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摘要: 液滴撞击超亲水表面铺展之后形成的薄液膜铺展直径是喷雾冷却、降膜蒸发等传热传质过程的一项关键控制参数. 以往模型在预测超亲水表面惯性力驱动下的最大铺展直径时, 存在低韦伯数下呈反常趋势、高韦伯数下预测值偏低等问题. 针对上述问题, 本文采用高速摄像技术研究液滴撞击过程中的铺展水力学特性, 发现了以往模型未完全考虑超亲水表面的铺展特性: 球冠状液膜、高黏性阻力及重力势能做功. 本文考虑了液膜球冠形态、重力势能、辅助耗散, 修正了以往最大铺展直径的预测模型, 并建立了适用于超亲水表面最大铺展直径的预测模型. 通过对铺展过程中各能量成分分析发现, 在超亲水表面上动能、表面能、重力势能均转化为黏性耗散能, 其中: 在低韦伯数下, 表面能转化为黏性耗散能占主要作用; 在高韦伯数下, 动能转化为黏性耗散能占主要作用. 并且, 在低韦伯数下, 重力势能和辅助耗散的引入对于准确预测超亲水表面最大铺展直径具有重要作用. 将模型预测结果与实验结果比较发现, 本模型成功消除了以往模型在低韦伯数下的反常趋势, 且能较好预测宽韦伯数范围下超亲水表面最大铺展直径. 同时, 本模型可以预测亲水和疏水固体表面的液滴最大铺展直径. 超亲水表面最大铺展直径的准确预测模型的提出对喷雾冷却, 降膜蒸发中提高和控制流体铺展距离和传热效率具有重要意义.Abstract: Liquid droplets impacting on the solid surface is an ubiquitous phenomenon in natural, agricultural, and industrial processes. The maximum spreading diameter of a liquid droplet impacting on a solid surface is a significant parameter in the industrial applications such as inkjet printing, spray coating, and spray cooling. However, former models cannot accurately predict the maximum spreading diameter on a superhydrophilic surface, especially under low Weber number (We). In this work, the spreading characteristics of a water droplet impacting on a superhydrophilic surface are explored by high-speed technique. The spherical cap of the spreading droplet, gravitational potential energy, and auxiliary dissipation are introduced into the modified theoretical model based on the energy balance. The model includes two viscous dissipation terms: the viscous dissipation of the initial kinetic energy and the auxiliary dissipation in spontaneous spreading. The energy component analysis in the spreading process shows that the kinetic energy, surface energy, and gravitational potential energy are all transformed into the viscous dissipation on the superhydrophilic surface. The transformation of surface energy into viscous dissipation is dominant at lower We while the transformation of kinetic energy into viscous dissipation is dominant at higher We. It is found that the gravitational potential energy and auxiliary dissipation play a significant role in spreading performance at low We according to the energy component analysis. Moreover, the energy components predicted by the modified model accord well with the experimental data. As a result, the proposed model can predict the maximum spreading diameter of a droplet impacting on the superhydrophilic surface accurately. Furthermore, the model proposed in this work can predict the maximum spreading diameter of the droplet impacting on the hydrophilic surface and hydrophobic surface. The results of this work are of great significance for controlling droplet spreading diameter in spray cooling and falling film evaporation.
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图 1 液滴铺展实验系统(A: 高速摄像机(Photron APX RS); B: 高速摄像机(Photron Mini UX100); C: 微量注射泵(LSP01-1 BH); D: 液滴生成装置; E: 恒温台; F: 高亮光源; G: 数据采集设备; G: 精确自动控制位移的直线位移滑台)
Figure 1. The droplet spreading experiment platform. (A: high speed camera (Photron APX RS); B: high speed camera (Photron Mini UX100); C: micro syringe pump (LSP01-1 BH); D: droplet generating device; E: heating platform; F: diffuse light source; G: data acquisition computer; H: linear displacement slide.
图 2 (a)光滑铜表面的SEM图; (b)超亲水表面SEM图
Figure 2. SEM images of (a) smooth copper surface and (b) superhydrophilic surface.
