Evaporation and Condensation

The Single Component and Multicomponent Droplet Evaporation Mass Transfer Rate models calculate the mass transfer rate from single and multi-component droplets to a multi-component gas. The Multicomponent model assumes that the liquid drops are internally homogeneous and that the liquid behaves like an ideal mixture.

Mass Transfer

Mass transfer (which can be evaporation or condensation) occurs when the system departs from the equilibrium condition:

Figure 1. EQUATION_DISPLAY

Y_{i}^{e q} = K_{i} (P, T) X_{i}^{e q}

(2048)

where:

$Y_{i}^{e q}$ is the interface equilibrium mole fraction (gas side) of the evaporating species $i$
$K_{i}$ is the equilibrium coefficient for the species $i$ . This coefficient is normally a function of pressure and temperature.
$X_{i}^{e q}$ is the interface equilibrium mole fraction (liquid side) of the evaporating species $i$

For the single-component case, this expression Eqn. (2048) simplifies to:

Figure 2. EQUATION_DISPLAY

Y_{e q} = K (P, T)

(2049)

Evaporation occurs when the gas bulk mole fraction of the evaporating component is below the equilibrium value $Y_{e q}$ , otherwise condensation occurs.

Using the model from Spalding [547], the total mass transfer rate $m_{i}$ is calculated according to:

Figure 3. EQUATION_DISPLAY

{\dot{m}}_{i} = - ϵ_{i} g^{*} A_{s} \ln (1 + B)

(2050)

where:

$B$ is the Spalding transfer number.
This value can be either positive or negative. If $B$ is negative, condensation (treated as ”reverse evaporation”) occurs.
$g^{*}$ is the mass transfer conductance (in the limit $B \to 0$ )
$A_{s}$ is the area of the droplet surface.
$ϵ_{i}$ is the fractional mass transfer rate, which has the property:
$\sum_{T} ϵ_{i} = 1$

with the summation over the $T$ transferred components.

For the single-component case, Eqn. (2050) simplifies to:

${\dot{m}}_{p} = - g^{*} A_{s} \ln (1 + B)$

The expression for $B$ and $g^{*}$ depends on the evaporation mode. Two evaporation modes are implemented: the vapor diffusion limited evaporation mode and the heat transfer limited evaporation mode. Simcenter STAR-CCM+ applies the appropriate evaporation mode automatically according to the value of $\sum_{T} Y_{i, e q}$ where $Y_{i, e q}$ is the equilibrium value of transferred species $i$ on the vapor side of the liquid-vapor interface (see Eqn. (2050)).

Vapor Diffusion Limited Evaporation

The vapor diffusion limited evaporation mode occurs when the droplet surface is not saturated (that is, $\sum_{T} Y_{i, e q} < 1$ ) and the evaporation rate depends on the rate at which the vapor species can diffuse away from the droplet. Under this condition, the fractional mass transfer rate and the mass transfer number are calculated as:

Figure 4. EQUATION_DISPLAY

ϵ_{i} = \frac{Y_{i, m} (1 + B) - Y_{i}}{B}

(2051)

Figure 5. EQUATION_DISPLAY

B = \frac{\sum_{T} Y_{i, m} - \sum_{T} Y_{i}}{1 - \sum_{T} Y_{i, m}}

(2052)

where:

$Y_{i}$ is the bulk gas vapor mass fraction of the evaporating component $i$ .
$Y_{i, m}$ is the equilibrium vapor mass fraction:

Figure 6. EQUATION_DISPLAY

Y_{i, m} = Y_{i, e q} \frac{W_{i}}{W_{s}}

(2053)

where:

$W_{i}$ is the molecular weight of the vapor at the drop surface.
$W_{s}$ is the molecular weight of the gas mixture at the drop surface.

