1. Introduction
In the modern world, the main tasks of technical design, such as weight reduction and object miniaturization, face implementation problems due to the disadvantages of conventional heat removal systems. Efficient dissipation of high heat fluxes requires new approaches to cooling [
1]. At present, various two-phase heat exchangers such as spray systems [
2], film cooling [
3], and microchannel heat exchangers [
4] are being widely investigated. In addition, some researchers are considering the possibility of changing the properties of the working fluid with the help of various additives [
5,
6].
One of the new directions in the study of the effectiveness of cooling systems is the use of aqueous solutions with surfactants as a working fluid [
7]. However, the addition of a surfactant can reduce the liquid evaporation rate, leading to a decrease in the equipment efficiency. The easiest way to find out the influence of this factor is to study the evaporation of an immobile sessile drop. The problem of an evaporating drop has attracted the attention of researchers since the end of the last century. A well-known problem is that of the “coffee drop” [
8]: the drop is deposited on a rough substrate and its contact line remains stationary during evaporation, when the capillary flow carries the material from the center to the edge and, on drying, results in a circular trail. Another case is when the drops are applied to a smooth, well-wettable surface. In this case, the drop spreads, i.e., the contact line is movable [
9,
10].
The theoretical description of evaporating droplets is complicated by a large number of physical effects that must be taken into account. The problem of droplet spreading and evaporation is the subject of many works, which are summarized in a sound review by Bonn et al. [
11]. In the classical mathematical formulation of the mass transfer model in single-component spherical droplets evaporating in still air, Maxwell’s approach was used [
12]. It was assumed that the ratio of diffusion time to evaporation time
is small, and then evaporation may be considered as a quasi-stationary process. In this case, the mass flow is limited by diffusion and for a spherical drop, it is equal to:
Assuming that the vapor obeys the ideal gas law, the vapor density can be expressed in terms of the partial pressure:
In the case of evaporation of a liquid drop in air, the resulting vapor on the droplet surface displaces the surrounding vapor-air mixture and forces it to move away from the drop at a certain velocity. This phenomenon is called the Stefan flow. The evaporation rate of a spherical droplet is limited by the Stefan flow [
13]:
Tonini and Cossali [
14] extended this model to the case of nonisothermal evaporation of a spherical drop in the surrounding atmosphere.
When considering small sessile drops with a contact radius less than the capillary length
, the gravitational forces are insignificant, so the liquid takes the form of a spherical cap. The height
hd, surface area
Sd and volume
Vd of a spherical cap can be calculated using the wet spot radius
Rd and the contact angle
θ:
Hu and Larsen [
15] developed an isothermal evaporation model for a sessile drop. An analytical dependence of the mass flow on the contact angle was obtained. The developed theory is in good agreement with the experimental data and the results of numerical calculations for contact angles from 0 to
π/2. According to this model, the mass flow is:
Experimental studies of the evaporation of a sessile drop have been carried out for a long time and have resulted in extensive and versatile data. The papers [
16,
17,
18] deal with water drop evaporation in the open air, showing the influence of such parameters as substrate material, angle of inclination, and environmental conditions on the evaporation process. By now, a relatively large amount of information on the evaporation of various liquids has been collected.
Tarasevich [
19] performed an analytical analysis of hydrodynamics inside a sessile drop at the initial stage of evaporation. The qualitative picture of the velocity field was shown to be independent of the ratio between the drop height and the radius of the contact line. Saada et al. [
20] numerically simulated the evaporation of a drop lying on a heated substrate. The simulation took into account the convection of the surrounding air as a result of heating from the substrate, but heat was considered to be transferred inside the drop only due to thermal conduction. The results revealed an underestimation of the evaporation rate compared to the experimental data, and the error increased with increasing substrate temperature. Timm et al. [
21] presented data on the evaporation rate in comparison with the diffusion dependence. This assumption was shown to be valid when the time scale of vapor concentration diffusion was much smaller than the total time of droplet evaporation. However, these conditions are valid for experiments in open air.
