Multi-Component Bulk Boiling

Consider a stationary coordinate reference frame and define $v_{i}$ as the velocity of component i in this frame. The molar flux of species i is expressed as:

Figure 1. EQUATION_DISPLAY

\begin{matrix} N_{i} = c_{i} v_{i} & [\frac{mol}{m^{2} s}] \end{matrix}

(2074)

Sum over all the species to get the total molar flux:

Figure 2. EQUATION_DISPLAY

N_{t} = \sum_{i = 1}^{n} N_{i} = c_{t} v

(2075)

where the molar average velocity $v$ is defined as:

Figure 3. EQUATION_DISPLAY

v = \sum_{i = 1}^{n} x_{i} v_{i}

(2076)

The diffusion flux, which is the flux of component i relative to the flux of the mixture, can now be defined. Choose $v$ as the mixture reference velocity and therefore define the molar diffusion flux as:

Figure 4. EQUATION_DISPLAY

J_{i} = c_{i} (v_{i} - v)

(2077)

with:

Figure 5. EQUATION_DISPLAY

\sum_{i = 1}^{n} J_{i} = 0

(2078)

The molar flux $N_{i}$ is related to the molar diffusion flux by:

Figure 6. EQUATION_DISPLAY

N_{i} = J_{i} + c_{i} v = J_{i} + x_{i} N_{t}

(2079)

Film Model

The assumptions of the Film Model can be summarized as follows:

All the resistance to mass transfer is localized in a thin film that is adjacent to the phase boundary.
Mass transfer occurs inside this film by steady-state molecular diffusion alone.
Mass transfer through the film occurs in the direction normal to the interface (that is, composition gradients along the interface are neglected).
There are no composition gradients outside the film.

Diffusion Flux Equation

With the previous assumptions, the species concentration gradients across the film can be calculated using finite differences and the equation for the diffusion flux becomes:

Figure 7. EQUATION_DISPLAY

J_{i} = N_{i} - x_{i} N_{t} = c_{t} \sum_{j = 1}^{n - 1} {k_{i j}}^{*} Δ x_{j}

(2080)

where we express the finite flux mass transfer coefficient matrix as:

Figure 8. EQUATION_DISPLAY

[k^{*}] = [k] [Ξ]

(2081)

with $[k]$ and $[Ξ]$ being the low-flux mass transfer coefficient matrix and the finite flux correction coefficient matrix respectively.

Mass Transfer Coefficient in the Zero Flux Limit

The low-flux mass transfer coefficient matrix $[k]$ is defined as:

Figure 9. EQUATION_DISPLAY

[k] = \frac{1}{l} {[B]}^{- 1}

(2082)

Figure 10. EQUATION_DISPLAY

B_{i i} = \frac{x_{i}}{D_{i n}} + \sum_{\begin{matrix} k = 1 \\ k \neq i \end{matrix}}^{n} \frac{x_{k}}{D_{i j}}

(2083)

Figure 11. EQUATION_DISPLAY

B_{i j} = - x_{i} (\frac{1}{D_{i j}} - \frac{1}{D_{i n}})

(2084)

The length scale $l$ that appears in Eqn. (2082) is derived from the film theory and is the thickness of the film layer. This quantity can be calculated from the Sherwood number, since:

Figure 12. EQUATION_DISPLAY

Sh = \frac{k d}{D} = \frac{d}{l}

(2085)

from which $l = \frac{d}{Sh}$ can be obtained.

The resulting film thickness is a function of flow conditions, geometry, and fluid properties. In Simcenter STAR-CCM+, only one characteristic length $d$ and one Sherwood number $Sh$ can be specified for each phase interaction. Therefore, the same value of $l$ is calculated and applied for all of the transferred components.

