Effects of Tube Radius and Surface Tension on Capillary Rise Dynamics of Water/Butanol Mixtures

applied sciences

Article Effects of Tube Radius and Surface Tension on Capillary Rise Dynamics of Water/Butanol Mixtures

Seungyeop Baek 1,†, Sungjin Jeong 2,†, Jaedeok Seo 1,3, Sanggon Lee 1, Seunghwan Park 4, Jaeyoun Choi 5, Hyomin Jeong 2 and Yonmo Sung 2,*

1 Graduate Program, Department of Energy and Mechanical Engineering, Gyeongsang National University, Tongyeonghaean-ro 2, Tongyeong-si 53064, Gyeongsangnam-do, Korea; [email protected] (S.B.); [email protected] (J.S.); [email protected] (S.L.) 2 Department of Energy and Mechanical Engineering, Gyeongsang National University, Tongyeonghaean-ro 2, Tongyeong-si 53064, Gyeongsangnam-do, Korea; [email protected] (S.J.); [email protected] (H.J.) 3 Department of Automobile, Korea Polytechnic VII, Changwon-si 51518, Gyeongsangnam-do, Korea 4 Hydrogen Technology Planning Team, Hyundai Steel Company, Dangjin-si 31719, Chungcheongnam-do, Korea; [email protected] 5 Environment and Energy Research Team, Hyundai Steel Company, Dangjin-si 31719, Chungcheongnam-do, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-55-772-9115 † Co-First authors.

Abstract: Capillary-driven action is an important phenomenon which aids the development of high-performance heat transfer devices, such as microscale heat pipes. This study examines the capillary rise dynamics of n-butanol/water mixture in a single vertical capillary tube with different

radii (0.4, 0.6, and 0.85 mm). For liquids, distilled water, n-butanol, and their blends with varying concentrations of butanol (0.3, 0.5, and 0.7 wt.%) were used. The results show that the height and

Citation: Baek, S.; Jeong, S.; Seo, J.; velocity of the capillary rise were dependent on the tube radius and liquid surface tension. The Lee, S.; Park, S.; Choi, J.; Jeong, H.; larger the radius and the higher the surface tension, the lower was the equilibrium height (he) and Sung, Y. Effects of Tube Radius and the velocity of rise. The process of capillary rise was segregated into three characteristic regions: Surface Tension on Capillary Rise purely inertial, inertial + viscous, and purely viscous regions. The early stages (purely inertial and

Dynamics of Water/Butanol Mixtures. inertial + viscous) represented the characteristic heights h1 and h2, which were dominant in the Appl. Sci. 2021, 11, 3533. https:// capillary rise process. There were linear correlations between the characteristic heights (h1, h2, and doi.org/10.3390/app11083533 he), tube radius, and surface tension. Based on these correlations, a linear function was established between each of the three characteristic heights and the consolidated value of tube radius and surface Academic Editor: Guan Heng Yeoh tension (σL/2πr2).

Received: 14 March 2021 Keywords: capillary tube; surface tension; butanol/water mixture; liquid rise; capillarity; meniscus Accepted: 14 April 2021 Published: 15 April 2021

Publisher’s Note: MDPI stays neutral 1. Introduction with regard to jurisdictional claims in published maps and institutional affil- The capillary penetration of wetting fluids into narrow spaces such as cylindrical (or iations. square) tube and porous media is important in many industrial and natural processes. For example, wettability changes in the oil production process like the chemical enhanced oil recovery have a significant effect on capillary pressure, irreducible water saturation, and residual oil saturation [1–4]. In addition, capillary-driven flow is a fundamental

Copyright: © 2021 by the authors. phenomenon in important applications such as the development of high-performance heat Licensee MDPI, Basel, Switzerland. transfer and thermal management systems. In particular, the cooling of modern electronic This article is an open access article devices requires a suitable configuration for high-performance devices with reduced-size distributed under the terms and because heat loads and heat fluxes increase exponentially [5,6]. In light of the continued conditions of the Creative Commons miniaturization of electronic components, micro heat pipes are promising micro-scale Attribution (CC BY) license (https:// cooling devices and their operation is sustained by two-phase heat transfer and capillary creativecommons.org/licenses/by/ forces [7]. Therefore, capillary-based flows can replace electric pumps that consume space 4.0/).

Appl. Sci. 2021, 11, 3533. https://doi.org/10.3390/app11083533 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 3533 2 of 13

and require electrical power to operate in microscale heat pipes. This implies that capillary flow does not require any additional power-consuming actuation devices [8,9]. Li et al. [10] investigated the capillary pumping performance of a porous wick for a looped heat pipe with water and acetone as the working fluid, and they found that the capillary pumping rate increased with increasing porosity of the wicks. Huang et al. [11] developed a novel stainless steel fiber-powder composite wick with high capillary pressure and good permeability for use in stainless steel heat pipes. Their work showed that the powder particle size has a significant influence on the capillary performance of the wick. Li et al. [12] showed that wicks using working fluids with lower viscosity, larger surface tension, and higher density demonstrated better capillary performance by comparing the results of three different working fluids: water, acetone, and ethanol. Generally, micro heat pipes have a wickless configuration, and hence, the capillary dynamics in the micro-pipes is utilized to drive the working fluids. Moreover, the operation of the micro-heat pipe is governed by transportation of the working fluids and aided by the capillaries or microscale pipes, and its overall thermal performance is predominantly determined by the thermal properties of the working fluids and the size of the capillaries. Thus, the selection of the type of working fluid and the physical size of the system, as well as their impacts on the capillary dynamics are of technological interest. Therefore, the capillary rise dynamics of mixtures with different surface tension and capillary tubes of several hundred micrometers were investigated. Cao et al. [13] investigated the dynamics of water/ethanol mixtures in capillary tubes and found that surface tension plays an important role in the travelling speed inside the capillary tubes. O’Loughlin et al. [14] investigated the capillary rise dynamics of glycerol and aqueous glycerol solutions in vertical glass capillaries of various radii. Extrand and Moon [15] measured the forces and energies generated by the rising liquids such as ethylene glycol, glycerol, and silicone oil in capillary tubes with diameters ranging from 0.5 to 1.4 mm. They found that larger diameter tubes produced a smaller height in the capillary rise and larger forces in terms of capillary dynamics. Quéré [16] found that the early stage of the capillary rise has a linear relationship with the position of the meniscus height versus time, and the meniscus oscillation around the equilibrium height occurs when the liquid viscosity is sufficiently low. Hamraoui et al. [17] investigated the capillary rise dynamics of pure water and ethanol as well as mixtures using a high-speed imaging technique. They found that the dynamic inertial effect is significant at the initial stages and may be important for capillaries of larger radii. Stange et al. [18] found that capillary-driven flow is divided into three successive phases: inertia, convective losses, and viscous forces which counteract the driving capillary force. Further, Fries and Dreyer [19] discussed the different time stages during early capillary rise, and they concluded that the purely inertial and purely viscous flow periods were separated by a viscoinertial stage, wherein both effects have to be considered. Zhong et al. [20] showed that the velocity of the capillary flow at the initial stage is determined by the Bo number (comparison of gravitational and surface tension forces), and the viscosity effect becomes significant in the final stage. In addition, Walls et al. [21] showed that the properties of liquids in capillary tube rise processes are significant in the early stages. The above studies showed that the properties such as surface tension and viscosity of liquids were important factors in the capillary rise dynamics. However, the dependency of the capillary rise dynamics on the liquid properties is not straightforward in microscale heat pipe systems because the process of capillary rise is a complex interaction between capillary size and fluid properties. Therefore, the influence of the various capillary sizes on the capillary rise dynamics is still indispensable. In particular, the inner diameter and the fluid properties cannot be considered independently, and the level of these conditions must be considered at the same time. To determine the effect of capillary size and liquid properties on the capillary dynamics for the performance enhancement of the microscale heat transfer systems, a consolidated correlation of those parameters is required. Therefore, in this study, effects of capillary tube Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 13

