Single-bubble dynamics in pool boiling of one-component fluids

We numerically investigate the pool boiling of one-component fluids with a focus on the effects of surface wettability on the single-bubble dynamics. We employed the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)], a diffuse-interface model for liquid-vapor flows involving liquid-vapor transition in nonuniform temperature fields. We first perform simulations for bubbles on homogeneous surfaces. We find that an increase in either the contact angle or the surface superheating can enhance the bubble spreading over the heating surface and increase the bubble departure diameter as well and therefore facilitate the transition into film boiling. We then examine the dynamics of bubbles on patterned surfaces, which incorporate the advantages of both hydrophobic and hydrophilic surfaces. The central hydrophobic region increases the thermodynamic probability of bubble nucleation while the surrounding hydrophilic region hinders the continuous bubble spreading by pinning the contact line at the hydrophobic-hydrophilic intersection. This leads to a small bubble departure diameter and therefore prevents the transition from nucleate boiling into film boiling. With the bubble nucleation probability increased and the bubble departure facilitated, the efficiency of heat transfer on such patterned surfaces is highly enhanced, as observed experimentally [Int. J. Heat Mass Transfer 57, 733 (2013)]. In addition, the stick-slip motion of contact line on patterned surfaces is demonstrated in one-component fluids, with the effect weakened by surface superheating.

Figure 1

Schematic illustration for simulated systems. (a) A bubble on a homogeneous surface, to be discussed in Sec. 3. (b) A bubble on a chemically patterned (hydrophilic-hydrophobic-hydrophilic) surface, to be discussed in Sec. 4. The black and white segments of the solid surface indicate the hydrophilic and hydrophobic parts, respectively. The system is closed in the $x$ direction by the periodic boundary conditions. Note that a vapor layer of thickness $h_v$, sandwiched by liquids in the $z$ direction, is introduced to maintain the supersaturated state of the liquid surrounding the bubble. This is essential to the simulation of boiling processes [30, 36].

Figure 2

A bubble under zero gravity on a homogeneous surface with static contact angle ${\theta }_s=0^{\circ }$ and surface temperatures $T_b=0.90T_c$ and $T_t=0.775T_c$, recorded at time $t=5000{\tau }_0$. The system measures $L_x=250\ell$ by $L_z=210\ell$ with $h_l=10\ell$, $h_v=100\ell$, and $H=100\ell$ as defined in Fig. 1. Upper panel: Density (color) and velocity (arrow) fields. Middle panel: Temperature contour (labeled thin lines) and liquid-vapor interfaces defined at $n=(n_g+n_l)/2$ (thick lines). Temperature values are in units of $T_c$. Lower panel: Reduced pressure $p/p_c$ with $p$ calculated from Eq. (6) (thin solid blue line) and reduced temperature $(T-T_t)/(T_b-T_t)$ along the $z$ direction at $x=125\ell$ (thick solid black line). The thin dashed blue line denotes the coexistence pressure $p_{\mathrm{cx}}\approx 0.48p_c$, and the thick dashed black line denotes the coexistence temperature $T_{\mathrm{cx}}\approx 0.84T_c$.

Figure 3

Bubble growth on homogeneous surfaces under zero gravity; effects of surface wettability. (a) Temporal evolution of a bubble on a surface with a given static contact angle ${\theta }_s$: (a1) ${\theta }_s=0^{\circ }$, (a2) ${\theta }_s={41}^{\circ }$, (a3) ${\theta }_s={76}^{\circ }$, and (a4) ${\theta }_s={120}^{\circ }$. The time step is $\Delta t=1000{\tau }_0$ in (a1)–(a3) and $\Delta t=200{\tau }_0$ in (a4). (b) Temporal evolution of the bubble base radius $R_b$ (half the length over which the bubble is in contact with the solid) in units of $\ell$. (c) Temporal evolution of the surface-integrated heat flux $Q_b$ (in units of $\epsilon \nu /v_0$), computed from Eq. (21). Here the system measures $L_x=250\ell$ by $L_z=210\ell$ with $h_l=10\ell$, $h_v=100\ell$, and $H=100\ell$ as defined in Fig. 1. The surface temperatures are $T_b=0.90T_c$ and $T_t=0.775T_c$, respectively.

