Maximum Spreading for Liquid Drop Impacting on Solid Surface

Article Cite This: Ind. Eng. Chem. Res. 2019, 58, 10053−10063

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Maximum Spreading for Liquid Drop Impacting on Solid Surface Gangtao Liang,* Yang Chen, Liuzhu Chen, and Shengqiang Shen Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China

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ABSTRACT: This paper investigates maximum spreading associated with liquid drop impact on a solid surface. By use of highspeed video, the transient spreading with fluids of water, butanol, and ethanol is observed and analyzed to further clarify the assumptions made in many modeling studies. Fluid properties have considerable effects on spreading, in which viscosity is relegated to a secondary role compared to surface tension in the spreading magnitude. Several commonly used predicting methods for the maximum spreading factor are assessed based on a consolidated database consisting of 213 data points amassed from 11 sources. A practical formula with quite good precision for predicting the maximum spreading scale of the fluids with low viscosity of μf < 0.01 Pa·s is obtained. A new formula for dimensionless time related to maximum spreading is also provided based on the measured data of fluids with high surface tension. generated shortly on the heated surface in the film boiling regime, followed by drop periodical bouncing, with the vapor layer thickness of dozens of micrometers, which separates the liquid drop from the solid surface. Shown in Figure 1 are schematics of drop spreading on different substrates with the corresponding subsequent outcomes. Despite that drop spreading on the hydrophobic surface and heated surface in the film boiling regime bear some similarities such as rebound after the maximum spreading, they are substantially different in both theoretical model constructions and simulation implementations, the detailed information of which has been reviewed in the authors’ previous work2 and by Josserand and Thoroddsen.3 1.2. Drop Maximum Spreading. Of most interest in drop spreading is the maximum spreading, which can be achieved at any substrate illustrated in Figure 1. The importance of both extent and time scales associated the maximum spreading have attracted much more attention since the 1970s, in pursuit of understanding of both fluid mechanics and heat transfer mechanisms involved. Particularly, the maximum spreading, representing the maximum heat transfer area, is significantly

1. INTRODUCTION 1.1. Drop Impact Applications and Classifications. The phenomenon of liquid drop impact is very common in nature, of which one typical example is rain drops impinging on the ground or liquid pool. Aside from its natural occurrence, this phenomenon is also witnessed in numerous industrial applications.1 These include inkjet printing, atomized sprays in humidifiers, spray cooling of electronic components and power devices, horizontal-tube falling film evaporation in saline water desalination and refrigeration, fire extinguishing, spray coating, and forensic bloodstain pattern analysis. Of these applications, drop breakup into numerous tiny droplets under relatively high impinging velocity, customarily termed as splashing, is desired in most applications. On the other hand, in applications such as inkjet printing and spray coating, the avoidance of splashing is often pursued, which is termed as spreading in the scientific community. On the basis of surface wettability and magnitude of heat flux under the surface, drop spreading can be grouped into (a) spreading on the hydrophilic surface, in which the liquid drop with small contact angle, α, spreads until an equilibrium state is achieved, (b) spreading on the hydrophobic surface, in which the liquid drop with large contact angle spreads and recoils, followed by drop oscillation without bouncing or rebound from the superhydrophilic surface, (c) spreading on a thin vapor layer © 2019 American Chemical Society

Received: Revised: Accepted: Published: 10053

April 13, 2019 May 20, 2019 May 22, 2019 May 22, 2019 DOI: 10.1021/acs.iecr.9b02014 Ind. Eng. Chem. Res. 2019, 58, 10053−10063

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Industrial & Engineering Chemistry Research

Figure 1. Schematics of drop spreading on different substrates and nomenclatures.

important to modeling of the heat transfer process in some applications, such as spray cooling.4,5 Recently, Ma et al.6 used the maximum spreading scale model by Roisman7 and the semiempirical correlation of dimensionless time corresponding to the maximum spreading by Liang et al.8 to build up their spray/wall impingement model successfully. Despite the inescapability of heat transfer during drop spreading for spray modeling efforts, the maximum spreading without heating condition is mainly concerned in this study, which is the first step toward better understanding of models that involve heat transfer. Progress of drop spreading on a heated solid surface, including maximum scale, residence time, and heat transfer models, can be found in a recent review.2 In efforts of drop maximum spreading without heating load, the main focus is to explore its underlying driving mechanisms and to improve the model predictive capability. In general, a ratio of the maximum spreading diameter, Dmax, to the liquid drop diameter prior to impact, ddrop, termed as maximum spreading factor, βmax = Dmax/ddrop, is often used to represent the maximum spreading scale. The dominant methodology for theoretically modeling maximum spreading is based on the energy balance approach, with regard to kinetic energy and surface energy of the drop prior to impact, and surface energy along with viscous dissipation at the maximum spreading scale. These include the earlier study of Jones,9 who neglected surface energy, and the theoretical efforts of Madejski10 and Collings et al.11 Bennett and Poulikakos12 critically reported that the predictions by Jones and by Collings et al. are inaccurate, and they also improved Madejski’s model by importing the equilibrium (static) contact angle in their formula. Similar modification was also achieved by Mao et al.,13 and they found that the maximum spreading has a strong dependency on liquid viscosity and impact velocity. Kurabayashi and Yang14 introduced a viscosity ratio of fluid to wall in their formula, which was later modified by Healy et al.15 using a correction factor accounting for surface wettability. Chandra and Avedisian16 arrived at a formula involving the advancing contact angle, based on their assumption of the simplified cylindrical disk shape at maximum spreading. Pasandideh-Fard et al.17