Misyura [
22] demonstrated that at high corrosion rates and intense gas evolution, the evaporation rate decreases due to a decrease in the rate of heat supply to the droplet surface. Modeling the corrosion kinetics also requires taking into account the statistical nature of the corrosion process, which depends on the wettability and droplet diameter. It was shown for the first time that the corrosion kinetics is determined by the wetting regimes. Misyura et al. [
23] also studied the effect of the thermocapillary convection and surfactant in a sessile droplet during nonisothermal evaporation.
Misyura [
24] investigated the influence of the free convection. The Marangoni thermal convection was shown to play a predominant role in the heat transfer in the liquid evaporation.
Truskett and Stebe [
25] investigated the effect of an insoluble surfactant monolayer on the evaporation rate. In this work, no difference from pure water was found. Kim et al. [
26] studied the flash evaporation of water droplets with the addition of a surfactant. The latter was shown to strongly affect the shape of the evaporating droplet due to Marangoni forces. Semenov et al. [
27] conducted an experimental and theoretical study of the contact angle during the evaporation of drops with surfactant solutions. Guti’errez et al. [
28] proved that during vacuum evaporation, a surfactant reduces the rate of water evaporation.
For normal liquids, in particular pure water and most aqueous surfactant solutions [
29], the surface tension decreases with increasing temperature. This means that the derivative of the surface tension coefficient with respect to temperature is negative. Hence, thermocapillary Marangoni forces arise with a nonuniform temperature distribution on the surface. The result is an outflow of fluid from warm area to cold area.
The term “self-wetting” was introduced by Abe et al. [
30], who studied the thermophysical properties of dilute aqueous solutions of high-carbon alcohols. Due to thermocapillary stresses and the shape of the surface tension–temperature curve, the investigated liquids spread in a “self-wetting” manner, spontaneously propagating towards hot areas, and thereby preventing hot surfaces from drying out and increasing the rate of heat transfer. Because of these properties, “self-wetting” liquids were associated with significantly higher critical heat fluxes compared to water [
31]. Savino et al. studied the behavior of vapor locks inside wickless heat pipes [
32]. They found that the plugs were significantly smaller than those of liquids such as water. The study of self-wetting liquids was carried out in microgravity for space applications on the International Space Station [
33]. Hu et al. [
34] demonstrated that the use of these fluids in micro-oscillatory heat pipes resulted in an increase in heat pipe efficiency.
The use of alcoholic solutions with water may be undesirable, since they can significantly differ in thermophysical properties from pure water. At the same time, a very small amount of this substance is used for creating a surfactant solution, which does not affect its thermophysical properties, but noticeably changes its surface properties.
The performed analysis of existing works has shown that there are very few experimental and theoretical data on the influence of various factors on the evaporation of a drop of an aqueous solution with surfactant in a wide range of wall temperatures. The complexity of these studies is related to the fact that at high-temperature evaporation, the isothermal model of diffusion is unacceptable and evaporation is influenced simultaneously by many key factors. In addition, the properties of the surfactant change strongly with increasing liquid temperature.
Previous experimental studies have demonstrated that the addition of a surfactant leads to a decrease in the evaporation rate of a liquid droplet. However, the specific factors influenced by the surfactant have not been established. In addition, there has been no comprehensive study of the effect of the surfactant in a wide range of wall temperatures and droplet base diameters.
Research objectives of this article are as follows. (1) Experimental determination of geometrical parameters of a droplet and measurement of temperature fields of a liquid droplet. The obtained experimental data are used to predict the evaporative behavior of droplets in a wide range of wall temperatures. (2) On the basis of experimental data, approximate calculations are made to estimate the influence of separate key factors on the droplet evaporation rate. The findings may be useful for the development of the existing models for determining the evaporation rate of droplets of solutions with surfactants, which is important for the creation of droplet irrigation technologies, widely used for heat exchange intensification in devices with high energy efficiency.
