Evaporation of ethanol-water sessile droplet of different compositions at an elevated substrate temperature

International Journal of Heat and Mass Transfer 145 (2019) 118770

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Evaporation of ethanol-water sessile droplet of different compositions at an elevated substrate temperature Pradeep Gurrala a, Pallavi Katre b, Saravanan Balusamy a, Sayak Banerjee a, Kirti Chandra Sahu b,⇑ a b

Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502 285, Telangana, India Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502 285, Telangana, India

a r t i c l e

i n f o

Article history: Received 23 July 2019 Received in revised form 21 September 2019 Accepted 21 September 2019

Keywords: Evaporation Binary mixture Sessile drop Experiment Theoretical modelling

a b s t r a c t We experimentally investigate the evaporation dynamics of sessile droplets with different compositions of ethanol-water binary mixture at different substrate temperatures. At room temperature, a binary droplet undergoes two distinct evaporation stages unlike only the pinned stage evaporation for pure droplets. In the binary droplets, the more volatile ethanol evaporates faster leading to a nonlinear trend in the evaporation process. At elevated substrate temperature, we observed an early spreading stage, an intermediate pinned stage and a late receding stage of evaporation. Increasing the substrate temperature decreases the lifetime of binary droplets rapidly. At high substrate temperature, we found that the lifetime of the droplet exhibits a non-monotonic trend with the increase in ethanol concentration in the binary mixture, which can be attributed to the non-ideal behaviour of water-ethanol binary mixtures. Interestingly, the evaporation dynamics for different compositions at high substrate temperature exhibits a self-similar trend showing a constant normalised volumetric evaporation rate for the entire evaporation process. This indicates that the evaporation dynamics of a binary droplet of a given composition at high substrate temperature is equivalent to that of another pure fluid with a higher volatility at room temperature. The evaporation rates of pure and binary droplets at different substrate temperatures are compared against a theoretical model developed for pure and binary mixture droplets. The model predictions are found to be quite satisfactory for the steady evaporation phase of the droplet lifetimes. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Evaporation of droplets is commonly encountered in atmospheric phenomena associated with clouds and raindrops, in biological systems, and also in industrial applications, such as combustion, ink-jet printing [1–7], hot-spot cooling, dropletbased microfluidics [8], coating technology, to name a few (see for instance, Refs. [9–12]). A proper understanding of the underlying physics of evaporating droplets plays an important role in wetting and surface characterization processes [13]. Many researchers have investigated the evaporation dynamics of a sessile droplet of pure fluids on hydrophilic and hydrophobic surfaces (see e.g. Refs. [14,15] and references therein). By conducting experiments, Birdi et al. [16] studied the evaporation of a sessile droplet on a hydrophilic substrate and observed a constant evaporation rate with a pinned contact line. Bourges-Monnier and Shanahan [17] experimentally investigated the evaporation of sessile droplets of water and n-decane placed on different ⇑ Corresponding author. E-mail address: [email protected] (K.C. Sahu). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118770 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

hydrophilic substrates and observed four distinct stages of evaporation depending on the roughness of the substrates. In the case of complete wetting, Picknett and Bexon [13] demonstrated the existence of two stages of evaporation, namely, the constant-contactangle stage and the constant-contact-line-area stage. Later, it was observed that for partially wetting substrates, there are four stages in the droplet evaporation process, which are the early short and rapid spreading stage, the slow spreading stage, the constantcontact-angle stage and the final stage where both the contact angle and the radius decrease until the drop completely evaporates [18]. An extensive review of the literatures on recent advances on wetting and evaporation of sessile droplets can be found in Brutin and Starov [19]. A sessile droplet placed on a heated substrate exhibits a temperature gradient in the vicinity of the solid surface and along the liquid-vapour interface, which in turn creates thermocapillary convection inside the droplet [20–22] and instability or hydrothermal waves (undulation) at the liquid-vapour interface for some liquids [23]. Sáenz et al. [24] investigated the evaporation of non-spherical droplets via numerical simulations and experiments. They demonstrated a universal scaling law for evaporation,

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which was valid even for droplets with complex shapes. Ristenpart et al. [21] investigated the effect of conductivity of the substrate on the evaporation dynamics of a droplet by conducting an asymptotic analysis. The evaporation of water droplets on hydrophobic and superhydrophobic surfaces at different temperatures have been investigated and compared their experimental results with the theoretical models [15,25]. Sobac and Brutin [26] have experimentally investigated an evaporating ethanol droplet at different temperatures on a hydrophilic and hydrophobic substrates. They have developed an evaporation model based on the quasi-steady, diffusion-driven assumptions and compared their experimental results with the theoretical predictions. Few authors [27,28] have also studied evaporation of pure liquids droplets at different substrate temperatures and developed empirical relationships by considering the combined influence of diffusion and convective transport. Next we discuss the literature on the evaporation of droplets of binary mixtures. Sefiane et al. [29] studied the evaporation of sessile droplets of ethanol-water mixtures of different compositions on a polytetrafluoroethylene (PTFE) substrate at room temperature and the dynamics was compared against that observed in pure droplets. It was found that the volume of a droplet of pure water or pure ethanol decreases monotonically. On the other hand, the evaporation for ethanol-water mixtures occurs through three distinct stages: the first and the last stages are mainly dominated by the evaporation of the more volatile component (ethanol) and the less volatile component (water), respectively, whereas at the intermediate stage the volume of the drop remains almost constant, but the contact angle varies significantly. They also concluded that the dynamics contact angle largely depends on the concentration of ethanol in the droplet. In order to explain the different stages, the flow field in an evaporating ethanol-water droplet by varying the composition of the binary mixture at the room temperature was studied using particle image velocimetry (PIV) [30,31]. Karpitschka et al. [32] conducted experiments and numerical simulations to study the Marangoni contraction of the droplet at 21
C. By conducting lubrication analyses, Karapetsas et al. [33–35] investigated the dynamics of a sessile drop of binary mixtures of non-azeotropic, high carbon alcohol solutions under the influence of thermo-capillary forces and evaporation flux. Due to the parabolic surface tension-temperature dependence of such solutions, the super-spreading behaviour of the droplet was observed. Sáenz et al. [24] also investigated complex shaped droplets consisting of ethanol-water binary mixtures by conducting theoretical modelling and infrared thermography [36] at different substrate temperatures and concentrations. However, most of the results presented in their study are at room temperature only. Several researchers have investigated the influence of different substrate on the evaporation dynamics of different binary mixtures placed on different substrates [37–43]. As the above-mentioned review shows, the evaporation dynamics of pure droplets have been investigated extensively for both low and high temperature substrate conditions; however, to the best of our knowledge, the evaporation dynamics of binary droplets have been studied for different compositions, but only at room temperature. Moreover, even at room temperature, the behaviour reported by the previous studies for different concentrations were also different as the dynamics depends on several factors, such as the property of the substrate (which have not been provided in most of the previous studies) and ambient conditions. Therefore, the present work focuses the wetting and the spreading dynamics of evaporation sessile droplets of ethanol-water mixtures at different substrate temperatures. The roughness of the substrate (cellulose-acetate tape) is studied using the atomic force microscopy (AFM) and the scanning electron microscope (SEM) techniques. The temperature of the substrate is varied from 25
C

to 60
C as the boiling temperature of pure ethanol is about 78
C [44]. The concentration of ethanol is varied from 0% ethanol (pure water) to 100 % ethanol (pure ethanol) on a volume basis. The rest of the paper is organised as follows. In Section 2, we discuss about the experimental set-up. The results obtained from our experiments are presented in Section 3. The theoretical models for the evaporation of a sessile droplet in different situations are developed and compared against the experimental findings. Finally, concluding remarks are given in Section 6.