obtained a predicting expression with considerations of both the advancing contact angle and boundary layer thickness, the latter of which was used to estimate viscous dissipation. This model was latter modified by Ukiwe and Kwok18 by accounting for the lateral sides of a cylinder disk. They reported that the advancing contact angle should be replaced by the Young contact angle because the advancing contact angle only represents approximation of the Young contact angle on a smooth or heterogeneous surface. They also declared that the use of equilibrium contact angle in the formulas is incorrect. Later, Vadillo et al.19 further modified the formula of Ukiwe and Kwok by replacing the Young contact angle with the dynamic contact angle. Lee et al.20 showed that the dynamic wetting process is important to the spreading at low velocity, characterized by the dynamic contact angle at maximum spreading. Follow-up study by Lee et al.21 also considered the dynamic contact angle in their energy balance model. One potential problem for those predicting approaches based on energy balance concept is the estimation of surface energy. The prevalent solution is using Young’s equation, which inevitably imports the critical parameter of contact angle. However, as reviewed above, several different definitions of contact angle were used in formulas, and the choice of an appropriate angle is still an open question. Since the advancing contact angle is greatly influenced by experimental conditions such as fluid viscosity,22 its accurate determination also poses another challenge.19 On the other hand, some studies15,20,23 showed that surface wettability has a significant effect on maximum spreading, while others24,25 revealed that wettability plays a minor role in drop spreading, implying that the effects of surface wettability or contact angle are still questionable. Roisman7 developed an analytical self-similar solution for the viscous flow in a liquid drop based on the full Navier−Stokes equations. The expression of boundary layer thickness was used to estimate the residual film thickness formed by drop impact, which was capable of developing a new scaling relation for the drop maximum spreading diameter. A semiempirical correlation was also provided to predict the maximum spreading factor. In an earlier work, Roisman et al.26 developed a model for the time10054

DOI: 10.1021/acs.iecr.9b02014 Ind. Eng. Chem. Res. 2019, 58, 10053−10063

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impact velocity, showing a fairly smooth evolution between 5 and 2.5 ms on a hydrophobic substrate but decreasing from 16 to 2.5 ms on a hydrophilic substrate. Lee et al.21 showed that tmax also has a dependence on surface tension. Besides, they developed an analytical model for tmax, using the ratio of surface tension of the given fluid to surface tension of the reference fluid, to replace the constant of 8/3 in Pasandideh-Fard et al. Note that the model by Lee et al. could not be solved directly due to its close link to the model of the maximum spreading factor. Thus, further clarifications and simple predicting methods for tmax are needed. Table S1 summarizes the historical models or correlations used to predict the maximum spreading factor. Note that almost all those models or empirical correlations with different forms were declared to have a good agreement with measurements.37 However, as reported by Marengo et al.,35 it is not easy to confirm their validity and to suggest a most suitable model based on those agreements. Therefore, it is necessary to assess those predicting methods with a consolidated database from various sources. Healy et al.36 assessed several old models before 1996 from refs 10, 14, and 16, using experimental data in the literature, and found that the Kurabayashi−Yang equation provides a good estimation, which was later modified in the follow-up study of Healy et al.15 Ukiwe and Kwok18 assessed models before 2005 from refs 10, 12, 14, 15, 17, and 26 using their own limited data. In the most recent decade, several new predicting methods for the maximum spreading have sprung up, such as those from refs 7, 19, 25, 32, 33, and 38. On the other hand, as a basic unit in many techniques, the fundamental knowledge of drop impact is being utilized to model the predicting methods of the overall heat and mass transfer performance for specific techniques, such as the earlier mentioned spray cooling, which points to an urgent need for reliable and easy-implemented approaches to estimate the maximum spreading factor and its corresponding time. 1.3. Objective of Study. This study provides an experimental treatment and assessment of the maximum spreading factor associated with single liquid drop impinging on a solid surface. First, an experimental system for drop impinging high-speed photography is built up. The maximum spreading factor and its corresponding time are carefully measured and discussed using different fluids. Second, four prevalent models or correlations for estimating the maximum spreading factor are assessed with a consolidated database amassed from 11 sources in the literature, consisting of 213 data points with broad parameter ranges. On the basis of this assessment, universal predicting methods with good applicability are recommended. Also presented are the results and scaling method of the time corresponding to the maximum spreading.

dependent drop spreading scale by solving mass and momentum equations of the rim bounding the spreading film, accounting for the effects of inertial, viscous, and surface forces and wettability. Aside from those theoretical models for predicting the maximum spreading scale, some scaling approaches and empirical correlations were also developed. Marmanis and Thoroddsen27 related the maximum spreading to a nondimensional group, Re1/2We1/4, which they called “impact Reynolds number”, where Reynolds number Re and Weber number We, are defined as Re =