Figure 4

Bubble growth on homogeneous surfaces under zero gravity; effects of surface superheating $\Delta T_{\sup }\equiv T_b-T_{\mathrm{cx}}$. (a) Temporal evolution of a bubble on a surfaces with a given bottom-surface temperature: (a1) $T_b=0.88T_c$ and (a2) $T_b=0.92T_c$. The top-surface temperature is $T_t=0.775T_c$. The time step is $\Delta t=1000{\tau }_0$. (b) Temporal evolution of the bubble base radius $R_b$ in units of $\ell$. (c) Temporal evolution of the surface-integrated heat flux $Q_b$ (in units of $\epsilon \nu /v_0$), computed from Eq. (21). Here the system measures $L_x=250\ell$ by $L_z=210\ell$ with $h_l=10\ell$, $h_v=100\ell$, and $H=100\ell$ as defined in Fig. 1. The static contact angle of the bottom surface is ${\theta }_s=0^{\circ }$.

Figure 5

Temporal evolution of the effective bubble radius $R_{\mathrm{eff}}$ for different static contact angles and bottom-surface temperatures. Here $R_{\mathrm{eff}}$ is defined by $\pi R_{\mathrm{eff}}^2=$ the instantaneous bubble area (volume in two dimensions).

Figure 6

A bubble under gravity ($\mathcal{G}={10}^{-4}$) on a homogeneous surface with static contact angle ${\theta }_s={60}^{\circ }$ and surface temperatures $T_b=0.90T_c$ and $T_t=0.775T_c$, recorded at time $t=3000{\tau }_0$. The system measures $L_x=200\ell$ by $L_z=185\ell$ with $h_l=10\ell$, $h_v=90\ell$, and $H=85\ell$ as defined in Fig. 1. Upper panel: Density (color) and velocity (arrow) fields. Middle panel: Temperature contour (labeled thin lines) and liquid-vapor interfaces defined at $n=(n_g+n_l)/2$ (thick lines). Temperature values are in units of $T_c$. Lower panel: Reduced pressure $p/p_c$ with $p$ calculated from Eq. (6) (thin solid blue line) and reduced temperature $(T-T_t)/(T_b-T_t)$ along the $z$ direction at $x=100\ell$ (thick solid black line). The thin dashed blue line denotes the coexistence pressure $p_{\mathrm{cx}}\approx 0.48p_c$, and the thick dashed black line denotes the coexistence temperature $T_{\mathrm{cx}}\approx 0.84T_c$.

Figure 7

Bubble departure under gravity ($\mathcal{G}={10}^{-4}$) from surfaces with different static contact angles: (a) ${\theta }_s={41}^{\circ }$. Bubbles are plotted in sequence from $t=0$ to $2000{\tau }_0$ with time interval $400{\tau }_0$. (b) ${\theta }_s={60}^{\circ }$. Bubbles are plotted in sequence from $t=0$ to $3600{\tau }_0$ with time interval $600{\tau }_0$ and at $t=3800{\tau }_0$. (c) ${\theta }_s={76}^{\circ }$. Bubbles are plotted in sequence from $t=0$ to $3600{\tau }_0$ with time interval $600{\tau }_0$ and at $t=4000{\tau }_0$. (d) ${\theta }_s={101}^{\circ }$. Bubbles are plotted in sequence from $t=0$ to $4800{\tau }_0$ with time interval $800{\tau }_0$ and at $t=5000{\tau }_0$. Here the surface temperatures are $T_b=0.90T_c$ and $T_t=0.775T_c$. The system measures $L_x=200\ell$ by $L_z=185\ell$ with $h_l=10\ell$, $h_v=90\ell$, and $H=85\ell$ as defined in Fig. 1. Inset: Snapshots of bubbles that are detached from the surface and reaching the liquid-vapor interface originally at $z=H$ (see Fig. 1). In all four plots, a semicircular bubble of radius $R_0=25\ell$ is introduced at the bottom surface at $t=0$. First, the bubble shrinks to a smaller one in a transient period ($\sim$hundreds of ${\tau }_0$) due to the mismatch between $R_0$ and the thermal and mechanical conditions around the bubble. Then the bubble grows continuously in both the horizontal and vertical directions. As the bubble radius becomes comparable with the capillary length $R_c$ ($\approx 16\ell$ here), the bubble evolution in the vertical direction becomes dominant over that in the horizontal direction due to buoyancy effect. Finally, the bubble detaches from the bottom surface.