2. Experimental set-up We investigate the evaporation dynamics of a sessile droplet of a binary mixture of water (W) and ethanol (E) on a heated substrate maintained at different temperatures. The experimental set-up consists of a heater, a motorised pump to create the droplet, cellulose acetate substrate placed on a multi-layered block, a light source and a complementary-metal-oxide-semiconductor (CMOS) camera (make: Do3Think, model: DS-CBY501E-H). The schematic of the experimental set-up is shown in Fig. 1. The substrate temperature is varied between experiments and maintained at a fixed temperature during each experiment using a proportional-inte gral-derivative (PID) controller. The entire setup is placed inside a big metallic box to minimise outside disturbances. This customised goniometer set-up was fabricated by Holmarc OptoMechatronics Pvt. Ltd. The experiments are conducted in an airconditioned room maintained at a temperature of 22 °C and a relative humidity of 36%, which have been measured with the help of an HTC 288-ATH hygrometer. The relative humidity inside the box is found to be constant throughout the experimentation. The multi-layered block contains a stainless steel plate of size 100 mm  80 mm  15 mm with two PID regulated electrical heaters situated at its base (see Fig. 1). The substrate is a celluloseacetate tape of thickness 63 lm placed on an aluminium block of thickness 5 mm coated with a black paint, which is placed over the stainless steel plate. The scanning electron microscope (make: Thermofisher Scientific, model: Phenom Prox) images of the cellulose-acetate tape at the room temperature and 60 °C are shown in Fig. 2(a) and (b), respectively. It can be seen that the surface roughness of cellulose-acetate tape is uniform even at 60 °C. Fig. 2(c) and (d) show the atomic force microscopy (make: Park System, model: NX10) images of the cellulose acetate substrate at room temperature and at 60 °C, respectively. The root-meansquare values of the roughness of the cellulose acetate substrate at 25 °C and 60 °C are 0.668 lm and 0.641 lm, respectively. The AFM images also confirm that the surface roughness of the cellulose acetate substrate is uniform. Thus, based on these findings, we choose the cellulose-acetate tape as the substrate in the present study. The SEM images of the used substrate showed that no adsorption or absorption of water-ethanol takes place on the substrate for all the temperatures considered. The substrate temperature is set in the PID controller, and the heater is switched on to stabilise the set temperature. The temperature on the aluminium plate is measured using an infrared heat gun (which measures the surface temperature with an error of about 0:5
C), and the controller is adjusted to achieve the required substrate temperature. To achieve a steady state condition, we wait for an hour and then apply the cellulose-acetate tape on the stainless steel plate. Again we wait for 10 min before placing the droplet on the substrate. We observe a negligible temperature gradient on the surface owing to the small thickness of tape. The binary solutions are prepared by varying the volume concentration of ethanol in purified deionised water (purity of 18:2 MX) such that the total volume of the solution is 100 ml. As water and ethanol are miscible at all concentrations, they are

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Fig. 1. Schematic diagram of the experimental set-up. It consists of a heater, a cellulose acetate substrate placed on a stainless steel plate, a light source and a camera. A typical image of an ethanol-water droplet recorded using a CMOS camera is also shown.

Fig. 2. (a, b) The scanning electron microscope images and (c, d) the atomic force microscopy images of the cellulose acetate substrate at 25 °C and 60 °C, respectively. The colour bars in panel (c) and (d) indicate the surface roughness in microns. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

thoroughly mixed using a stirrer, and homogeneous solutions of different compositions are prepared. An U-TeK chromatography syringe of size 100 ll with a needle of outer diameter 1.59 mm (supplied by Unitek Scientific Corporation) is connected with a motorised pump to create a droplet, which is gently placed on the substrate. The motorised pump controls the volume flow rate of the solution with an error < 1% at the desired flow rate. The droplet volume is kept constant at 5 ll for all the experiments conducted in the present study. The composition of the ethanolwater solution and the temperature of the substrate are varied, and the dynamics of the droplet is recorded using the CMOS camera at 10 frames-per-second (fps) with a spatial resolution of 1280  960 pixels. After each experiment, the syringe is cleaned with acetone and is allowed to dry, and the cellulose-acetate tape is also replaced. For each set of parameters, more than four repetitions of the experiment are conducted. The acquired droplet images are processed using the Matlab software to calculate the wetting diameter (D), the height (h), the

contact angle and the volume of the droplet (V) as a function of time normalised by the lifetime of droplet, t e (i.e. time taken by the droplet to evaporate completely). The time, t is measured from the instant when the droplet reaches its initial equilibrium state. The detail about the image processing is given in the supporting information. 3. Experimental results and discussion 3.1. Effect of substrate temperature We begin the presentation by investigating the effect of varying the substrate temperature, T s on the evaporation dynamics of a (E 50% + W 50%) binary droplet. The lifetimes ðte Þ of the (E 50% + W 50%) binary droplet of initial volume 5 ll at different substrate temperatures are shown in the second column of Table 1 and plotted in Fig. 3. We found that, as expected, te decreases with the increase in the temperature of the substrate, T s .

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Table 1 Lifetime of the droplets, t e (s) of pure water (E 0% + W 100%), binary mixture of (E 50% + W 50%) and pure ethanol (E 100% + W 0%) at different temperatures of the substrate.

T s (°C)

te (s) for E 0% + W 100%

te (s) for E 50% + W 50%

t e (s) for E 100% + W 0%

25 40 50 60

1488 ± 63 – – 190 ± 5

1035 ± 13 147 ± 6 64 ± 2 38 ± 1

183 ± 3 – – 12 ± 1

Fig. 3. The variation of the lifetime time, te (in s) of a droplet of (E 50% + W 50%) binary mixture with temperature of the substrate, T s .