ρf vdropddrop μf

(1a)

and We =

ρf vdrop2ddrop σ

(1b)

respectively. This scaling is similar to the later widely used approach, ReWe1/2, by Bayer and Megaridis,25 Scheller and Bousfield,28 and Seo et al.29 Clanet et al.30 divided the spreading into the capillary regime, corresponding to low velocity impact, and the viscous regime, which corresponds to relatively high velocity impact. They also reported that the maximum spreading in the capillary regime can be scaled by We1/4, based on their momentum balance by taking an effective capillary length into account, while the other scaling of We1/2 based on energy balance was also reported by Collings et al.11 and Bennett and Poulikakos.12 On the other hand, in the viscous regime, according to the assumption of pancake-shaped drop and energy balance approach, Madejski,10 Bennett and Poulikakos,12 and Fedorchenko et al.31 reported that the maximum spreading extent can be scaled by Re1/5. Meanwhile, an alternative scaling method of Re1/4 in the viscous regime was recommended by Pasandideh-Fard et al.17 However, Eggers et al.32 and Laan et al.33 found that these scaling methodologies with single dimensionless parameters fail to scale the maximum spreading factor with the broad parameter ranges. Using a combination of viscous regime with Re1/5 and capillary regime with We1/2 or We1/4, Eggers et al. and Clanet et al. showed that the maximum spreading factor can be scaled by Re1/5f(We Re−2/5) (energy balance based) or Re1/5f(We Re−4/5) (momentum balance based), respectively, where f is a function of the parameter We Re−2/5 or We Re−4/5, varying between zero, representing capillary regime, and infinity, representing viscous regime. Eggers et al. revealed that the f function, based on the energy balance rather than that based on the momentum balance, is capable of scaling the maximum spreading factor effectively, which was also confirmed by Lagubeau et al.34 Later, Laan et al. successfully smoothed the two asymptotics of viscous and capillary regimes, the latter of which is based on the energy balance, using a Padé approximant signifying a ratio of two firstorder polynomials that approximates the f function. This approach has achieved great success in predicting the maximum spreading factor for both Newtonian and non-Newtonian fluids. For the time scale corresponding to the maximum spreading, tmax, Pasandideh-Fard et al.17 theoretically derived tmax = (8/3) ddrop/vdrop based on their simple geometric assumptions of the drop, from spherical cap into cylindrical disk. However, Vadillo et al.19 proved that this model fails to estimate their measurements due to absence of spreading hydrodynamics in the model of Pasandideh-Fard et al. Vadillo et al. also found that tmax has a strong dependence on both surface wettability and

2. EXPERIMENTAL METHODS 2.1. High-Speed Photography System. Shown in Figure 2 is the schematic of experimental apparatus used in this study. The main components include a syringe along with a hypodermic needle to produce the liquid drop under controlled pressure, an adjustable test platform to sustain a stainless steel substrate, a high-speed camera, a xenon lamp used to provide illumination for high-speed photography, a light diffuser, and a data acquisition computer. Production of single liquid drop through the stainless steel flat-tipped needle is realized by imposing a certain pressure to the fluid in the syringe. The pressure and distance from the needle to the impact surface are adjusted using the control 10055


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3. RESULTS AND DISCUSSION 3.1. Impact Evolutions of Different Fluids. Shown in Figure 3 are evolution images after single drop impact on a solid

Figure 2. Schematic of experimental apparatus.

system of Kruss DSA 30, which is also used to measure the equilibrium contact angle and provide the test platform with adjustable positioning. The impact process is recorded by a Phantom V12.1 high-speed digital camera with capacity of 106 frames per second, which is equipped with a 100 mm, f-2.8 Tokina macrolens. The camera is aligned with a tilted angle of 5° to the impact surface. In order to obtain photographs with sufficient image resolution, the shooting speed is set as 10 000 frames per second, with 1024 × 512 pixels in each image. The back light method is employed in the experiments to expose the impact images, and the cold light source is provided by a xenon lamp XD-300 with a power of 350 W. A light diffuser is used between the surface and the xenon lamp to make the light be distributed uniformly. Before each experiment, several liquid drops are purposely cleared away to ensure that the liquid remains free of any air bubbles. The entire impinging process occurs in a very short time, so the trigger mode of the camera is selected as post-trigger in its control software; namely, the camera captures video all the time until it is interrupted after impact, and then the impact video can be located in buffer memory. 2.2. Data Acquisition. In this study, three fluids of water, butanol, and ethanol are adopted to study the effects of liquid properties. The drop diameter is acquired by pixel analyzing, which can be fulfilled by using the commercial software MATLAB 7.1, and calibration is performed by using a reference. The drop is approximated to an ellipse and its diameter is measured in both horizontal and vertical directions, and the drop diameter is defined as ddrop = (dh2dv)1/3, where dh and dv are the measured values in horizontal and vertical, respectively. The spreading diameter is acquired by programming in the software LabVIEW, based on pixel analyzing with automatic identification of interface. The uncertainty in pixel analyzing is 1 pixel, which corresponds to an error of 0.025 mm. The distance between needle tip and surface is adjusted to vary drop impinging velocity, vdrop, which is derived by tracking the location of the drop centroid in two images with 0.5 ms time spacing prior to impact, with an accuracy of ±0.05 m/s. The experiment for each impact velocity is repeated at least two times. Experimental conditions and liquid properties are summarized in Table S2. Average surface roughness of the stainless steel substrate used in this study is less than 0.1 μm.

