**4. Consideration of Kelvin equation**

We consider the equilibrium state where the gas and liquid phases coexist and temperature is constant. The gas–liquid interface is a flat surface. In this situation, the amount of evaporated liquid from the gas phase to the liquid phase is determined by the relationship between the saturated vapor pressure and ambient pressure if the temperature is constant. The vapor in the gas phase condenses to the liquid phase when the ambient pressure is increased to greater than the saturated vapor pressure. On the other hand, the liquid evaporates to the gas phase when the ambient pressure is decreased to less than the saturated vapor pressure. This is formulated by considering the changes in chemical potential thermodynamically. The same discussion is applied to a bubble and droplet. However, a bubble or droplet has a curvature. The vapor pressure of a droplet takes a different value when the interface has a curvature. The concrete equation is as follows (Butt et al., 2003).

$$\ln\left(\frac{P\_\kappa}{P^\circ}\right) = \frac{2H\sigma V\_m}{\mathcal{R}T} \tag{32}$$

In this equation, \* *P* and *P*κ represent the vapor pressures where the gas–liquid interface is flat and with a curvature, respectively. *H* [1/m3], σ [N/m], *Vm* [m3/kg], R [J/kg K], and *T* [K] are the mean curvature, surface tension coefficient, specific volume, gas constant, and temperature, respectively. Equation (32) is mainly derived based on both the Gibbs–Duhem equation and Young-Laplace equation from the thermodynamic point of view. In this section, we show that the Kelvin equation can be derived from the multi-scale multiphase flow equation.

#### **4.1 Derivation of Kelvin equation from multi-scale multiphase flow equation**

The Chapman–Enskog expansion is applied to the conventional Navier–Stokes equation to derive the multi-scale multiphase flow equation from which the thermodynamic interfacial jump condition is finally derived. In this section, the Kelvin equation is derived using the same procedure.

Figure 9 shows a schematic of a multi-scale concept around the interface. *O*(1) represents the order of a macroscopic scale. ε and <sup>2</sup> ε represent mesoscopic scales: the scale of ε is smaller than that of <sup>2</sup> ε . However, we assume that the continuum approximation holds in each scale. Based on this assumption, we consider the Kelvin equation.

By considering the Chapman–Enskog expansion in Eq. (7) and ε until the second order, the following equation is derived.

$$
\rho \frac{\text{D } \mathfrak{u}}{\text{D } t^{(0)}} + \varepsilon \rho \frac{\text{D } \mathfrak{u}'}{\text{D } t^{(1)}} + \varepsilon^2 \rho \frac{\text{D } \mathfrak{u}''}{\text{D } t^{(2)}} = -\nabla^{(0)} \cdot \mathbf{T} - \varepsilon \nabla^{(1)} \cdot \mathbf{T}' - \varepsilon^2 \nabla^{(2)} \cdot \mathbf{T}'' + \rho \mathbf{g}, \tag{33}
$$

Macroscopic Gas-Liquid Interfacial Equation

( )

*d*

 

1 1 2 2

 σ

ψ

 

 πε

+∞ +∞ −∞ −∞

∫ ∫

12 1 2 (1) (1)

*t t*

ε

++ + ⎜ ⎟ ∂ ∂∂ ⎝ ⎠

*<sup>d</sup> s s*

+∞ −∞

d

In this equation, the third term is transformed into

μ ψ

( )

*j*

 

− + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂∂ ∂∂ ⎝⎠ ⎝⎠ ⎣ ⎦

 

*l*

*P P*

τ

*n n*

σ

2

π

 *h*

∫

( )

+∞ −∞

∫

 κ

( )

*m*

 

∞ −∞

∫

− +

επ

 

*tt n*

0

ε

ε

εκ

2

ε ρ

as follows:

follows:

2

ε

ε

Based on Thermodynamic and Mathematical Approaches 75

<sup>L</sup> (2) d*n n* μ

ψψ

d d

 

*i*

 ψ ( )

*e*

 

τ

*n* (40)

εκ κ σ

ψψ

 

2 2

1 12

μ

<sup>∂</sup> ∫ *n n <sup>n</sup>* (42)

=− + + *<sup>s</sup> T VP n* ∑ (44)

 μ

<sup>∂</sup> <sup>∂</sup> ∂ ∂ <sup>∂</sup> ∫∫ ∫ (43)

ψ ψ

*<sup>n</sup> s ss <sup>t</sup>*

⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

 