2. Experimental Setup
Preparation for the experimental study of droplet evaporation involved obtaining aqueous solutions with surfactants. In the presented work, three different surfactants were selected: polyoxyethylene sorbitan monooleate (TWEEN 80), cetrimonium bromide (CTAB), and sodium lauryl sulfate (SLS). They differed in properties depending on the charge component of their hydrophilic components: nonionic (TWIN80), cationic (CTAB), or anionic (SLS).
Using a high-precision micro analytical balance AND BM-252 with a readability of 0.01 mg, the weight of the surfactant was measured. It was then mixed with deionized nanofiltered Milli-Q pure water, and the solution was thoroughly stirred at room temperature in a closed cuvette, using a magnetic stirrer with medium speed.
Attempts to obtain an aqueous solution of TWIN80 have demonstrated that producing the solutions of nonionic surfactants requires a special procedure, described in chemical protocols, including special conditions and additional reagents. Concluding that TWIN80 is poorly soluble in water, it was decided not to use it for our experiments. The study of the remaining solutions is carried out using two experimental setups described below.
2.1. Tensiometer KRÜSS K100
The analysis of the properties of aqueous solutions with surfactants was carried out on a precision tensiometer KRÜSS K100 with KRÜSS laboratory software. Using a Huber thermostat, a cuvette with a working liquid was heated or cooled in the temperature range from 294 K to 343 K. Measurements were performed using the Wilhelmy plate method. A standard 10 mm × 20 mm × 0.2 mm platinum plate was used to measure surface tension since the contact angle of such a plate with most liquids is approximately zero. The second plate with dimensions of 10 mm × 20 mm × 2.8 mm was made of copper of industrial grade A1 and was used to measure contact angles. The copper surface was technically treated. In most cases, copper with high thermal conductivity and low cost is used as a main material for heat pipes. In the case of using a nonstandard plate, a special holder was used to ensure a strictly vertical position for the copper plate.
The tensiometer placed the platinum strip in the solution, then slowly removed it at a constant rate until it was completely out of the solution. During this procedure, the tensiometer measured the force f with which the plate was removed. Furthermore, the value of the surface tension coefficient σ was calculated by the formula:
In this case, the contact angle θ between the platinum plate and the investigated fluid was taken to be zero. To measure the value of the contact angle with the copper plate, a similar procedure was followed. To verify the obtained data, measurements were carried out in series of 5 runs. As a result, statistically averaged data for the surface tension coefficient σ(T) and the contact angle θ were obtained.
2.2. Drop Shape Analyzer KRÜSS DSA100
Experiments for the sessile liquid drop evaporation were performed with the use of the DSA100 drop shape analysis system, produced by KRÜSS and presented in
Figure 1. This device consists of three main parts: a high-precision dosing system with the dosing step of 0.1 μL, a motorized object table with program-driven movement in 2 horizontal axes, and the optical shadow system which includes a 50 W light source and a CCD camera with a resolution of 780 × 580 pixels (field of view varied from 3.7 mm × 2.7 mm to 23.2 mm × 17.2 mm). A Peltier chamber open to the atmosphere was used for heating substrates. It allowed the specified temperature of the bottom wall to be maintained with an accuracy of about 0.1 °C. The thermostat was used to cool the chamber and maintain the temperature of the substrate. The needle with the selected solution was moved into the chamber and a drop was deposited onto the substrate surface.
Measurement of the surface temperature of the copper substrate was carried out using a platinum resistance thermometer, fixed at the distance of 1–2 mm from the liquid drop. A special platinum resistance thermometer pt100 with a diameter of 2.5 mm, produced by KRÜSS, was applied. The accuracy of pt100 was 1/3 DIN class B (±0.1 °C at 0 °C to 0.8 °C at 400 °C). To increase the reliability of temperature measurements, the wall temperature was also measured by a thermocouple, located at a distance of 0.3–0.5 mm from the surface of the copper substrate on which a droplet was placed. The measurement error for the wall temperature (Tw) made by this thermocouple did not exceed 0.5–1 °C (taking into account the estimated distance of the thermocouple from the wall). The substrate surface temperature in each experiment was kept constant. The temperature and relative humidity of the ambient air during the experiments was 20–21 °C and 30–32%, respectively.