The temporal evolutions of the photographic images of the droplet of binary mixture are shown in Fig. 4 at T s ¼ 25
C, 40
C, 50
C and 60
C. The non-dimensional time, t=t e is used for the comparisons. A similar plot was given by Sterlyagov et al. [45] for an ethanol-water droplet on a Teflon substrate for different compositions at 24
C. The dynamics observed in the present study at T s ¼ 25
C qualitatively agrees with that presented by Sterlyagov et al. [45]. The videos showing the evaporation dynamics of a droplet of (E 50% + W 50%) mixture at T s ¼ 25
C, 40
C, 50
C and 60
C

are provided as supplementary movies I (300 fps), II (300 fps), III (300 fps), and IV (100 fps), respectively. The contours of the droplets obtained by post-processing the droplet images shown in Fig. 4, are presented in Fig. 5. The different stages of the evaporation process are highlighted in the right hand side of Fig. 5. It can be seen in Fig. 5 that for T s ¼ 25
(top panel), the droplet remains pinned till t=t e ffi 0:2 and begins to recede for t=te > 0:2. At the later stage the wetting diameter of the droplet reduces gradually. At T s ¼ 40
C (second panel of Fig. 5), it can be seen that during 0 6 t=t e 6 0:2, the droplet of binary mixture spreads a little; the spreading is observed only in the right contact line, while the left contact line is pinned. Note that the reference to the left and right contact angles is with respect to the camera field of view. In contrast, at high substrate temperatures, this binary droplet exhibits three distinct stages at T s ¼ 50
C (third panel in Fig. 5) and T s ¼ 60
C (fourth panel in Fig. 5). They are, the early spreading stage ðt=te < 0:2Þ, the intermediate pinned stage ð0:2 6 t=te 6 0:6Þ and the late receding stage ðt=te > 0:6Þ. In the early spreading stage, the wetting diameter of the droplet increases and reaches to a value comparable to that for a pure ethanol droplet. Subsequently, up to t=t e  0:6, the binary droplet remains pinned and finally undergoes a slow contact line recession till the completion of the evaporation process. It is to be noted that the above behaviour was reproducible and was observed in all experimental runs (6 times). Thus, the complex behaviour of binary mixtures, particularly those where both component concentrations are initially comparable, is underscored by the above observations. In Fig. 6(a)–(d), we present the variations of the droplet height (h), the droplet wetting diameter (D), the contact angle (h) and the normalised volume of the droplet (V=V 0 ) obtained at different values of T s versus t=t e , respectively for a binary droplet. It can be seen in Fig. 6(a) that the height of the droplet decreases rapidly at the early time for the T s ¼ 50 and T s ¼ 60 cases due to the initial spreading of the droplet on the substrate. After the initial spreading stage, the droplet height decreases at a slower rate due to evaporation. Inspection of Fig. 6(b) reveals that at T s ¼ 25
C, the

Fig. 4. Temporal evolutions of a (E 50% + W 50%) droplet at different substrate temperatures. The length of the scale bar shown in each panel is 200 lm.

P. Gurrala et al. / International Journal of Heat and Mass Transfer 145 (2019) 118770

Fig. 5. Comparison of the droplet spreading behaviour at different substrate temperatures for (E 50% + W 50%) composition.

droplet does not undergo the initial spreading stage and the wetting diameter decreases during the entire evaporation process. At T s ¼ 40
C, the wetting diameter remains almost constant, which is responsible for the near-monotonic behaviour observed at T s ¼ 40
C in Fig. 6(a) and (c). In contrast, at T s ¼ 50
C and 60
C, the binary droplet spreads faster at early times (D increases) and then D becomes constant (pinned phase) as shown in Fig. 6(b). In Fig. 6(c), it is observed that a binary droplet exhibits hydrophilic behaviour (contact angle less than 90
) at all the substrate temperatures. In Fig. 6(d) the variations of V=V 0 versus t=te are shown at different values of T s . Close inspection of Fig. 6(d) reveals that, at the high substrate temperature (T s ¼ 60
C), the normalised volume of the binary droplet decreases linearly with normalised time. However, the curve becomes nonlinear as we decrease the substrate temperature. Thus, it can be concluded that a binary droplet at T s ¼ 60
C behaves like a droplet of another pure fluid with a higher volatility. 3.2. Effect of composition The behaviour of evaporating droplets (initial volume 5 ll) of different compositions deposited on a heated substrate at T s ¼ 60
C is investigated in this section. The experiments are con-

5

ducted for seven different compositions of ethanol-water binary mixtures, namely, (E 0% + W 100%), (E 20% + W 80%), (E 40% + W 60%), (E 50% + W 50%), (E 60% + W 40%), (E 80% + W 20%) and (E 100% + W 0%). The lifetimes of the droplets of different compositions are presented in Table 2. It can be seen that the droplet lifetimes, t e range from 12 s for a pure ethanol droplet to 190 s for a pure water droplet. The variation of te with the percentage volumetric concentration of ethanol (E %) in the binary mixture at T s ¼ 60
C is shown in Fig. 7. It can be seen that there is a rapid linear decrease in the lifetime of the droplet with an increase in the concentration of ethanol till E = 50%, after which the lifetime of the droplet is seen to vary only slightly with further increase in ethanol concentration till E = 80%. This behaviour can be attributed to the non-ideal vapour pressure phase diagram of waterethanol binary mixtures [46]. In Fig. 8, the photographic images of the droplets of pure water (E 0% + W 100%), a binary mixture (E 50% + W 50%) and pure ethanol (E 100% + W 0%) are presented at different values of t=te at T s ¼ 60
C. It can be seen that the droplets of pure water and pure ethanol remain pinned for the majority of their steady lifetimes. However, for a (E 50% + W 50%) binary droplet, we see that the droplet has an initial spreading phase after its deposition on the heated substrate that lasts up to 20% of its total evaporation time, after which it transitions into a pinned evaporation stage. At T s ¼ 60
C, significantly undulation (interfacial instability driven by the Marangoni convection) is observed in Fig. 8 for the pure ethanol droplet near the end stages of the evaporation (t=t e P 0:7). The late stage evaporation dynamics even becomes asymmetrical, as is evident at t=t e ¼ 0:9 in Fig. 8. The photographic movies associated with pure water, (E 50% + W 50%) binary mixture and pure ethanol at T s ¼ 60
C obtained from our experiments are provided as supplementary movies V (at 300 fps), VI (at 100 fps) and VII (at 10 fps), respectively. A detailed comparison of temporal contour evolutions of the droplets of pure water (E 0% + W 100%), binary mixtures of compositions (E 20% + W 80%), (E 50% + W 50%) and (E 80% + W 20%), and pure ethanol (E 100% + W 0%) at T s ¼ 60
C is presented in Fig. 9. It can be seen in the top panel of Fig. 9 that the contact angle at the early stage (t=t e 6 0:4) is greater than 90
. As the droplet evaporates, the contact line remains pinned, but the contact angle decreases gradually. For droplets of (E 20% + W 80%), (E 80% + W 20%) and pure ethanol (E 100% + W 0%), the contact angle is always less than 90
, and the pinned contact line behaviour is observed for most of the evaporation process. The droplet interface undulations for higher ethanol compositions ((E 80% + W 20%) and pure ethanol) are also evident in the bottom two panels of Fig. 9. The undulations invariably begin as a deviation from the spherical cap profile such that there appears a pinching constriction at the middle of the droplet. The undulation waves progressively intensify and at the end of the evaporation stage, the droplet breaks up into several satellite droplets. As also discussed in Section 3.1, the earlytime dynamics of a (E 50% + W 50%) droplet is also interesting. In this case, the initial spreading of the droplet is more pronounced, which is not prominent for other compositions in which one component is present in a greater proportion than the other component. The plots for the droplet height (h), the droplet wetting diameter (D), the contact angles (h) and the normalised volume of the droplet (V=V 0 ) against the normalised evaporation time ðt=t e Þ for different compositions of ethanol and water mixture at T s ¼ 60
C are presented in Fig. 10(a)–(d), respectively. It can be seen in Fig. 10(a) that the droplet height decreases almost linearly with the increase in t=te for all the compositions, except in case of the droplet of (E 50% + W 50%) binary mixture. In Fig. 10(b), no monotonic trend on the variations of D versus t=te is observed. For the droplets of (E 50% + W 50%) and (E 100% + W 0%), the dro-

