The film vaporization law is applied when the film particle temperature is above the vaporization temperature . In Ansys Fluent, the vaporization rate is modeled using following models:

 

12.9.5.2.1.1. Diffusion Controlled Model

The film vaporization rate is governed by gradient diffusion from the surface exposed to the gas phase. The gradient of vapor concentration between the film surface and the gas phase is

(12–275)

where is the molar flux of vapor (kmol/m²-s), is the mass transfer coefficient (m/s), and and are the vapor concentrations on the film surface and in the bulk gas, respectively (kmol/m³).

The vapor concentration at the film surface is evaluated using the saturated vapor pressure at the film temperature, and the bulk gas concentration is obtained from the flow field solution. The vaporization rate is sensitive to the saturated vapor pressure, similar to the droplet vaporization.

The film mass transfer coefficient is obtained as:

(12–276)

where is the binary diffusivity of species , is the Nusselt correlation, and the is the characteristic length (in meters) computed as the square root of the film parcel area :

(12–277)

is taken to be a mass weighted percentage of the wall face area :

(12–278)

In Equation 12–276 for the mass transfer coefficient, the Nusselt correlation is used after replacing the Prandtl number with the Schmidt number as follows:

• Laminar Flow:

  (12–279)

• Turbulent Flow:

  (12–280)

where the Reynolds number is calculated from the particle film characteristic length and the particle relative velocity component that is parallel to the wall.

For multicomponent vaporization, the diffusivity of each species is used in Equation 12–276 and Equation 12–279 to obtain the vaporization rate of each component in the liquid film mixture.

The mass of the particle parcel is decreased according to:

(12–281)

where is the molecular weight of the gas phase species to which the vapor from the liquid is added. The diameter of the film particle is decreased to account for the mass loss in the individual parcel. This keeps the number of drops in the parcel constant.

 

12.9.5.2.1.2. Convection/Diffusion Controlled Model

The film vaporization rate is governed by:

(12–282)

where
	= vapor mass flux of species (kg/m2/s)
	and = mass fraction of species i on the film surface and the bulk respectively
	= film mass transfer coefficient (m/s) calculated from Equation 12–276
	= is density of the bulk gas (kg/m3)

 

12.9.5.2.1.5. Film Boiling Model with Conduction Heat Transfer Model

When the film temperature reaches the boiling point, the film boiling rate is calculated by setting the right-hand side of the energy balance equation (Equation 12–301) to 0 in and solving for the film boiling rate .

The resulting boiling rate expressions are as follows:

Constant Temperature Walls

(12–283)

Heat Flux Walls

(12–284)

with:

(12–285)

where
	= film parcel boiling rate (kg/s)
	= the film parcel area (m2)
	= latent heat (J/kg)
	= film parcel temperature (K)
	= wall temperature (K)
	= temperature in the bulk phase (K)
	= wall heat flux (W/m2)
	= the film height (m)
	= thermal conductivity of the liquid film (W/m/K)

If the Gas-Side Boundary Layer model is enabled, the heat transfer coefficient is computed as follows.

• Laminar flow: , where the length is the normal distance from film surface to the bulk.

• Turbulent flow: is determined from the wall functions as described in Energy Transfer from the Film.

In other cases, Fluent uses the Nusselt number expression:

(12–286)

where
	= thermal conductivity of the continuous phase (W/m/K)
	= the characteristic length (m)

The Nusselt number is computed from Equation 12–299. If the film is cooled below the boiling point by contacting a cold wall, the vaporization rate calculation will revert to the vaporization equations (Equation 12–275 or Equation 12–282).

 

12.9.5.2.1.6. Film Boiling Model with Convective Heat Transfer Model

The boiling rate expressions for the convective heat transfer model are as follows:

For constant temperature walls:

(12–287)

For heat flux walls:

(12–288)

with

(12–289)

In the above equations,

= mass of the film particle (kg)

= film particle temperature (K)

= bulk temperature (K)

= wall temperature (K)

= latent heat of vaporization (J/kg)

= heat transfer coefficient from wall to liquid film (W/K-m2)

= heat transfer coefficient from liquid film to bulk phase (W/K-m2)

= film parcel area, taken to be a mass weighted percentage of the wall face area (m2)

= heat flux imposed on the wall (W)

= liquid Prandtl number

The heat transfer coefficient from liquid film to bulk phase () is obtained as described in Film Boiling Model with Conduction Heat Transfer Model.

The heat transfer coefficient from wall to liquid film is obtained from Equation 12–312.

If the film is cooled below the boiling point by contacting a cold wall, the vaporization rate calculation will revert to the vaporization equations (Equation 12–275 or Equation 12–282).