Figure 3. Evolution images after drop impact with vdrop = 0.39 m/s: (a) water, (b) butanol, and (c) ethanol.

surface for fluids of water, butanol, and ethanol, respectively, in which impact velocity is 0.39 m/s and 0.0 ms is set as the drop exactly contacts the solid. The select fluids of water and ethanol are to study the surface tension effects, which bear comparable viscosity, while the fluids of ethanol and butanol having similar surface tension are selected to study the viscosity effects. Figure 3 shows that the evolution morphologies have a strong dependence on fluid properties, especially surface tension. The drop begins to spread and wet the solid surface after impact, in which its potential energy and kinetic energy convert to surface energy of the splat with viscous dissipation. It spreads to a maximum extent when the kinetic energy approximates zero and surface energy reaches maximum, after which the water drop starts to recoil, shown at 10.5 ms, and undergoes a series of damped oscillations before it achieves an equilibrium state. In contrast, no oscillation is observed for the butanol and ethanol drops due to their low surface tension forces. The second feature of interest is that the shape of water drop approximates a pancake, shown at 5.5 ms. Despite a crater on the drop top surface, this observation confirms the assumptions in many theoretical models for maximum spreading using the energy balance approach. In fact, most of those models were verified using water as the fluid. However, those models will lose their validation for the butanol and ethanol drops, as shown in Figure 3b and Figure 3c, in which the drops no longer show the pancake shape, especially at the rim of the splat. Another feature is the tiny bubble entrainment during the recoiling process of the 10056


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Figure 4. Effects of We on spreading factor for (a) water, (b) butanol, and (c) ethanol.

water drop, shown at 10.5 ms, caused by regain of the crater on the top surface of the drop. Figure 4 shows the effects of We (varied by changing impact velocity) on the evolution of spreading factor for the three fluids. Two curves are plotted for each fluid to make the changing trends clear. It shows that the qualitative observations in Figure 3 are verified well. The water spreading factor first increases, then decreases after a maximum value, which increases slightly after a minimum value, while no variations can be observed for the butanol and ethanol spreading factors once the maximum spreading is achieved. Also, the spreading factor shows a strong dependence on We. With the increasing of We, the spreading factor increases appreciably because of increased kinetic energy. However, the time corresponding to the maximum spreading factor decreases. One observation of interest for the water minimum spreading factor is that the time corresponding to the minimum increases slightly with increasing We. Shown in Figure 5 is the comparison of the spreading factor for water, butanol, and ethanol. Impact velocities for the three fluids are the same, vdrop = 0.78 m/s, so variations of the spreading factor are only determined by liquid properties. It shows that both surface tension and viscosity can influence the spreading magnitude greatly. The water spreading factor is smallest due to its much high surface tension, which is followed by butanol, and the ethanol spreading factor is largest because of its lowest viscosity. And the discrepancy of spreading factor between water and ethanol is higher than that between butanol and ethanol. This indicates that viscosity will be relegated to a secondary role if surface tension is considered. It is also clear that the time corresponding to the maximum spreading appears with the same trend to spreading factor for the three fluids in terms of fluid properties. Although only the velocity of 0.78 m/s is

Figure 5. Comparison of spreading factor for the three fluids with vdrop = 0.78 m/s.

exhibited in Figure 5, the effects of surface tension and viscosity also can be observed for other velocities performed in this study. In the following sections, the maximum spreading factor and its corresponding time of these three fluids will be mainly focused. Along with data of other fluids in the literature, the authors will provide a set of universal predicting methods for the maximum spreading through assessment of existing models or correlations. 3.2. Assessment of Maximum Spreading Factor. 3.2.1. Previous Predicting Methods. As discussed in section 1.2, several old models or correlations have been assessed, which therefore are not included in this study. Despite good capabilities of some previous predicting methods involving 10057


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of the non-Newtonian fluids and molten metal liquids are excluded. Overall, this consolidated database consists of 213 maximum spreading data points from 11 sources with the coverages of 4 ≤ Re ≤ 14191, 3 ≤ We ≤ 959, and 0° < αe ≤ 111°. Major fluids included are water, glycerol, water−glycerol mixture, ethanol, and butanol. 3.2.3. Assessment of Previous Methods. The parameter used to assess the accuracy of individual models or correlations is the mean absolute error, MAE, defined as

contact angle, validated using very limited measured data, they are excluded because of the confusion and inconsistency of both the effects of contact angle and its various definitions. Another cause for the exclusion of those methods is that values of contact angle are unfortunately not mentioned in many data sources. Even though some researchers used the estimated values of contact angle to verify those methods, it is not advisible due to complexities of contact angle as well as its various influencing parameters. In the present assessment, four most commonly used models or correlations for predicting the drop maximum spreading factor are selected, including those of Scheller and Bousfield,28 Bayer and Megaridis,25 Roisman,7 and Laan et al.,33 summarized in Table S3. Figure 6 compares the variations of the correlations

