Numerical and Theoretical Analysis of Sessile Droplet Evaporation in a Pure Vapor Environment

Abstract

The evaporation of sessile droplets is not only a common occurrence in daily life, but it also plays a vital role in many scientific and industrial fields. However, most of the current research is focused on the evaporation of droplets in the air environment, where vapor transport is controlled by the diffusion model, but when the droplet evaporation is in its own pure vapor environment, the above model will no longer apply, and the evaporation will be dominated by kinetic theory. Thus the Hertz–Knudsen model can be applied to describe the evaporation kinetics. However, in most of the studies, it is assumed that the temperature distribution is uniform along the vapor-liquid interface of the droplet, but due to the evaporative cooling effect, this assumption is not correct in actual evaporation. In this paper, theoretical analysis and numerical simulation were combined to study the characteristics of droplet evaporation with multiphysics coupling. In the theoretical model, heat conduction in the droplet and substrate was coupled with vapor transport at the droplet surface. In the numerical simulation, internal thermocapillary flow and heat transfer of the droplet were coupled with vapor transport at the droplet surface. The effects of contact angle, thermocapillary convection, ambient pressure ratio, and substrate superheat on the droplet evaporation characteristics were quantitatively analyzed. It was found that the high substrate superheat or low ambient pressure ratio will enhance the droplet thermocapillary convection as well as evaporation rate. Furthermore, a critical contact angle was found; below this value, the droplet evaporation rate was inversely proportional to the contact angle, but upon this value, the trend was reversed. These findings have important implications for revealing the physical mechanism of kinetics-controlled droplet evaporation in a pure vapor environment.

1. Introduction

The evaporation of the sessile droplet is a common phenomenon in scientific research and industrial production [1], and it is widely used in inkjet printing [2], spray cooling [3,4], material fabrication [5,6], disease diagnostics [7], etc. Most of current research is focused on the evaporation of droplets in the air environment and the vapor transport outside the droplet that is diffusion-controlled [8,9,10]. For sessile droplets on the substrate, two extreme evaporation modes were experimentally found [11], namely, the constant contact radius (CCR) mode and the constant contact angle (CCA) mode. In the CCR mode, during the evaporation of the droplet, the contact radius remains unchanged, and the contact angle decreases from the initial contact angle to 0. In the CCA mode, the contact angle remains unchanged, and the contact radius decreases from the initial contact radius to 0. So far, extensive studies have been carried out for diffusion-controlled droplet evaporation through theoretical analysis and numerical simulation [8,9,10].

However, in some applications, the diffusion-controlled droplet evaporation is not applicable, such as in desalination systems [12] and vacuum spray flash evaporation cooling systems (VSFEC) for the thermal control of spacecraft [13], etc. In order to reduce vapor diffusion resistance and to improve system efficiency, non-condensable gases need to be removed so that the droplets evaporate in a pure vapor environment; thus, the vapor transport will be controlled by the kinetic theory of gases (KTG) [14] instead of the diffusion-controlled model. The Hertz–Knudsen model is usually adopted to describe evaporation kinetics, in which the velocity distribution of the vapor molecules at the droplet surface is assumed to satisfy the Maxwell distribution [15]. The evaporation flux is calculated according to the difference between the vapor molecule number escaping from the gas-liquid interface and the gas molecule number entering the liquid from the surrounding vapor. Since the seminal contributions of Hertz [16], Knudsen [17], and Bond et al. [18], the importance of kinetic models has gradually been recognized. Kryukov et al. [19] used a diffusion model and a kinetic model to analyze the diesel fuel droplet evaporation problem, respectively, and they proposed that the kinetic model predicted a longer droplet evaporation time than the diffusion model. Similarly, Pati et al. [20] also studied the effects of the diffusion model and kinetic model on droplet evaporation; they proposed that the lifetime of the droplet in the kinetic model was longer than that of the diffusion model. For a droplet with an initial radius of 20 μm, both models underestimated the lifetime of evaporation; for droplets with an initial radius of 5 μm, the diffusion model underestimated the droplet lifetime of evaporation, and the kinetic model overestimated it.