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Fig. 6. Variations of (a) the height (h) in mm, (b) the wetting diameter of the droplet (D) in mm, (c) the contact angle (h) in degree and (d) the normalised volume with initial volume of the droplet ðV=V 0 Þ versus time normalised with the life time of the droplet ðt=te Þ at different substrate temperatures for (E 50% + W 50%) solution.

Table 2 Lifetime of the droplets, te (s) at T s ¼ 60
C for different compositions of ethanolwater binary mixture. E (%) 0 20 40 50 60 80 100

W (%) 100 80 60 50 40 20 0

te (s) at 60 °C 190 ± 5 124 ± 4 62 ± 1 38 ± 1 44 ± 2 38 ± 1 12 ± 1

pure water (E 0% + W 100%), h  96
at t=t e ¼ 0. This initial hydrophobic behaviour of the droplet on the substrate at T s ¼ 60
C is visually evident in Fig. 9. The droplets of other compositions exhibit hydrophilic behaviour during the entire evaporation process. It can also be seen in Fig. 10(c) that the contact angle dynamics of the droplets of (W 50% + E 50%) mixture and pure ethanol becomes nonlinear; the nonlinearity is more for the droplet of (W 50% + E 50%) mixture. Note that due to the increase in interfacial undulations at the late stages for droplets with high ethanol percentages, the post-processing of the images becomes difficult. Thus in Fig. 10(a)–(d), we present the results till the time the undulation is minimum. Another interesting feature is observed in Fig. 10(d), where the normalised volume ðV=V 0 Þ against the normalised time ðt=te Þ plots for all the compositions considered are seen to nearly collapse into a single monotonically decreasing line. This suggests that despite the complexities just alluded to, the global evaporation flux rates for all the cases have an inherent self-similar nature that may be modelled through simple analytical methods. 4. Theoretical modelling of droplet evaporation rate

Fig. 7. The variation of the lifetime time, t e (in s) of a droplet with the percentage concentration of ethanol (E) in the binary mixture at T s ¼ 60
C.

plet wetting diameter ðDÞ increases with the increase in t=t e (spreading stage) and reaches a plateau designating the intermediate pinned stage (0:2 6 t=te 6 0:6). A modest initial spreading is observed for the droplet of pure ethanol (E 100% + W 0%), whereas a significantly high contact line spreading is observed in case of a droplet of (E 50% + W 50%) binary mixture. At the later times, the droplet undergoes a receding stage of evaporation as explained in Fig. 9. For other compositions, the wetting diameter, D remains almost constant (no spreading), while the height of the droplet, h decreases continuously. Inspection of Fig. 10(c) reveals that for

In order to gain a greater insight into the physics of the evaporation dynamics, we have developed theoretical evaporation models for sessile droplets of pure and binary fluids at room and elevated temperatures. The diffusion and free convection processes from the droplet interface are the primary mechanisms governing the droplet evaporation rates in an initially quiescent and unsaturated atmosphere. Four effects are responsible for the final evaporation flux of a sessile droplet. These are: (i) the mass flux due to the diffusion of vapour in the ambient air, (ii) the mass flux due to the Stefan flow, (iii) the mass flux due to the free convection of vapour from the saturated interface to the unsaturated ambient, and (iv) the passive mass transport flux due to the natural convection of air from the hot substrate to the cool ambient. Moreover, these effects are coupled to each other creating a complex mass flux field around the droplet interface. Thus, it is challenging to develop a satisfactory analytical correlation to calculate the evaporation flux for such a complex coupled phenomenon and

P. Gurrala et al. / International Journal of Heat and Mass Transfer 145 (2019) 118770

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Fig. 8. Temporal evolution of droplet shape for pure water (E 0% + W 100%), (E 50% + W 50% solution) and pure ethanol (E 100% + W 0%) at 60 °C substrate temperature. The length of the scale bar shown in each panel is 200 lm.

numerical methods are usually used for incorporating the effect of evaporation fluxes other than the steady state diffusion model (see, for instance, Ref. [24]). However, analytical correlations are extremely useful for engineering and design applications since they are easier to use, and in this work, we attempt to develop such a correlation for the vaporization rate of binary water-ethanol sessile droplets evaporating from a heated substrate. 4.1. Evaporation of pure fluid droplets For substrates at the room or nearly the room temperature, the diffusion based vapour transport mechanism is expected to dominate the evaporation process [47]. However, for heated substrates, the free convection driven fluxes are observed to play a significant role [48]. Here, first, we discuss the diffusion limited evaporation model and then extend it to incorporate the convection flux for pure liquids. 4.1.1. The diffusion limited model The diffusion limited droplet evaporation model is relatively well-developed and widely used by several research groups to model evaporation flux of pure droplets at room temperatures [12]. In our case, all the droplets can be assumed to have a spherical cap profile during the evaporation process. Thus we can calculate the experimental droplet volume as

VðtÞ ¼

pR3 ð1  cos hÞ2 ð2 þ cos hÞ 3

3

sin h

;

ð1Þ

where R is the wetting radius of the droplet as shown in Fig. 11, that provides a schematic diagram of a sessile droplet and other parameters used in the theoretical modelling. We assume that the effect of internal temperature gradients on the evaporation flux is small. Hence our analysis assumes the droplet to be isothermal and at the same temperature as that of the substrate throughout the evaporation process, which is good assumption for high thermal conductivity metallic substrates [26,28]. In our experiments, the contact angle, h is less than 90
at T s ¼ 25
C. Further, it is assumed that the vapour concentration at the liquid-vapour interface is at the saturated condition, csat ðT s Þ. The vapour concentration in the region far away from the droplet c1 ðT 1 Þ is Hcsat ðT s Þ, where H is the relative humidity of ambient air. In the case of pure ethanol, H ¼ 0. Under the abovementioned assumptions, the rate of evaporation for a pure droplet due to diffusion is given by [26,28]


 dm ¼ pRDM½csat ðT s Þ  c1 ðT 1 Þf ðhÞ; dt d

ð2Þ

  where dm is the mass evaporation rate due to diffusion, M is the dt d molecular weight of the fluid, D is the vapour diffusion coefficient at the mean temperature ðT s þ T 1 Þ=2 and the expression for f ðhÞ for h 6 90
is given by f ðhÞ ¼ 1:3 þ 0:27h2 [49]. 4.1.2. Model for free convection While the evaporation of a droplet on an unheated substrate is primarily governed by the diffusive fluxes, the evaporation

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Fig. 11. Schematic diagram of a sessile droplet on a substrate maintained at temperature, T s . H and RðtÞ are the height and the wetting radius of the droplet; h is the contact angle; T i and T 1 are the temperature at the liquid-vapour interface and the temperature of the ambient far away from the droplet, respectively.

