A continuum approach to study vapor vaporization within textured surface

Abstract

To keep a high heat transfer rate, it is important to retain the water in nucleate boiling regime and prevent film boiling. In this research, the continuum approach is applied to study liquid vaporization within a textured surface. The volume-of-fluid method and cavitation model are used in this simulation. The primary focus is on reproducing qualitative trends using continuum as molecular dynamic studies and gaining insights into the underlying mechanisms. From the simulation results, it is found that the superhydrophilic surface stabilizes the liquid existing inside the pore.

Introduction

Heat transfer influences technologies in various fields. In particular, boiling occurs in numerous industrial applications such as high heat flux electronic devices, ¹ chemical processes, ² and power plants. ³ During the nucleate boiling process, a heated surface is placed adjacent to a liquid which then vaporizes. The vaporization produces large latent heat, making it an efficient mode of heat transfer. Film boiling, on the other hand, leads to inefficient heat transfer and potentially damages the heated surface, because, unlike nucleate boiling, it lacks a stable insulating vapor layer between the liquid and the surface. Thus, for maintaining a high efficiency of heat transfer, nucleate boiling is preferred and film boiling should be avoided.

To enhance nucleate boiling and prevent film boiling, textured or chemical treatment of the surface is suggested. Surface characteristics are known to be a critical factor in determining the efficiency of heat transfer.^4–11 Utilizing optimal surfaces can increase energy efficiency and have a profound impact on industrial applications. For example, superhydrophilic surfaces have been designed to promote wetting and enhance the heat transfer efficiency by changing the surface interfacial chemistry property or altering the surface texture spacing and geometry. Surface texture can increase the wettability, and capillary pressure can drive the pooled liquid into the pore which is created by the surface texture.^5,6 The invading liquid can lead to vapor evacuation from roughness so vapor film formation is inhibited as well. This roughness-based wettability is more effective for pillar-type geometries with large spacing. With small spacing between pillars, resistance makes lateral evacuation of vapor difficult. Hence, it is essential to define the critical roughness scale.

Molecular dynamic (MD) simulations have been applied to study the sustaining dry surfaces under water. ¹² However, MD simulations are expensive and in limited scale. In this work, we explore whether it is possible to study vapor vaporization within textured surface using continuum approach. Instead of MDs and Boltzmann equation, continuum approach uses continuity equation, momentum equation, and energy equation to describe the fluid physics.

Energy analysis

Superheated liquid is seen as metastable state at a liquid pressure p_l which is lower than the saturation pressure p_sat for a given temperature T_l (see blue inverse triangle in Figure 1).

Figure 1.

Typical liquid–vapor phase diagram for water.

Figure 2.

The schematic of textured surface: the roughness valley (pocket) are in between square microposts.

In this liquid–vapor phase diagram, the co-existence curve can be represented by the Clausius–Clapeyron equation

$$p_{\mathrm{sat}}-p_l=\frac{h_{\mathrm{lv}}}{T_{\mathrm{sat}}(p_l)(v_v-v_l)}(T_l-T_{\mathrm{sat}}(p_l))$$

(1)

where p_sat is the saturation pressure, p_l is the liquid pressure, h_lv is the mass-specific change in enthalpy during phase change, v_v is the specific volume of the vapor, v_l is the specific volume of the liquid, T_l is the liquid temperature, and T_sat is the saturation temperature.

For a liquid–vapor interface at the top of the pore, if it is a mechanical equilibrium state, the Young–Laplace equation gives the radius of curvature r_e

$$r_e=\frac{2{\sigma }_{\mathrm{lv}}}{(p_l-p_v)}$$

(2)

where p_v and p_l are the pressures of the vapor and liquid phases, respectively; ${\sigma }_{\mathrm{lv}}$ is the liquid–vapor surface tension. Thus, the liquid–vapor radius of curvature can be

$$r_e=\frac{2{\sigma }_{\mathrm{lv}}{T}_{\mathrm{sat}}(p_l)(v_v-v_l)}{h_{\mathrm{lv}}(T_l-T_{\mathrm{sat}}(p_l))}$$

(3)

Another way to express phase change is by equating the chemical potential at that temperature; the pressure of the liquid and the vapor during the phase change is

$$p_v=p_{\mathrm{sat}}\exp [\frac{v_l}{R{T}_l}(p_l-p_{\mathrm{sat}})]$$

(4)

where p_l and p_v are the liquid and vapor pressures during the phase change, respectively. p_sat is the saturation pressure (the pressure on the co-existence curve at temperature T_l). For a typical ambient condition, $p_v\approx p_{\mathrm{sat}}$ . This enables equation (2) become

$$r_e=\frac{2{\sigma }_{\mathrm{lv}}}{(p_{\mathrm{sat}}-p_v)}$$

(5)

