Evaporation and Condensation

Simcenter STAR-CCM+ models evaporation of the fluid film into a gas phase and condensation from a gas into the fluid film.

Mass Conservation

When modelling evaporation from multi-component liquid films, the vapor pressure of each individual component and therefore its evaporation rate is dependent on the concentration of the different components in the mixture.

The species mass flux for every component $i$ is conserved at the interface between the gas and the fluid film, such that:

$$\rho Y_i(v-\dot{h})-\rho D_i\frac{{dY}_i}{dy}={\rho }_f{Y}_{f,i}(v_f-\dot{h})-{\rho }_f{D}_{f,i}\frac{d{Y}_i}{\mathrm{dy}}{|}_f$$

(2770)

where, evaluated at the interface:

• $\rho$ and ${\rho }_f$ are the gas and liquid film densities
• $Y_i$ and $Y_{f,i}$ are the mass fractions for the gas and liquid film
• $v$ and $v_f$ are the normal velocity components for the gas and liquid film
• $D_i$ and $D_{f,i}$ are the gas and liquid film molecular diffusion coefficients
• $\dot{h}$ is the rate of change of film thickness

The total mass flux is also conserved:

$$\rho (v-\dot{h})={\rho }_f(v_f-\dot{h})$$

(2771)

Combining Eqn. (2770) and Eqn. (2771) the following differential equation for vapor mass fraction $Y_i$ is obtained:

$$\sum _j^{N_V}\rho D_i\frac{d{Y}_i\left(y\right)}{dy}={\dot{m}}_v(\sum _j^{N_V}{Y}_i\left(y\right)-1)$$

(2772)

where $N_V$ is the number of interacting components, and the evaporation rate is defined as:

$${\dot{m}}_v=-{\rho }_f\dot{h}$$

(2773)

Note that Eqn. (2770) is valid only below saturation conditions.

The normal derivative is treated through the species transfer coefficients $s_{t,i}$ and the Spalding transfer number $B$ such that:

$${\dot{m}}_v=-\frac{\sum _j^{N_v}{s}_{t,j}\left(Y_{c,j}-Y_j\right)}{1-\sum _j^{N_v}{Y}_j}\cdot \frac{\ln (1+B)}{B}$$

(2774)

The Spalding number, $B$ is defined as:

$$B=\frac{\sum _j^{N_v}{Y}_j-\sum _j^{N_v}{Y}_{\infty }}{1-\sum _j^{N_v}{Y}_j}$$

(2775)

The subscript $c$ indicates cell values and it can be assumed that $Y_{\infty }\approx Y_c$ .

The interfacial gas mass fraction $Y_j$ in Eqn. (2774) is related to $Y_{f,j}$ through a background molar weight $W_{bg}$ , and the interfacial vapor mass fraction $Y_i$ :

$$W_{bg}=\frac{\sum _j^{N_{P,G}}{X}_j{W}_j}{\sum _j^{N_{P,G}}{X}_j}$$

(2776)

$$Y_i\approx \frac{X_i{W}_i}{X_{bg}{W}_{bg}+\sum _j^{N_v}{X}_j{W}_j}$$

(2777)

The evaporation of each component depends on the other components, because its vapor pressure is a function of the concentration of these components. The vapor pressure $p_i$ of a component i in the liquid is expressed as:

$$p_i={\gamma }_i{X}_{f,i}{p}_i^{*}\left(T\right)$$

(2778)

where:

• ${\gamma }_i$ is the activity coefficient, which accounts for the interactions between the different components in a mixture. Simcenter STAR-CCM+ provides two models to calculate the activity coefficient: Raoult's Law and the Modified UNIFAC model.
• $p_i^{*}$ is the vapor pressure of the pure component i.
• $X_{f,i}$ is the mole fraction of the component i in the liquid mixture.

The gas side mole fraction can be calculated as:

$$X_i=\frac{p_i}{p}$$

(2779)

where $p$ is the ambient pressure.

Raoult's Law

Raoult’s law assumes an ideal mixture with a similar molecular structure for the liquid film components. The activity coefficient for Raoult’s law is approximately 1 for all components.

Modified UNIFAC

The modified UNIFAC model is an adapted version of the UNIFAC (UNIQUAC Functional-group Activity Coefficients) method. The UNIFAC database [631] contains two tables namely the surface area and volume contributions listed by structural groups and the energy interaction parameters between different groups. The modified UNIFAC model [621] regards a molecule as an aggregate of functional groups and assumes that certain thermodynamic properties can be calculated by summing the group contributions. With this method, the molecular activity coefficient of component i in is Eqn. (2778) split into two parts:

$$\ln \left({\gamma }_i\right)=\ln \left({\gamma }_i^C\right)+\ln \left({\gamma }_i^R\right)$$

(2780)

where:

• ${\gamma }_i^C$ is the combinatorial term representing contributions due to differences in molecular size.
• ${\gamma }_i^R$ is the residual term representing contributions due to differences to molecular interactions.