MAE =

1 N

∑

|βmax,pred − βmax,exp| βmax,exp

× 100% (2)

where βmax,pred and βmax,exp are predicted and measured maximum spreading factors, respectively. Computations of We and Re are based on the properties of fluids from different sources. For the fluids’ absence of properties in the sources, those properties are obtained using NIST’s REFPROP 8.0 software. Shown in Figure 7 are the consolidated data of the drop maximum spreading factor from 11 sources versus We, Re, and ReWe1/2, respectively. It shows that Re fails to scale the maximum spreading factor, but the scaling by the group number, ReWe1/2, is relatively good. For the scaling of We, partial data points have an apparent departure from others, as shown in the circled region in Figure 7a, which is caused by the high viscosity fluids of glycerol in An and Lee41 and Lee et al.,21 with the viscosity higher than 0.01 Pa·s. If these data points in the circled region are removed, We alone can also scale the maximum spreading factor successfully, which will be discussed later. Shown in Figure 8 is the comparison of measured data of maximum spreading factor with predictions of the four models or correlations listed in Table S3. Overall, all of these four methods can fairly estimate the maximum spreading scale. Methods of Scheller and Bousfield28 and Bayer and Megaridis25 slightly overestimate the consolidated data of small values at the small impact velocity, with comparable MAE of 15.9% and 15.8%, respectively, while the method of Roisman7 underestimates this part of the small values, with lower MAE of 13.3%. Instead, predictions by the most recent method of Laan et al.33 bear a more uniform distribution, with MAE of 9.5%, having good capabilities. One major reason for this is that the Laan et al. method not only accounted for viscous and capillary regimes during drop impact but also considered the smooth transition between the two regimes with an energy-conservation scheme. Note that the measured data in the consolidated database originated from 11 sources with very broad parameter ranges, which explains why MAEs of these four methods are higher than that from the single or limited sources with MAE of even less than 5%. In fact, one tends to select the measured data in the literature in accordance with their predictions, to validate their predicting methods and to lower the MAE. Thus, MAE of 9.5% computed by 213 data points from 11 sources can be deemed very good. As discussed in Figure 7c, the relatively good scaling method by the combination of We and Re, i.e., ReWe1/2, is capable of fitting the experimental data. Through a proper regression of consolidated data of the maximum spreading factor, the following formula is obtained,

Figure 6. Variations of maximum spreading factor against impinging velocity for different models or correlations, using water as fluid.

or models in Table S3 versus impinging velocity, using water as fluid. The correlation of Bayer and Megaridis is not plotted in this figure because it has the same form with Scheller and Bousfield, the purpose of which is to show the comparison clearly. As shown in Figure 6, the discrepancy between different curves decreases with increasing impact velocity, which implies that the effects of surface tension and viscosity become minor at the high velocity. However, at the low velocity, these three predicting methods differ greatly. In particular to the model of Roisman, the maximum spreading factor even shows incorrect negative values when the impact velocity is less than 0.3 m/s. This indicates that the errors of different methods are mainly caused by the prediction in the low velocity region. 3.2.2. Consolidated Database. In this study, a total of 213 data points of the maximum spreading factor for Newtonian liquid drop impact on a solid surface are amassed from 10 sources13,17,19,21,39−44 in the literature and from this study. The authors avoid the data points from sources listed in Table S3, the purpose of which is to provide the most objective assessment for those methods. All these data are obtained from tables in the above 11 sources or extracted from digitalized figures using commercial software. Table S4 provides the key information on the individual databases incorporated in the consolidated database. Some of the data are purposely excluded from the individual databases. These include those of shear-thinning liquids of An and Lee41 ink and silicon oil of Kim and Chun.44 Note that all data points

βmax = 0.505(Re We1/2)0.178

(3)

The comparison of consolidated data with predictions by eq 3 is shown in Figure 9. It shows that the distribution of data points is 10058


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Figure 7. Consolidated data of maximum spreading factor from 11 sources against (a) We, (b) Re, and (c) ReWe1/2. (d) is the legends.

Figure 8. Comparison of experimental data of maximum spreading factor with predictions of previous methods: (a) Scheller and Bousfield,28 (b) Bayer and Megaridis,25 (c) Roisman,7 and (d) Laan et al.33

similar to those of Scheller and Bousfield28 and Bayer and Megaridis25 because of the same form for the three formulas, but

eq 3 has lower MAE of 13.6%. Unfortunately, MAE of eq 3 is still higher than that of Laan et al.33 10059


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Table S5 provides detailed MAE of the previous individual models or correlations and the new proposed formulas in this study. 3.3. Time Scale Corresponding to Maximum Spreading. In the studies of drop impact,45−47 the dimensionless time, τ = vdropt/ddrop, is usually defined to compare dynamic parameters at different experimental conditions. Accordingly, the time corresponding to the maximum spreading can be defined as τmax = vdroptmax/ddrop. Figure 11a and Figure 11b show

Figure 9. Comparison of experimental data of maximum spreading factor with eq 3.