In the studies, the temperature along the vapor-liquid interface of the droplet is always assumed to be uniform when the kinetic model is applied to analyze the droplet evaporation [16,17,18,19,20]. The heat transfer inside the droplet and the mass transfer at the interface, due to the phase change, are not considered. Recently, Zhang et al. [21] used numerical simulation to analyze the effect of internal thermocapillary convection on evaporation when the sessile droplet evaporated into its own pure vapor environment; they found that thermocapillary convection was enhanced as the substrate temperature increased and the ambient pressure ratio decreased. It was also found that, at the droplet contact angle ${\theta }_{\mathrm{c}}>60°$, the evaporation rate increased with the increasing contact angle, but at contact angle ${\theta }_{\mathrm{c}}<60°$, the evaporation rate decreased with the increasing contact angle. The mechanism behind this result was not revealed. In addition, the effect of the substrate on the droplet evaporation was also neglected. Therefore, in order to gain insight into the evaporation of the sessile droplet in its own vapor environment, the multiphysics coupled model needs to be established. In this paper, theoretical analysis was carried out to study the droplet evaporation at the contact angle ${\theta }_{\mathrm{c}}<40°$ with a simplified one-dimensional model. The heat transfer in the droplet and substrate was coupled with the vapor transport at the interfacial vapor-liquid interface. In order to prove its feasibility, a fully coupled numerical simulation was carried out. The effects of the ambient pressure ratio, substrate superheat, contact angle, and thermocapillary convection on droplet evaporation were quantitatively analyzed. Furthermore, an adaptive algorithm was adopted to calculate the droplet evaporation lifetimes in both the CCR and CCA modes. Finally, the work can provide guidance for kinetics-controlled droplet evaporation in pure vapor environments.

2. Mathematical Model

2.1. Physical Model

Figure 1 shows the schematic diagram of droplet evaporation on a hot substrate. The droplet radius $R_{\mathrm{c}}$ is considered to be much smaller than its capillary size (it is 2.7 mm for water) [22], hence the effect of gravity on its shape is negligible, and the surface tension keeps the droplet in the shape of spherical cap, so the two-dimensional axisymmetric cylindrical coordinate system (r, z) is adopted.

2.2. Theoretical Analysis

The ambient temperature and pressure in the vapor environment are $T_{\infty },P_{\infty }$, respectively. The bottom temperature of the substrate is $T_{\mathrm{s}}$ and the height is h_s. According to the geometric characteristics of the spherical cap shape, the droplet surface $S=\{h\left(r\right)|r\le R_c\}$ satisfies

$$h(r)=\sqrt{\frac{R_{\mathrm{c}}{}^2}{{\sin }^2({\theta }_{\mathrm{c}})}-r^2}-\frac{R_{\mathrm{c}}}{\tan ({\theta }_{\mathrm{c}})}$$

(1)

where h(r) represents the droplet height.

Hu and Larson [23] used the lubrication theory to derive the internal velocity field of sessile evaporating droplets and proposed that when the droplet contact angle $0<{\theta }_{\mathrm{c}}\le 40°$, the center height is much smaller than the droplet contact radius $h(0)\ll R_{\mathrm{c}}$, the spherical cap can be assumed as flat, and the internal heat transfer can be simplified to one-dimensional heat conduction from the droplet bottom to the surface. The thermal resistances in substrate and droplet are ${\Re }_{\mathrm{s}}$, ${\Re }_{\mathrm{l}}$, respectively, where ${\Re }_{\mathrm{s}}=h_{\mathrm{s}}/k_{\mathrm{s}}$; ${\Re }_{\mathrm{l}}=h(r)/k_{\mathrm{l}}$. Through the substrate and the droplet, the heat flux can be expressed as

$$Q=\frac{T_{\mathrm{s}}-T_i(r)}{{\Re }_{\mathrm{s}}+{\Re }_{\mathrm{l}}}$$

(2)

where $T_i(r)$ is the local temperature at the droplet surface with a distance r in the radial direction; $k_{\mathrm{s}}$ and $k_{\mathrm{l}}$ are the substrate and droplet thermal conductivities, respectively.

At the vapor-liquid interface, the local evaporation flux is calculated by the Hertz-Knudsen equation [15]

$$J(r)=\frac{\sigma }{\sqrt{2\pi R}}\left(\frac{P_i(r)}{\sqrt{T_i(r)}}-\frac{\eta P_{\propto }}{\sqrt{T_{\propto }}}\right)$$

(3)

where $\sigma$ is the accommodation coefficient, and the average and minimum values of $\sigma$ for water are 0.5 and 0.04 [16,17], respectively. In this paper, 0.5 was adopted. R is the gas constant of vapor, $\eta$ is the ambient pressure ratio, which is the ratio of the actual pressure of the surrounding environment to the saturation value corresponding to the temperature $T_{\infty }$, and $P_i\left(r\right)$ is the saturation pressure at the local temperature $T_i\left(r\right)$ at the vapor-liquid interface.

