Transport phenomena in the Knudsen layer near an evaporating surface

Using the combination of the kinetic theory of gases (KTG), Boltzmann transport equation (BTE), and molecular dynamics (MD) simulations, we study the transport phenomena in the Knudsen layer near a planar evaporating surface. The MD simulation is first used to validate the assumption regarding the anisotropic velocity distribution of vapor molecules in the Knudsen layer. Based on this assumption, we use the KTG to formulate the temperature and density of vapor at the evaporating surface as a function of the evaporation rate and the mass accommodation coefficient (MAC), and we use these vapor properties as the boundary conditions to find the solution to the BTE for the anisotropic vapor flow in the Knudsen layer. From the study of the evaporation into a vacuum, we show the ratio of the macroscopic speed of vapor to the most probable thermal speed of vapor molecules in the flow direction will always reach the maximum value of $\sqrt{1.5}$ at the vacuum boundary. The BTE solutions predict that the maximum evaporation flux from a liquid surface at a given temperature depends on both the MAC and the distance between the evaporating surface and the vacuum boundary. From the study of the evaporation and condensation between two parallel plates, we show the BTE solutions give good predictions of transport phenomena in both the anisotropic vapor flow within the Knudsen layer and the isotropic flow out of the Knudsen layer. All the predictions from the BTE are verified by the MD simulation results.

Figure 1

The schematic diagram of the one-dimensional (1D) steady-state evaporation of a monoatomic liquid to its own vapor. $v_a$ is the average local velocity of vapor evaporated from the liquid surface.

Figure 2

The dimensionless (a) density, (b) temperature, and (c) molar flux of vapor at the evaporating surface as a function of dimensionless macroscopic velocity of vapor at the evaporating surface for different values of MAC.

Figure 3

(Top panel) A snapshot of the model system for the MD study of evaporation of liquid Ar on an Au surface at $T_h=85\mathrm{K}$ into a vacuum, and (bottom panel) the temperature profile in the vapor flow direction. The uncertainty of $T_x$ and $T_y$ is smaller than the size of symbols.

Figure 4

The (a) molar, (b) momentum, and (c) energy flux in each bin of the vapor region. The horizontal dashed lines show the average value of molar, momentum and energy flux in the vapor region. The uncertainties are smaller than the size of symbols.

Figure 5

The velocity distribution of vapor molecules in the second bin (∼18 nm from the evaporating surface) in the vapor region. The red circles and blue diamonds are MD simulation results for the velocity components along the vapor flow direction and perpendicular to the flow direction, respectively. The dashed line is the Maxwell velocity distribution (MVD) of Ar molecules at $T=69\mathrm{K}$. The solid line is the shifted MVD (SMVD) of Ar molecules at $T=51\mathrm{K}$.

Figure 6

(Top panel) A snapshot of the model system for the MD simulation of evaporation of liquid Ar on a hot Au surface at $T_h=85\mathrm{K}$ and condensation of vapor Ar on a cold Au surface at $T_l=35\mathrm{K}$, and (bottom panel) the temperature profile in the vapor flow direction. The uncertainty of $T_x$ and $T_y$ is smaller than the size of symbols.

Figure 7

(a) Dependence of $T_x$ and $T_y$ on $t$ obtained from EMD simulation of vapor Ar with a density of ${\rho }_v=0.0636\mathrm{mol}/\mathrm{L}$ and an initial temperature of $T_{x,i}=51.5\mathrm{K}$ and $T_{y,i}=68.5\mathrm{K}$. The inset shows a linear fit of $\ln \left[\left(T_x\left(t\right)\text{–}{T}_e\right)/\left(T_{x,i}\text{–}{T}_e\right)\right]$ vs. $t$. (b) The dependence of molecular collision frequency, 1/τ, on ${\rho }_v\sqrt{T_e}$ obtained from MD simulations. The dashed line shows a trend line for the linear dependence between 1/τ and ${\rho }_v\sqrt{T_e}$.

Figure 8

The distributions of (a) and (b) temperature, (c) density, and (d) dimensionless macroscopic velocity of vapor near an evaporating liquid Ar surface (located at $x=0$) and at a surface temperature of $T_L=82.8\mathrm{K}$. The results are the prediction from the four-moment solution to the BTE. $v_{R,0}$ is the $v_R$ of vapor at $x=0$, i.e., the liquid-vapor interface.

Figure 9

(Top panel) A snapshot of the model system for the study of evaporation of liquid Ar at $T_L=82.8\mathrm{K}$ into a vacuum. The vacuum boundary is 103 nm from the evaporating surface. (Bottom panels) The (a) temperature, (b) density, and (c) dimensionless macroscopic velocity in the vapor region. The scatters are MD simulation results. The lines are predictions from the BTE.

Figure 10

Same as described in the caption of Fig. 9 except that the temperature at the evaporating surface is increased to $T_L=90.5\mathrm{K}$.

Figure 11

The predictions from the BTE and the MD model on $v_{R,0}$ and $J_{\max }$ as a function of $L_{\mathrm{vap}}$ for the case of evaporation of liquid Ar at $T_L=82.8\mathrm{K}$ into a vacuum. The top horizontal dash-dot line indicates $\alpha {J_L}^{+}$, i.e., the molar flux of molecules that are emitted from the liquid surface and change to vapor phase. The bottom dash-dot line indicates $v_{R,0}$($L_{\mathrm{vap}}\to \infty$), i.e., the $v_{R,0}$ predicted by the KTG shown in Table 1.

Figure 12

(Top panel) A snapshot of the model system for the study of evaporation and condensation of fluid Ar between two parallel plates. The separation between the evaporating and the condensing surfaces is 106 nm. (Bottom panels) The (a) temperature, (b) density, and (c) dimensionless macroscopic velocity in the vapor region. The scatters are MD simulation results. The lines are predictions from the BTE.

Figure 13

(a) The velocity distribution of vapor molecules in the second bin of the vapor region. The red circles and blue diamonds are MD simulation results for the velocity components along the vapor flow direction and perpendicular to the flow direction, respectively. The dashed line is the Maxwell velocity distribution (MVD) of Ar molecules at $T=71\mathrm{K}$. The solid line is the shifted MVD (SMVD) of Ar molecules at $T=61\mathrm{K}$. (b) $R_{\mathrm{JME}}$ vs. $J_{\mathrm{net}}$ at the evaporating and condensing surfaces.

Figure 14

Same as described in the caption of Fig. 12 except that the length of vapor region is doubled.