图 3 液滴铺展过程 (a)光滑铜亲水表面; (b)超亲水表面We = 1.91; (c)超亲水表面We = 25.59
Figure 3. Droplet spreading process: (a) Smooth copper hydrophilic surface; (b) superhydrophilic surface at We = 1.91; (c) superhydrophilic surface at We = 25.59.
图 4 液滴在不同We下撞击超亲水表面铺展因子随时间的变化过程
Figure 4. The variation of spreading factor β with time
图 5 液滴在超亲水表面最大铺展直径形态
Figure 5. Sketch of droplet shape at its maximum spread on superhydrophilic surface.
图 6 (a)模型计算的ΔEk, ΔEs, W, ΔEp随We的变化; (b)低We下ΔEk, ΔEs, W, ΔEp占总能量的占比(青色区域放大)
Figure 6. (a) Variation of the energy component: ΔEk, ΔEs, W, ΔEp with We; (b) comparison of the energy component: ΔEk, ΔEs, W, ΔEp at low We
图 7 (a)超亲水表面液滴铺展过程各项黏性耗散实验和模型计算值对比; b)低We下总黏性耗散中Wvis和Wad的占比(青色区域放大)
Figure 7. (a) Variation of the viscous dissipation components value with We; (b) comparison of the Wvis, Wad at low We.
图 8 液滴在不同We下撞击超亲水表面最大铺展因子的实验和模型预测结果对比(模型包括去除重力势能或辅助耗散的模型及全部考虑的模型)
Figure 8. Comparison of the current experimental measurements of βm with the theoretical prediction from model (models includes without Ep, without Wad and present model).
表 1 基于能量守恒的最大铺展因子的预测模型
Table 1. Theoretical models for predicting the maximum spreading factor.
文献 最大铺展模型预测表达式 表面润湿
性/(°)We 液滴形态 Lee等[14] $\begin{aligned} \rho {V_0}{D_0} + 12\sigma\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad \\= 3\sigma (1 - \cos \theta )\beta _{\rm{m} }^2 + 8\sigma \dfrac{1}{ { {\beta _{\rm{m} } } } } + 3\sqrt { {b / c} } \rho V_{\rm{0} }^2{D_0}\beta _{\rm{m} }^{ {5 / 2} }\dfrac{1}{ {\sqrt {Re} } }\end{aligned}$ 60—115 1—290 圆饼 Chandra等[27] $\dfrac{3}{2}\dfrac{ {We} }{ {Re} }\beta _{\rm{m} }^4 + \left( {1 - \cos \theta } \right)\beta _{\rm{m} }^2 - \left( {\dfrac{1}{3}We + 4} \right) = 0$ ～32 ～43 圆饼 Pasandideh-
Fard等[28]${\beta _{\rm{m} } } = \sqrt {\dfrac{ {We + 12} }{ {3\left( {1 - \cos {\theta _{\rm{a} } } } \right) + 4\left( { { {We} / {\sqrt {Re} } } } \right)} } }$ 27—140 27—447 圆饼 Mao等[29] $\left( {\dfrac{ {1 - \cos \theta } }{4} + 0.35\dfrac{ {We} }{ {\sqrt {Re} } } } \right)\beta _{\rm{m} }^4 - \left( {\dfrac{ {We} }{ {12} } + 1} \right)\beta + \dfrac{2}{3} = 0$ 30—120 5—1000 圆饼 Ukiwe等[30] $\left( {We + 12} \right){\beta _{\rm{m} } } = 8 + \beta _{\rm{m} }^3\left[ {3\left( {1 - \cos \theta } \right) + 4\dfrac{ {We} }{ {\sqrt {Re} } } } \right]$ 57—90 18—370 圆饼 Huang等[31] $\begin{aligned}\frac{3}{4}\left( {\frac{ {We} }{ {\sqrt {Re} } } + \frac{ {We^*} }{ {\sqrt {Re^*} } } } \right)\beta _{\rm{m} }^4 + 3\left( {1 - \cos {\theta _{\rm{a} } } } \right)\beta _{\rm{m} }^3 \qquad\\ - \left( {We + 12} \right){\beta _{\rm{m} } } + 8 = 0, ~~{V_0} < V^* \qquad\qquad\qquad\end{aligned}$