Mass Transfer Correction Coefficient Matrix

The form of the correction coefficients depends on the mass transfer model that is adopted. For the two fluid film model, following the convention of positive flux from the vapor to the liquid, the correction matrices are:

Figure 13. EQUATION_DISPLAY

{[Ξ]}^{V} = {[Φ]}^{V} {[\exp {[Φ]}^{V} - [I]]}^{- 1}

(2086)

and

Figure 14. EQUATION_DISPLAY

{[Ξ]}^{L} = {[Φ]}^{L} \exp {[Φ]}^{L} {[\exp {[Φ]}^{L} - [I]]}^{- 1}

(2087)

for the vapor and the liquid respectively, where:

Figure 15. EQUATION_DISPLAY

Φ_{i i} = \frac{N_{i}}{c_{t} κ_{i n}} + \sum_{\begin{matrix} k = 1 \\ k \neq i \end{matrix}}^{n} \frac{N_{k}}{c_{t} κ_{i k}}

(2088)

Figure 16. EQUATION_DISPLAY

Φ_{i j} = - N_{i} (\frac{1}{c_{t} κ_{i j}} - \frac{1}{c_{t} κ_{i n}})

(2089)

Interface Energy Balance

Continuity of the energy flux across the vapor - liquid interface is imposed:

Figure 17. EQUATION_DISPLAY

E^{V} = E^{L} = E

(2090)

where $E^{V}$ and $E^{L}$ are the normal components of the energy flux at the interface. Eqn. (2090) can be rewritten as:

Figure 18. EQUATION_DISPLAY

q^{V} + \sum_{i = 1}^{n} N_{i}^{V} {\bar{H}}_{i}^{V} (T^{I}) = q^{L} + \sum_{i = 1}^{n} N_{i}^{L} {\bar{H}}_{i}^{L} (T^{I})

(2091)

with the heat fluxes given by:

Figure 19. EQUATION_DISPLAY

q^{V} = {h_{V}}^{*} (T^{V} - T^{I})

(2092)

Figure 20. EQUATION_DISPLAY

q^{L} = {h_{L}}^{*} (T^{I} - T^{L})

(2093)

The finite flux heat transfer coefficient in the vapor phase is calculated according to the film model as:

Figure 21. EQUATION_DISPLAY

{h_{V}}^{*} = h^{V} (Φ_{H}^{V}) / (exp Φ_{H}^{V} - 1) = Ξ_{H} h^{V}

(2094)

where $h^{V}$ is the zero-flux heat transfer coefficient, $Ξ_{H} = (Φ_{H}^{V}) / (exp Φ_{H}^{V} - 1)$ is the Ackermann correction factor and

Figure 22. EQUATION_DISPLAY

Φ_{H} = \sum_{i = 1}^{n} N_{i} W_{i} c_{p i} / h

(2095)

Similarly, for the liquid:

Figure 23. EQUATION_DISPLAY

{h_{L}}^{*} = h^{L} (Φ_{H}^{L}) exp Φ_{H}^{L} / (exp Φ_{H}^{L} - 1) = Ξ_{H}^{L} h^{L}

(2096)

with $Ξ_{H}^{L} = (Φ_{H}^{L}) exp Φ_{H}^{L} / (exp Φ_{H}^{L} - 1)$ .

The high flux correction for $h^{L}$ is expected to be close to unity due to the high value of the liquid phase heat transfer coefficient.

Species Equilibrium at the Interface

The composition of the phases on either side of the interface between the two phases are assumed to be in instant local equilibrium with each other and with the interface temperature and the system pressure:

Figure 24. EQUATION_DISPLAY

y_{i}^{I} = K_{i}^{0} (P, T^{I}) x_{i}^{I}

(2097)

where ideal behavior of the mixtures is assumed. The mole fractions also obey the constraints:

Figure 25. EQUATION_DISPLAY

\sum x_{i}^{I} = 1

(2098)

Figure 26. EQUATION_DISPLAY

\sum y_{i}^{I} = 1

(2099)

Note that a solution of these interface state relations is possible only if at least one of the vapor-liquid equilibrium coefficients $K_{i}^{0}$ is less than unity while at least one of the $K_{i}^{0}$ is greater than unity.