To prevent the formation of vapor nuclei, or promote condensation, pore diameter D should satisfy the following equation

$$D\le 2r_e\cos {\theta }_e$$

(6)

where ${\theta }_e$ is the surface contact angle. Using the radius of curvature r_e derived from equation (2)

$$D\le \frac{4{\sigma }_{\mathrm{lv}}\cos {\theta }_e}{(p_v-p_l)}$$

(7)

Jones et al. ¹² performed MD simulation to study the sustaining dry surfaces under liquid. However, MD simulation is timely and computationally expensive. Also, the calculation domain size is limited to nano scale. The objective of this study is to perform a micro-scale continuum numerical simulation of superheated liquid sustaining inside a pore to investigate the effect of contact angle and surface texture. The primary focus in this research is on using continuum and liquid–vapor mass transfer model and reproducing qualitative trends as MD studies and gaining insights into the underlying mechanisms.

Numerical method

There are several simulation methods proposed to study the vapor–liquid phase change problem, such as the volume-of-fluid (VOF) method, ¹³ lattice Boltzmann method, ¹⁴ immersed boundary method, ¹⁵ direct front tracking method,^16,17 and level set (LS) method. ¹⁸ The VOF method has been used for nucleate, film boiling, and dropwise condensation simulations. We choose VOF method to study this pore vaporization problem.

VOF

VOF method is implemented in this research; the two phases are represented by phase volume and are defined as follows: in VOF, the two phases are represented by phase volume fractions such that

$${\alpha }_l+{\alpha }_v=1$$

(8)

where $\alpha$ is the volume fraction; subscripts l and v represent the liquid and vapor phases, respectively. The governing equations consist of the continuity equation for the two phases and a one-fluid model for the momentum, energy equation, and equation of state

$$\{\begin{array}{c}\frac{\partial }{\partial t}({\alpha }_v{\rho }_v)+\nabla ·({\rho }_v{\alpha }_v\overrightarrow{u})={\stackrel{·}{m}}_{\mathrm{lv}}\hfill \\ \frac{\partial }{\partial t}({\alpha }_l{\rho }_l)+\nabla ·({\rho }_l{\alpha }_l\overrightarrow{u})=-{\stackrel{·}{m}}_{\mathrm{lv}}\hfill \\ \frac{\partial }{\partial t}(\rho \overrightarrow{u})+\nabla ·(\rho \overrightarrow{u}\overrightarrow{u})=-\nabla p+\nabla ·(\mu \nabla \overrightarrow{u})+\rho \overrightarrow{g}+\overrightarrow{F}\hfill \\ \frac{\partial }{\partial t}(\rho C_{p}T)+\nabla ·(\rho \overrightarrow{u}{C}_{p}T)=-\nabla ·(k\nabla T)+S_h\hfill \\ p=\rho \mathrm{RT}\hfill \end{array}$$

(9)

where ${\rho }_v$ and ${\rho }_l$ are the vapor and liquid densities, respectively; t is the time, $\vec u$ is the average fluid velocity; and ${\stackrel{·}{m}}_{\mathrm{lv}}$ is the mass source due to liquid-to-vapor phase change. In the momentum (second) equation, p is the pressure, and $\overrightarrow{g}$ is the gravitational force. $\rho$ and $\mu$ are the density and viscosity of the mixture of the liquid and vapor, respectively. In the energy (third) equation, T is the temperature, C_p is the mixture-specific heat, k is the mixture thermal conductivity, and S_h is the heat source. $\rho$ , $\mu$ , C_p, and k are

$$\{\begin{array}{c}\rho ={\rho }_l{\alpha }_l+{\rho }_v{\alpha }_v\hfill \\ \mu ={\mu }_l{\alpha }_l+{\mu }_v{\alpha }_v\hfill \\ C_p=C_{\mathrm{pl}}{\alpha }_l+C_{\mathrm{pv}}{\alpha }_v\hfill \\ k=k_l{\alpha }_l+k_v{\alpha }_v\hfill \end{array}$$

(10)

where subscripts l and v represent the liquid and vapor phases, respectively. In the momentum equation, $\overrightarrow{F}$ is the surface tension force between the two phases that is expressed as a volume force density using the continuum surface force (CSF) model ¹⁹

$$\overrightarrow{F}=\sigma \frac{{\alpha }_l{\rho }_l{\kappa }_l\nabla {\alpha }_l+{\alpha }_v{\rho }_v{\kappa }_v\nabla {\alpha }_v}{\frac{1}{2}({\rho }_l+{\rho }_v)}$$