According to the Antoine equation, the saturation pressure for water vapor is:

$$\ln \left(P_i\left(r\right)\right)=9.3876-\frac{3826.36}{T_i\left(r\right)-45.47}$$

(4)

At the vapor-liquid interface of droplet, the heat conduction from the droplet to the interface is equal to the heat absorbed by evaporation

$$\overrightarrow{n}\cdot (k_{\mathrm{l}}\nabla T_{\mathrm{l}})=J(r)L$$

(5)

where L represents the latent heat of phase change.

Combining the above equations, the droplet surface temperature can be obtained by:

$$T_s-T_i(r)=\frac{L\sigma }{\sqrt{2\pi R}}\left(\frac{P_i(r)}{\sqrt{T_i(r)}}-\frac{\eta P_{\propto }}{\sqrt{T_{\propto }}}\right){\left(1-{\left(\frac{r}{R_{\mathrm{c}}}\right)}^2{\sin }^2({\theta }_{\mathrm{c}})\right)}^{-0.5}\frac{R_{\mathrm{c}}}{k_l}(\frac{h_s}{R_c}\frac{k_l}{k_s}+\frac{h\left(r\right)}{R_{\mathrm{c}}})$$

(6)

Combining Equations (3) and (6), the local evaporation flux at the droplet interface can be obtained. However, since the analytical solution of evaporation flux is not available in the open literature, the evaporation rate cannot be obtained by integrating the evaporation flux over the area of the interface, as in the diffusion-controlled model. Hereby, the numerical integration was adopted to calculate the overall evaporation rate

$$-\stackrel{·}{m}(t)={\displaystyle \underset{S}{\int }J(r)dS}=\underset{n\to \propto }{\lim }\frac{1}{n}{\displaystyle \sum _{i=1}^{n}J(r_i})2\mathsf{\pi }{r}_i{R}_{\mathrm{c}}{}^2{\left(1-r_i{}^2{\sin }^2({\theta }_{\mathrm{c}})\right)}^{-0.5}$$

(7)

where 0 = r₀ < r₁ < r₂ < r_n = 1. In order to ensure that the numerical result of the evaporation rate is independent of n, the step-length independence study is conducted and it is considered as convergent when the deviation is satisfied.

$$\epsilon =\left|\frac{\stackrel{·}{m}|{}_n-\stackrel{·}{m}|{}_{n-1}}{\stackrel{·}{m}|{}_n}\right|\le 0.01$$

(8)

2.3. Numerical Simulation

Numerical simulation was carried out to verify the theoretical results and to prove the applicability range for the simplified theoretical model. The following assumptions were made:

(1)

As compared with the droplet evaporation lifetime, the time scales for vapor and heat transfer are orders of lower magnitude, so the droplet evaporation is assumed to be in a quasi-steady state [8,9,10].

(2)

The flow inside of a droplet is the incompressible laminar flow of a Newtonian fluid [8,22]. In addition, the physical properties of the droplet remain unchanged, except for the surface tension at the vapor-liquid interface, which is inversely proportional to temperature [9].

2.3.1. Numerical Model

The governing equation of the evaporating droplet can be expressed by the following equations

Continuity equation:

$$\nabla \cdot \overrightarrow{u}=0$$

(9)

Momentum equation:

$$\rho (\overrightarrow{u}\cdot \nabla )\overrightarrow{u}=\nabla \cdot [-P\overrightarrow{I}+\mu (\nabla \overrightarrow{u}+{(\nabla \overrightarrow{u})}^T)]$$

(10)

where $\overrightarrow{u}$, P, $\overrightarrow{I}$, $\mu$ are the fluid velocity, fluid pressure, identity tensor, and fluid dynamic viscosity, respectively.

Inside the droplet, the buoyancy is neglected due to little density variation over the temperature range.

Energy equation inside the evaporating droplet:

$$\rho C_p(\overrightarrow{u}\cdot \nabla )T=\nabla \cdot (k\nabla T)$$

(11)

2.3.2. Boundary Conditions

At the droplet vapor-liquid interface, the temperature distribution is not uniform, so thermocapillary convection will occur. The liquid shear stress and the thermocapillary force are balanced along the droplet surface as

$${\overrightarrow{\overrightarrow{\tau }}}_{\mathrm{l}}\cdot \overrightarrow{t}={\sigma }_{\mathrm{T}}{\nabla }_{\mathrm{s}}T$$

(12)

where ${\overrightarrow{\overrightarrow{\tau }}}_{\mathrm{l}}$, $\overrightarrow{t}$, ${\sigma }_{\mathrm{T}}$, and ${\nabla }_{\mathrm{s}}$ are the stress tensor of the liquid phases, the tangential vector along the droplet surface, the surface tension coefficient, and the surface gradient operator.