In applications such as spray cooling and falling film evaporation, the low viscosity fluids are commonly used, including dielectric fluorocarbon liquids of FC-72, HFE-7100, and FC-87, refrigerants of R-141b and R-113, and water, etc. Thus, it is very necessary to provide a more practical predicting method for the maximum spreading factor of these low viscosity fluids. As discussed earlier, We has a good capability in scaling the maximum factor. By fitting of the 173 data points for fluids with viscosity lower than 0.01 Pa·s, the following formula can be obtained, βmax = We1/4

(4)

Shown in Figure 10 is the comparison of experimental data of the maximum spreading factor for the low viscosity fluids with predictions of eq 4, indicating that this formula has a quite good agreement with the consolidated data of μf < 0.01 Pa·s, with MAE of 7.3%. Note that the form of eq 4 is consistent with the model of Clanet et al.30 based on the momentum balance.

Figure 11. Scaling of the time corresponding to maximum spreading with (a) Re and (b) We.

τmax versus Re and We, respectively, in which experimental data of water and glycerol from Lee et al.21 are also plotted to make a comparison. Note that the measured data of butanol and ethanol are not plotted because their measuring errors of tmax are very large. This can be ascribed to the capillary effect at the rim of the splat for the fluids with low surface tension. When the spreading reaches its maximum, the splat rim still extends slowly, resulting in large uncertainties in determining tmax. These uncertainties were also reflected for the fluid of ethanol with large scatters in Lee et al. Thus, only tmax of the fluids with high surface tension could be measured accurately. As shown in Figure 11a, the water data agree well with that in Lee et al. Also it shows that Re fails to scale τmax. Instead, when τmax is scaled by We, all data collapse

Figure 10. Comparison of experimental data of maximum spreading factor for low viscosity fluids with predictions of eq 4. 10060


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Industrial & Engineering Chemistry Research onto a single curve, as shown in Figure 11b. This indicates that τmax strongly depends on surface tension, and the effects of liquid viscosity on τmax is very minor. In fact, the viscosity of glycerol in Lee et al. (10−2 Pa·s) is about 1 order of magnitude higher than water (10−3 Pa·s). Pasandideh-Fard et al.17 theoretically derived τmax = 8/3 using their simplified assumptions. From Figure 11 it is clear that τmax is a function of impinging velocity rather than a constant. In the authors’ previous study,8 a power law of τmax = aWeb, derived from the period of a freely oscillating drop, is used to fitting measure data of τmax on a heated wall in the film boiling regime. This approach is applied for τmax on a solid surface at room temperature in this study, and then the following formula is obtained, τmax = 0.44We 0.3

(5)

The comparison of predictions by eq 5 with measured data of water and glycerol in the present study and Lee et al.21 is shown in Figure 12, indicating that this formula agrees generally well

Figure 13. Comparison of the time corresponding to maximum spreading between eq 5 and eq 6.

should be selected carefully when modeling the heat and mass transfer processes in applications.

4. CONCLUSIONS This paper investigates single liquid drop impinging on a solid surface, focusing mainly on the maximum spreading after impact. The transient spreading processes with fluids of water, butanol, and ethanol are observed using a high-speed video camera. Four prevalent models or correlations for predicting the maximum spreading factor are assessed based on a consolidated database consisting of 213 data points from 11 sources. Finally, the time corresponding to the maximum spreading is also studied. Key conclusions can be summarized as follows. (1) Drop spreading morphologies have a strong dependence on the fluid properties, especially surface tension. The fluids with high surface tension will undergo spreading, recoiling, and a series of damped oscillation, and the phenomenon of tiny bubble entrainment also can be observed. Both surface tension and viscosity influence spreading magnitude appreciably, while viscosity is relegated to a secondary role when considering surface tension. (2) The model of Laan et al.33 has the best agreement with the consolidated database, with overall MAE of 9.5%, followed by the formula through a regression of the consolidated database using the dimensionless group number of ReWe1/2 as the scaling approach and by the model of Roisman.7 A more practical formula for the fluids with low viscosity of μf < 10−2 Pa·s is also proposed for modeling studies in specific applications, with overall MAE of 7.3%. (3) The time corresponding to maximum spreading has a strong dependence on surface tension, in contrast to the negligible effects of viscosity. The dimensionless time, τmax, increases with increasing the impact velocity, which can be scaled by We successfully. A formula for the fluids with high surface tension is obtained for predicting τmax, with MAE of 13.2%, based on the data in this study along with the data in the literature. This new formula is independent on surface wettability.