 ψψ

 ψ

( )

ψ ψ 3

(41)

d

*n*

(1)

2 (2) 2

<sup>∂</sup> − =− <sup>∂</sup> <sup>∫</sup>

Finally, Equation (37) is transformed into the interfacial coordinates system as follows:

( G GG GG L LL LL G 1 2 ) ( ) ( ) ( ) ( ) ( )

επ

*a b c*

 

*nn n*

12 G (1) (1) (1) (1) (1) (1) (1)

∂ ∂ ∂ ∂∂ ∂ ∂∂ ⎝ ⎠ ⎝⎠

+ + ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ <sup>−</sup> <sup>+</sup> <sup>⎟</sup> <sup>−</sup> <sup>+</sup> <sup>⎟</sup> ∂∂ ∂ ⎢ ⎥ ∂ ∂ ⎝ ⎠⎝ ⎠ ⎝ <sup>⎠</sup> ⎣ ⎦

i i 1 2 (1) (1) <sup>1</sup> (1) (1) <sup>2</sup> (1) (1) 1 2 11 22

2 12

Here, for the sake of simplicity, we focus on the terms (a), (b), (c), and (m) (this is new term.)

G LG G 1 2 G (2) 0 d *P P <sup>n</sup> <sup>n</sup>*

<sup>∂</sup> ∂∂ ∂∂ <sup>=</sup> <sup>+</sup>

In the present discussion, change in temperature is not considered. Thus, the second term on the right-hand side of Eq. (43) is omitted. Here, the thermodynamic relation is considered as

d dd d

*ii i ij j*

(2) (2) (2) ddd *P T nnn nnn P T*

*N j c*

*j*

 μ

*n t* <sup>2</sup> (1) (1) (1)

( ) () <sup>2</sup>

∞∞ ∞

−∞ −∞ −∞

μ

=− − + −

εκ κ σ ερ <sup>∞</sup> −∞ ∂

ψ ψ

⎡ ⎤ ⎛ ⎞ ∂ ∂ ∂∂ ∂∂ ⎛⎞ ⎛⎞ + ++ ⎜ ⎟ ⎢ ⎥ ⎜⎟ ⎜⎟ + ∂ ∂ ⎢ ⎥ ∂∂ ∂∂ ⎝ ⎠ ⎝⎠ ⎝⎠ ⎣ ⎦

> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

⎛ ⎞ ⎛⎞ ∂ ∂ ∂∂∂ ∂∂∂

1 2 1 12 2

1 2 1 2 ( ) ( )

*<sup>d</sup> n n s s ss ss*

d

*<sup>n</sup> <sup>n</sup>*

d

( )

*k*

<sup>d</sup> +∞

⎡ ⎤ ⎛ ⎞⎛ ⎞ ∂∂∂ ⎢ ⎥ ⎜ ⎟⎜ ⎟ <sup>+</sup> ∂ ∂∂ ⎣ ⎦ ⎝ ⎠⎝ ⎠ <sup>∫</sup> 

ε

<sup>1</sup> d d

*<sup>d</sup> s s s ns s ns*

⎡ ⎤ ∂∂ ∂ ⎛ ⎞⎛ ⎞ ⎛ ∂ ∂ <sup>⎞</sup>

*<sup>g</sup> <sup>f</sup>*

*<sup>d</sup> <sup>n</sup> ab n ns s s s <sup>n</sup> t t*

 μ ερ <sup>∞</sup> −∞

ερ

∇

= − − − + ⋅ −⎡ − − + ⋅ ⎤− + ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

2 2

∫

G (1) (1) (1) 1 2 (1) (1)

+∞ −∞

 ψ

( ) ( )

2 2

2

*tt t t*

 ψψ

*t t* (1)

ψψ

 ε

− +− ⎜ ⎟ ⎜⎟ +

+∞ +∞ −∞ −∞

1 2

1 2 (1) (1) (1) (1) 1 2 21 ( )

*<sup>d</sup> <sup>n</sup> ss ss*

⎡ ⎤ ∂∂ ∂∂ ⎛⎞ ⎛⎞

G 1 (2) (1) (1) (1)

−∞

μ

μ

∂ ∂ ∂∂

∂ ∂∂∂

*n d <sup>n</sup> s ss*

⎛ ⎞ ∂ ∂∂

∫ ∫

Here, the diffusion process is proportional to the gradient of the chemical potential (Onsager, 1931a, 1931b), which yields the thermodynamic driving force as follows:

$$\begin{aligned} F'' &= -\rho \nabla \frac{\delta F}{\delta \vec{\eta}}\\ &= -\rho \nabla \mu \end{aligned} \tag{34}$$