To determine the effect of capillary size and liquid properties on the capillary Appl. Sci. 2021, 11, 3533dynamics for the performance enhancement of the microscale heat transfer systems, a 3 of 13 consolidated correlation of those parameters is required. Therefore, in this study, effects of capillary tube radius (0.4, 0.6, and 0.85 mm) and the surface tension of water, n-butanol, and their blends on the capillary rise dynamics were investigated. The height of capillary rise and velocityradius were (0.4, measured 0.6, and 0.85using mm) a high-speed and the surface camera. tension The of capillary water, n-butanol, rise process and their blends was divided intoon the three capillary characteristic rise dynamics stages: wereI, II, and investigated. III, corresponding The height to purely of capillary inertial, rise and velocity inertial + viscous,were measuredand purely using viscous a high-speed region, respectively. camera. The The capillary characteristic rise process heights, was divided into including the threeequilibrium characteristic height stages:as functions I, II, and of the III, tube corresponding radius and to surface purely tension, inertial, wereinertial + viscous, then analyzedand and purely correlated. viscous region, respectively. The characteristic heights, including the equilibrium height as functions of the tube radius and surface tension, were then analyzed 2. Materials andand Methods correlated. 2.1. Materials 2. Materials and Methods In this study,2.1.Materials two types of liquids, distilled water (DW) and n-butanol (B), were used, including their blends.In this The study, blending two typesratio base of liquids,d on the distilled mass fraction water (DW) (wt.%) and was n-butanol 0%, 0.3%, (B), were used, 0.5%, 0.7% ofincluding n-butanol, their namely, blends. B0% The (DW), blending B0.3%, ratio B0.5%, based and on theB0.7%, mass respectively. fraction (wt.%) was 0%, ® Capillary tubes0.3%, (DURAN 0.5%, 0.7% High-Precision of n-butanol, namely,Glass Capillary, B0% (DW), SCHOTT B0.3%, B0.5%, AG, andLandshut, B0.7%, respectively. Germany) wereCapillary made of tubes borosilicate (DURAN glass.® High-Precision To investigate Glass the effect Capillary, of tube SCHOTT diameters AG, on Landshut, Ger- the capillary dynamicsmany) were considering made of borosilicate future researches glass. To on investigate the development the effect of ofmicroscale tube diameters on the heat pipe systems,capillary capillaries dynamics with considering three different future inner researches radii (r on = the0.4, development0.6, and 0.85 ofmm) microscale heat were employedpipe at systems,a fixed outer capillaries diameter with of three6 mm different and tube inner length radii (L)( rof= 150 0.4, mm 0.6, in and this 0.85 mm) were study. employed at a ﬁxed outer diameter of 6 mm and tube length (L) of 150 mm in this study.

2.2. Methods 2.2. Methods Figure 1 showsFigure the 1experimental shows the experimental setup designed setup for designedmeasuring for the measuring surface tension the surface tension (σ) and capillary(σ) andrise dynamics. capillary rise The dynamics. measurem Theent measurementsystem comprises system of three comprises subsystems: of three subsystems: a PC/monitoringa PC/monitoring part, capillary part, rising capillary part, and rising high-speed part, and imaging high-speed part, imaging as shown part, in as shown in Figure 1a–c, respectively.Figure1a–c, respectively.In the experiments, In the experiments, we raised the we height raised of thethe heightliquid using of the a liquid lab using a lab jack to connectjack the to bottom connect of thethe bottomcapillaries of the with capillaries the liquid, with hence the liquid,facilitating hence the facilitating entry of the entry of liquid into theliquid capillaries. into the The capillaries. height of Therise height(h) of the of riseliquid (h) was of the measured liquid was as measuredthe point as the point where the capillarywhere thetube capillary just touches tube justthe touchesliquid to the the liquid point to where the point the meniscus where the stops meniscus at stops at the the equilibriumequilibrium height (h heighte). These (he ).rising These processes rising processes were recorded were recorded using usinga high-speed a high-speed camera camera (FASTCAM(FASTCAM SA4, SA4,Photron Photron Ltd., Ltd., Tokyo Tokyo, Japan) Japan) connected connected to a to PC. a PC. The The σ firstσ ﬁrst measured measured usingusing dual dual capillary capillary tubes tubes based based on ondifferential differential capillary-rise-height capillary-rise-height method method [22], and [22], and the thecapillary capillary dynamics dynamics using using a asingle single capillary capillary tube waswas thenthen analyzed analyzed by by measuring the measuring theheight height of of rise rise and and its its velocity velocity [13 [13,14,23].,14,23].

Figure 1. SchematicFigure 1. of Schematic the experimental of the experimental setup. (a) PC setup. and monitor, (a) PC (andb) capillary monitor, tube (b) capillary and liquid, tube and and (c) liquid, high-speed camera and lens. and (c) high-speed camera and lens.