$\begin{aligned} \frac{3}{4}\left( {\frac{ {We} }{ {\sqrt {Re} } } + \frac{ {We^*} }{ {\sqrt {Re^*} } }\frac{ {Re^*} }{ {Re} } } \right)\beta _{\rm{m} }^4 + 3\left( {1 - \cos {\theta _{\rm{a} } } } \right)\beta _{\rm{m} }^3 \\ - \left( {We + 12} \right){\beta _{\rm{m} } } + 8 = 0,~~ {V_0} > V^*\qquad\qquad\qquad \end{aligned}$64—110 2—500 圆饼 Park等[32] $\begin{aligned} \left( {0.33\frac{ {We} }{ {\sqrt {Re} } } - \frac{1}{4}\cos \theta + \frac{1}{2}\left( {\frac{ {1 - \cos {\theta _{\rm{a} } } } }{ { { {\sin }^2}{\theta _{\rm{a} } } } } } \right)} \right)\beta _{\rm{m} }^2 \\ - 1 - \frac{ {We} }{ {12} } + \frac{ {\Delta {E_{\rm{s} } } } }{ { {\text{π} }D_0^2\sigma } } = 0 \qquad\qquad\qquad\qquad\quad\end{aligned}$ 31—113 0.2—180 球冠 Li等[33] $\dfrac{ {We} }{ {12} }\left( {1 - {C_{\rm{k} } } - \dfrac{3}{ {2\sqrt {Re} } }\displaystyle\int_{ {H_{\rm{m} } } }^{ {H_{\rm{s} } } } { {d^2}{\rm{d} }h} } \right) = {C_{\rm{S} } }P\left( { {D_{\rm{e} } } } \right) - P\left( { {D_{ {\rm{max} } } } } \right)$ 30—150 0—10 球冠 Gao等[34] $\begin{aligned} 1 + \frac{ {We} }{ {12} } = \frac{1}{6}\left[ {\frac{1}{ { { {\hat r}_{\rm{c} } } } } + \frac{1}{ { { {\hat R}_{\rm{c} } } } } } \right] + 4{\theta _{\rm{a} } }{ {\hat r}_{\rm{c} } }{ {\hat R}_{\rm{c} } } + {\left( { { {\hat R}_{\rm{c} } } - { {\hat r}_{\rm{c} } }\sin {\theta _{\rm{a} } } } \right)^2} \\ + {\left( { { {\hat R}_{\rm{c} } } + { {\hat r}_{\rm{c} } }\sin {\theta _{\rm{a} } } } \right)^2}\left( {\frac{4}{3}\frac{ {We} }{ {\sqrt {Re} } } - \cos {\theta _{\rm{a} } } } \right) \qquad\quad\end{aligned}$ 74—155 135—210 圆环 Wang等[35] $\begin{aligned} We + 12 =\qquad \qquad\qquad \qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad \qquad\\ \frac{3}{4}\left( {\frac{ {We} }{ {\sqrt {Re} } } + \alpha \frac{ {W{e_{\rm{c} } } } }{ {\sqrt {R{e_{\rm{c} } } } } } } \right)\beta _{\rm{m} }^3 + 3\left( {1 - \cos {\theta _{\rm{a} } } } \right)\beta _{\rm{m} }^2 + 12\bigg\{ \frac{ {\xi _{\rm{r} }^2} }{ { { {\left( {1 - \cos {\theta _{\rm{m} } } } \right)}^2} } } \\ \times \bigg[ {\sin ^2}{\theta _{\rm{m} } } - \frac{ { {\beta _{\rm{m} } } } }{ { {\xi _{\rm{r} } } } }\sin {\theta _{\rm{m} } }(1 - \cos {\theta _{\rm{m} } }) + 2(1 - \cos {\theta _{\rm{m} } }) \bigg] \qquad \qquad \qquad \\ \left. + 2{\xi _{\rm{r} } }\left( {\frac{ { {\beta _{\rm{m} } } } }{2} - {\xi _{\rm{r} } }\frac{ {\sin {\theta _{\rm{m} } } } }{ {1 - \cos {\theta _{\rm{m} } } } } } \right)\left( {\left| {1 - \kappa } \right| + \frac{ { {\theta _{\rm{m} } } } }{ {1 - \cos {\theta _{\rm{m} } } } } } \right) \right\}\qquad\qquad \quad \end{aligned}$ 34—100 0.1—427 环状-薄片 