The diffusion flux (*J*) is then represented by

$$J = -m\mathbf{C}\overline{\mathbf{V}}\mu\tag{35}$$

where, *m* and *C* are the mobility [m·mol/N·s] and mol concentration[mol/m3], respectively. In the present study, we assumed that the thermodynamic force (Eq. (34)) corresponds to *T*'' in Eq. (33) as follows:

$$-\varepsilon^{2}\mathbf{V}^{(2)} \cdot \mathbf{T}'' = \varepsilon^{2}\mathbf{F}'' \tag{36}$$

In Eq. (33), the order of ε is smaller than that of *O*(1) . Thus, the order of <sup>2</sup> ε is smaller than that of ε . Therefore, an interfacial phenomena characterized by <sup>2</sup> ε can be negligible compared to that of *O*(1) .

However, the driving force (Eq. (34)) characterized by small scales such as <sup>2</sup> ε will affect phenomena at large scales such as *O*(1) after enough time, even if there are differences between the scales in Eq. (33).

Considering Eqs. (33), (34), and (36) and performing a simple tensor analysis, such as the derivation of Eq. (10), gives

$$\begin{split} \rho \frac{\operatorname{D} \,\boldsymbol{\mu}}{\operatorname{D} \,t^{(0)}} + \varepsilon \rho \frac{\operatorname{D} \,\boldsymbol{\mu}'}{\operatorname{D} \,t^{(0)}} + \varepsilon^2 \rho \frac{\operatorname{D} \,\boldsymbol{\mu}''}{\operatorname{D} \,t^{(2)}} &= -\boldsymbol{\nabla}^{(0)} \cdot \boldsymbol{T} + \varepsilon \boldsymbol{\nabla}^{(0)} \cdot \left(f\_0(\boldsymbol{\nu})\boldsymbol{I}\right) \\ &- \varepsilon d \boldsymbol{\nabla}^{(0)} \psi \left(\mathbf{V}^{(0)} \cdot \boldsymbol{\nabla}^{(0)} \boldsymbol{\nu}\right) - \varepsilon^2 \rho \boldsymbol{\nabla}^{(2)} \boldsymbol{\mu} + \rho \mathbf{g}, \end{split} \tag{37}$$

Equation (37) is transformed into the interfacial coordinates system in a similar manner to that discussed in section 3.2. For the sake of simplicity, we show the transformation of the fourth term on the right-hand side of Eq. (37). Referring to Eq. (13) in section 3.2, the fourth term on the right-hand side of Eq. (37) is transformed as follows:

$$-\varepsilon^{2}\rho\nabla^{(2)}\mu \rightarrow -\varepsilon^{2}\rho\left(\mathbf{t}\_{1}\frac{\partial\mu}{\partial\mathbf{s}\_{1}^{(2)}} + \mathbf{t}\_{2}\frac{\partial\mu}{\partial\mathbf{s}\_{2}^{(2)}} + \mathbf{n}\_{1}\frac{\partial\mu}{\partial n^{(2)}}\right) \tag{38}$$

Integration of this equation over the interface gives

$$-\varepsilon^{2}\,\rho\int\_{-\alpha}^{\alpha}\left(\mathbf{t}\_{1}\frac{\partial\mu}{\partial\mathbf{s}\_{1}^{(2)}}+\mathbf{t}\_{2}\frac{\partial\mu}{\partial\mathbf{s}\_{2}^{(2)}}\right)\mathrm{d}n-\varepsilon^{2}\,\rho\mathbf{n}\_{1}\int\_{-\alpha}^{\alpha}\frac{\partial\mu}{\partial\mathbf{n}^{(2)}}\,\mathrm{d}n\tag{39}$$

In this calculation, chemical potential is assumed to be constant along the interface in tangential direction. Therefore, the first term in Eq. (39) is omitted. Thus, Equation (38) is calculated as follows:

Here, the diffusion process is proportional to the gradient of the chemical potential

ρ

<sup>δ</sup> ′′ = − <sup>δ</sup>

ρ μ

∇

∇μ

 ε

is smaller than that of *O*(1) . Thus, the order of <sup>2</sup>

= − *<sup>F</sup>* ∇

*J* = −*mC*

2 (2) <sup>2</sup> − ⋅=

phenomena at large scales such as *O*(1) after enough time, even if there are differences