(11)

where $\sigma$ is the surface tension, and the interface curvature is given by

$${\kappa }_l=-{\kappa }_v=-\nabla ·\left(\frac{\nabla {\alpha }_l}{|\nabla {\alpha }_l|}\right)$$

(12)

Cavitation: Merkel mass transfer model

To study vaporization in the isothermal condition, a cavitation model is used here.^20,21 There are various cavitation models, and Merkel model was chosen in our simulation. The mass transfer ${\stackrel{·}{m}}_{\mathrm{lv}}$ is based on production ${\stackrel{·}{m}}^{+}$ and reduction ${\stackrel{·}{m}}^{-}$ of the liquid. The transfer from liquid to vapor is evaluated as being proportional to the amount by which the pressure is below the saturation pressure. The transformation of vapor to liquid is based on the pressure difference as well

$$\begin{array}{c}{\stackrel{·}{m}}^{+}=\frac{C_v{\rho }_v\alpha }{\frac{1}{2}{\rho }_l{U}_{\infty }^2{t}_{\infty }}\mathrm{MAX}(0,p_{\mathrm{sat}}-p)p<p_{\mathrm{sat}}\hfill \\ {\stackrel{·}{m}}^{-}=\frac{C_c(1-{\rho }_v)\alpha }{\frac{1}{2}{U}_{\infty }^2{t}_{\infty }}\mathrm{MIN}(0,p_{\mathrm{sat}}-p)p>p_{\mathrm{sat}}\hfill \end{array}$$

(13)

where $\alpha$ is the volume fraction; C_v, C_c, $U_{\infty }$ , and $t_{\infty }$ are empirical constants. The specific mass transfer rate is then defined as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{m}}_{\mathrm{lv}}={\stackrel{·}{m}}^{+}+{\stackrel{·}{m}}^{-}>0,\mathrm{vaporization}}\\ \multicolumn{1}{c}{{\stackrel{·}{m}}_{\mathrm{lv}}={\stackrel{·}{m}}^{+}+{\stackrel{·}{m}}^{-}<0,\mathrm{condensation}}\end{array}$$

Problem description

To define the critical pore diameter, equation (7) is used here. The liquid–vapor surface tension is chosen as ${\sigma }_{\mathrm{lv}}=0.1\mathrm{N}/\mathrm{m}$ and the saturation pressure is equal to $p_{\mathrm{sat}}=27.19\mathrm{bar}$ and $p_v\approx p_{\mathrm{sat}}=27.19\mathrm{bar}$ ; hence, the critical pore diameter is

$$D\le \frac{4\times 0.1\times \cos {\theta }_e}{(27.19-p_l)\times 1.01325\times {10}^5}$$

(15)

D is approximately equal to $4\mu \mathrm{m}$ . Hence, we choose the diameter of the pore as $3\mu \mathrm{m}$ (Figure 3). The geometry size of whole computational domain is shown in Figure 3. The upper boundary of computational domain is set as a constant pressure outlet $p=22.12-26.12\mathrm{bar}$ which is smaller than the saturation temperature and constant temperature $T=525\mathrm{K}$ at all boundaries. All fluid properties are listed in Table 1. The contact angle of the bottom boundaries will be defined as hydrophilic ${\theta }_e=10\circ$ and hydrophobic ${\theta }_e=150\circ$ . The initial vapor–liquid interface is at location $\mathrm{h}=2\mu \mathrm{m}$ .

Figure 3.

The schematic of 2D problem description.

Table 1.

Properties of the fluid used in simulation.

Parameter	Liquid	Vapor
Density $\rho$ (kg/m3)	200	Ideal gas
Viscosity $\mu$ (Pa s)	0.1	0.005
Thermal conductivity k (W/m K)	40	1
Specific heat Cp (J/kg K)	400	200
Latent heat hlv (J/kg)	$1\times {10}^4$
Surface tension $\sigma$ (N/m)	0.1

Results and conclusion

Using the numerical methods and governing equations discussed in previous section, we model the vaporization in surface roughness under different outlet pressure and contact angle at bottom wall. In the following results of phase contour, red color represents the vapor, and the blue one represents the liquid.