Figure 2 showsFigure the principle2 shows theof the principle different ofial the capillary differential rise capillarymethod and rise the method captured and the captured image for theimage difference for the in differencethe rising height. in the rising When height. the liquid When contacts the liquid the bottom contacts of the the bottom of the capillary tube,capillary the contact tube, forces the contact with the forces wall with of the wallcapillaries of the capillariesaid in the capillary aid in the rise capillary rise (he) for each of the capillary tubes. The height of the meniscus is a function of the capillary Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 13

Appl. Sci. 2021, 11, x FOR PEER REVIEW (he) for each of the capillary tubes. The height of the meniscus is4 aof function 13 of the capillary Appl. Sci. 2021, 11, 3533 4 of 13 radius. σ can be calculated from the difference in the height of the capillary tubes. σ in the captured image on the right side of Figure 2 is calculated by the following equation [24]:

(he) for each of the capillary tubes. The height of the meniscus is a function of the capillary radius. σ can be calculated from the difference𝜌𝑔Δℎ in the height𝑟𝑟 of the capillary tubes. σ in the radius. σ can be calculated from the difference in the 𝜎height = of the capillary tubes., σ in the (1) captured image on the right side of Figure2 is calculated by the following equation [24]: captured image on the right side of Figure 2 is calculated by the2 following𝑟 −𝑟 equation [24]: studies where σ is the surface tensionρg∆h (mN/m), r r g is the standard gravity (m/s2), ρ is the 𝜌𝑔Δℎσ =𝑟𝑟 e 1 2 , (1) 𝜎 = 3 , (1) density of the liquid (kg/m2 ), r 𝑟is −𝑟the2 capillaryr1 − r2 radius (mm), and Δhe is the difference in → equilibrium height (mm), which is experimentally measured at the rise2 height (h) (t ∞); studies wherestudies σ is the where surfaceσ is tension the surface (mN/m), tension g is (mN/m), the standardg is thegravity standard (m/s2 gravity), ρ is the (m/s ), ρ is the r1 is larger than r2. The σ for3 DW was measured with three capillary tube combinations (r1/ density of thedensity liquid of(kg/m the3 liquid), r is the (kg/m capillary), r is radius the capillary (mm), radiusand Δh (mm),e is the and difference∆he is the in difference in r2 = 0.8/1.2, 0.8/1.7, and 1.2 mm/1.7 mm). The measurements were conducted at room equilibrium heightequilibrium (mm), heightwhich (mm),is experimentally which is experimentally measured at the measured rise height at the (h rise) (t → height ∞); (h)(t → ∞); temperature (23 ± 1 °C), and the results are presented in Figure 3. The mean value of each r1 is larger thanr1 ris2. largerThe σ thanfor DWr2. Thewas σmeasuredfor DW was with measured three capillary with three tube capillary combinations tube combinations (r1/ (r1/ test case was 72.17, 72.27, and 71.99 mN/m at r1/ r2 = 0.8/1.2, 0.8/1.7, and 1.2 mm/1.7 mm, r2 = 0.8/1.2, 0.8/1.7,r2 = 0.8/1.2, and 1.2 0.8/1.7, mm/1.7 and mm). 1.2 mm/1.7The measurements mm). The measurements were conducted were at conductedroom at room ◦ temperature (23temperaturerespectively. ± 1 °C), and (23 theThus,± 1resultsC), the and aremean thepres resultsvalueented areofin Figureσ presented for DW 3. The inwas Figuremean 72.13 value3. ThemN/m of mean each at value 23 ± 1 of °C, each which was test case was test72.17,consistent case 72.27, was withand 72.17, 71.99 a 72.27, known mN/m and value 71.99 at r1/ mN/m ofr2 =72.29 0.8/1.2, at mN/mr1/ 0.8/1.7,r2 = at 0.8/1.2, and23 °C 1.2 0.8/1.7, [25]. mm/1.7 and mm, 1.2 mm/1.7 mm, ◦ respectively. Thus,respectively. the mean Thus, value the of mean σ for value DW ofwasσ for72.13 DW mN/m was 72.13at 23 mN/m± 1 °C, which at 23 ± was1 C, which was ◦ consistent withconsistent a known with value a knownof 72.29 value mN/m of at 72.29 23 °C mN/m [25]. at 23 C[25].

Figure 2. Principle of the measurement method and instantaneous image of the capillary rise Figure 2. Principle of the measurement method and instantaneous image of the capillary rise Figure 2. Principleheight. of the measurement method and instantaneous image of the capillary rise height. height.

Figure 3. Surface tension of DW (distilled water) with different tube radius combinations in the differential capillary-rise-heightFigureFigure 3. 3.Surface Surface method. tension tension of of DW DW (distilled (distilled water) water) with with different different tube tube radius radius combinations combinations in the in the differentialdifferential capillary-rise-height capillary-rise-height method. method. To measure the height of rise and velocity, individual images of the liquid invasion inside a single capillaryToTo measure measuretube were the the heightcaptured height of at rise of a frame andrise velocity,and rate velocity, of 1000 individual frames individual imagesper second images of the (fps) liquid of the invasion liquid invasion and a pixel resolutioninsideinside a a singleof single 640 capillary× capillary 640 using tube tube a hi weregh-speed were captured captured camera at aat in frame a conjunction frame rate rate of 1000 withof 1000 framesa 105 frames per per second second (fps) × mm lens (F2.8(fps) EXand DG anda pixelOS a pixelHSM resolution resolutionMACRO, of SIGMA, of 640 640 × 640Japan),640 using using as shown a a high-speedhigh-speed in Figure camera camera 1b. The in pixelin conjunction conjunction with with a 105 size (220 μm/pixel)a 105 mmcalibration lens (F2.8 was EX based DG on OS the HSM known MACRO, size of SIGMA, the tapeline. Japan), Every as shown image in Figure1b. mm lens (F2.8 EX DG OS HSM MACRO, SIGMA, Japan), as shown in Figure 1b. The pixel for the heightThe of rise pixel of sizethe liquid (220 µ wasm/pixel) obtained calibration with a time was resolution based on theof 1 knownms (1000 size fps), of the tapeline. size (220 μm/pixel) calibration was based on the known size of the tapeline. Every image for the height of rise of the liquid was obtained with a time resolution of 1 ms (1000 fps), Appl. Sci. 2021, 11, 3533 5 of 13

Every image for the height of rise of the liquid was obtained with a time resolution of 1 ms (1000 fps), and then the rise velocity was calculated using the frame-to-frame method (∆h/∆t). All capillary rise measurements were carried out at room temperature (23 ± 1 ◦C). Each experiment was repeated ﬁve times, and the mean value was calculated, along with standard deviation to measure the error. In addition, there was an additional error of ±1 pixel (220 µm) for the vertical observation because the capillary tip was not perfectly submerged inline with liquid surface in each experiment.