Considering Eqs. (33), (34), and (36) and performing a simple tensor analysis, such as the

2 (0) (1)

∇

ε

*d*

∇

Equation (37) is transformed into the interfacial coordinates system in a similar manner to that discussed in section 3.2. For the sake of simplicity, we show the transformation of the fourth term on the right-hand side of Eq. (37). Referring to Eq. (13) in section 3.2, the fourth

> μ

1 2 (2) (2) L (2)

*ss n*

ε ρ

− →− + + ⎜ ⎟ ∂∂ ∂ ⎝ ⎠

 ∞ ∞ −∞ −∞ ⎛ ⎞ ∂∂ ∂

In this calculation, chemical potential is assumed to be constant along the interface in tangential direction. Therefore, the first term in Eq. (39) is omitted. Thus, Equation (38) is

ε

∇

. Therefore, an interfacial phenomena characterized by <sup>2</sup>

However, the driving force (Eq. (34)) characterized by small scales such as <sup>2</sup>

(0) (1) (2) 0

 ε ρ

term on the right-hand side of Eq. (37) is transformed as follows:

2 (2) 2

 ερ

2 2

μμ

1 2

 μ

*tt t*

ερ

Integration of this equation over the interface gives

ε ρ

∇

DD D ( ) DD D

*uu u T I*

′ ′′ + + =− ⋅ + ⋅

where, *m* and *C* are the mobility [m·mol/N·s] and mol concentration[mol/m3], respectively. In the present study, we assumed that the thermodynamic force (Eq. (34)) corresponds to *T*''

*F*

η

(34)

(35)

ε

ε

is smaller than

will affect

(37)

can be negligible

ε

,

*g*

 μ ρ

∇

*T F* ′′ ′′ (36)

( )

 μ

 μ

*ttn* (38)

 ψ

*f*

( )

∇

ε

∇ ∇

12 L (2) (2) (2) 1 2 *ss n*

⎛ ⎞ ∂∂ ∂

μ

d d *n n*

− +− ⎜ ⎟ ∂∂ ∂ ⎝ ⎠ ∫ ∫ *tt n* (39)

ψ

(1) (1) (1) 2 (2)

 ψ ερ

− ⋅− +

(Onsager, 1931a, 1931b), which yields the thermodynamic driving force as follows:

The diffusion flux (*J*) is then represented by

ε

 ερ

in Eq. (33) as follows:

In Eq. (33), the order of

compared to that of *O*(1) .

between the scales in Eq. (33).

derivation of Eq. (10), gives

ρ

calculated as follows:

ε

that of

$$-\varepsilon^{2}\,\rho\,\mathbf{V}^{(2)}\,\mu = -\varepsilon^{2}\,\rho\mathfrak{m}\_{\perp}\int\_{-\diamond}^{\diamond} \frac{\partial\mu}{\partial n^{(2)}}\,\mathrm{d}n\tag{40}$$

Finally, Equation (37) is transformed into the interfacial coordinates system as follows:

( G GG GG L LL LL G 1 2 ) ( ) ( ) ( ) ( ) ( ) 12 G (1) (1) (1) (1) (1) (1) (1) 1 2 1 12 2 ( ) 0 1 1 2 2 επ επ εκ κ σ σ σ ψψ ψψ ε ε = − − − + ⋅ −⎡ − − + ⋅ ⎤− + ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎛ ⎞ ⎛⎞ ∂ ∂ ∂∂∂ ∂∂∂ − +− ⎜ ⎟ ⎜⎟ + ∂ ∂ ∂ ∂∂ ∂ ∂∂ ⎝ ⎠ ⎝⎠ *a b c d P P <sup>d</sup> s s s ns s ns n n* τ *nn n* τ *tt n* ( ) (1) ( ) 2 2 3 G (1) (1) (1) 1 2 (1) (1) 1 2 1 2 ( ) ( ) d <sup>1</sup> d d 2 ψ ψ ψψ ε ε ψ ψ +∞ −∞ +∞ +∞ −∞ −∞ ⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎡ ⎤ ∂∂ ∂ ⎛ ⎞⎛ ⎞ ⎛ ∂ ∂ <sup>⎞</sup> + + ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ <sup>−</sup> <sup>+</sup> <sup>⎟</sup> <sup>−</sup> <sup>+</sup> <sup>⎟</sup> ∂∂ ∂ ⎢ ⎥ ∂ ∂ ⎝ ⎠⎝ ⎠ ⎝ <sup>⎠</sup> ⎣ ⎦ ∫ ∫ ∫ *e <sup>g</sup> <sup>f</sup> n <sup>d</sup> <sup>n</sup> ab n ns s s s <sup>n</sup> t t* ( ) 2 2 i i 1 2 (1) (1) <sup>1</sup> (1) (1) <sup>2</sup> (1) (1) 1 2 11 22 ( ) ( ) 12 1 2 (1) (1) 1 2 d d 2 π πε ψ ψ ε ψ ψψ εκ κ +∞ +∞ −∞ −∞ ⎡ ⎤ ⎛ ⎞ ∂ ∂ ∂∂ ∂∂ ⎛⎞ ⎛⎞ + ++ ⎜ ⎟ ⎢ ⎥ ⎜⎟ ⎜⎟ + ∂ ∂ ⎢ ⎥ ∂∂ ∂∂ ⎝ ⎠ ⎝⎠ ⎝⎠ ⎣ ⎦ ⎛ ⎞ ∂ ∂∂ ++ + ⎜ ⎟ ∂ ∂∂ ⎝ ⎠ ∫ ∫ *h i <sup>d</sup> n n s s ss ss <sup>d</sup> s s tt t t t t* (1) ( ) 2 2 1 2 (1) (1) (1) (1) 1 2 21 ( ) 2 G 1 (2) (1) (1) (1) 2 12 ( ) d d 2 d ε ψψ μ ψ ψ ε ρ ε +∞ −∞ +∞ −∞ ∞ −∞ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎡ ⎤ ∂∂ ∂∂ ⎛⎞ ⎛⎞ − + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂∂ ∂∂ ⎝⎠ ⎝⎠ ⎣ ⎦ ∂ ∂∂∂ − + ∂ ∂ ∂∂ ∫ ∫ ∫ *j l m <sup>n</sup> <sup>n</sup> <sup>d</sup> <sup>n</sup> ss ss n d <sup>n</sup> s ss t t n t* <sup>2</sup> (1) (1) (1) 1 12 ( ) <sup>d</sup> +∞ ψ ψ −∞ ⎡ ⎤ ⎛ ⎞⎛ ⎞ ∂∂∂ ⎢ ⎥ ⎜ ⎟⎜ ⎟ <sup>+</sup> ∂ ∂∂ ⎣ ⎦ ⎝ ⎠⎝ ⎠ <sup>∫</sup> *k <sup>n</sup> s ss <sup>t</sup>* (41)

Here, for the sake of simplicity, we focus on the terms (a), (b), (c), and (m) (this is new term.) as follows:

$$0 = \left(P\_{\rm G} - P\_{\rm L}\right) \mathfrak{n}\_{\rm G} - \mathfrak{n}\_{\rm G} \varepsilon \left(\kappa\_{\rm l} + \kappa\_{\rm z}\right) \sigma - \varepsilon^{2} \,\rho \mathfrak{n}\_{\rm G} \int\_{-\infty}^{\infty} \frac{\partial \mu}{\partial n^{(2)}} \mathrm{d}n \tag{42}$$

In this equation, the third term is transformed into

$$\int\_{-\infty}^{\infty} \frac{\partial \mu}{\partial n^{(2)}} \, \mathrm{d}n = \int\_{-\infty}^{\infty} \frac{\partial \mu}{\partial P} \frac{\partial P}{\partial n^{(2)}} \, \mathrm{d}n + \int\_{-\infty}^{\infty} \frac{\partial \mu}{\partial T} \frac{\partial T}{\partial n^{(2)}} \, \mathrm{d}n \tag{43}$$

In the present discussion, change in temperature is not considered. Thus, the second term on the right-hand side of Eq. (43) is omitted. Here, the thermodynamic relation is considered as follows:

$$\mathbf{d}\mu\_i = -\overline{\mathbf{s}}\_i \mathbf{d}T + \overline{V}\_i \mathbf{d}P + \sum\_{j}^{N\_{\cdot j}} \mu\_{ij}^{\mathcal{E}} \mathbf{d}n\_j \tag{44}$$

Macroscopic Gas-Liquid Interfacial Equation

( *H* ≠ 0 ). The mean curvature *H* = 0 in Eq. (47) gives

Thus, the subtraction of Eq. (48) from Eq. (47) is

zero. Finally, the following equation is obtained.

where 1/ *Vm* =

**5. Conclusion** 

ρ

multiphase flow equation.

