During Law 2, when the rate of vaporization is slow, it can be assumed to be governed by gradient diffusion, with the flux of droplet vapor into the gas phase related to the difference in vapor concentration at the droplet surface and the bulk gas:

(12–80)

Note that Ansys Fluent’s vaporization law assumes that is positive (evaporation). If conditions exist in which is negative (that is, the droplet temperature falls below the dew point and condensation conditions exist), Ansys Fluent treats the droplet as inert ().

The concentration of vapor at the droplet surface is evaluated by assuming that the partial pressure of vapor at the interface is equal to the saturated vapor pressure, , at the droplet temperature, :

(12–81)

where is the universal gas constant.

The concentration of vapor in the bulk gas is known from solution of the transport equation for species as:

(12–82)

where is the local bulk mole fraction of species , is the local pressure, and is the local bulk temperature in the gas. The mass transfer coefficient in Equation 12–80 is calculated from the Sherwood number correlation [542], [543]:

(12–83)

The vapor flux given by Equation 12–80 becomes a source of species in the gas phase species transport equation, (see Setting Material Properties for the Discrete Phase in the User's Guide) or in the mixture fraction equation for nonpremixed combustion calculations.

The mass of the droplet is reduced according to

(12–84)

Ansys Fluent can also solve Equation 12–84 in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Including Coupled Heat-Mass Solution Effects on the Particles in the User’s Guide for details.

For high vaporization rates, the effect of the convective flow of the evaporating material from the droplet surface to the bulk gas phase (Stefan Flow) becomes important.

In Ansys Fluent, the following expression has been adopted following the work of Miller [442] and Sazhin [573]:

(12–85)

is the Spalding mass number given by:

(12–86)

The mass transfer coefficient is given by Equation 12–83.

The single-rate thermolysis model [67] uses the following Arrhenius expression to compute the mass transfer rate from the droplet to the bulk gas phase:

(12–87)

A simplified constant rate thermolysis model is represented as:

(12–88)

You must define the saturation vapor pressure as a polynomial or piecewise linear function of temperature () during the problem definition. Note that the saturation vapor pressure definition is critical, as is used to obtain the driving force for the evaporation process (Equation 12–80 and Equation 12–81). You should provide accurate vapor pressure values for temperatures over the entire range of possible droplet temperatures in your problem. Saturation vapor pressure data can be obtained from a physics or engineering handbook (for example, [511]).

You must also input the diffusion coefficient, , during the setup of the discrete phase material properties. Note that the diffusion coefficient inputs that you supply for the continuous phase are not used in the discrete phase model.

You can define the binary diffusion coefficient (or binary diffusivity) to be either constant, or a function of the continuous phase temperature. Alternatively, you can define the binary diffusion coefficient as function of a film-averaged temperature , which is computed by

(12–89)

You can also choose to have Fluent compute the diffusion coefficient based on the assumption of unity Lewis number:

(12–90)

According to Polling et al. [525] , for low to moderate pressures (for example, pressures < 0.9), the binary diffusivity is inversely proportional to the pressure , so the following relation applies:

(12–91)

where is the binary diffusivity of species at the reference pressure .

The boiling point and the latent heat are defined as constant property inputs for the droplet-particle materials. The default boiling point data in the Ansys Fluent property database correspond to a pressure of 1 atm (normal boiling point) and the latent heat data correspond to the normal boiling point of the droplets.

During the evaporation process, as the particle changes its temperature, the latent heat will vary according to Equation 12–92

(12–92)

For simulations at or near atmospheric pressure, the latent heat variation with droplet temperature is generally small and can be ignored, so . To include the droplet temperature effects on the latent heat according to Equation 12–92, see Including the Effect of Droplet Temperature on Latent Heat in the Fluent User's Guide.

If the pressure in your simulation differs from atmospheric, you will have to revise the default boiling point data in the Ansys Fluent property database. Refer to Considering Pressure Dependence in Boiling in the Fluent User's Guide for details.

When the Real Gas Model (RGM) (see Real Gas Models in the Fluent User's Guide) is used in a DPM simulation, the appropriate modeling approach depends on the operating condition regimes. Refer to Using the Cubic Equation of State Models with the Lagrangian Dispersed Phase Models in the Fluent User's Guide for guidelines and restrictions of this approach. Note that the NIST RGM is not compatible with the DPM, except for the inert and massless particle types.

Finally, the droplet temperature is updated according to a heat balance that relates the sensible heat change in the droplet to the convective and latent heat transfer between the droplet and the continuous phase:

(12–93)

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model Dialog Box.

The heat transferred to or from the gas phase becomes a source/sink of energy during subsequent calculations of the continuous phase energy equation.

When the vaporization rate is computed by the Convection/Diffusion Controlled model (Equation 12–85), the convective heat transfer coefficient in Equation 12–93 is calculated with a modified number as follows [573]:

(12–94)

is the Spalding heat transfer number defined as:

(12–95)

is computed from the Spalding mass number by:

(12–96)

For the unity Lewis number and with the further assumption that , Equation 12–96 reduces to:

(12–97)

When temperature differences between droplet and bulk gas are large, transient effects become important. The assumption of uniform droplet temperature and the assumption of emanating vapor in temperature equilibrium with the bulk gas may be questioned. In such conditions, an averaging of the Spalding heat transfer term may produce more realistic results.

ANSYS Fluent provides the option to average the term in Equation 12–94 with the evaporating species surface mass fraction as follows:

(12–98)