Residual term

The group molar fraction $X_m$ in a group m in the mixture is defined as:

$$X_m\approx \frac{{\sum _j^{N_c}X}_j{v}_m^j}{\sum _j^{N_c}{X}_j\sum _k^{N_{g,j}}{v}_k^j}$$

(2781)

where:

• $N_{g,j}$ is the number of groups in component j.
• $N_g$ is the total number of groups in the mixture.
• $N_c$ is the total number of components in the mixture.
• $v_k^i$ is the structural group k number in component j.
• $X_j$ is the molar fraction of component j in mixture.

The relative surface area of group m in the mixture is defined as:

$${\mathrm{\Theta }}_m=\frac{X_m{Q}_m}{{\displaystyle \underset{n}{\sum ^{N_g}}{X}_n{Q}_n}}$$

(2782)

The group interaction parameter ${\mathrm{\Psi }}_{mn}$ according to the modified UNIFAC model is calculated as follows:

$${\mathrm{\Psi }}_{mn}=\exp (-\frac{a_{mn}+b_{mn}T+c_{mn}{T}^2}{T})$$

(2783)

where:

• T is the temperature.
• $a_{mn}$ , $b_{mn}$ and $c_{mn}$ are the group interaction parameters between groups m and n, taken from the database.

The residual contribution from each group k to the activity coefficient is then calculated as:

$$\ln {\mathrm{\Gamma }}_k=Q_k\{1-\ln {\displaystyle \sum _m^{N_g}}{\mathrm{\Theta }}_m{\mathrm{\Psi }}_{mk}-{\displaystyle \sum _m^{N_g}}\frac{{\mathrm{\Theta }}_m{\mathrm{\Psi }}_{km}}{{\displaystyle \sum _m^{N_g}{\mathrm{\Theta }}_n{\mathrm{\Psi }}_{nm}}}\}$$

(2784)

where $\ln \left({\mathrm{\Gamma }}_k\right)$ is computed by assuming the component exists in a pure form ( $X_j=1$ ).

The residual contribution to the activity coefficient is calculated as:

$$\ln \left({\gamma }_i^R\right)={\displaystyle \sum _k^{N_{g,i}}}{v}_k^i(\ln ({\mathrm{\Gamma }}_k)-\ln ({\mathrm{\Gamma }}_k^i\left)\right)$$

(2785)

Combination term

The combinatorial contribution due to differences in molecular size is obtained from:

$$\ln \left({\gamma }_i^c\right)=1-V_i^{\prime }+\ln \left(V_i^{\prime }\right)-\frac{z}{2}{q}_i[1-\frac{V_i}{F_i}+\ln (\frac{V_i}{F_i}\left)\right]$$

(2786)

The surface area $q_i$ and volume $r_i$ of component i are estimated as:

$$q_i={\displaystyle \sum _k^{N_{g,i}}{Q}_k{v}_k^i}$$

(2787)

$$r_i={\displaystyle \sum _k^{N_{g,i}}{R}_k{v}_k^i}$$

(2788)

where $R_k$ and $Q_k$ represent group k volume and surface area contributions, taken from the UNIFAC database. The following parameters are defined as:

$$V_i=\frac{r_i}{{\displaystyle \sum _j^{N_c}{X}_j{r}_j}}$$

(2789)

$$V_i\prime =\frac{r_i^{\frac{3}{4}}}{{\displaystyle \sum _j^{N_c}{X}_j{r}_j^{\frac{3}{4}}}}$$

(2790)

$$F_i=\frac{q_i}{{\displaystyle \sum _j^{N_c}{X}_j{q}_j}}$$

(2791)

where z is the molecule coordination number, commonly defined in literature as a constant value of 10.

Due to the assumption that film is thin, the mass transfer resistance on the gas side is considered to be considerably larger than on the liquid side. As a result, the film interfacial molar fractions concentration is approximated to follow from the associated cell center value.

Using the activity coefficient, the background molar fraction and a constant species distribution in the liquid film, the component evaporation rate can be expressed as:

$${\dot{m}}_{v,i}=Y_i{\dot{m}}_v-\rho D_i\frac{d{Y}_i}{dy}$$

(2792)

which is valid under all conditions.

In addition to the hydrodynamic effects, the evaporation rate can also be influenced by thermal effects.

Energy Conservation

The interfacial heat flux conservation is expressed as:

$$k\frac{dT}{\mathrm{dy}}-k_f\frac{dT}{\mathrm{dy}}{|}_f-{\stackrel{\cdot }{Q}}_v=0$$

(2793)

where, $k$ and $k_f$ denote gas and liquid film thermal conductivity and:

$${\stackrel{\cdot }{Q}}_v=\sum _j^{N_v}\mathrm{\Delta }{H}_i^{vap}\dot{m_{v,i}}$$

(2794)

which is a condition for the interfacial surface temperature $T_s$ .

A combination of Eqn. (2792) and Eqn. (2794) leads to an expression for the total evaporation rate:

$${\dot{m}}_v=\frac{{\dot{Q}}_v+\sum _i^{N_v}\mathrm{\Delta }{H}_i^{\mathrm{vap}}\rho D_i\frac{d{Y}_i}{dy}}{\sum _i^{N_v}\mathrm{\Delta }{H}_i^{\mathrm{vap}}{Y}_i}$$

(2795)

which also takes thermal effects into account and is valid under all conditions.