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表 2 超亲水表面最大铺展因子βm实验结果与以往典型模型预测值[27-32]之间的比较
Table 2. Comparison of previous model[27-32] prediction value of βm with experimental data
V0/(m·s–1) We βm-exp Chandra 等[27] Pasandideh-Fard 等[28] Mao等[29] Park等[32] Ukiwe 等[30] Huang等[31] 0.25 1.91 3.41 2.4 6.37 3.05 11.31 5.84 0.58 0.44 5.90 3.46 2.1 4.82 2.55 7.93 4.42 0.44 0.60 10.77 3.60 1.97 4.37 2.41 6.67 4.02 0.35 0.71 15.26 3.82 1.93 4.20 2.37 6.09 3.90 0.29 0.93 25.59 3.93 1.89 4.07 2.35 5.41 3.82 0.20 130 51.17 4.08 1.90 4.08 2.38 4.9 3.93 0.12 1.50 68.59 4.26 1.91 4.13 2.41 4.78 3.93 0.10 1.89 109.3 4.43 1.93 4.26 2.47 4.67 4.07 0.06 2.35 168.98 4.70 1.97 4.42 2.54 4.66 4.24 0.04 2.8 239.89 4.90 2.00 4.57 2.60 4.72 4.39 0.03 3.08 290.08 5.00 2.02 4.67 2.64 4.76 4.48 0.02
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表 3 本文模型预测值与文献[28,29]中不同润湿性表面的最大铺展因子的实验值对比
Table 3. Comparison of current theoretical model of βm with experimental data in literature[28,29].
固体/液体 D0, mm V0/(m·s–1) We θ/(°) βm-exp βm-model (βm-exp–βm-model)/ βm-model 玻璃/水 2.7 0.55 11.21 37 1.77 2.41 0.26 玻璃/水 2.7 0.82 24.91 37 2.20 2.74 0.19 玻璃/水 2.7 1.00 37.05 37 2.53 2.94 0.14 玻璃/水 2.7 1.58 92.48 37 3.11 3.51 0.11 玻璃/水 2.7 1.86 128.17 37 3.70 3.81 0.03 玻璃/水 2.7 2.77 284.26 37 4.50 4.48 0.00 玻璃/水 2.7 3.72 512.67 37 4.94 4.89 0.01 不锈钢/水 2.7 0.55 11.21 67 1.67 1.95 0.14 不锈钢/水 2.7 0.82 24.91 67 2.16 2.28 0.05 不锈钢/水 2.7 1.00 37.05 67 2.34 2.51 0.06 不锈钢/水 2.7 1.58 92.48 67 3.09 3.13 0.01 不锈钢/水 2.7 1.86 128.17 67 3.67 3.38 0.08 不锈钢/水 2.7 2.77 284.26 67 4.42 4.15 0.06 不锈钢/水 2.7 3.72 512.67 67 4.88 4.65 0.05 石蜡/水 2.7 0.55 11.21 97 1.65 1.58 0.04 石蜡/水 2.7 0.82 24.91 97 2.10 1.91 0.10 石蜡/水 2.7 1.00 37.05 97 2.26 2.13 0.06 石蜡/水 2.7 1.58 92.48 97 3.01 2.79 0.07 石蜡/水 2.7 1.86 128.17 97 3.60 3.09 0.16 石蜡/水 2.7 2.77 284.26 97 4.32 3.89 0.11 石蜡/水 2.7 3.72 512.67 97 4.78 4.44 0.08 蜂蜡/水 0.62 2.61 59 111 2.65 2.19 0.21 蜂蜡/水 0.78 3.29 118 111 3.18 2.76 0.15 蜂蜡/水 0.89 3.71 171 111 3.45 3.09 0.11 蜂蜡/水 0.98 4.00 219 111 3.79 3.33 0.14 蜂蜡/水 1.05 4.28 271 111 3.91 3.53 0.10
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