Fig. 9. Comparison of the droplet spreading behaviour for different compositions. The contours are plotted in an interval of 0.2. The substrate temperature is T s ¼ 60
C.

dynamics of a droplet on a heated substrate depends on both the Stefan flow and the natural convection, particularly for the more volatile ethanol liquid that boils at 78 °C (see Ref. [44]). Sobac and Brutin [26] found that for light alcohols and hydrocarbons, the pure diffusion model may underestimate the evaporation flux by over 50%. In our analysis, we find that the steady state diffusion model underpredicts the evaporation rate of pure ethanol droplet

at T s ¼ 60
C (see Fig. 12), even though the model remains satisfactory for the evaporation of the water droplet. Carle et al. [28] developed a thermal Rayleigh number based correlation by fitting the data of the mean evaporation rate over the entire droplet lifetime at different substrate temperatures for several alcohols and hydrocarbons. While the correlation fit is extremely useful, the time evolution behaviour was not modelled, and further, a temperature gradient based Rayleigh number does not take into account the convection due to the concentration gradients. Recently, KellyZion et al. [50] have developed a correlation that takes into account both the diffusive and the concentration driven natural convective mass fluxes from an evaporating sessile droplet for several types of pure hydrocarbon species. Since these mass flux correlations are expressed as a function of the droplet wetting radius, they may be used to model the decrease of the droplet mass (and hence volume) over its lifetime. We have incorporated this model to calculate the droplet volume VðtÞ during the evaporation process and compared with our experimental results. Unfortunately, all the correlations developed by Kelly-Zion et al. [50] are for unheated substrates. Therefore, in the present study, a modified version of Kelly-Zion et al. [50] correlation has been implemented in order to account for the evaporation of an ethanol droplet from a heated substrate. Our modified evaporation model is discussed below.

Fig. 10. Variations of (a) the height (h) in mm, (b) the wetting diameter of the droplet (D) in mm, (c) the left contact angle (hl ) in degree, and (d) the normalised volume with initial volume of the droplet ðV=V 0 Þ versus time normalized with the lifetime of the droplet ðt=te Þ for different ethanol (E) – water (W) concentration at T s ¼ 60
C.

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The correlations developed for the modified Sherwood numbers are adapted from [50], and they are given by

2

Shd

gR30 ¼ Shd 41 þ a



The evaporation mass transport rate due to the free convection can be expressed as


 dm ¼ hm As ðqv ;s  qv ;1 Þ; dt c

ð3Þ

where hm is the convective mass transfer coefficient, qv ;s is the density of the air-vapour mixture just above the droplet free surface and qv ;1 is the density of the ambient medium. The liquid-vapour interface area, As is given by (assuming a spherical cap profile)

As ¼

2pR2 : 1 þ cos h

ð4Þ

Neglecting the Stefan flow, the total mass transport rate can be calculated as





 dm dm dm dm ¼ þ þ ; dt dt d dt c dt t

ð5Þ

where the first, second and third terms in the right hand side of Eq. (5) are the diffusion mass transfer rate, the free convective mass transfer rate and the vapour mass transport rate due to air convection, respectively. The convective mass transfer coefficient is usually expressed in terms of the convective Sherwood number, Shc hm R=D, where D is the vapour diffusion coefficient. To incorporate the convective and the diffusive mass transfers in a single expression, we can define a diffusion Sherwood number, Shd hd R=D. The diffusive mass transfer coefficient, hd can then be calculated from the following relation

dm dt d

¼

pRDME ðcsat ðT s Þ  c1 ðT 1 ÞÞf ðhÞ;

¼ hd As ME ðcsat ðT s Þ  c1 ðT 1 ÞÞ;

ð6Þ

where ME is the molecular weight of ethanol vapour, csat ðT s Þ and c1 ðT 1 Þ are the saturated vapour concentration at the substrate temperature and the vapour concentration in the ambient. Noting that the concentration of ethanol is zero in the ambient, the expression for the diffusion Sherwood number reduces to

Shd ¼

f ðhÞð1 þ cos hÞ : 2

ð7Þ

The correlation for the effective Sherwood number is given by



Shcor ¼ Shd þ Shc ;

Shd

Shc

ð8Þ

where and are the modified diffusion and the modified convective Sherwood numbers, respectively after taking into account of the coupling between the diffusion flux and the convective flux when both the evaporation modes are active.

qm  qa Sc qa

m2o

Shc ¼ d

Fig. 12. Comparison of the experimental and theoretically obtained ðV=V 0 Þ versus t=t e at T s ¼ 60
C for pure ethanol (E 100% + W 0%) calculated using diffusion (Df ), diffusion + convection (Df þ C v ) and diffusion + convection + transport (Df þ C v þ T m ) models.

!b


gR30

!i

m2o

m 2ðijÞ
q  q f j o m a

m

qa

3

cb Ra

b5

;

Scej Raj ;

ð9Þ

ð10Þ

where R0 is a nominal drop radius (¼ 1 mm); qm and qa denote the density of the air-vapour mixture near the interface and of ambient air, respectively; mo is the ambient air viscosity at the standard condition, i.e. 25
C and one atmospheric pressure; m is the viscosity of the ethanol vapour at ðT s þ T 1 Þ=2; g is the acceleration due to gravity; Scð m=DÞ is the Schmidt number of the vapour at ðT s þ T 1 Þ=2. The Rayleigh number, Ra associated with the convective mass transfer is defined as




qm  qa Ra ¼ GrSc ¼ qa



gR3

m

2

!



m D

;