The measurement error of the droplet diameter did not exceed 5%. In the range of contact angle of 90–20°, the inaccuracy of the contact angle of the drop ranged within 3–5%. The maximum error of the evaporation rate did not exceed 5–7%.
The initial liquid drop volume was about 2–3 μL. The KRÜSSDSA Advance software allowed the processing of the shadow images of evaporating liquid droplets in fully automatic mode 2 (real-time determination of the droplet volume, base diameter and contact angles).
Figure 2 shows a photograph of the liquid droplet shape analysis.
3. Experimental Results
The presented results of the experimental studies are divided into two main parts. To begin with, the analysis of aqueous solutions of SLS and CTAB on the KRÜSS tensiometer is considered. The second part is devoted to the results of the study of droplet evaporation.
3.1. Surface Tension
The wide temperature range specific for the operation of a pulsating heat pipe imposes restrictions on the applicability of working fluids. One of the determining roles in heat transfer and efficient operation of a pulsating heat pipe is played by the properties of the working fluid used.
At the outset, researchers obtained data on the dependence of surface tension on the concentration of surfactants in an aqueous solution at a room temperature of about 20 °C. For this purpose, a wide range of concentrations of surfactant solutions was used. The maximum concentration exceeded the CMC (critical micelle concentration), while the minimum concentration was similar in terms of surface tension to that of pure water. The components of the experimental setup were carefully prepared before each measurement. To reduce contamination caused by the surfactants used, the cuvette for the investigated working fluid was cleaned with a 70% isopropanol solution and washed several times with clean water.
Aqueous solutions with four different concentrations of SLS were investigated and pure water was examined for comparison.
Figure 3 shows temperature dependence of surface tension of aqueous solutions with surfactants.
The general dependence
σ(
T) for CTAB is similar to that for water. That is, the first derivative of surface tension with respect to temperature is less than zero. In a heat exchange facility, for example, such as a pulsating heat pipe, liquid flows from the more heated area to the less heated one occurring on the surface due to Marangoni forces. This leads to the formation of dry spots and a decrease in the heat transfer coefficient, which may lead to overheating of the equipment. However, the SLS solution does not exhibit regular dependence, as CTAB, water and other normal liquids do [
35].
Figure 3 shows that
dσ/
dT is positive over a wide range of SLS concentrations. At the same time, it should be noted that at low SLS concentrations, a rapid change in the sign of the derivative is observed with increasing temperature. The temperature of the break point depends on the concentration. At higher temperature, the surface tension of the solution completely coincides with that of water.
Thus, aqueous solutions of SLS have the properties of a self-wetting liquid in a wide range of concentrations and temperatures. However, self-wetting liquids are quite often known to be unstable, which is also confirmed by the results of the current study.
From the obtained data, it may be inferred that SLS solutions are useful in solving the problem of dry spots and expanding the working range into the region of higher heat fluxes—with the proviso, of course, that for each task it is necessary to pay attention to the permissible range of applicability for each specific concentration of the solution.
3.2. Evaporation
In droplet evaporation, it is important to take into account the contact angle, since this determines the shape of the droplet surface. Therefore, the contact angles of evaporating droplets were measured for different concentrations of surfactant solutions and different substrate temperatures.