3. Results and Discussion 3.1. Effect of Tube Inner Radius on Capillary Dynamics Figure4 shows time series of images for DW, demonstrating the movements of the meniscus during 0.25 s, with different capillary tube radii. In every result obtained at the tube diameter—at 0.4 mm (Figure4a), 1.2 mm (Figure4b), and 1.7 mm (Figure4c)—the location of the meniscus exhibited a common trend with time. In the very early stage of the capillary rise near 0.05 s, as shown in Figure4c, the rise profile in the vertically oriented tube increased nonlinearly with time. Then, the rise degraded over time. The rise height corresponding to the meniscus location at 0.25 s decreases from 29 to 16 mm, with increase in the tube radius from 0.4 to 0.85 mm. As shown in Figures3 and4a, the maximum rise height at r = 0.4 and 0.6 mm was 29 and 23 mm at 0.25 s, respectively. However, the maximum value (17 mm) was observed at 0.08 s and not at 0.25 s, as shown in Figure4c. This phenomenon, called meniscus oscillation, has been previously reported by researchers [13,26,27]. The main reason for the oscillation is that the kinetic force may lead to the oscillation. More precisely, meniscus oscillation occurs when the boundary layer of viscous grows at a slow pace with low viscous forces, and keeping liquid inertia to balance the capillary and gravitational force only. In other words, the viscous force restrains the meniscus oscillation energy out while fluids rise up in the tube having radius [23]. In addition, this oscillation is affected by liquid temperature and capillary tube diameters [28]. According to the study by Zhmud et al. [29], such oscillations occur when the aspect ratio (tube radius/equilibrium height), r/he, is approximately greater than 0.05. In this study, aspect ratios for tube radius/equilibrium height, 0.4/36.1, 0.6/24.1, and 0.85 mm/16.1 mm, were 0.011, 0.025, and 0.053, respectively. Another criterion proposed by Masoodi et al. [30] describes the occurrence of oscillations and involves using a dimensionless number, ω, as follows: ρ2gr4 = ω 2 , (2) 64µ he where ρ is the density of the liquid (kg/m3), g is the standard gravity (m/s2), r is the capillary radius (mm), µ is the viscosity of the liquid (m Pa·s), and he is the equilibrium height (mm). The critical value of ω was 0.25, according to Masoodi’s results (oscillation at ω > 0.25). The test case for r = 0.85, in which oscillation occurs in this study, has ω = 5.7. Figure5 shows the rise height ( h) of DW in time (t), which is plotted as log 10 to magnify the differences in the h at the test case of r = 0.4 mm. The capillary rise has several stages [31]: the first stage is purely dominated by inertial forces, the second is where the influence of viscosity increases (visco-inertial flow), and the third is when the effect of inertia becomes nil and the flow becomes purely viscous. According to proposed analytical solutions in the previous studies [16,31–33], the capillary rise for the stages I–III can be expressed by Equations (3)–(5), respectively: s 2σ cos θ h = t , (3) ρr

s σr cos θ ρr2 − 8µ t h = t − 1 − e ρr2 , (4) 2µ 8µ Appl. Sci. 2021, 11, 3533 6 of 13

s σr cos θ Appl. Sci. 2021, 11, x FOR PEER REVIEW h = t.6 of (5) 13 2µ

Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 13

Figure 5a shows both h and t expressed in log scales for identifying the early stage of the rise by inertial forces, as found in previous studies [16,19,34]. In Figure 5a, the red dotted line represents a linear variation of h versus t in the inertial stage of the capillary rise dynamics. However, it was difficult to divide the stages after the first stage using inertial forces. In this study, we modified Figure 5a as Figure 5b, where h is plotted as a function of t and expressed in millimeters to clearly describe the various stages of a capillary rise phenomenon. The capillary rise can be divided into three characteristic stages: Stage I is the purely inertial region, Stage II is inertial + viscous region, and Stage III is purely viscous region). Similar studies in terms of segregation of the capillary rise process have been reported [18,35]. Stage II shows a linear increase in h with t. Meniscus height increases as per the parabolic law were observed at both Stages I and III. The h1 represents the transition between Stages I and II, and h2 represents the transition between Stages II and III. The he was calculated as the position at 0.99he at the start of initiating the capillary rise. End positions of individual stages (Stages I, II, and III), such as heights (h1, h2, and he), can also be expressed for their corresponding times t1, t2, and te, respectively. As shown in Figure 6, comparing the experimental results of this study with Equations (3)–(5) shows that the capillary rise height of water is relatively well described with the models in the early stage. Therefore, the transition time between Stages I and II was 2 calculated by t1 = ρr /4μ [31,34] based on the Washburn’s equation. The value of t1 was 0.045 in rather good agreement with this study, as shown in Figure 5b. For the transition time between Stages II and III, t2 was determined at the point where the linear region ends at the start point of t1. The lengths (including percentages of corresponding Stages I, II, Figure 4. TimeTime series series of of direct direct photographs photographs for for DW DW sh showingowing the the height height with with different different tube tube radii. radii. and III) of the inertial, inertial + viscous, and viscous stages, and their end positions are ((a)) 0.40.4 mm,mm, ((b)) 0.60.6 mm,mm, andand ((cc)) 0.850.85 mm.mm. given in Table 1 with different tube radii. Figure 5 shows the rise height (h) of DW in time (t), which is plotted as log 10 to magnify the differences in the h at the test case of r = 0.4 mm. The capillary rise has several stages [31]: the first stage is purely dominated by inertial forces, the second is where the influence of viscosity increases (visco-inertial flow), and the third is when the effect of inertia becomes nil and the flow becomes purely viscous. According to proposed analytical solutions in the previous studies [16,31–33], the capillary rise for the stages I–III can be expressed by Equations (3)–(5), respectively:

2𝜎cos𝜃 ℎ=𝑡 , (3) 𝜌𝑟

FigureFigure 5.5.Variation Variation of of height height (h ()h versus) versus time time (t )(t for) for the the DW DW rise rise in in capillary capillary tube tube at testat test case case of 2 ofr = 2r 0.8 = . 0.8. (a) h (log 10 mm) versus t (log 10𝜎𝑟cos𝜃 s) and (b) h (mm)𝜌𝑟 versus t (log 10 s). (a) h (log 10 mm) versus t (logℎ= 10 s) and (b) h (mm)𝑡− versus1t (log − 𝑒 10 s). , (4) 2𝜇 8𝜇 Figure5a shows both h and t expressed in log scales for identifying the early stage of the rise by inertial forces, as found in previous studies [16,19,34]. In Figure5a, the red 𝜎𝑟cos𝜃 ℎ= 𝑡. (5) 2𝜇