Pressure outlet 26.12 bar under different contact angles

In the following results of phase contour, red color represents the vapor, and the blue one represents the liquid. Figure 4 shows the phase change at pressure outlet $p=26.12\mathrm{bar}$ and contact angle ${\theta }_e=10\circ$ . Initially, the vapor–liquid interface is flat. At t = 0.05 ms, the vapor bubbles start to form because the liquid pressure is below the saturation pressure, and a curved vapor–liquid interface proceeds to form near the entrance of the pore. A curve-up inside the pore indicates a hydrophilic surface which prefers the contact of liquid. Figure 5 and Figure 6 are corresponding pressure contour plot and temperature contour plot.

Figure 4.

Phase contour at p = 26.12 bar, ${\theta }_e=10\circ$ .

Figure 5.

Pressure contour at p = 26.12 bar, ${\theta }_e=10\circ$ .

Figure 6.

Temperature contour at p = 26.12 bar, ${\theta }_e=10\circ$ .

Figure 7 shows the phase change at pressure outlet $p=26.12\mathrm{bar}$ and contact angle ${\theta }_e=150\circ$ . Initially, the vapor–liquid interface is flat. At t = 0.05 ms, the vapor bubbles start to form because the liquid pressure is below the saturation pressure, and a curved vapor–liquid interface proceeds to form near the entrance of the pore. A curve-down inside the pore indicates a hydrophobic surface which expels the contact of liquid.

Figure 7.

Phase contour at p = 26.12 bar, ${\theta }_e=150\circ$ .

Pressure outlet 24.12 bar under different contact angles

As the outside boundary pressure decreases as shown in Figure 8, it displays the phase change at pressure outlet $p=24.12\mathrm{bar}$ and contact angle ${\theta }_e=10\circ$ . Initially, the vapor–liquid interface remains flat. At t = 0.05 ms, the vapor bubbles start to form because the liquid pressure is below the saturation pressure, and a curved vapor–liquid interface proceeds to form near the entrance of the pore. It is found that there are more vapors with larger size formed compared with Figure 4. A curve-up inside the pore indicates a hydrophilic surface which prefers the contact of liquid. As time progressed, the vapor bubble formed in a large size.

Figure 8.

Phase contour at p = 24.12 bar, ${\theta }_e=10\circ$ .

Figure 9 shows the phase change at pressure outlet $p=24.12\mathrm{bar}$ and contact angle ${\theta }_e=150\circ$ . At t = 0.05 ms, the vapor bubbles start to form because the liquid pressure is below the saturation pressure, and a curved vapor–liquid interface proceeds to form near the entrance of the pore. And it is found that there are more vapors with larger size formed compared with Figure 4. A curve-down inside the pore indicates a hydrophobic surface which disfavors the contact of liquid.

Figure 9.

Phase contour at p = 24.12 bar, ${\theta }_e=150\circ$ .

Pressure outlet 22.12 bar under different contact angles

The outside boundary pressure is set as $p=22.12\mathrm{bar}$ and contact angle ${\theta }_e=10\circ$ . In Figure 10, the vapor–liquid interface is initially flat. At t = 0.05 ms, the vapor bubbles start to form because the liquid pressure is below the saturation pressure, and a curved vapor–liquid interface proceeds to form near the entrance of the pore. And it is found that there are more vapors with larger size. A curve-up inside the pore indicates a hydrophilic surface which prefers the contact of liquid. As time progressed, the vapor bubble formed in a large size. Also, because of the formation of vapor bubble, the pressure inside the pore increased; a curve-down interface ended at the end.

Figure 10.

Phase contour at p = 22.12 bar, ${\theta }_e=10\circ$ .

The outside boundary pressure is set as $p=22.12\mathrm{bar}$ and contact angle ${\theta }_e=150\circ$ . In Figure 11, the vapor–liquid interface is initially flat. At t = 0.05 ms, the vapor bubbles start to form because the liquid pressure is below the saturation pressure, and a curved vapor–liquid interface proceeds to form near the entrance of the pore. And it is found that there are more vapors with larger size. A curve-down inside the pore indicates a hydrophobic surface which expels the contact of liquid. As time progressed, the liquid expelled from the surface and formed a liquid droplet within vapor bubbles inside.

Figure 11.

Phase contour at p = 22.12 bar, ${\theta }_e=150\circ$ .

Conclusion

To summarize, we have demonstrated a numerical simulation using continuum to study superheated liquid vaporization within the surface texture. While the pressure outlet is far below the saturation pressure, there is more vaporization produced. Also, it is found that the superhydrophilic surface stabilizes the liquid existing inside the pore which is in agreement with the experimental results. Our numerical simulation successfully combines the VOF method and cavitation model to reproduce superheated liquid vaporization in surface roughness and has reasonable simulation results comparing with theory and experiments.