Figure 4 shows the evolution of a pure water drop during evaporation under isothermal conditions at a room temperature of about 21 °C. Since the drop has a small volume of the order of 1 μL, its shape is a spherical cap. In this case, evaporation apparently occurs at a fixed contact line, with the so-called “pinning” of the contact line. The evaporation is as a result of the diffusion of vapor from the drop surface, where it is saturated, into the atmosphere with a constant vapor concentration (in the experimental conditions, the humidity was about 30–40%). The evaporation is almost uniform over the entire surface area and is well described by Stefan’s theoretical model [
13]. Droplet evaporation on the heated surface qualitatively coincides with isothermal evaporation, but there are important differences. In this case, the mass flow from the droplet surface is substantially nonuniform. A significant mass flow is provided by intense evaporation in the area of the contact line, called the microregion [
36].
In a further step, the evaporation of drops of surfactant solutions was considered in order to find out the effect of the surfactant concentration on the evaporation rate (heat exchange with the substrate) and the behavior of the contact line during evaporation.
The data for a small set of drops are presented below, whereby all the given graphs of diameter and contact angle correspond to the same drops.
Figure 5 and
Figure 6 show values of the droplet contact angle and the droplet height versus time during evaporation. At the initial moment of time, the contact angle of droplets of surfactant solutions is in the range from 35° to 100°. The initial droplet height is 0.4–1 mm. Upon evaporation, the contact angle decreases, which is associated with a change in the droplet volume, but the contact line remains almost immovable. It is clearly seen that pure water corresponds to the maximum contact angle for both low (30 °C) and higher (80 °C) temperatures.
Meanwhile, at a low surfactant concentration, the contact angle does not differ from that of pure water, and with an increase in the surfactant concentration, it monotonically decreases. There are two reasons for this. First, the surface tension reduction leads to a decrease in capillary forces and spreading of the droplet. Secondly, at the initial stage, the temperature of the liquid is close to room temperature and, upon contact with the heated substrate, it begins to heat up rapidly. It is supposed that the drop heating in the vicinity of the contact line occurs faster than in the main volume of the drop, which gives rise to the Marangoni force. The force vector is directed along the surface from lower to higher temperature. Estimating the time required to heat the drop to the substrate temperature, previous researchers obtained the formula τ = R2ρcp/λ ≈ 6 s. Thus, the droplet diameter should grow noticeably in the first seconds of heating.
Figure 7 shows the droplet diameters as a function of time. At low surfactant concentrations in pure water, the diameter is seen to be constant during the entire evaporation time, except for the last seconds of the droplet life.
With increasing concentration, the area where the change in diameter occurs is clearly observed to stretch over time. Moreover, for a high concentration (above CMC), the droplet diameter changes throughout the entire evaporation process: in the first seconds, the drop diameter increases as the drop itself spreads, and then decreases as it dries.
4. Predicting the Evaporative Behavior of a Water Droplet with a Surfactant
Direct numerical simulation of the unsteady and nonisothermal evaporation of a water droplet with a surfactant is very complicated due to the large number of uncertain factors that affect each other. In the presence of a surfactant, it is hard to estimate the equilibrium diameter of the droplet, which can change markedly over time when the droplet evaporates. Any change in the diameter, contact angle or height of the droplet alters free convection in the gas and liquid phase. The surfactant affects the partial pressure of water vapor and, accordingly, the evaporation rate. To date, there are no reliable methods for computing all these factors. Calculating the temperature of the free surface of the droplet is extremely difficult, since the Marangoni flow and convection in a droplet depend on the surfactant concentration. Therefore, assessment of the influence of various key factors on the evaporative behavior of the droplet becomes crucial at the first stage. For this purpose, it is advisable to use experimental data on the geometry of the droplet, as well as on the thermal field measured by a thermal imager (Ts) and a thermocouple (Tw).
4.1. Key Factors Effecting the Droplet Evaporation
In general, the rate of evaporation depends on the geometry of the droplet, the conditions of heat transfer and on free convection. It is possible to distinguish the following key parameters that determine the intensity of evaporation of a water droplet with the addition of a surfactant:
- (1)
Drop diameter;
- (2)
Natural convection of gas;
- (3)
Natural convection of liquid;
- (4)
Nonisothermality;
- (5)
Buoyancy of water vapor in air and the Stefan flow;
- (6)
Evaporation rate of surfactant solution.