Figure 6. Comparison of analytic solutions with experimental results of this study (water in a capillary tube with r = 0.4 mm). The variable value of cosθ, as extracted from Equation (6), was used. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 13

Figure 5a shows both h and t expressed in log scales for identifying the early stage of the rise by inertial forces, as found in previous studies [16,19,34]. In Figure 5a, the red dotted line represents a linear variation of h versus t in the inertial stage of the capillary rise dynamics. However, it was difficult to divide the stages after the first stage using inertial forces. In this study, we modified Figure 5a as Figure 5b, where h is plotted as a function of t and expressed in millimeters to clearly describe the various stages of a capillary rise phenomenon. The capillary rise can be divided into three characteristic stages: Stage I is the purely inertial region, Stage II is inertial + viscous region, and Stage III is purely viscous region). Similar studies in terms of segregation of the capillary rise process have been reported [18,35]. Stage II shows a linear increase in h with t. Meniscus height increases as per the parabolic law were observed at both Stages I and III. The h1 represents the transition between Stages I and II, and h2 represents the transition between Stages II and III. The he was calculated as the position at 0.99he at the start of initiating the capillary rise. End positions of individual stages (Stages I, II, and III), such as heights (h1, h2, and he), can also be expressed for their corresponding times t1, t2, and te, respectively. Appl. Sci. 2021, 11, 3533 7 of 13 As shown in Figure 6, comparing the experimental results of this study with Equations (3)–(5) shows that the capillary rise height of water is relatively well described with the models in the early stage. Therefore, the transition time between Stages I and II was 2 dottedcalculated line represents by t1 = ρ ar linear/4μ [31,34] variation based of h versuson thet inWashburn’s the inertial stage equation. of the capillaryThe value rise of t1 was dynamics.0.045 in rather However, good it wasagreement difficult with to divide this thestudy, stages as aftershown the in first Figure stage using5b. For inertial the transition forces. In this study, we modified Figure5a as Figure5b, where h is plotted as a function time between Stages II and III, t2 was determined at the point where the linear region ends of t and expressed in millimeters to clearly describe the various stages of a capillary rise phenomenon.at the start point The capillary of t1. The rise lengths can be divided(including into percentage three characteristics of corresponding stages: Stage IStages is I, II, theand purely III) of inertial the inertial, region, Stageinertial II is + inertial viscous, + viscous and viscous region, andstages, Stage and III is their purely end viscous positions are region).given in Similar Table 1 studies with different in terms oftube segregation radii. of the capillary rise process have been reported [18,35]. Stage II shows a linear increase in h with t. Meniscus height increases as per the parabolic law were observed at both Stages I and III. The h1 represents the transition between Stages I and II, and h2 represents the transition between Stages II and III. The he was calculated as the position at 0.99he at the start of initiating the capillary rise. End positions of individual stages (Stages I, II, and III), such as heights (h1, h2, and he), can also be expressed for their corresponding times t1, t2, and te, respectively. As shown in Figure6, comparing the experimental results of this study with Equations (3)–(5) shows that the capillary rise height of water is relatively well described with the models in the early stage. Therefore, the transition time between Stages I and II was calculated by 2 t1 = ρr /4µ [31,34] based on the Washburn’s equation. The value of t1 was 0.045 in rather good agreement with this study, as shown in Figure5b. For the transition time between Stages II and III, t2 was determined at the point where the linear region ends at the start

point of t1. The lengths (including percentages of corresponding Stages I, II, and III) of the inertial,Figure 5.inertial Variation + viscous, of heightand (h viscous) versus stages, time (t and) for theirthe DW end rise positions in capillary are given tube inat Tabletest case1 of 2r = with0.8. ( differenta) h (log 10 tube mm) radii. versus t (log 10 s) and (b) h (mm) versus t (log 10 s).

FigureFigure 6. 6.Comparison Comparison of of analytic analytic solutions solutions with with experimental experimental results results of this of this study study (water (water in a in a capillarycapillary tube tube with withr = r 0.4 = mm).0.4 mm). The variableThe variable value value of cosθ of, as cos extractedθ, as extracted from Equation from (6),Equation was used. (6), was used. Table 1. Lengths of characteristic stages including percentages of corresponding stages and their end positions for DW with different capillary tube radii.

r (mm) Stage I (mm) Stage II (mm) Stage III (mm) h1 (mm) h2 (mm) he (mm) 0.4 11.2 (31.2%) 16.4 (45.7%) 8.3 (23.1%) 11.2 27.6 35.9 0.6 8.7 (36.1%) 10.1 (41.9%) 5.3 (22%) 8.7 18.8 24.1 0.8 7.2 (44.7%) 7.4 (46%) 1.5 (9.3%) 7.2 14.6 16.1

The results in Table1 were obtained from the data ( h versus t) in Figure7a and using the method shown in Figure5b. As shown in Figure5b and Table1 at the test case of r = 0.4 mm, the sequence for the length of each stage was as follows: Stage II (inertial + viscous) > Stage I (inertial) > Stage III (viscos region). In the capillary rise processes for r = 0.4 mm, the lengths of Stages I, II, and III were 11.2, 16.4, and 8.3 mm, corresponding to 31.2%, 45.7%, and 23.1%, respectively. Stage II was completed at time point (t2) of approximately 0.23 s. This period is a relatively short time for the entire rise Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 13