Based on Thermodynamic and Mathematical Approaches 77

( ) () <sup>2</sup> <sup>L</sup> G LG G 1 2 G

=− − + − ⎜ ⎟ ⎝ ⎠

In this equation, *P*L and *P*G are the pressure in bulk phase. The Kelvin equation explains the difference in vapor pressure between the flat surface ( *H* = 0 ) and curved surface

> ( ) <sup>2</sup> 0L G LG G

⎛ ⎞ =− − ⎜ ⎟ ⎝ ⎠

( ) <sup>2</sup> L 0G

= +− ⎜ ⎟ ⎝ ⎠

( ) <sup>2</sup> <sup>L</sup> G12 G

= +− ⎜ ⎟ ⎝ ⎠

Considering the equilibrium state in Eq. (50), the coefficient of the normal vector is set to

*m i*

ε

In a previous study, a new interfacial model of the gas–liquid interface was developed based on thermodynamics, assuming that the interface has a finite thickness, similar to a thin fluid membrane. In particular, the free energy was derived based on a lattice-gas model that includes the electrostatic potential due to contamination. The free energy was incorporated into the NS equation by using the Chapman–Enskog expansion. Finally, a multi-scale multiphase flow equation was derived that characterizes the mesoscopic scale. The interfacial equation for a macroscopic-scale gas–liquid interface is characterized by a jump condition. In the present study, the jump condition at the gas–liquid interface treated by thermodynamics was derived by using the multi-scale multiphase flow equation and compared with the conventional jump condition. Finally, we developed the multi-scale gas– liquid interfacial model; this model supports the interfacial phenomena from the

1. The thermodynamic interfacial jump condition was derived by using the multi-scale multiphase flow equation. The present study indicated the relationship between the

*HV P T P*

L 0L

. This equation is the Kelvin equation derived from the multi-scale

*i*

⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠

*<sup>P</sup> PP T*

ε ρ

0 R ln *<sup>i</sup>*

0 R ln *i i*

In Eq. (49), we assume *P P i i* G 0G ≈ and focus on the liquid phase. Equation (49) becomes

0 R ln *<sup>i</sup>*

<sup>2</sup> ln R

σ

microscopic to macroscopic scale theoretically to give the following results:

G12 G

εκ κ σ ερ

> εκ κ σ ερ

*<sup>P</sup> P P <sup>T</sup>*

0 R ln *<sup>i</sup>*

εκ κ σ ερ

G

*i*

⎛ ⎞

*P*

*n n* 48)

*n n n* (47)

0G

G 0L

0L

(51)

*n n* (50)

*i <sup>P</sup> <sup>T</sup> P*

⎛ ⎞

*i i P P <sup>T</sup> P P*

⎛ ⎞

*n n* (49)

*i*

*P*

Then, the following equation is obtained by considering Eqs. (46) and (42).

Fig. 9. Multi-scale schematic around the interface region

In this equation, *is* and *Vi* are the enthalpy and volume per unit mol of component *i*. In the derivation, adding or reducing new components from the external system is not considered. Thus, d 0 *<sup>j</sup> n* = . The following equation is derived from Eq. (44) under constant temperature.

$$\begin{aligned} \left(\frac{\partial \mu\_i}{\partial P}\right)\_{T, u\_j} &= \overline{V}\_i \\ &= \frac{\mathbf{R}T}{P\_i} \end{aligned} \tag{45}$$

Then, the substitution of Eq. (45) into the third term in Eq. (42) by considering Eq. (43) and transformation of the equation yields

$$\begin{aligned} \int\_{-\infty}^{\infty} \frac{\partial \mu}{\partial n^{(2)}} \, \mathrm{d}n &= \int\_{-\infty}^{\infty} \frac{\partial \mu}{\partial P} \frac{\partial P}{\partial n^{(2)}} \, \mathrm{d}n \\ &= \int\_{-\infty}^{\infty} \frac{\partial \mu}{\partial P} \mathrm{d}P \\ &= \int\_{P\_{\mathrm{G}}}^{P\_{\mathrm{L}}} \frac{\mathrm{R}T}{P\_{\mathrm{i}}} \, \mathrm{d}P \\ &= \mathrm{R}T \ln \left(\frac{P\_{\mathrm{i}\mathrm{I}}}{P\_{\mathrm{i6}}}\right) \end{aligned} \tag{46}$$

Gas

Liquid

(a) Macroscopic scale

Fig. 9. Multi-scale schematic around the interface region

transformation of the equation yields

Gas

Interface region

<sup>2</sup> ∼ *O*( ) ε

In this equation, *is* and *Vi* are the enthalpy and volume per unit mol of component *i*. In the derivation, adding or reducing new components from the external system is not considered. Thus, d 0 *<sup>j</sup> n* = . The following equation is derived from Eq. (44) under constant temperature.

