Evaporation and Condensation of Droplets

When simulating flows of liquid droplets dispersed in the gas phase, the Spalding Evaporation and Condensation model accounts for super-critical, heat transfer limited, and vapor diffusion limited evaporation. The maximum evaporation rate is determined by the conditions of the gas near the evaporating surface.

The following physical assumptions are made:

The droplets are internally homogeneous.
The droplets consist of an ideal mixture of liquid/gaseous components, some of which are transferred to the gas phase.
Inert components can exist in the droplets and/or gas.

The following conditions can be identified for transferred components $i$ (where the sum over transferred components is written as $\sum_{T}$ ):


Condition	Evaporation Computation
$T_{m} \geq \max T_{c, i}$	Super-critical evaporation
$\sum_{T} X_{i s} \geq 1$ or $\sum_{T} Y_{i} = 1$	Heat-limited evaporation
All other conditions	Vapor diffusion-limited evaporation

where:

$T_{m}$ is the mixture temperature
$T_{c, i}$ is the critical temperature of the transferred component $i$
$Y_{i}$ are the mass fractions

$X_{i s}$ is the equilibrium mole fraction at the droplet surface:

Figure 1. EQUATION_DISPLAY

X_{i s} = \frac{p_{s a t}^{i} (T_{m})}{p} X_{i p}

(2884)

where $X_{i p}$ is the component mole fraction in the droplet and $p_{s a t}^{i} (T)$ is the saturation pressure of transferred component $i$ at temperature $T$ .

For super-critical evaporation, all transferred components vaporize immediately. Otherwise, the rate of change of each transferred component $i$ due to quasi-steady evaporation ${\dot{m}}_{p i}$ is:

Figure 2. EQUATION_DISPLAY

{\dot{m}}_{p i} = - ε_{i} g^{*} A_{s} \log (1 + B_{i})

(2885)

where:

$A_{s}$ is the droplet surface area
$ε_{i}$ is the fractional mass transfer rate of component $i$
$B_{i}$ is the Spalding transfer number
$g^{*}$ is the mass transfer conductance in the case $B \to 0$

For iteration $n + 1$ , the final evaporation rate ${\dot{m}}_{p i}$ is calculated as:

${\dot{m}}_{p i}^{n + 1} = ω {\dot{m}}_{p i} + (1 - ω) {\dot{m}}_{p i}^{n}$

where $ω$ is the user-specified Under-Relaxation Factor.


	Evaporation Type
	Heat Transfer Limited	Diffusion Limited
$ε_{i} =$	$\frac{Y_{i p}}{\sum_{T} Y_{i p}}$	$\frac{Y_{i s} (1 + B_{i}) - Y_{i}}{B_{i}}$
$B_{i} =$	$\frac{C_{p} (T_{s a t}^{i} - T_{m})}{\sum_{T} ε_{i} L_{i}}$	$\frac{\sum_{T} Y_{i s} - \sum_{T} Y_{i}}{1 - \sum_{T} Y_{i s}}$
$g^{*} =$	$\frac{k {Nu}_{p}}{C_{p} D_{p}}$	$\frac{ρ_{g} D_{ν} {Sh}_{p}}{D_{p}}$

where:

$Y_{i p}$ is the mass fraction of component $i$ in the particle (liquid phase)
$C_{p}$ is the specific heat capacity
$ρ_{g}$ is the gas phase density
$T_{s a t}^{i} (p)$ is the saturation temperature of transferred component $i$ at pressure $p$
${Nu}_{p}$ is the particle Nusselt number
${Sh}_{p}$ is the particle Sherwood number
$D_{v}$ molecular diffusivity of the gas phase
$L_{i}$ is the latent heat of vaporization of component
$k$ is the thermal conductivity
$D_{p}$ is the droplet diameter

$Y_{i s} = X_{i s} \frac{W_{i}}{W_{s}}$ is the equilibrium mass fraction of transferred component $i$ at the droplet surface.

Armenante-Kirwan Correlation

The particle Nusselt number and the particle Sherwood number are calculated as:

Figure 3. EQUATION_DISPLAY

{Nu}_{p} = 2.0 + α {Re}_{T}^{β} \Pr^{γ}

(2886)

Figure 4. EQUATION_DISPLAY

{Sh}_{p} = 2.0 + α {Re}_{T}^{β} {S c}^{γ}

(2887)

where:

Figure 5. EQUATION_DISPLAY

{Re}_{T} = \frac{ϵ^{\frac{1}{3}} D_{p}^{\frac{4}{3}}}{ν}

(2888)

where ${Re}_{T}$ is the turbulent Reynolds number. The turbulent dissipation rate $ϵ$ is computed from the turbulent length and time scales for all models that provide it (all turbulence models except for the Spalart-Allmaras model). For the Spalart-Allmaras model, and in the laminar and inviscid case, this correlation falls back to the constant 2.0. Default values for the parameters $α$ , $β$ , $γ$ are:

$\begin{matrix} α = 0.6, & β = \frac{1}{2}, & γ = \frac{1}{3} \end{matrix}$