Appl. Sci. 2021, 11, 3533 8 of 13

where r is the radius of capillary tube, σ is the surface tension between the gas and liquid phases, and θ is the contact angle of the gas-liquid interface with respect to the solid process,surface. therefore,In Equation it is(8), an the early surface stage. tension These of results and the indicate tube radius that the are capillary the parameters rise at thethat earlyaffect stage capillary is dominantly pressure at influenced fixed value by of thecosθ pure = 1. At inertia r = 0.4, and 0.6, hybrid and 0.85 (inertia mm in + viscous)radii, the regionsPC values rather were than 360, by 240, the and pure 169 viscous N/m2, respectively. region. The length The trend of Stages of capillary I + II and rise III height was 27.6with anda variation 8.3 mm, in corresponding tube radii was to consistent 76.9%, and with 23.1%, the results respectively. of capillary This influencepressures. was As shown more noticeablein Figure at7b, higher the larger tube radius. the radius The lengthof the of capillary Stage III, tube, which the is alower purely is viscousthe maximum region, wasinstantaneous reduced from rise 23.1% velocity. to 9.3% At r with= 0.4, the 0.6, increase and 0.85 in mm the capillaryin radii, the tube peak radius velocity from values 0.4 to 0.85were mm. 0.46, Additionally, 0.36, and 0.27 the m/s, time respectively. expressed as Tht2ecorresponding velocity decreases to h2 afterwas reducesreaching from the 0.23peak tovalue 0.1 s and with tends the capillary to converge tube radiusto zero increasing near he. fromThese 0.4 trends to 0.85 are mm, in asqualitative shown in agreement Figure7a. Basedwith a on previous these results, publication the capillary [20,23,39,40]. rise dynamics Regarding can the be controlledeffects of changes by the tube in the diameter, radius, regardlesswe demonstrated of whether that the the flow capillary is inertial rise is dominant sensitive or to viscous the radius; dominant thus, the in allcapillary the capillary radius riseis an applications. important control parameter in the capillary rise dynamics.

FigureFigure 7.7. Effect of of tube tube diameter diameter (2 (2r r= =0.8, 0.8, 1.2, 1.2, and and 1.7 1.7mm) mm) on capillary on capillary rise dynamics rise dynamics of DW. of ( DW.a) (aheight) height and and (b) ( bvelocity.) velocity.

3.2. EffectThe effect of Surface of the Tension tube radius of the onFluids the on capillary Capillary rise Rise height Dynamics is shown in Figure7a. At the end ofExperiments Stage III, gravitational were performed forces become in a single signiﬁcant capillary owing tube to the (r balance = 0.4 ofmm) gravity at room and capillarity.temperature The (23 meniscus ± 1 °C) to stops evaluate at the the equilibrium effect of liquid capillary surface rise tension height on (h thee) expressed capillary rise as followsdynamics. [15, 36Based]: on the measurement method shown in Figure 2, the σ values of the test 2σ cos θ materials for B0% (DW), B0.3%, B0.5%,h = and B0.7%e ,were approximately 72, 64, 60, and (6) 56 e ρgr mN/m, respectively. The values of σ were consistent with the measurement results whereobtainedσ is from the surface the pendant tension drop (mN/m), methodg is [41,42] the standard with SmartDrop gravity (m/s (SmartDrop2), ρ is the Plus density HS, 3 ofFemtobiomed the liquid (kg/m Inc., Gyeonggi-do,), r is the capillary South Korea). radius (mm),The measurement and θe is the σ contactvalues for angle. B0% From(DW), FigureB0.3%,7 aB0.5%, and Table and 1B0.7%, he was in the 35.9, pendent 24.1, anddrop 16.1 method mm atwere tube 72.32 radii ± 0.4,0.42, 0.6, 64.25 and ± 0.73, 0.85 mm,60.02 respectively.± 0.67, and The56.19 theoretical ± 0.57 mN/m, prediction respectively. values were The 36.8 addition at r = 0.4 mm,of n-butanol 24.5 at r =in 0.6 a mm,low andconcentration 17.3 at r = 0.85causes mm. a Inconsiderable this theoretical decrease prediction in the for interfacial Equation tension (6), we usedof water/butanol cosθe = 1 in allmixtures. the test conditionsThis phenomenon because thestems contact from angle the forproperty water atof hpuree was n-butanol measured tothat be has very a smallrelatively by [17 low]. In surface Figure tension7a, it is (24 expected mN/m) that compared the contact to DW angle (72θ mN/m).varies in For time, this and reason, the valuethe surface of cosθ tensionextracted of from the Equationbutanol-added (3) increases solutions with sharply increasing decreases the time with (or rise increasing height). Atbutanol the equilibrium concentration state, until the valueit reaches of cos aθ criticalreaches concentration, unity (cosθe = after 1) and which its value the insurface this studytension was decreases 0.98, 0.98, monotonically. and 0.93 at tube The radii critic 0.4, 0.6,al mass and 0.85 concentrations mm, respectively. of alcohol/water In addition, thesolutions effect of with capillary butanol, tube pentanol, radii on and the risehexanol height were depends 3%, 0.5% on theand capillary 0.3%, respectively pressure ( P[43].C). TheIn general, isothermal the pressures σ of n-butanol/water in gas (PG) and mixt liquidures (PasL )well phases as areother deﬁned alcohol/water as the PC mixturesthat is: depend on their corresponding temperature and composition, and even at low n-butanol = − concentrations of 0 to 1 wt%, the σ valuePC variesPG P [43,44].L. Figure 8 shows the capillary (7)rise dynamics for four different concentrations of n-butanol/water mixtures with different The equilibrium relation between the pressures on the two sides of an interface can be values of σ. Generally, as shown in Figure 8, all measurements show that the height of rise expressed by well-known Young–Laplace equation as given below for a spherical meniscus increases with time; however, the velocity of rise at which the meniscus moves through in a cylindrical tube [37,38]: the capillary decreases over time, as shown2σ in Figure 8b. In addition, the height of rise P = cos θ, (8) decreased with an increase in the concentrationC r of n-butanol. In particular, he was 35.9, 32.1, 28.7, and 26.6 mm at B0%, B0.3%, B0.5%, and B0.7%, respectively. The viscosity, where r is the radius of capillary tube, σ is the surface tension between the gas and liquid phases, and θ is the contact angle of the gas-liquid interface with respect to the solid surface. Appl. Sci. 2021, 11, 3533 9 of 13

In Equation (8), the surface tension of and the tube radius are the parameters that affect capillary pressure at ﬁxed value of cosθ = 1. At r = 0.4, 0.6, and 0.85 mm in radii, the 2 PC values were 360, 240, and 169 N/m , respectively. The trend of capillary rise height with a variation in tube radii was consistent with the results of capillary pressures. As shown in Figure7b, the larger the radius of the capillary tube, the lower is the maximum instantaneous rise velocity. At r = 0.4, 0.6, and 0.85 mm in radii, the peak velocity values were 0.46, 0.36, and 0.27 m/s, respectively. The velocity decreases after reaching the peak value and tends to converge to zero near he. These trends are in qualitative agreement with a previous publication [20,23,39,40]. Regarding the effects of changes in the radius, we demonstrated that the capillary rise is sensitive to the radius; thus, the capillary radius is an important control parameter in the capillary rise dynamics.