The condensation model is triggered when the partial pressure of species in the bulk exceeds its vapor pressure at the film surface. The Fluent Lagrangian wall film model uses two different expressions for wall film condensation under laminar and turbulent flow conditions. The derivation follows [739], assuming that the condensation of species from the gas phase takes place in the presence of noncondensables.

In the Ansys Fluent Lagrangian Wall Film Condensation model, component condenses on the wall film in the presence of non-condensable components. For the condensation rate of component through a plane at a distance from the film surface, Fick’s law can be written as:

(12–290)

For laminar flow, the mass fraction of species is integrated from the film surface to the bulk gas conditions yielding the logarithmic term, while in case of turbulent flow, finite differences are taken directly. The final expressions are:

For laminar flow, the mass transfer coefficient in Equation 12–291 is computed as:

(12–293)

where is the normal distance from the film surface to the bulk. is computed as the normal distance from the wall adjacent cell center to the wall.

For turbulent flow, the turbulent mass transfer coefficient in Equation 12–292 is determined from the wall functions. The current implementation uses wall functions as described by Equation 4–348. When the enhanced wall treatment is enabled in the turbulence model, the implementation follows the calculations in Enhanced Wall Treatment ε-Equation (EWT-ε).

For multicomponent particles, Equation 12–291 or Equation 12–292 are applied to each liquid component.

See Film Condensation Model in the Fluent User's Guide for more information on how to use the film condensation model.

For the multicomponent Lagrangian wall film, the vaporization rate is calculated as the sum of the vaporization rates of the individual components. The vaporization rate of each component in the liquid film are computed from:

If the Thermolysis model has been enabled for any component in the film, Ansys Fluent applies Equation 12–164 to obtain the thermolysis rate of the individual component for the single rate or Equation 12–165 to obtain the constant rate.

If the Lagrangian wall film is considered, the secondary rate model can be used. This model uses Equation 12–164 to calculate separately the vaporization rate of the free-stream particle (using the particle pre-exponential factor and the particle activation energy ) and the vaporization rate of the film (using the film pre-exponential factor and the film activation energy ).

When the total vapor pressure at the multicomponent film surface exceeds the cell pressure, Ansys Fluent applies a multicomponent film boiling equation by using averaged forms of Equation 12–283 or Equation 12–284 in accordance with the thermal boundary condition applied at the wall. The average latent heat is computed according to Equation 12–170 from the latent heats of the individual liquid components and the overall boiling rate of the film is calculated by averaging the rates computed from Equation 12–283 or Equation 12–284 for the conduction heat transfer model, and from Equation 12–287 or Equation 12–288 for the convective heat transfer model, for each liquid component for each liquid component with the fractional vaporization computed by Equation 12–171.

To obtain an equation for the temperature in the film, energy flux from the gas side as well as energy flux from the wall side must be considered. In the absence of film evaporation/boiling, the energy transfer from/to the film particle is then given by:

(12–294)

Figure 12.13: Assumption of a Bilinear Temperature Profile in the Film

When the thermal boundary condition on the wall is set to a constant temperature, is calculated as:

(12–295)

For walls where the heat flux, is imposed by the boundary condition, Equation 12–295 is replaced by:

(12–296)

The convective heat flux, , is calculated as:

(12–297)

If the film condensation or the gas-side boundary layer model is enabled, the heat transfer coefficient is computed as follows.

In other cases, Fluent uses a Nusselt number expression for :

The Nusselt number is calculated as:

(12–299)

The film surface temperature in Equation 12–297 is determined from the interface conservation relation:

(12–300)

When the particle changes its mass during vaporization, an additional term is added to Equation 12–294 to account for the enthalpy of vaporization:

(12–301)

where is the film parcel vaporization rate (kg/s), and is the latent heat of vaporization (J/kg).

As the film parcel trajectory is computed, Ansys Fluent integrates Equation 12–294 or Equation 12–301 and obtains the temperature at the next time step.

The implementation does not account for film radiation effects. The energy source terms are calculated for the film parcels from the energy difference along the film trajectory. For the cases where the film resides on a wall with heat-flux imposed, the heat losses or gains from the film to the wall equal the heat-flux values. For all other thermal boundary conditions, except heat flux, Fluent determines the film temperature based on the wall temperature and the heat transferred from the bulk fluid to the film. The total heat flux to the wall is then calculated by adding the heat flux from the bulk fluid to the wall and the heat flux from the film to the wall.

The convective wall-film heat transfer model provides a more accurate calculation of heat transfer between the Lagrangian wall film and the wall surface where it is formed. It is especially useful for thicker wall films, films moving at high velocity and where the film liquid material has a high Prandtl number. Refer to Setting the Lagrangian Wall Film Model in the Fluent User's Guide to learn how to use this model.