Figure 8

Bubble departure under gravity ($\mathcal{G}={10}^{-4}$) from surfaces with different bottom-surface temperatures: (a) $T_b=0.92T_c$. Bubbles are plotted in sequence from $t=0$ to $2400{\tau }_0$ with time interval $400{\tau }_0$. (b) $T_b=0.93T_c$. Bubbles are plotted in sequence from $t=0$ to $2000{\tau }_0$ with time interval $400{\tau }_0$ and at $t=2200{\tau }_0$. Here the static contact angle is ${\theta }_s={41}^{\circ }$. The system measures $L_x=200\ell$ by $L_z=185\ell$ with $h_l=10\ell$, $h_v=90\ell$, and $H=85\ell$ as defined in Fig. 1. Inset: Snapshots of bubbles that are detached from the surface and reaching the liquid-vapor interface originally at $z=H$ (see Fig. 1). The process of bubble growth followed by departure is similar to that described in the caption to Fig. 7.

Figure 9

Stick-slip motion of contact line on chemically patterned surfaces. (a) Schematic illustration of the system. The black and white segments of the surface indicate the hydrophilic and the hydrophobic parts, respectively. The system is closed in the $x$ direction by the periodic boundary conditions. A body force is applied in the $+x$ direction to induce Poiseuille-type flow. (b) Temporal evolution of two liquid-vapor interfaces. The time ranges from $t=0$ (with the two interfaces denoted by two thick lines) to $1200{\tau }_0$ with time step $\Delta t=80{\tau }_0$. The system measures $L_x=200\ell$ by $L_z=60\ell$ with $D_i=D_i^{\prime }=D_o=40\ell$. (c) Temporal evolution of the contact-line coordinates $x_{\mathrm{cl}}$ of the two liquid-vapor interfaces. The contact line of the left interface is represented by empty symbols and exhibits a fast slip near the intersection at $x=80\ell$. The contact line of the right interface is represented by solid symbols and exhibits a pinning (stick) near the intersection at $x=160\ell$.

Figure 10

Bubble growth under zero gravity on chemically patterned surfaces with different values of $D_o$ as defined in Fig. 1: (a) $D_o=50\ell$, (b) $D_o=80\ell$, and (c) $D_o=110\ell$. The time step is $400{\tau }_0$. The bold lines indicate the onset of the depinning of the contact line at the hydrophobic-hydrophilic intersection. (d) Bubble profiles selected from (a)–(c) with thick solid lines from (a), thin solid lines from (b), and thin dashed lines from (c). The collapse of bubble profiles at $t=8000{\tau }_0$ indicates the separation of length and time scales. Here the static contact angle of the hydrophilic segment is ${\theta }_s=0^{\circ }$ and that of the hydrophobic segment is ${\theta }_s={180}^{\circ }$. The surface temperatures are $T_b=0.90T_c$ and $T_t=0.775T_c$. The system measures $L_x=200\ell$ by $L_z=280\ell$ with $h_l=10\ell$, $h_v=120\ell$, and $H=150\ell$ as defined in Fig. 1.

Figure 11

Bubble departure under gravity ($\mathcal{G}=5.0\times {10}^{-5}$) on chemically patterned surfaces with different values of $D_o$ as defined in Fig. 1: (a) $D_o=10\ell$. Bubbles are plotted in sequence at $t=0$, $1000{\tau }_0$, $3000{\tau }_0$, $4000{\tau }_0$, $5000{\tau }_0$, $6000{\tau }_0$, $6400{\tau }_0$, and $6760{\tau }_0$. (b) $D_o=60\ell$. Bubbles are plotted in sequence at $t=0$, $1000{\tau }_0$, $3000{\tau }_0$, $5000{\tau }_0$, $7000{\tau }_0$, $7600{\tau }_0$, $8000{\tau }_0$, and $8360{\tau }_0$. Here the static contact angle of the hydrophilic segment is ${\theta }_s=0^{\circ }$ and that of the hydrophobic segment is ${\theta }_s={180}^{\circ }$. The surface temperatures are $T_b=0.90T_c$ and $T_t=0.775T_c$. The system measures $L_x=200\ell$ by $L_z=250\ell$ with $h_l=10\ell$, $h_v=90\ell$, and $H=150\ell$ as defined in Fig. 1. The process of bubble growth followed by departure is similar to that described in the caption to Fig. 7.