With a small droplet diameter (1–3 mm) and in the absence of wall or air heating, convection in the liquid and gaseous phases can be neglected. When a droplet evaporates on a hot wall, free convection has to be taken into account. The influence of these key factors on the evaporation of liquid are considered separately below.
(1) Let us take a detailed look at the influence of the wetted droplet diameter on the evaporation rate. In most experimental curves, there is an attached contact line for most of the evaporation time (evaporation mode of constant contact radius, or CCR). With the growth of the surfactant concentration, the droplet radius increases. Despite the decrease in the contact angle with time, for most of the curves, a quasi-linear decrease in the droplet volume is realized (i.e., at a fixed radius, the evaporation rate (dV/dt) is constant, despite a significant decrease in θ). Thus, the effect of the surfactant on evaporation due to the change in radius will be proportional to the ratio of the radii of the two drops (J1/J2 ∼ Rd1/Rd2). To analyze the influence of other key factors, it is necessary to exclude the influence of the radius being a part of the ratio J/Rd.
(2) Let us estimate the effect of gas convection on the evaporation of a drop. When a liquid layer evaporates, the convective mass flow of water vapor through air can be estimated using Equation (1) [
37],
where the Sherwood number
Sh =
βRd/
Dv, the Schmidt number
Sc =
νg/
Dv, the Reynolds number
Re =
UcaRd/
νg, in which
β is the mass transfer coefficient,
Rd is the radius of the droplet,
Dv is the diffusion coefficient of water vapor in air,
Uca is the rate of thermogravitational convection of air at the temperature difference (Δ
Ts) between the temperature of the free surface of the droplet
Ts and the temperature of the ambient air
Ta, and
νg is the kinematic viscosity of the gas. The evaporation rate
J is related to the mass transfer coefficient through the concentration difference
ρvs −
ρva given in Equation (2),
where
F is the interface area of the layer.
For intense convection, when the Sherwood number is much greater than 1,
, then in accordance with Equations (1) and (2),
The air velocity due to thermogravitational convection can be approximated from Equation (4),
and the evaporation rate is related to the droplet radius according to Equation (5):
As indicated in Equation (5), the presence of the surfactant leads to an increase in the droplet radius; and an increase in the radius contributes to an increase in evaporation.
(3, 4) Let us evaluate the joint influence of factors 3 and 4, since they are interrelated. For convenience of analysis, the effect of these factors will be denoted as the effect of nonisothermality. The heat flux in the liquid
ql is supplied to the interface of the drop and spent on evaporation (
qe =
rJ,
J = Δ
m/Δ
t), radiation (
qr =
εσ(
T4s −
T4a)) and free air convection (
qc =
αa(
Ts −
Ta)). The heat balance for the droplet surface corresponds to Equation (6),
where
r is the latent heat of vaporization, and
F is the area of the liquid interface. The air heat transfer coefficient
αa is determined by the Nusselt number
Nu = α
aRd/
λa = 0.54
Ra0.2, where
Ra is the Rayleigh number [
37]. The heat supplied to the liquid from the wall is determined by the conductive and convective heat transfer in the liquid according to Equation (7) [
38,
39,
40],
where
αl is the liquid heat transfer coefficient,
Ucl ∼
c(
Ma +
Ra),
Ul is the average convective velocity in the liquid droplet, the Marangoni number
Ma = (Δ
Twh/
μlal)(
dσ/
dTs), the Rayleigh number
Ra =
gβTΔ
Twh3/
νlal,
βT is the thermal liquid expansion,
g is the gravity acceleration, Δ
Tw =
Tw −
Ts,
νl is the water kinematic viscosity,
al is the liquid thermal diffusivity, and μL is the water dynamic viscosity. Empirical expressions for evaporation intensification with an increase in the
Ra number are given in [
41]. For very small droplets (droplet height below 1 mm), low values of
Ra and
Ma and low values of
Ul are realized, and the condition
αl ∼ 1/
h (conductive heat transfer) is satisfied. Therefore, the ratio of heat fluxes for a pure water drop (
ql1) to a water drop with surfactant (
ql2) will be inversely proportional to the ratio of heights
ql1/
ql2 ∼
h2/
h1. To determine the relationship between Δ
Tw and the heat flux, it is necessary to solve a system of differential equations and it is difficult to make a simplified estimate for the change in Δ
ρs due to nonisothermality. Therefore, the effect of nonisothermality can only be determined after the evaluation of other parameters (key factors 1, 2, 5, 6) and taking into account experimental data. At different wall temperatures, the droplet free surface temperature and
ρs were determined using thermal imaging measurements and the equilibrium curve for the equilibrium partial pressure of water vapor.
