynamic diffusivity of the component in the continuum and is the Sherwood number (see Equation 6–44). and are the molecular weights of the vapor and the mixture in the continuous phase, is the equilibrium vapor mole fraction of the evaporating component at the droplet surface, and is the mole fraction of the evaporating component in the gas phase.

The mass source to the continuous fluid is obtained from:

| (6–49)

#### 6.3.3.1. Extension of the Liquid Evaporation Model

The Liquid Evaporation model described in the previous section is limited to being applied to a single component of a particle. Under the following assumptions, the Liquid Evaporation model can be extended to multi-component evaporation:

1. Particle liquids are entirely miscible in all proportions

2. Intermolecular forces between different particle components are of equal strength

These assumptions imply that the liquids in a multi-component particle form an ideal mixture. By definition an ideal mixture is one which obeys Raoult's law (where the vapor pressure of an ideal mixture is dependent on the vapor pressure of each chemical component and the mole fraction of the component present in the mixture).

##### 6.3.3.1.1. Examples of ideal mixtures

There is actually no such thing as an ideal mixture. However, some liquid mixtures get fairly close to being ideal. These are mixtures of closely similar substances.

Commonly quoted examples include:

- hexane and heptane

- benzene and methylbenzene

The more dissimilar the nature of the substances, the more strongly the solution is expected to deviate from ideality.

#### 6.3.3.2. Determination of the Total Vapor Pressure of an Ideal Mixture

Based on the assumption of an ideal mixture, the total vapor pressure, , can be calculated based on the assumption of Raoult’s law

| (6–50)

where is the vapor pressure of a pure component and is the mole fraction of component in the liquid phase. The vapor pressure of a pure component, is determined from an Antoine equation. This equation has the following form:

| (6–51)

where , and are the Antoine coefficients of binary mixture i.

Inserting Equation 6–51 into Equation 6–50 allows the calculation of the total vapor pressure of the mixture as follows:

| (6–52)

It can be seen that the total vapor pressure is a function of the material properties of the components, the particle temperature, as well as the instantaneous composition of the evaporating droplet.

A particle is assumed to be boiling if the total vapor pressure, , of all evaporating components is equal to or larger than the ambient gas pressure, :

If the total vapor pressure is below the ambient gas pressure, the particle is assumed to not boil.

#### 6.3.3.3. Diffusion Regime (Non-Boiling Particles)

The particle evaporates in the diffusion regime if the particle is below the boiling point.

The mass transfer of a single particle component can be derived from the mass conservation equation and can be written in the following form (see, for example, Abramzon and Sirignano [201] and Sazhin [202])

| (6–53)

where is the droplet diameter, is the dynamic diffusivity of the component in the continuum, is the Sherwood number, and where and are the molecular weights of the vapor of component and the mixture in the continuous phase. is the mole fraction of the vapor of component in the continuum, is the equilibrium vapor mole fraction of component at the droplet surface.

For the calculation of the molar concentration of component on the particle surface, , again the assumption of Raoult’s law is applied:

| (6–54)

The total mass flow rate is computed as the sum of the mass flow rates of all evaporating components:

| (6–55)

#### 6.3.3.4. Boiling Particles

A particle is boiling if the total vapor pressure, , is greater than or equal to the ambient gas pressure .

When the particle reaches the boiling point, the mass transfer is determined by the convective and radiative heat transfer:

| (6–56)

where is the latent heat of evaporation of component , and where and are the convective and radiative heat transfer, respectively.

In order to determine the evaporation rate of a particular particle component, an additional assumption is required. It is assumed that the evaporation rate of each component is proportional to the ratio of the vapor pressure of this component to the total vapor pressure:

| (6–57)

The component vapor pressure is computed from Raoult’s law:

| (6–58)

Rearranging Equation 6–57:

| (6–59)

and substituting it into Equation 6–56 gives the total mass transfer rate:

| (6–60)

Using Equation 6–59 and Equation 6–60 the following equation for the mass transfer of component can be derived:

| (6–61)

with the modified latent heat (averaged with the molar concentration):

| (6–62)

Note that this modified latent heat is only used for the computation of the mass transfer rates in Equation 6–61.

Release 18.2 - © ANSYS, Inc. All rights reserved.