According to the Hertz-Knudsen-Schrage model [21], the evaporation flux along the droplet surface is:

$$J\left(r\right)=\frac{2\sigma }{2-\sigma }\sqrt{\frac{1}{2\pi R}}\left(\frac{P_i(r)}{\sqrt{T_i(r)}}-\frac{\eta P_{\propto }}{\sqrt{T_{\propto }}}\right)$$

(13)

At the vapor-liquid interface, the energy equation is the same as Equation (5)

$$\overrightarrow{n}\cdot (k_{\mathrm{l}}\nabla T_{\mathrm{l}})=J(r)L$$

(14)

At the liquid-solid interface (r, 0), the temperature is kept constant and the non-slip boundary condition is applied:

$$T(r,0)=T_{\mathrm{s}}; u_r=u_z=0$$

At the axis of symmetry (0, z), symmetric boundary condition is applied:

$$\frac{\partial u_z}{\partial r}=0, u_r=0, \frac{\partial T}{\partial r}=0,$$

At infinite distance from the droplet, the ambient temperature and pressure are $T=T_{\infty }$, $P=\eta P_{\mathrm{sat}}\left(T_s\right)$.

2.3.3. Grid Independence Study

COMSOL Multiphysics 5.5 was adopted to perform the numerical simulations. Unstructured triangular mesh was used, and fine mesh was used at the vapor-liquid interface and the three-phase line. The grid independence study was carried out for droplet evaporation at the contact angle ${\theta }_{\mathrm{c}}=90°$, contact radius $R_{\mathrm{c}}=1 \mathrm{mm}$, and substrate superheat $\mathsf{\Delta }T=7 \mathrm{K}$, and the results are shown in

Table 1. Grid independence analysis.

Grid Quantity	Evaporative Rate (kg/s)	Deviation (%)
5706	7.15 × 10−8
9796	7.05 × 10−8	1.3529
14,570	6.97 × 10−8	1.2196
28,293	6.95 × 10−8	0.2214
56,924	6.94 × 10−8	0.0993

.

It can be seen that, when the grid number is 28,293, the relative deviation is less than 0.1%; thus, this grid density was adopted in our numerical simulation. Under this density, the grid number ranged from 12,000 to 54,000, depending on the cases. In our study, the ambient temperature was $T_{\infty }=25 °\mathrm{C}$, and the droplet contact radius was $R_{\mathrm{c}}=1 \mathrm{mm}$.

3. Result and Discussion

3.1. Model Validation

To verify the accuracy of the theoretical and numerical models, the interfacial evaporative flux and temperature in both models were compared with numerical data by Zhang et al. [21] at the initial contact angle ${\theta }_0=43.6°$ and at the substrate temperature 298.15 K. From Figure 2, it can be seen that the theoretical and numerical results were in good agreement with their results, indicating that our theoretical and numerical models were reliable.

In addition, it was found that interfacial temperature will increase with the increasing pressure ratio, but the interfacial evaporation flux will decrease, because with the increasing pressure ratio, the pressure difference will be reduced between the saturation pressure at the interface and ambient pressure, according to the kinetic theory of evaporation.

3.2. Theoretical Results

The variation of interfacial temperatures and evaporative flux along the droplet surface under different contact angles is shown in Figure 3, at the substrate temperature $T_{\mathrm{s}}=305 \mathrm{K}$ and ambient temperature $T_{\infty }=298 \mathrm{K}$. It can be found that, with the increasing radial distance, both the interfacial temperature and evaporation flux increased gradually first and increased sharply near the contact line. Because the distance between the interface and the substrate increased with the increasing radial distance, the thermal resistance between the substrate and droplet surface was reduced, resulting in a higher interface temperature. Meanwhile, the interfacial temperatures and evaporative flux decreased with the increasing contact angle, because at low contact angles, the heat transfer distance was reduced from the substrate to the droplet interface, resulting in a larger evaporation flux.