ð11Þ

where Gr is the Grashof number. The fitting parameter values are taken from [50]. The values of the relevant physical properties of ethanol vapour, water vapour, liquid ethanol, liquid water and air are provided in Table S1 (supporting information). Once the value of Shcor is evaluated from Eq. (8), the combined evaporation mass transfer rate due to diffusion and convection from the droplet interface can be evaluated from the following relation



 dm dm þ ¼ hdþc As Mðcsat ðT s Þ  c1 ðT 1 ÞÞ; dt d dt c

ð12Þ

where the combined diffusion and convection mass transfer coefficient, hdþc is given by

hdþc ¼

Shcor D : R

ð13Þ

The above expression (Eq. (13)) can be used for the evaporation of both pure ethanol and pure water droplets at room temperature. 4.1.3. Passive transport due to free convection of air The relation considered so far incorporates the effects of both the diffusive mass flux and the convective mass flux due to the gradients in vapour concentration. But in the case of a sessile droplet on a heated substrate, additional terms arise due to the temperature gradient driven free convection of air. While water vapour can spontaneously rise up due to its natural buoyancy in air, the heavier ethanol vapour is passively transported along with the free convective air flow in the upward direction. This transport mass flux, denoted by ðdm=dtÞt , can be expressed in terms of the mass flow rate of air as



 dm dm ¼ Y sv ; dt t dt a

ð14Þ

where Y sv is the mass fraction of (ethanol) vapour above the free surface of the droplet. The mass convection of air over the area of the heated substrate covered by the droplet can be expressed as



 dm Ma pa1 pas a : ¼ hm pR2  dt a Ru T 1 T s

ð15Þ

Here, air has been approximated as an ideal gas; Ru is the universal a gas constant; hm denotes the mass transfer coefficient for air; Ma is the molecular weight of air; pa1 and pas are the partial pressures at the ambient and the plate surface, respectively. The Sherwood numa ber for air is given by Sha ¼ ðhm RÞ=Da , wherein Da is the diffusion

10

P. Gurrala et al. / International Journal of Heat and Mass Transfer 145 (2019) 118770

coefficient of air. For natural convection over a horizontal flat surface, Sha is related with the air Rayleigh number, Raa as follows [51],

Sha ¼ 0:54Raa1=4 ;

ð16Þ

where




qa ðT s Þ  qa ðT 1 Þ Raa ¼ Gra Sca ¼ qa ðT 1 Þ



gL3

!

m2a




ma

Da

 :

ð17Þ

Here all the properties are with respect to air; the characteristic length, L ¼ Ap =P p , wherein Ap and P p are the area and perimeter of the heated plate, respectively. The final mass evaporation rate for ethanol from the substrates at elevated temperatures is thus the sum of diffusion, convection and passive transport terms given by Eq. (5) which can now be expressed as

dm dt

¼

dm

dt d

þ

dm

dt c

þ

dm

dt t

¼ hdþc As Mðcsat ðT s Þ  c1 ðT 1 ÞÞ þ Y sv

dm

ð18Þ

dt a

and is evaluated for the pure ethanol droplet at a given substrate temperature, T s . The density of ethanol is calculated by assuming that the temperature of the droplet is equal to the substrate temperature. The calculated density is used to compute V=V 0 at various substrate temperatures. In Fig. 12, we plot the relative contributions of the three evaporation mechanisms for a pure ethanol droplet at T s ¼ 60
C. It is seen that pure diffusion alone (Df ) grossly underpredicts the rate of evaporation mass loss. The ðV=V o Þ slope remains underpredicted even when convection is coupled with diffusion (Df þ C v ) using the correlations developed by [50]. Only after adding the natural convection of air induced passive mass transport (Df þ C v þ T m ), do we see a good match with the experimental results (Fig. 12). Thus our experiments provide good validation of the importance of the three above mentioned mechanisms in jointly determining the evaporation history of ethanol from a sessile ethanol droplet deposited on a heated substrate. For water, the coupling between the diffusion and the convection alone is found to be sufficient as water vapour, being lighter than air, rises up by its inherent buoyancy. 4.2. Evaporation of binary fluid droplets For binary mixtures, the drop volatility and the component mole-fractions in the evaporating vapour now varies with the variation of the concentration of liquid water and liquid ethanol inside the evaporating droplet. Thus, the evaporation dynamics is governed by the vapour-liquid equilibrium of the binary mixtures. Details of the vapour-liquid equilibrium diagrams for the ideal and non-ideal solutions can be referenced from a chemical thermodynamics book (e.g. Ref. [52]). The water-ethanol binary mixture in the liquid state deviates significantly from the ideal solution model and hence does not follow Raoult’s law. However the vapour-liquid

equilibrium (VLE) diagram for non-ideal solutions can be evaluated using the functional group based semi-empirical UNIFAC and modified UNIFAC models [53]. Sample plots of the VLE diagrams of water-ethanol binary mixture at T s ¼ 25
C and T s ¼ 60
C are provided in Fig. 13(a) and (b), respectively. In Fig. 13(a) and (b), the ethanol mole-fraction ðve Þ and the vapour pressure are plotted along the x and y-axes, respectively. The saturated liquid line (bubble line) and the saturated vapour line (dew line) are shown, which separate the pure vapour, the pure liquid and the two-phase region in between. For a given initial mole-fraction based on the droplet composition and at a given substrate temperature, the saturated liquid line provides the vapour pressure of the evaporating binary mixture. Then the tie line intercept with the saturated vapour line provides the molar composition of the newly evaporated vapour [52]. These vapour pressure and vapour phase mixture composition data from the VLE plot are used to calculate the instantaneous mass evaporation rate of the individual components (water and ethanol). The new molar composition of the liquid in the droplet for the next time step is calculated subsequently and is used in conjunction with the VLE diagram to evaluate the new vapour pressure and the bubble point composition. This iterative process is continued till the end of evaporation. The liquid solution density is evaluated at every time step, which is used to calculate the instantaneous droplet volume as shown below. The total mass of the droplet at any instant, mdroplet ðtÞ is given by

mdroplet ðtÞ ¼ mw ðtÞ þ me ðtÞ;

ð19Þ

where mw ðtÞ and me ðtÞ are the masses of water and ethanol present in the droplet of the binary mixture at any instant t. The mass fractions of water Y w ðtÞ and ethanol Y e ðtÞ in the droplet at any time, t are given by

Y w ðtÞ ¼

mw ðtÞ ; mw ðtÞ þ me ðtÞ

and Y e ðtÞ ¼ 1  Y w ðtÞ;

ð20Þ

respectively. The mole fractions of water, vw ðtÞ and of ethanol, ve ðtÞ can also be evaluated from the following relations

vw ðtÞ ¼

mw ðtÞ=Mw ; mw ðtÞ=Mw þ me ðtÞ=Me

and

ve ðtÞ ¼ 1  vw ðtÞ:

ð21Þ

The ethanol-water mixture is a non-ideal solution and requires an estimation of the excess molar volume of mixing V  [55]. The density of the non-ideal mixture, qm can be evaluated from the expression,

qm ðtÞ ¼

vw ðtÞMw þ ve ðtÞMe w e V  þ vw ðtÞM þ ve ðtÞM q q w

:

ð22Þ

e

where qw and qe are the densities of water and ethanol, respectively. The values of the excess volume vary with mixture composition and mixture temperature, and are either tabulated [55] or

Fig. 13. The vapour-liquid pressure curves for the binary mixture at (a) 25
C and (b) 60
C [54].