Equations of the Point Solver

Consider two mixtures $x$ and $y$ divided by an interface. The superscripts $L$ and $V$ are used for bulk quantities in the liquid and in the vapor respectively, while $I$ indicates interface quantities. The interface is considered as a surface with no resistance to mass transfer and where equilibrium conditions prevail at any time. It is assumed that no chemical reactions occur at the interface or in the film.

When the interface is not stationary, the fluxes $N_{i}$ refer to the interface and not to the laboratory fixed frame of reference (see [554] for more details). It is assumed that the mass transfer takes place in a direction normal to the interface:

Figure 27. EQUATION_DISPLAY

\begin{matrix} N_{i}^{x} = N_{i}^{y} = N_{i} & i = 1, 2, ..., n \end{matrix}

(2100)

Figure 28. EQUATION_DISPLAY

i = 1 n

(2101)

where $N_{i}^{x}$ is the normal component of $N_{i}$ and is directed from the bulk phase $x$ to the interface $I$ while $N_{i}^{y}$ is directed from the interface $I$ to the bulk phase $y$ .

In the general case, there can be $m$ species in the liquid and $n$ in the gas, not all of them volatile or condensable. Since there are $k$ couples between the volatile and condensable species, then $m - k$ species are non-volatile and $n - k$ species are non-condensable.

Thus, the variables to calculate are $m$ interface mole fractions $x_{i}^{I}$ , $n$ interface mole fractions $y_{i}^{I}$ , $k$ interface molar fluxes $N_{i}$ , and the interface temperature $T^{I}$ .

Therefore, the total number of variables is $m + n + k + 1$ and the equations of the model are:

$m - 1$ equations from the interface molar fluxes (liquid side)

Figure 29. EQUATION_DISPLAY

\begin{matrix} N_{i} - x_{i}^{I} N_{t} = c_{t}^{L} \sum_{i = 1}^{n - 1} {k_{i l}^{*}}^{L} (x_{j}^{I} - x_{j}) & i = 1, k + 1, ..., m - 1 \end{matrix}

(2102)

$n - 1$ equations from the interface molar fluxes (vapor side)

Figure 30. EQUATION_DISPLAY

\begin{matrix} N_{i} - y_{i}^{I} N_{t} = c_{t}^{V} \sum_{i = 1}^{n - 1} {k_{i j}^{*}}^{V} (y_{j} - y_{j}^{I}) & i = 1, k + 1, ..., n - 1 \end{matrix}

(2103)

1 equation from the energy balance at the interface

Figure 31. EQUATION_DISPLAY

h_{V^{*}} (T^{V} - T^{I}) + \sum_{i = 1}^{n} N_{i}^{V} {\bar{H}}_{i}^{V} W_{i} (T^{I}) = h_{L^{*}} (T^{I} - T^{L}) + \sum_{i = 1}^{m} N_{i}^{L} {\bar{H}}_{i}^{L} W_{i} (T^{I})

(2104)

$k$ equations from the equilibrium relations at the interface

Figure 32. EQUATION_DISPLAY

\begin{matrix} y_{i}^{I} = K_{i} x_{i}^{I} & i = 1, 2, ..., k \end{matrix}

(2105)

1 equation from the constraint on mole fractions at the interface (liquid side)

Figure 33. EQUATION_DISPLAY

\sum_{i = 1}^{m} x_{i}^{I} = 1

(2106)

1 equation from the constraint on mole fractions at the interface (vapor side)

Figure 34. EQUATION_DISPLAY

\sum_{i = 1}^{n} y_{i}^{I} = 1

(2107)

The number of equations is, therefore, $m + n + k + 1$ as the number of variables. If species $i$ is not volatile or non-condensable, then $N_{i} = 0$ in Eqn. (2102) and Eqn. (2103). If the liquid is a pure substance ( $m = 1$ ), there are no equations of the form Eqn. (2102) and Eqn. (2105) is always satisfied.

Likewise, if the vapor is a pure substance ( $n = 1$ ) then there are no equations of the form Eqn. (2103) and Eqn. (2106) is always satisfied.