Figure 4 shows the effect of the contact angle on the overall evaporation rate under different pressure ratios and substrate superheats. It was found that the overall evaporation rate decreased with the increasing contact angle. As the contact angle increased from $10°$ to $40°$, the evaporation rate was reduced by almost two times. The evaporative rate was large at the higher superheat and lower pressure ratio because, according to the Hertz-Knudsen equation, when the ambient pressure is lower, there are fewer vapor molecules around the droplet, so evaporation is enhanced due to low resistance to interfacial evaporation.

3.3. Numerical Results

When the droplet evaporated, the evaporative cooling effect at the vapor-liquid interface made the temperature distribution of the interface uneven, resulting in thermocapillary forces along the surface caused by temperature-dependent surface tension. The thermocapillary convection has great influence on temperature and velocity fields, evaporative rate, and lifetime.

3.3.1. Temperature and Velocity Fields

The temperature field and velocity field inside the evaporating droplet at the contact angles of $40°$ and $90°$ are shown in Figure 5, and the ambient pressure ratio and substrate superheat are $\eta =1$ and $\mathsf{\Delta }T=7 \mathrm{K}$, respectively. Because the surface tension was inversely proportional to temperature, the surface tension was lower near the contact line due to the higher temperature. Thus, the surface tension gradient along the droplet interface was formed, leading to a counter-clockwise thermocapillary flow. Meanwhile, it can also be seen that, near the axis and at the vapor-liquid interface, the fluid velocity was larger, resulting in a larger deformation of the local isotherm lines. Furthermore, it is worth noting that the thermocapillary convection inside the droplet was enhanced with the increasing contact angle. For example, at the contact angle ${\theta }_{\mathrm{c}}=40°$, the maximum value of the velocity inside the droplet was 0.035 m/s, but at the contact angle ${\theta }_{\mathrm{c}}=90°$, the velocity was twice higher.

3.3.2. Evaporative Rate

The effect of thermocapillary convection on the evaporation rate at different pressure ratios and contact angles is shown in Figure 6, at the substrate superheat $\mathsf{\Delta }T=7 \mathrm{K}$. When the thermocapillary convection inside the droplet was ignored, the droplet evaporation rate decreased with the increasing contact angle because, with the increasing distance from the substrate to the droplet for increasing contact angles, the interface temperature decreased, resulting in a reduction in the evaporation rate. However, when thermocapillary convection was considered, there were two trends in the evaporation rate with the increasing contact angle. First, the evaporation rate would gradually decrease, and, after reaching the minimum value at the critical contact angle around ${\theta }_{\mathrm{cri}}=40°$, it would gradually increase. At the contact angle ${\theta }_{\mathrm{c}}<{\theta }_{\mathrm{cri}}$, the heat conduction path was the main factor affecting the evaporation rate, and the low contact angle resulted in low heat transfer resistance between the substrate and the droplet, resulting in a larger evaporation rate. However, at the contact angle ${\theta }_{\mathrm{c}}>{\theta }_{\mathrm{cri}}$, the evaporation area and thermocapillary convection were the main factors affecting the evaporation rate. With the increasing contact angle, the thermocapillary convection inside the droplet gradually strengthened, and the evaporation area also increased gradually, and, combined with the above reasons, the evaporation rate also increased.

The variation of interfacial temperature and evaporation flux at contact angles $40°$ and $90°$ is shown in Figure 7. It can be seen that, when thermocapillary convection was ignored, due to the effect of thermal resistance, interfacial temperature and evaporative flux at the contact angle of 40° was larger than 90°, and it was larger than the effect of the evaporation area on evaporation, so that the evaporation rate decreased with the increasing contact angle. However, when thermocapillary convection is considered, the thermocapillary convection increased with the increasing contact angle, so that at large angles, thermocapillary convection promoted evaporation more strongly, and in addition to this, it also increased the evaporation area. Combining the above reasons, the evaporation rate at the contact angle of 90° was larger than 40°.

The effect of the pressure ratio and contact angle on the evaporation rate is shown in Figure 8. It can be seen that, when the pressure ratio was constant, with the increase of the contact angle, the evaporation rate first decreased and then increased, which is similar to Figure 6. However, when the contact angle was constant, with the change of the pressure ratio, the change of the evaporation rate was not obvious, especially for the large contact angle. For example, when the contact angle ${\theta }_{\mathrm{c}}=20°$, the ambient pressure ratio increased from 0.92 to 1, and the evaporation rate decreased by 18%, but when the contact angle ${\theta }_{\mathrm{c}}=120°$, the evaporation rate only decreased by 9%.