The following assumptions are made for the flow and heat transfer in the liquid film:

Figure 12.14: Convective Wall-Film Heat Transfer: Film Flow Over a Heated Surface

The energy balance of a film particle at temperature exchanging heat with a wall surface and a bulk phase can be expressed as:

(12–302)

with

(12–303)

and

(12–304)

where

= mass of the film particle (kg)

= heat capacity of the liquid (J/K-kg)

= film particle temperature (K)

= latent heat of vaporization (J/kg)

= heat transfer coefficient from wall to liquid film (W/K-m2)

= heat transfer coefficient from liquid film to bulk phase (W/K-m2)

= wall face area (m2)

= film parcel area, taken to be a mass weighted percentage of the wall face area (m2)

= bulk temperature (K)

= film surface temperature (K)

= wall temperature (K)

= heat flux imposed on the wall (W)

The temperature profile inside the thermal boundary layer in the direction normal to the wall is expressed as:

(12–305)

where is the boundary layer height.

The film temperature above the thermal boundary remains constant at .

The average temperature inside the boundary layer can be obtained by integrating along the direction:

(12–306)

which results in:

(12–307)

The average film parcel temperature can be obtained by weighting and with the heights and () under the assumptions that the film velocity and density remain constant:

(12–308)

where is the liquid film height.

For > 0.6, the ratio of the thermal boundary layer thickness to the velocity boundary layer thickness can be represented as:

(12–309)

The surface temperature can be calculated by combining equations Equation 12–307 - Equation 12–309 for the constant temperature walls:

(12–310)

For walls where the constant heat flux is imposed, the corresponding equation for the surface temperature is expressed as:

(12–311)

As the film parcel trajectory is computed, Ansys Fluent integrates Equation 12–302 and obtains the temperature at the next time step.

The model implementation does not account for film radiation effects. The energy source terms are calculated for the film parcels from the energy difference along the film trajectory. For cases where the film resides on a wall with heat-flux imposed, the heat losses or gains from the film to the wall equal the heat-flux values. For all other thermal boundary conditions, except heat flux, Ansys Fluent determines the film temperature based on the wall temperature and the heat transferred from the bulk fluid to the film. The total heat flux to the wall is then calculated from Equation 12–304 .

Calculation of the heat transfer coefficient

The heat transfer coefficient from film to wall is expressed as:

(12–312)

where is the thermal conductivity of the liquid film, and is the characteristic length.

The Nusselt number is calculated from the laminar boundary-layer theory for > 0.6:

In the above equations, the Reynolds number and the Prandtl number are calculated as:

(12–315)

(12–316)

where

, , , and = density, viscosity, thermal conductivity and heat capacity of the liquid film, respectively.

= velocity of the liquid film in the bulk, that is, above the film flow boundary layer (m/s)

Calculation of

The velocity profile for fully developed flow over a flat plate in the laminar boundary layer is described in the direction normal to the wall by:

(12–317)

The flow boundary layer height can be obtained as function of distance from the edge of the flat plate [372]:

(12–318)

The bulk velocity is calculated as a function of the average wall film velocity for the following regimes:

Considerations on the characteristic length

is the distance of the current particle location from the impingement point. The impingement location is known for new film particles only and can only be recorded during the impingement event. It cannot be back-calculated when reading film particles from case/data files created in earlier versions. For those old particles, the film height is used as the characteristic length together with the below expressions for the Nusselt number providing the averaged heat transfer coefficients over a specified length.

The Nusselt number for > 0.6 is expressed as:

Considerations on the heat transfer coefficient

The heat transfer coefficient is computed by Equation 12–312 as a function of the Nusselt number. However, it reverts to the conduction heat transfer coefficient under the below conditions:

In those cases, the temperature profile in the film is assumed to be linear, and the characteristic length will be equal to the film height .

The conduction heat transfer coefficient is computed as:

(12–322)

Turbulent film approximation for heat transfer

For high film Reynolds numbers, the film-to-wall heat transfer is significantly enhanced because of turbulence phenomena. Ansys Fluent provides a simplified approach that improves the heat transfer predictions when the film is in the turbulent regime.

For the heat transfer calculations only, the following assumptions are made:

With the Reynolds number expressed by

(12–323)

where is the film average velocity, and the remaining nomenclature is the same as in Equation 12–315, the Nusselt number for the wall-to-film heat transfer is defined as follows:

Calculation of the heat transfer coefficient from wall to film on the impingement point

The boundary layer theory is not applicable at the particle impingement point. Here, the following expression for the Nusselt number is used [373]:

(12–329)

Here, the Reynolds number is expressed as:

(12–330)

where is the jet velocity (m/s).

The characteristic length used in the above expression is the jet diameter, which is approximated by the reference diameter for the relevant injection type. The reference diameter is defined as the hydraulic diameter corresponding to the surface area of the surface injection, or the maximum particle diameter for injections involving particle size distributions.