For the single-component case, the expression for the mass transfer number, Eqn. (2052), simplifies to:

Figure 7. EQUATION_DISPLAY

B = \frac{Y_{v, s} - Y_{v}}{1 - Y_{v, s}}

(2054)

The conductance $g^{*}$ in this evaporation mode is:

Figure 8. EQUATION_DISPLAY

g^{*} = \frac{ρ D_{v} {S h}_{p}}{D_{p}}

(2055)

where:

$D_{v}$ is the molecular diffusivity of the vapor
$D_{p}$ is the drop diameter
${S h}_{p}$ is the Sherwood number.

Heat Transfer Limited Evaporation

The heat transfer limited evaporation mode is active when the vapor at the droplet surface is saturated (that is, $\sum_{T} Y_{i, e q} \geq 1$ ). This evaporation mode is also used when there are no inert species in the gas.

Under this condition, the volatile components vaporize in proportion to their mass fraction according to ([546]):

Figure 9. EQUATION_DISPLAY

ϵ_{i} = \frac{Y_{i p}}{\sum_{T} Y_{i p}}

(2056)

It is assumed that heat does not penetrate the droplet and that the evaporation rate results from the balance between heat transfer and latent heat due to evaporation. This mode requires thermal non-equilibrium between the phases for mass transfer to occur.

The transfer number is:

Figure 10. EQUATION_DISPLAY

B = \frac{C_{p} (T_{g} - T_{d})}{\sum_{T} ϵ_{i} L_{i}}

(2057)

where:

$T_{g}$ is the gas temperature
$T_{d}$ is the drop temperature
$c_{p}$ is the specific heat of the gas
$L_{i}$ is the latent heat of vaporization of species $i$ , which is calculated as the difference between the vapor and liquid enthalpies at the drop temperature.

For the single-component case, the expression for the transfer number, Eqn. (2057), simplifies to:

Figure 11. EQUATION_DISPLAY

B = \frac{c_{p} (T_{g} - T_{d})}{L}

(2058)

The conductance is:

Figure 12. EQUATION_DISPLAY

g^{*} = \frac{k N u}{c_{p} D_{p}}

(2059)

where:

$N u$ is the Nusselt number
$k$ is the thermal conductivity of the gas.

Heat Transfer

The droplet evaporation mass transfer rate model assumes thermal equilibrium inside the drops: the model neglects temperature gradients inside the drop, so the interface temperature is assumed to be the same as the inner drop temperature. The energy balance at the drop interface is:

Figure 13. EQUATION_DISPLAY

q_{L} - q_{G} = \sum_{T} {\dot{m}}_{i} L_{i}

(2060)

where:

$q_{L}$ is the heat that is transferred by the liquid
$q_{G} = h (T_{g} - T_{d})$ is the heat that is transferred by gas to the interface
$h$ is the heat transfer coefficient
$L_{i}$ is the latent heat of vaporization of species $i$ .

The value of $q_{L}$ is derived from Eqn. (2060) after the calculation of ${\dot{m}}_{i}$ and $q_{G}$ .

Multi-Component Mass Transfer

If a phase is chosen to be a multi-component gas, liquid, or solid (particle), then the segregated species solver can be selected for that phase.

This solver allows multiple species transport equations to be solved:

Figure 14. EQUATION_DISPLAY

\frac{δ ρ_{j} α_{j} Y_{j, i}}{δ t} + \nabla \cdot (ρ_{j} α_{j} Y_{j, i} v_{j}) = \nabla \cdot (ρ_{j} α_{j} D_{j} \nabla Y_{j, i}) + S_{j, i} + \overset{'}{m_{j, i}}

(2061)

where:

$ρ_{j}$ , $α_{j}$ and $v_{j}$ are the density, volume fraction and velocity of phase j
$Y_{j, i}$ is the mass fraction of species i in phase j, that is, the fraction of the total mass of that phase
$D_{j}$ is the mass diffusivity; combining molecular and turbulent diffusivity
$S_{j, i}$ is a general mass source
$\overset{'}{m_{j, i}}$ is used here to refer specifically to the transfer of species from one phase to another.

Note	If there is a single phase with $α_{j}$ equal to 1 and no interphase mass transfer, this equation reverts to the species transport equation for a single phase.

Similar to a single phase, boundary, and initial conditions are required for all species for all multi-component phases.