3.2. Effect of Surface Tension of the Fluids on Capillary Rise Dynamics Experiments were performed in a single capillary tube (r = 0.4 mm) at room temperature (23 ± 1 ◦C) to evaluate the effect of liquid surface tension on the capillary rise dynamics. Based on the measurement method shown in Figure2, the σ values of the test materials for B0% (DW), B0.3%, B0.5%, and B0.7% were approximately 72, 64, 60, and 56 mN/m, respectively. The values of σ were consistent with the measurement results obtained from the pendant drop method [41,42] with SmartDrop (SmartDrop Plus HS, Femtobiomed Inc., Gyeonggi-do, South Korea). The measurement σ values for B0% (DW), B0.3%, B0.5%, and B0.7% in the pendent drop method were 72.32 ± 0.42, 64.25 ± 0.73, 60.02 ± 0.67, and 56.19 ± 0.57 mN/m, respectively. The addition of n-butanol in a low concentration causes a considerable decrease in the interfacial tension of water/butanol mixtures. This phenomenon stems from the property of pure n-butanol that has a relatively low surface tension (24 mN/m) compared to DW (72 mN/m). For this reason, the surface tension of the butanol-added solutions sharply decreases with increasing butanol concentration until it reaches a critical concentration, after which the surface tension decreases monotonically. The critical mass concentrations of alcohol/water solutions with butanol, pentanol, and hexanol were 3%, 0.5% and 0.3%, respectively [43]. In general, the σ of n-butanol/water mixtures as well as other alcohol/water mixtures depend on their corresponding temperature and composition, and even at low n-butanol concentrations of 0 to 1 wt%, the σ value varies [43,44]. Figure8 shows the capillary rise dynamics for four different concentrations of n-butanol/water mixtures with different values of σ. Generally, as shown in Figure8, all measurements show that the height of rise increases with time; however, the velocity of rise at which the meniscus moves through the capillary decreases over time, as shown in Figure8b. In addition, the height of rise decreased with an increase in the concentration of n-butanol. In particular, he was 35.9, 32.1, 28.7, and 26.6 mm at B0%, B0.3%, B0.5%, and B0.7%, respectively. The viscosity, which depends on the concentration of blends, can be one of the parameters that plays an important role in capillary dynamic parameter because the viscosity can be changed due to the blends during the rising period. To calculate the equilibrium capillary rise height (he) that is considered as steady state characteristics of blends, Equation (3), using the surface tension of the blends and well-established and widely used by the balance of gravity and capillarity, was used [45]. As shown in Figure8b, the higher the σ of the mixtures, the lower is the maximum instantaneous rise velocity. At B0%, B0.3%, B0.5%, and B0.7%, the peak velocity values were 0.45, 0.41, 0.34, and 0.3 m/s, respectively. The peak velocity when comparing σ values was observed at the same position on time basis (0.005 s), as shown in Figure8b, unlike the peak values observed at different positions when comparing the capillary radii shown in Figure7b. In Figure8b, the peak value decreases systematically with σ because there is no oscillation phenomenon, as shown in Figure4. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 13

which depends on the concentration of blends, can be one of the parameters that plays an important role in capillary dynamic parameter because the viscosity can be changed due to the blends during the rising period. To calculate the equilibrium capillary rise height (he) that is considered as steady state characteristics of blends, Equation (3), using the surface tension of the blends and well-established and widely used by the balance of gravity and capillarity, was used [45]. As shown in Figure 8b, the higher the σ of the mixtures, the lower is the maximum instantaneous rise velocity. At B0%, B0.3%, B0.5%, and B0.7%, the peak velocity values were 0.45, 0.41, 0.34, and 0.3 m/s, respectively. The peak velocity when comparing σ values was observed at the same position on time basis (0.005 s), as shown in Figure 8b, unlike the peak values observed at different positions when comparing the capillary radii shown in Figure 7b. In Figure 8b, the peak value Appl. Sci. 2021, 11, 3533 10 of 13 decreases systematically with σ because there is no oscillation phenomenon, as shown in Figure 4.

FigureFigure 8. Effect 8. Effect of surface of surface tension tension of liquids of ( σliquids= 56, 60, (σ 64,= 56, and 60, 72 64, mN/m) and 72 on mN/m) capillary on rise capillary dynamics rise at r = 0.4 mm. (a) heightdynamics and (b) at velocity. r = 0.4 mm. (a) height and (b) velocity.

Table 2 shows theTable lengths2 shows of the characteristic lengths of characteristic stages (as stagesdefined (as deﬁnedin Figure in Figure5) of 5n-) of n- butanol/water mixtures at r = 0.4 mm. In each of Stage I, II, and III length, the overall trend butanol/water mixtureswas dependent at r = 0.4 upon mm. the In increase each of in Stage n-butanol I, II, and blending III length, ratios. Thethe lengthoverall increases trend with was dependent uponthe increase the increase of n-butanol in n-butanol ratio in Stageblending II, whereas ratios. there The waslength no similarincreases trend with observed the increase of n-butanolin Stages ratio I and in III, Stage for the II, same whereas condition. there In was the comparison no similar betweentrend observedTables1 and in2 , the Stages I and III, forcharacteristic the same hcondition.in lengths ofIn Stages the comparison I, II, III shows between a clear trend Tables with 1 increasingand 2, ther and σ. characteristic h inTo lengths clarify of these Stages relations I, II, forIII parametersshows a clear such trend as capillary with increasing tube radius r and and liquid σ. To surface tension on the capillary rise characteristics, we analyzed a consolidated quantity between r clarify these relations for parameters such as capillary tube radius and liquid surface and σ on characteristic heights. The quantity, which is a formulation analogous to pressure tension on the capillary(σL/2π rrise2), was characteristics, a consolidated expressionwe analyzed of liquid a consolidated surface tension quantity (σ), capillary between tube length 2 r and σ on characteristic(L), and tube heights. surface The area (2quantity,πr ). Figure which9 shows is thea formulation characteristic heightsanalogoush1, h 2to, and he pressure (σL/2πr2as), was a function a consolidated of r, σ, and σexpressionL/2πr2. Straight of liquid lines surface represent tension linear ﬁts (σ between), capillary character- tube length (L), andistic tube heights surface and each area of (2 theπr comparative2). Figure 9 parameters.shows the Incharacteristic Figure9a, characteristic heights h heights1, showed a linear decrease with capillary tube radius. The characteristic heights are also h2, and he as a function of r, σ, and σL/2πr2. Straight lines represent linear fits between linearly dependent on σ, as shown in Figure9b. Each of characteristic heights becomes a characteristic heights and each of the comparative parameters. In Figure 9a, characteristic linear function of σL/2πr2 as shown in Figure9c. In Figure9c, the linearity (R-square) of heights showed a linear decrease with capillary tube radius. The characteristic heights are h1, h2, and he with r and σ was 0.78, 0.93, and 0.95. Hence, all the characteristic heights also linearly dependentincrease withon anσ, increaseas shown of the in said Figure function. 9b. This Each linear of increment characteristic was more heights noticeable in becomes a linear thefunction sequence of σ ofL/2h1π