Figure 12. Comparison of experimental data of the time corresponding to maximum spreading with predictions of eq 5.

with the measured data, in which overall MAE is 13.2%. Note that the τmax data of water and glycerol from Lee et al. included in Figure 12 were measured on both hydrophilic and hydrophobic surfaces with equilibrium contact angles from 52.4° to 109.6°, which implies that wettability has negligible effects on τmax. As discussed in section 1.1, the mechanisms of liquid drop spreading on a heated surface in the film boiling regime and spreading on a solid surface at room temperature differ greatly, in the former of which the drop is separated from the surface by a vapor layer. Regardless these facts, Figure 13 compares the dimensionless time in relation to the maximum spreading for the two cases arbitrarily. The formula of τmax for liquid drop spreading on the heated surface in Liang et al.8 is expressed as τmax = 0.247We 0.477

(6)

It shows that the difference in predictions of eqs 5 and 6 is minor for We < 30. However, the prediction of eq 6 is higher than that of eq 5 for We > 30, and this discrepancy is enlarged with increasing We. This information indicates that the drop contacting with the solid or not could greatly influence the time to achieve a maximum extent, and an appropriate formula 10061


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(4) Liang, G.; Mudawar, I. Review of spray cooling−Part 1: Singlephase and nucleate boiling regimes, and critical heat flux. Int. J. Heat Mass Transfer 2017, 115, 1174. (5) Liang, G.; Mudawar, I. Review of spray cooling−Part 2: High temperature boiling regimes and quenching applications. Int. J. Heat Mass Transfer 2017, 115, 1206. (6) Ma, T.; Feng, L.; Wang, H.; Liu, H.; Yao, M. A numerical study of spray/wall impingement based on droplet impact phenomenon. Int. J. Heat Mass Transfer 2017, 112, 401. (7) Roisman, I. V. Inertia dominated drop collisions. II. An analytical solution of the Navier−Stokes equations for a spreading viscous film. Phys. Fluids 2009, 21, No. 052104. (8) Liang, G.; Shen, S.; Guo, Y.; Zhang, J. Boiling from liquid drops impact on a heated wall. Int. J. Heat Mass Transfer 2016, 100, 48. (9) Jones, H. Cooling, freezing and substrate impact of droplets formed by rotary atomization. J. Phys. D: Appl. Phys. 1971, 4, 1657. (10) Madejski, J. Solidification of droplets on a cold surface. Int. J. Heat Mass Transfer 1976, 19, 1009. (11) Collings, E. W.; Markworth, A. J.; McCoy, J. K.; Saunders, J. H. Splat-quench solidification of freely falling liquid-metal drops by impact on a planar substrate. J. Mater. Sci. 1990, 25, 3677. (12) Bennett, T.; Poulikakos, D. Splat-quench solidification: estimating the maximum spreading of a droplet impacting a solid surface. J. Mater. Sci. 1993, 28, 963. (13) Mao, T.; Kuhn, D.; Tran, H. Spread and rebound of liquid droplets upon impact on flat surfaces. AIChE J. 1997, 43, 2169. (14) Yang, W. J. Theory on vaporization and combustion of liquid drops of pure substances and binary mixtures on heated surfaces. No. 535 Report; Institute of Space and Aeronautical Science, University of Tokyo, 1975. (15) Healy, W. M.; Hartley, J. G.; Abdel-Khalik, S. I. Surface wetting effects on the spreading of liquid droplets impacting a solid surface at low Weber numbers. Int. J. Heat Mass Transfer 2001, 44, 235. (16) Chandra, S.; Avedisian, C. T. On the collision of a droplet with a solid surface. Proc. R. Soc. London, Ser. A 1991, 432, 13. (17) Pasandideh-Fard, M.; Qiao, Y. M.; Chandra, S.; Mostaghimi, J. Capillary effects during droplet impact on a solid surface. Phys. Fluids 1996, 8, 650. (18) Ukiwe, C.; Kwok, D. Y. On the maximum spreading diameter of impacting droplets on well-prepared solid surfaces. Langmuir 2005, 21, 666. (19) Vadillo, D. C.; Soucemarianadin, A.; Delattre, C.; Roux, D. C. D. Dynamic contact angle effects onto the maximum drop impact spreading on solid surfaces. Phys. Fluids 2009, 21, 122002. (20) Lee, J. B.; Laan, N.; de Bruin, K. G.; Skantzaris, G.; Shahidzadeh, N.; Derome, D.; Carmeliet, J.; Bonn, D. Universal rescaling of drop impact on smooth and rough surfaces. J. Fluid Mech. 2016, 786, R4. (21) Lee, J. B.; Derome, D.; Guyer, R.; Carmeliet, J. Modeling the maximum spreading of liquid droplets impacting wetting and nonwetting surfaces. Langmuir 2016, 32, 1299. (22) Š ikalo, Š ; Tropea, C.; Ganić, E. N. Dynamic wetting angle of a spreading droplet. Exp. Therm. Fluid Sci. 2005, 29, 795. (23) Jung, S.; Hutchings, I. M. The impact and spreading of a small liquid drop on a non-porous substrate over an extended time scale. Soft Matter 2012, 8, 2686. (24) Rioboo, R.; Marengo, M.; Tropea, C. Time evolution of liquid drop impact onto solid, dry surfaces. Exp. Fluids 2002, 33, 112. (25) Bayer, I. S.; Megaridis, C. M. Contact angle dynamics in droplets impacting on flat surfaces with different wetting characteristics. J. Fluid Mech. 2006, 558, 415. (26) Roisman, I. V.; Rioboo, R.; Tropea, C. Normal impact of a liquid drop on a dry surface: model for spreading and receding. Proc. R. Soc. London, Ser. A 2002, 458, 1411. (27) Marmanis, H.; Thoroddsen, S. T. Scaling of the fingering pattern of an impacting drop. Phys. Fluids 1996, 8, 1344. (28) Scheller, B. L.; Bousfield, D. W. Newtonian drop impact with a solid surface. AIChE J. 1995, 41, 1357.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b02014. Tables in this study (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Gangtao Liang: 0000-0002-4126-207X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support by the National Natural Science Foundation of China under Grant 51876025 and the Fundamental Research Funds for the Central Universities under Grant DUT19JC10 is gratefully acknowledged.