Eqn. (2102) to Eqn. (2106) are a system of non-linear algebraic equations that are solved for each cell of the computational domain with a Newton solver.

Reduced Model for Dilute Dispersions

When the Maxwell-Stefan diffusion model is not active, the assumption of dilute dispersion is made, that is $x_{i} \approx 0$ for $i = 1, 2, ..., n - 1$ and $x_{n} \approx 1$ . With this assumption, only the interactions between each disperse species and the last species in the list (the solvent) are accounted for. The interactions between the disperse species are neglected. Consequently, Eqn. (2102) and Eqn. (2103) reduce to:

Figure 35. EQUATION_DISPLAY

\begin{matrix} N_{i} - x_{i}^{I} N_{t} = c_{t}^{L} κ_{i n}^{L} \frac{Φ_{i i}^{L} e^{Φ_{i i}^{L}}}{e^{Φ_{i i}^{L}} - 1} (x_{i}^{I} - x_{i}) \end{matrix}

(2108)

and

Figure 36. EQUATION_DISPLAY

\begin{matrix} N_{i} - y_{i}^{I} N_{t} = c_{t}^{V} κ_{i n}^{V} \frac{Φ_{i i}^{V}}{e^{Φ_{i i}^{L}} - 1} (y_{i} - y_{i}^{I}) \end{matrix}

(2109)

respectively, with $κ_{i n} = \frac{D_{i n}}{l}$ and $Φ_{i i} = \frac{N_{t}}{c_{t} κ_{i n}}$

The other equations are not affected by the assumption of dilute dispersion.

Nomenclature

The terms that are used in the equations of the Multi-component Boiling Mass Transfer model are defined in the following table.


$c_{i}$	Molar density of $i = ρ_{i} / M_{i}$	$[\frac{mol}{m^{3}}]$
$c_{t}$	Mixture molar density $= \sum_{i = 1}^{n} c_{i}$	$[\frac{mol}{m^{3}}]$
$c_{p}$	Specific heat	$[\frac{J}{kgK}]$
$d$	Characteristic length scale of the system	$m$
$\| D \|$	Matrix of Fick diffusion coefficients	$[\frac{m^{2}}{s}]$
$\| D \|$	Matrix of Maxwell-Stefan diffusion coefficients	$[\frac{m^{2}}{s}]$
$h$	Heat transfer coefficient	$[\frac{W}{m^{2} K}]$
$H$	Specific enthalpy	$[\frac{J}{kg}]$
$\| I \|$	Unit tensor
$J_{i}$	Molar diffusion flux $= c_{i} (v_{i} - v)$	$[\frac{mol}{m^{2} s}]$
$l$	Film thickness	$m$
$M_{i}$	Molar mass of component $i$	$[\frac{kg}{mol}]$
$N_{i}$	Molar flux of component $i$	$[\frac{mol}{m^{2} s}]$
$Sh$	Sherwood number
$v$	Molar averaged referenced velocity $= \sum_{i = 1}^{n} x_{i} v_{i}$	$[\frac{m}{s}]$
$x_{i}$ , $y_{i}$	Bulk mole fraction of component $i$ (liquid side, vapor side)
$x_{i}^{I}$ , $y_{i}^{I}$	Interface mole fraction of component $i$ (liquid side, vapor side)
$W_{i}$	Molecular weight of $i$	$[\frac{g}{mol}]$
$μ_{i}$	Molar chemical potential of $i$	$[\frac{J}{mol}]$
$ρ_{i}$	Mass density of $i = c_{i} M_{i}$	$[\frac{kg}{m^{3}}]$
$ρ_{t}$	Mixture mass density $= \sum_{i = 1}^{n} ρ_{i}$	$[\frac{kg}{m^{3}}]$
$ω_{i}$	Mass fraction of $i = ρ_{i} / ρ_{t}$
$[Φ]$	Matrix of mass transfer rate factors
$[Ξ]$	Matrix of high flux correction factors