P. Gurrala et al. / International Journal of Heat and Mass Transfer 145 (2019) 118770

expressed in terms of Redlich-Kister (R-K) correlations [56]. For our calculations of V  , we have used the R-K polynomial expansion coefficients provided in a recent study [57]. Using Eqs. (19) and (22), we obtain the volume of the droplet of the ethanol-water mixture at any instant as

VðtÞ ¼

mdroplet ðtÞ : qm ðtÞ

ð23Þ

5. Comparison of the theoretical model with experimental results In this section, the theoretically obtained ðV=V 0 Þ versus t=te for droplets of pure ethanol and water as well as ethanol-water binary solutions at a near-ambient and at elevated substrate temperatures are compared against the experimental results. We have compared the theoretical and the experimental ðV=V 0 Þ versus t=t e curves for a binary droplet at 25
C, T s ¼ 40
C, 50
C and 60
C in Fig. 14(a)–(d), respectively. It is seen that the implementation of the binary VLE based iterative algorithm provides an excellent theoretical match with the observed droplet volume curve for the binary mixture for all the substrate temperatures. It is to be noted here that the experimental result for binary mixture is on the volume/volume basis and is converted to the mole/mole basis for the implementation in the VLE based calculation. At elevated substrate temperatures, the contribution of free convection and passive transport due to air convection is significant and the evaporation mass flux is severely underpredicted when only pure diffusive flux is considered. In contrast, for unheated substrates, diffusion mass transfer is the dominant mode of evaporation. The theoretically evaluated mass-fraction histories of the individual constituents along with the decrease in the normalised mass ðm=m0 Þ for the binary droplet with t=t e are shown in Fig. 15(a)–(d) at the corresponding values of T s . The figure demonstrates the continuous decrease in the concentration of ethanol in the liquid solution with time, such that the droplet is almost entirely pure water for t=te > 0:6. From the VLE diagrams shown in Fig. 13 (a) (for 25
C) and Fig. 13(b) (for 60
C), one sees that a decreasing fraction of ethanol in the solution with dilute ethanol concentration

11

(ve < 0:1) leads to a steep fall in the vapour pressure of the solution which explains the experimentally observed flattening curvature of the ðV=V 0 Þ versus t=t e curve in the later stages of the droplet lifetime. Thus, apart from providing reliable predictions, the theoretical model is also capable of providing physical insights into the system dynamics that could not be gleaned from the experimental data alone. It is to be noted that the normalisation of the time axis for ease of visualisation obscures the fact that the absolute evaporation times of the binary droplet between the substrate at 25
C and the substrate at 60
C are widely divergent, spanning over two orders of magnitude, as can be seen in Fig. 3. The ability of the model to accurately predict the evaporation rates over such a wide range of evaporation times and different substrate temperatures points to its inherent robustness. As the theory also incorporates the correlation from Ref. [50] that had been well validated for a wide range of hydrocarbons, we believe that the current methodology can be easily extended to other types of binary droplets as well, though that is left as a topic of future research. We compare the theoretical predictions for droplet volume with time for different composition at an elevated substrate temperature (T s ¼ 60
C). Fig. 16(a)–(f) show the comparison of the theoretical and experimental ðV=V 0 Þ against t=t e at T s ¼ 60
C for the droplets pure water (E 0% + W 100%), (E 20% + W 80%), (E 50% + W 50%), (E 60% + W 40%), (E 80% + W 20%) and pure ethanol (E 100% + W 0%), respectively. The theoretical model is seen to match the experimental V=V 0 data quite well for all the compositions, though some over-prediction of the evaporation rate is noticed for the (E 80% + W 20%) case. As noted in Section 3.1, the observed evaporation rates at the elevated substrate temperature cannot be modelled unless the contribution to mass transfer due to free convection and passive transport are taken into account. Fig. 12 gives the relative contribution of the three evaporation modes for pure ethanol droplet at T s ¼ 60
C. It can be seen in Fig. 16 that reasonably good agreements between the theoretical predictions and the experimental results are achieved for all compositions considered. However, there is significant late stage divergence between the experimental and the theoretical V=V 0 values for the (E 80% + W 20%) droplet

Fig. 14. Comparison of the experimental and theoretically obtained ðV=V 0 Þ versus t=t e for a droplet of (E 50% + W 50%). (a) T s ¼ 25
C, (b) T s ¼ 40
C, (c) T s ¼ 50
C and (d) T s ¼ 60
C.

12

P. Gurrala et al. / International Journal of Heat and Mass Transfer 145 (2019) 118770

Fig. 15. The variations of the normalised mass of the droplet with the initial mass of the droplet ðm=m0 Þ, water and ethanol mass fractions, Y w and Y e versus t=t e at different substrate temperatures for a droplet of (E 50% + W 50%) binary mixture. (a) T s ¼ 25
C, (b) T s ¼ 40
C, (c) T s ¼ 50
C and (d) T s ¼ 60
C.

Fig. 16. Comparison of the experimental and theoretically obtained ðV=V 0 Þ versus t=t e at T s ¼ 60
C for droplet of binary mixtures (a) Pure water (E 0% + W 100%), (b) (E 20% + W 80%), (c) (E 50% + W 50%), (d) (E 60% + W 40%), (e) (E 80% + W 20%) and (f) pure ethanol (E 100% + W 0%).

P. Gurrala et al. / International Journal of Heat and Mass Transfer 145 (2019) 118770

evaporation case (Fig. 16(e)). We suspect that this divergence is associated with the early onset of vigorous interfacial instability waves and eventual droplet break-up that were observed for this composition as has been noted earlier. We observed that even before the droplet break-up, the profile of the (E 80% + W 20%) droplet is visibly deviating from the spherical cap shape, while the (E 60% + W 40%) droplet retains the spherical cap profile till the very end of evaporation. Thus it is not surprising that the theoretical evaluations begin to deviate away from the experimental results as the droplet departs from the assumed spherical cap profile due to the increasingly intense waves and undulations on its free surface. Thus, while the simple analytical model remains adequate for the predictions of the steady-state droplet evaporation dynamics of the binary solutions on heated substrates, instability induced oscillations do have a non-negligible impact on the evaporation behaviour for some (though not all) binary compositions. More experimental and theoretical studies are required to predict the onset of such instabilities and assess their impact on the binary droplet evaporation. Fig. 17(a)–(d) compare the evolution of the mass fractions of the individual components as well as the total normalised droplet mass ðm=m0 Þ with respect to the normalised evaporation time, t=te for the (E 20% + W 80%), (E 50% + W 50%), (E 60% + W 40%) and (E 80% + W 20%) compositions, respectively. For the (E 20% + W 80%) case, the droplet is predicted to completely lose all its ethanol by t=t e ¼ 0:15, which is associated with the inflection point seen in the theoretical ðV=V 0 Þ versus t=t e curve at this value of t=t e (Fig. 16(b)). The real behaviour however shows a much gradual change in the slope curvature (experimental curve of Fig. 16 (b)), indicating that some ethanol remains in the droplet at a very dilute concentration for a time period considerably longer than that predicted by the mass loss calculation alone. The presence of the low concentrations of ‘‘residual” alcohol during the late stage evaporation of binary alcohol-water mixtures has been indicated in earlier studies as well [39]. This observation is related to the fact that at very low alcohol concentrations the thermo-solutal convection flows are switched off inside the droplet [30] and the evaporation of ethanol becomes limited by the slow diffusion of ethanol