3.3.3. Lifetime of Droplet Evaporation

With the adaptive algorithm [24] on quasi-steady droplet evaporation, the lifetimes of the evaporative droplet were studied in the CCA and CCR modes.

Figure 9 shows the evolution of the contact angle with time in the CCR mode under different substrate superheats and ambient pressure ratios at the initial contact angles 40°, 90°, and 120°, respectively. The initial droplet volume was set to $2.09 \mathsf{\mu }\mathrm{L}$, which was calculated based on the droplet with contact radius $R_{\mathrm{c}}=1 \mathrm{mm}$ and the contact angle ${\theta }_{\mathrm{c}}=90°$. It can be found that the evaporative lifetime will be short at low contact angles under the same substrate superheat and pressure ratio. For example, at the pressure ratio $\eta =1$ and substrate superheat $\mathsf{\Delta }T=7 \mathrm{K}$, the evaporation lifetime was about 45 s at contact angle ${\theta }_{\mathrm{c}}=120°$, while the evaporation lifetime was reduced by three times at contact angle ${\theta }_{\mathrm{c}}=40°$.

The effect of pressure ratio and substrate superheat on the evaporative lifetime is shown in Figure 10, where the volume and droplet angle are fixed values, $V_0=2.09 \mathsf{\mu }\mathrm{L}$ and ${\theta }_{\mathrm{c}}=90°$, respectively. It can be seen from the variation trend of the contour line, the effect of the substrate temperature on evaporation was greater than that of the ambient pressure ratio, especially when the superheat degree was low. For example, when $\mathsf{\Delta }T<12 \mathrm{K}$, the contour line distribution was denser, indicating that the effect of the superheat on evaporation was relatively high. However, when the substrate superheat $\mathsf{\Delta }T>18 \mathrm{K}$, the contour line distribution was relatively sparse, indicating that the effect of the superheat on evaporation was relatively small. Besides, it also can be seen that, with the increase of superheat and the decrease of pressure ratio, the droplet evaporation lifetime decreased, and the difference between the maximum and minimum evaporation lifetime was 82%.

Figure 11 shows the evolution of droplet contact radius with time in the CCA mode; the initial contact angle of the droplet is 90°. It can be found that the slope was larger with the droplet evaporating, which indicates that the evaporative rate was faster. Meanwhile, as ambient pressure ratio decreased and superheat increased, the lifetimes decreased. For example, at pressure ratio $\eta =1$ and superheat $\mathsf{\Delta }T=7 \mathrm{K}$, the evaporation lifetime was about 59 s, while the lifetime decreased to 47 s at the pressure ratio $\eta =0.92$. Moreover, the lifetime will be reduced by three times to about 18 s at the pressure ratio $\eta =0.92$ and substrate superheat $\mathsf{\Delta }T=17 \mathrm{K}$.

Similarly, the variation range of superheat was enlarged (7 K~27 K), as well as the pressure ratio (0.92~1), to investigate the evaporative lifetime in the CCA mode. Figure 12 shows the evaporation lifetime in the CCA mode under different variations of pressure ratios and substrate superheating temperatures.

It can be seen that the lifetime in the CCA mode was longer than that in the CCR mode under the same conditions. Besides, it can be seen from the variation trend of the contour line that the effect of substrate superheat on evaporation time in the CCA mode was greater than that of pressure ratio, especially when the substrate superheat was low. As the superheat increased, the contour line distribution was relatively sparse, indicating that the effect of superheat on evaporation was relatively small.

4. Conclusions

Evaporation of sessile droplets in their own vapor environment is important for many applications. The traditional diffusion model is no longer applicable, and the evaporation process is controlled by kinetic theory of gases (KTG). In our study, theoretical analysis and numerical simulations were performed to reveal the evaporation mechanisms of a droplet. An improved theoretical model and fully coupled numerical model were established, and the effects of key parameters on evaporation were quantitatively analyzed. The main conclusions are as follows

• There is a critical contact angle ${\theta }_{\mathrm{cri}}=40°$; when the contact angle ${\theta }_{\mathrm{c}}<{\theta }_{\mathrm{cri}}$, the evaporation rate was inversely proportional to the contact angle; conversely, the evaporation rate was proportional to the angle.
• High substrate superheat or low-pressure ratio enhanced thermocapillary convection inside the droplet, resulting in an increased evaporation rate, and it shortened the evaporative lifetime in both the CCA and CCR modes.
• Substrate superheat affected the evaporation rate and evaporation lifetime more than ambient pressure ratio, especially when the superheat was low.