heightsTable increase 2. Lengths with of characteristic an increase stages of for the butanol/water said function. mixtures This at the linear test case increment of r = 0.4 mm. was more noticeable in the sequence of h1 < h2 < he. As such, the slope value of h1, h2, and he with the Mixture (Butanolquantity wt.%) was 0.00139, Stage I (mm) 0.00506, Stage and I (mm) 0.00796. Stage I (mm) h1 (mm) h2 (mm) he (mm) σ (mN/m) B0% (DW) 11.2 (31.2%) 16.4 (45.7%) 8.3 (23.1%) 11.2 27.6 35.9 72 B0.3% 9.3 (29%) 14.3 (44.5%) 8.5 (26.5%) 9.3 23.6 32.1 64 B0.5% 8.9 (31%) 12.5 (43.6%) 7.3 (25.4%) 8.9 21.4 28.7 60 B0.7% 8.6 (32.3%) 11.2 (42.1%) 6.8 (25.6%) 8.6 19.8 26.6 56 Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 13

Table 2. Lengths of characteristic stages for butanol/water mixtures at the test case of r = 0.4 mm.

Mixture (Butanol wt.%) Stage I (mm) Stage I (mm) Stage I (mm) h1 (mm) h2 (mm) he (mm) σ (mN/m) B0% (DW) 11.2 (31.2%) 16.4 (45.7%) 8.3 (23.1%) 11.2 27.6 35.9 72 B0.3%Appl. Sci. 2021, 11, 3533 9.3 (29%) 14.3 (44.5%) 8.5 (26.5%) 9.3 23.6 32.1 64 11 of 13 B0.5% 8.9 (31%) 12.5 (43.6%) 7.3 (25.4%) 8.9 21.4 28.7 60 B0.7% 8.6 (32.3%) 11.2 (42.1%) 6.8 (25.6%) 8.6 19.8 26.6 56

Figure 9. (a) Characteristic heights (h1, h2, and he) as a function of the capillary tube radius (r). (b) Figure 9. (a) Characteristic heights (h1, h2, and he) as a function of the capillary tube radius (r). (b) Characteristic heights as Characteristic heights as a function of the surface tension of liquids (σ). (c) Characteristic heights a function of the surface tension of liquids (σ). (c) Characteristic heights as a function of the quantity in terms of r and σ as a function of the quantity in terms of r and σ (σL/2πr2). Solid lines represent linear fits between (σL/2πr2). Solid lines represent linear ﬁts between characteristic heights and each comparative parameter. characteristic heights and each comparative parameter.

4. Conclusions4. Conclusions The experimentalThe correlation experimental between correlation the ca betweenpillary tube the capillaryradius and tube the radius liquid and surface the liquid surface tension on thetension capillary on rise the capillarydynamics rise was dynamics investigated. was investigated.The conclusions The of conclusions the present of the present study are summarizedstudy are as summarized follows: as follows: (1) The height(1) of riseThe and height the of velocity rise and of the rise velocity are dependent of rise are on dependentthe capillary on tube the capillaryradii. tube radii. The larger the radius, the lower is the equilibrium height and the velocity of rise. The The larger the radius, the lower is the equilibrium height and the velocity of rise. The meniscus oscillation was observed only at the test case of r = 0.85. meniscus oscillation was observed only at the test case of r = 0.85. (2) The capillary rise process is divided into three characteristic stages: the inertial (2) The capillary rise process is divided into three characteristic stages: the inertial (Stage (Stage I), inertial + viscous (Stage II), and viscous (Stage III) ﬂow region. The cap- I), inertial + viscous (Stage II), and viscous (Stage III) flow region. The capillary rise illary rise was predominantly inﬂuenced by the pure inertia and hybrid region was predominantly influenced by the pure inertia and hybrid region (inertia + (inertia + viscous). viscous). (3) The capillary rise height decreased with an increase in the concentration of n-butanol (3) The capillary rise height decreased with an increase in the concentration of n-butanol and with a decrease in the surface tension of n-butanol/water mixtures. With respect and with a decrease in the surface tension of n-butanol/water mixtures. With respect to the velocity of rise, the higher the surface tension of the mixtures, the higher to the velocity of rise, the higher the surface tension of the mixtures, the higher the the velocity. velocity. (4) There were some correlations between the characteristic heights (h1, h2, and he) and 1 2 e (4) There were somethe capillarycorrelations tube between radius andthe liquidcharacteristic surface tension.heights ( Characteristich , h , and h ) and heights showed a the capillary tubelinear radius decrease and liquid in the capillarysurface tension. tube radius Characteristic and a linear heights decrease showed in the a liquid surface linear decreasetension. in the capillary Consequentially, tube radius each and characteristic a linear decrease height in exhibited the liquid a linearsurface function of the consolidated quantity of σL/2πr2.

Author Contributions: Conceptualization, H.J.; methodology, S.P.; software, J.C.; investigation, S.B.; data curation, J.S.; writing—original draft preparation, S.J. and S.B.; writing—review and editing, S.B. and Y.S.; visualization, S.L.; project administration, Y.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript. Appl. Sci. 2021, 11, 3533 12 of 13

Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1049268). Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

B n-butanol DW Distilled water g Standard gravity [m/s2] h Capillary rise height [mm] h1 Height at the transition time between the purely inertial and the visco-inertial stage [mm] h2 Height at the transition time between the visco-inertial and the purely viscous stage [mm] he Equilibrium capillary rise height [mm] L Capillary tube length [mm] PC Capillary pressure [Pa] PG Pressure in gas [Pa] PL Pressure in liquid [Pa] r Capillary tube radius [mm] t1 Transition time between the purely inertial and the visco-inertial stage [s] t2 Transition time between the visco-inertial and the purely viscous stage [s] ◦ θe Equilibrium contact angle [ ] µ Viscosity of liquid [mPa·s] ρ Density of liquid [kg/m3] σ Surface tension of liquid [mN/m]

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