,

*P*

μ

∫ ∫

∞ ∞ −∞ −∞

⎛ ⎞ ∂μ = ⎜ ⎟ ⎝ ⎠ ∂

*j <sup>i</sup> <sup>i</sup> T n*

Then, the substitution of Eq. (45) into the third term in Eq. (42) by considering Eq. (43) and

L G

*i i P <sup>P</sup> <sup>i</sup>*

R ln

L G

*i i*

(2) (2)

∂ ∂∂ <sup>=</sup> ∂ ∂ <sup>∂</sup>

=

∞ −∞

<sup>∂</sup> <sup>=</sup> <sup>∂</sup>

∫

*<sup>P</sup> n n n n <sup>P</sup>*

d d

 μ

d

μ

*P P <sup>T</sup> <sup>P</sup> P*

<sup>R</sup> <sup>d</sup>

*<sup>P</sup> <sup>T</sup> P*

⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠

R

=

*V*

*i*

*T P*

∼ *O*(1)

Liquid

(a) Mesoscopic scale

Bulk

Bulk

(45)

∫ (46)

δ ( ∼ *O*( ) ε) Then, the following equation is obtained by considering Eqs. (46) and (42).

$$0 = \left(P\_{\rm G} - P\_{\rm L}\right) \mathfrak{n}\_{\rm G} - \mathfrak{n}\_{\rm G} \varepsilon \left(\kappa\_{\rm l} + \kappa\_{\rm 2}\right) \sigma - \varepsilon^2 \rho \mathfrak{n}\_{\rm G} \mathbb{R}T \ln\left(\frac{P\_{\rm L}}{P\_{\rm G}}\right) \tag{47}$$

In this equation, *P*L and *P*G are the pressure in bulk phase. The Kelvin equation explains the difference in vapor pressure between the flat surface ( *H* = 0 ) and curved surface ( *H* ≠ 0 ). The mean curvature *H* = 0 in Eq. (47) gives

$$0 = \left(P\_{\rm G} - P\_{\rm L}\right) \mathfrak{n}\_{\rm G} - \varepsilon^2 \rho \mathfrak{n}\_{\rm G} \mathbb{R}T \ln\left(\frac{P\_{i0\rm L}}{P\_{i0\rm G}}\right) \tag{48}$$

Thus, the subtraction of Eq. (48) from Eq. (47) is

$$0 = \mathfrak{n}\_{\rm G} \varepsilon \left(\kappa\_{\rm l} + \kappa\_{\rm 2}\right) \sigma - \varepsilon^2 \rho \mathfrak{n}\_{\rm G} \mathbb{R}T \ln\left(\frac{P\_{\rm iL}}{P\_{i\rm G}} \frac{P\_{i0\rm G}}{P\_{i0\rm L}}\right) \tag{49}$$

In Eq. (49), we assume *P P i i* G 0G ≈ and focus on the liquid phase. Equation (49) becomes

$$0 = \mathfrak{n}\_{\mathbb{G}} \varepsilon \left(\kappa\_{\mathbb{I}} + \kappa\_{\mathbb{Z}}\right) \sigma - \varepsilon^2 \rho \mathfrak{n}\_{\mathbb{G}} \mathbb{R}T \ln\left(\frac{P\_{\text{IL}}}{P\_{i0\mathbb{L}}}\right) \tag{50}$$

Considering the equilibrium state in Eq. (50), the coefficient of the normal vector is set to zero. Finally, the following equation is obtained.

$$\frac{2H\sigma V\_m}{\mathbf{R}T} = \varepsilon \ln\left(\frac{P\_{i\text{L}}}{P\_{i0\text{L}}}\right) \tag{51}$$

where 1/ *Vm* = ρ . This equation is the Kelvin equation derived from the multi-scale multiphase flow equation.