13

molecules to the interface from deep inside the droplet as has been discussed in greater detail by Liu et al. [58]. The ethanol component lasts longer for higher initial concentrations, though Fig. 17 (a)–(d) show that mass fraction of water increases inside the droplet as evaporation proceeds. Interestingly, for the (E 80% + W 20%) case, Fig. 17(d) shows that ethanol remains as the dominant component in the mixture till near the end stages of evaporation, with Y e  0:5 when m=m0  0:2. Whether the large concentration of ethanol and the corresponding low surface tension of the droplet interface near the end stages of evaporation are conducive to the observed larger end-stage instability and breakup propensity of the (E 80% + W 20%) evaporating droplet warrants further investigation. Nevertheless, once again, the simple analytical theory is seen to provide invaluable insights into the physics behind some of the more complex dynamics observed in our experiments for binary sessile droplets on heated substrates.

6. Concluding remarks We experimentally investigate the effects of substrate temperature and composition on the evaporation dynamics of sessile droplet (fixed volume of 5 ll) of ethanol-water binary mixtures. Seven different compositions of the ethanol-water binary mixture: pure water (E 0% + W 100%), (E 20% + W 80%), (E 40% + W 60%), (E 50% + W 50%), (E 60% + W 40%), (E 80% + W 20%) and pure ethanol (E 100% + W 0%), and four different values of substrate temperatures: T s ¼ 25
C, 40
C, 50
C and 60
C, have been considered in our experiments. The roughness of the substrate is studied using atomic force microscope (AFM) and scanning electron microscope (SEM). It is observed that the lifetime of the droplet, t e decreases logarithmically with the increase in the substrate temperature from 25
C to 60
C. At high substrate temperature (T s ¼ 60
C), the normalised volume of binary (E 50% + W 50%) droplet decreases linearly with the normalised time, but the trend becomes nonlinear at low substrate temperatures. This behaviour indicates that the evaporation dynamics of a E = 50% droplet at T s ¼ 60
C is similar to a droplet of another pure fluid with a higher

Fig. 17. The variations of normalised mass of the droplet with the initial mass of the droplet ðm=m0 Þ; Y w and Y e versus t=te at T s ¼ 60
C for droplets of (a) (E 20% + W 80%), (b) (E 50% + W 50%), (c) (E 60% + W 40%) and (d) (E 80% + W 20%).

14

P. Gurrala et al. / International Journal of Heat and Mass Transfer 145 (2019) 118770

volatility at room temperature. In the evaporation process, the more volatile ethanol evaporates faster leading to this nonlinear trend at the early stage. At T s ¼ 60
C, the lifetime of the droplet, te shows a non-monotonic trend with the increase in ethanol concentration in the binary mixture. For E < 50%, t e decreases linearly, but for E P 50%, te does not change much, which we believe is due to the non-ideal vapour pressure phase behaviour of the waterethanol binary mixtures. At this elevated substrate temperature (T s ¼ 60
C), except for the E = 50%, droplets of other compositions remain pinned at the early stage, followed by a receding stage at the later times. In case of the E = 50% droplet, we observe an early spreading stage, an intermediate pinned stage and a late receding stage. Unlike T s ¼ 25
C, at T s ¼ 60
C, the contact angle of the droplet of pure water at the early times is greater than 90° (hydrophobic), but for other compositions, the contact angle is less than 90° (hydrophilic) during the entire evaporation process. A self-similar nature in the variations of the normalised volume ðV=V 0 Þ against the normalised time ðt=te Þ curves is seen for different compositions at T s ¼ 60
C. We have also shown that the theoretical mass flux models developed by combining the effects of diffusion, concentration gradient driven convection, and passive vapour transport with the freely convecting air flow agree well with the experimentally observed trends of ðV=V 0 Þ versus ðt=te Þ. We have also observed undulations (interfacial instabilities) at the liquid-vapour interface of the droplets with high ethanol concentrations at elevated temperatures, which has not been understood clearly yet and can be studied in future. Declaration of Competing Interest No competing interests to declare from the authors. Acknowledgement: K.C.S. thanks Science & Engineering Research Board, India for their financial support (Grant No.: MTR/2017/000029). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2019.118770. References [1] B.J. de Gans, U.S. Schubert, Inkjet printing of well-defined polymer dots and arrays, Langmuir 20 (18) (2004) 7789–7793. [2] E. Tekin, B.J. de Gans, U.S. Schubert, Ink-jet printing of polymers–from single dots to thin film libraries, J. Mater. Chem. 14 (17) (2004) 2627–2632. [3] H. Koo, M. Chen, P. Pan, L. Chou, F. Wu, S. Chang, T. Kawai, Fabrication and chromatic characteristics of the greenish lcd colour-filter layer with nanoparticle ink using inkjet printing technique, Displays 27 (3) (2006) 124–129. [4] T. Lim, S. Han, J. Chung, J.T. Chung, S. Ko, C.P. Grigoropoulos, Experimental study on spreading and evaporation of inkjet printed pico-liter droplet on a heated substrate, Int. J. Heat Mass Transf. 52 (1–2) (2009) 431–441. [5] J. Park, J. Moon, Control of colloidal particle deposit patterns within picoliter droplets ejected by ink-jet printing, Langmuir 22 (8) (2006) 3506–3513. [6] D. Kim, S. Jeong, B.K. Park, J. Moon, Direct writing of silver conductive patterns: Improvement of film morphology and conductance by controlling solvent compositions, Appl. Phys. Lett. 89 (26) (2006) 264101. [7] T. Lim, J. Yang, S. Lee, J. Chung, D. Hong, Deposit pattern of inkjet printed picoliter droplet, Int. J. Precision Eng. Manuf. 13 (6) (2012) 827–833. [8] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Capillary flow as the cause of ring stains from dried liquid drops, Nature 389 (6653) (1997) 827. [9] M.K. Yau, R.R. Rogers, A Short Course in Cloud Physics, Elsevier, 1996. [10] S.T. Chang, O.D. Velev, Evaporation-induced particle microseparations inside droplets floating on a chip, Langmuir 22 (4) (2006) 1459–1468. [11] M.K. Tripathi, K.C. Sahu, Evaporating falling drop, Proc. IUTAM 15 (2015) 201– 206. [12] D. Brutin, Droplet Wetting and Evaporation: From Pure to Complex Fluids, Academic Press, 2015.

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