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a b D d f g N Re t v We

NOMENCLATURE constant exponent spreading diameter diameter function gravitational acceleration number of data points Reynolds number time velocity Weber number

Greek Symbols

α β μ ρ σ τ

contact angle spreading factor dynamic viscosity density surface tension dimensionless time

Subscripts

a d drop e exp f h max pred v w Y

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advancing dynamic liquid drop equilibrium experiment liquid horizontal maximum predicted vertical surface Young

REFERENCES

(1) Liang, G.; Mudawar, I. Review of mass and momentum interactions during drop impact on a liquid film. Int. J. Heat Mass Transfer 2016, 101, 577. (2) Liang, G.; Mudawar, I. Review of drop impact on heated walls. Int. J. Heat Mass Transfer 2017, 106, 103. (3) Josserand, C.; Thoroddsen, S. T. Drop impact on solid surface. Annu. Rev. Fluid Mech. 2016, 48, 365. 10062


Article

Industrial & Engineering Chemistry Research (29) Seo, J.; Lee, J. S.; Kim, H. Y.; Yoon, S. S. Empirical model for the maximum spreading diameter of low-viscosity droplets on a dry wall. Exp. Therm. Fluid Sci. 2015, 61, 121. (30) Clanet, C.; Béguin, C.; Richard, D.; Quéré, D. Maximal deformation of an impacting drop. J. Fluid Mech. 1999, 517, 199. (31) Fedorchenko, A. I.; Wang, A. B.; Wang, Y. H. Effect of capillary and viscous forces on spreading of a liquid drop impinging on a solid surface. Phys. Fluids 2005, 17, No. 093104. (32) Eggers, J.; Fontelos, M. A.; Josserand, C.; Zaleski, S. Drop dynamics after impact on a solid wall: theory and simulations. Phys. Fluids 2010, 22, No. 062101. (33) Laan, N.; de Bruin, K. G.; Bartolo, D.; Josserand, C.; Bonn, D. Maximum diameter of impacting liquid droplets. Phys. Rev. Appl. 2014, 2, No. 044018. (34) Lagubeau, G.; Fontelos, M. A.; Josserand, C.; Maurel, A.; Pagneux, V.; Petitjeans, P. Spreading dynamics of drop impacts. J. Fluid Mech. 2012, 713, 50. (35) Marengo, M.; Antonini, C.; Roisman, I. V.; Tropea, C. Drop collisions with simple and complex surfaces. Curr. Opin. Colloid Interface Sci. 2011, 16, 292. (36) Healy, W. M.; Hartley, J. G.; Abdel-Khalik, S. I. Comparison between theoretical models and experimental data for the spreading of liquid droplets impacting a solid surface. Int. J. Heat Mass Transfer 1996, 39, 3079. (37) Asai, A.; Shioya, M.; Hirasawa, S.; Okazaki, T. Impact of an ink drop on paper. J. Imaging Sci. Technol. 1993, 37, 205. (38) Wildeman, S.; Visser, C. W.; Sun, C.; Lohse, D. On the spreading of impacting drops. J. Fluid Mech. 2016, 805, 636. (39) Ford, R. E.; Furmidge, C. G. L. Impact and spreading of spray drops on foliar surfaces. Wetting; Society of Chemical Industry: London, 1967. (40) Cheng, L. Dynamic spreading of drops impacting onto a solid surface. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 192. (41) An, S. M.; Lee, S. Y. Maximum spreading of a shear-thinning liquid drop impacting on dry solid surfaces. Exp. Therm. Fluid Sci. 2012, 38, 140. (42) Zhang, X.; Basaran, O. A. Dynamic surface tension effects in impact of a drop with a solid surface. J. Colloid Interface Sci. 1997, 187, 166. (43) Kim, H.-Y.; Park, S.-Y.; Min, K. Imaging the high-speed impact of microdrop on solid surface. Rev. Sci. Instrum. 2003, 74, 4930. (44) Kim, H.-Y.; Chun, J.-H. The recoiling of liquid droplets upon collision with solid surfaces. Phys. Fluids 2001, 13, 643. (45) Liang, G.; Yang, Y.; Guo, Y.; Zhen, N.; Shen, S. Rebound and spreading during a drop impact on wetted cylinders. Exp. Therm. Fluid Sci. 2014, 52, 97. (46) Rieber, M.; Frohn, A. A numerical study on the mechanism of splashing. Int. J. Heat Fluid Flow 1999, 20, 455. (47) Liang, G.; Guo, Y.; Shen, S.; Yang, Y. Crown behavior and bubble entrainment during a drop impact on a liquid film. Theor. Comput. Fluid Dyn. 2014, 28, 159.

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Maximum Spreading for Liquid Drop Impacting on Solid Surface

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