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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 Wewhile 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 Weaccording 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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Corresponding author:Ma Xue-Hu,xuehuma@dlut.edu.cn

Funds:Project supported by the National Natural Science Foundation of China (Grant Nos. 51836002, 52006025) and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. DUT20RC(3)016))

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Cited By

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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

DownLoad: CSV

V₀/(m·s^–1)	We	β_m-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

DownLoad: CSV

固体/液体	D₀, mm	V₀/(m·s^–1)	We	θ/(°)	β_m-exp	β_m-model	(β_m-exp–β_m-model)/β_m-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

DownLoad: CSV

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Vol. 70, Issue 10 - 2021-05-20

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