Appendix A.1. LDEF Region Governing Equations
The mass conservation equation of the LDEF region is given by the following equation:
where
is a mass source term generated by the evaporation of spray particles.
The momentum conservation in the LDEF region is given by the following equation:
where the effective viscosity
is the sum of the molecular and turbulent viscosity and the rate of strain (deformation) tensor
is defined as
The pressure gradient and gravity force terms are rearranged numerically in the following form:
, where
and
r is the position vector from the wall.
is a momentum source term due to the drag force exerted by the fluid (ambient air) on liquid particles.
The internal field is designed to have an initial chemical composition similar to the atmosphere. The evaporation of spray particles introduces a third gaseous species. These three gaseous entities are modelled by their mass fraction
(gaseous), and are calculated from the equation of species in the internal field:
with
being a diffusion coefficient of the
k species and
a source term describing the generation of a species. The spray and the liquid film contain only the
species, so
and
Energy conservation is ensured by solving the following enthalpy equation:
where
is the kinetic energy per unit mass,
the enthalpy per unit mass, and
is the effective thermal diffusivity, which is the sum of laminar and turbulent thermal diffusivities:
where
T is the sensible enthalpy,
and
are the molar fraction and the standard enthalpy of formation of the
k species, respectively.
is the thermal conductivity,
is the specific heat at constant pressure,
is the dynamic viscosity,
is the turbulent (kinematic) viscosity,
is the Prandtl number, and
is the turbulent Prandtl number.
models the particle evaporation and heat convection.
Appendix A.2. EDEF Region Governing Equations
The mass conservation equation of the EDEF region is analogue to the VOF method:
where
, , are the volume fraction, density, and velocity, respectively, of entity
. Then, the specie transport [
27] equation is solved to update concentrations and the liquid thermophysical properties:
where
is the mutual diffusion coefficients of species 1 and 2 defined as:
with
the kinematic viscosity and
the Schmidt number of the liquid phase. Immediately after solving equation (A8) the liquid phase thermophysical properties are updated using the mixture rules.
After resolving the mass conservation equations, the phases are considered (in each cell) as continuous and dispersed following two threshold values: (i) the minimum volume fraction of a phase to be considered fully continuous, and by default the other phase will become a dispersed phase. The dispersed phase is modelled as a set of droplets characterized by a diameter and a shape. For the sake of simplicity, the shape of the droplets is considered as a perfect sphere. (ii) the minimum volume fraction of a phase for which it can be treated as partly continuous, i.e., the cell is part of an interface ().
Two momentum equations of the same form are solved for each phase (gaseous and liquid) [
28]:
where
is the viscosity of phase
and the rate of strain (deformation) tensor
is defined as
.
is an added source term modeling the melting or solidification of phase
, as discussed in [
8].
is the inter-phase momentum transfer with the convention of
The two phases exchange momentum in the form of drag and virtual mass
. The drag and virtual mass forces are due, respectively, to the velocity and acceleration differences between the two phases. In the following equations, quantity differences presented by the symbol
are defined as:
.
For cells with dispersed phase [
29]:
For interface [
30]:
where
and
are the drag coefficients and
the dispersed phase surface. The drag coefficient models implemented in OpenFOAM are presented in [
8]. Since droplets are supposed to have a spherical shape, the virtual mass coefficient was fixed to
[
29].
Two energy equations are solved for gaseous and liquid phases:
where
,
, and
are the enthalpy, kinetic energy, and effective thermal diffusivity, respectively.
is a sink term modeling phase change,
is the heat transfer coefficient between the two phases. Two models have already been implemented in OpenFOAM; the first is an analytical model for a perfect sphere and the second is a correlation for turbulent heat transfer, also for a sphere. The heat transfer coefficients are presented in [
8].
The source terms and are the solid phase presence effect, calculated following the enthalpy–porosity technique. In this technique, the solid–liquid free boundary is not tracked explicitly. Instead, a phase fraction indicator field (0 for solid and 1 for liquid) is updated at each iteration based on an enthalpy balance. The enthalpy–porosity technique was originally developed and used for single-phase flows, in this paper a modified version of the technique is proposed.
The momentum sink
, introduced in Equation (3), classically models the buoyancy effect
due to the thermal expansion of ice and the drag force
exerted by ice on the liquid. However, for the gas phase the buoyancy term should be removed since the implicit solid phase belongs to the liquid phase. Additionally, in the proposed problem, the buoyancy effect for the liquid phase can be neglected since the flow is under forced convection (impinging spray):
In the original version of the enthalpy–porosity technique [
31], the solid phase was not designed to move. The present work focusses on the decontamination of a surface, and since no ice breakup was modelled, this characteristic is preserved:
The Darcy law for flow through porous media is used. For isothermal solidification/melting, permeability has no physical significance; however, it is used classically as a numerical technique to estimate the velocity at the mushy region [
31]:
where
and
are model coefficients, as discussed in the numerical method paragraph.
where
, with
is the sensible enthalpy and
is the enthalpy of fusion (or latent heat of fusion).
Moving the
on the right side of the energy equation and developing it gives:
which by identification with Equation (6) gives:
where
is the energy sink term due to phase change of the liquid phase (solid–liquid). It represents also the latent heat released during solidification. The gaseous phase does not undergo a phase change (the deposition–sublimation phenomenon is not modelled), thus the term
is null.