(5) Let us consider the effect of water vapor buoyancy. Since water vapor is lighter than air, the effect of the additional flow must be considered. The vapor convection velocity
Uρ corresponds to Equations (8) and (9) [
42],
where
ρa is the density of the air,
ρv is the density of the vapor,
ρm is the density of the air + vapor mixture, and
L is the characteristic length of the diffusion vapor layer. Experimental and calculated data in [
42] show that the increase in the evaporation rate due to the buoyancy of water vapor (for a wall with high thermal conductivity) is about 40%. It does not depend on the droplet radius (
Rd > 0.8 mm). Thus, despite the change in droplet diameter due to the surfactant, the effect of buoyancy on
J will be the same for a water droplet with and without surfactant.
(6) Let us consider the influence of the surfactant on the droplet evaporation rate. Since the surfactant alters the radius of the drop, the simultaneous influence of several factors appears. To eliminate the effect of the droplet radius, one may consider the evaporation of a water layer (with and without surfactant). According to Equation (10),
at zero contact angle, the evaporation rate does not depend on the angle: in this case, the expression for evaporation of the liquid layer thus coincides with the expression for a droplet with a very small height, Equation (11):
4.2. Evaporation of a Layer of Water with Surfactant
Figure 8 presents experimental data on the evaporation of a layer of water with (10,000 ppm) and without surfactant.
Figure 8a,b gives experimental curves for layer evaporation of
Tw = 21 °C, and
Tw = 90 °C, respectively. All curves demonstrate a linear dependence of the layer mass on time.
The measurement of the water layer mass is carried out using the gravimetric method. The working section made of copper with a layer of liquid is placed on the scales. The wall temperature is measured with a thermocouple. Although the cell radius is constant (40 mm), the characteristic radius Rd in Equation (11) is different according to the scenario considered. In the case of water and surfactant, the length of the meniscus on the side wall of the cuvette is greater than for pure water, and the total radius Rd1 is thus larger than for the water layer by about 11–12% at Tw = 21 °C and by 9–10% at Tw = 90 °C. Thus, to simplify the variables to a single radius when calculating the evaporation rate of the water with surfactant layer, it is necessary to divide the value of J/Rd by 1.11–1.12 and 1.09–1.1 (taking into account the length of the meniscus).
Figure 9 shows thermal images of the liquid layer and the profiles of
Ts along the line 0
L. The average surface temperature
Ts of the layer increases during evaporation due to the decrease in layer height. The central region temperature is lower than at the edges. For a water layer with surfactant, the average temperature over the layer surface is higher than the temperature for a pure water layer.
4.3. Thermal Imaging Measurements of the Free Surface of a Water Droplet with and without Surfactant
A droplet of water with surfactant shows a 15–20% decrease in
Jρ (compared to pure water) at
Tw = 21 °C and a 9–11% decrease at
Tw = 90 °C. The obtained data correspond to the previously obtained measurements in other works. In [
43], the maximum decrease in the rate of droplet evaporation with a surfactant is 25–30%. The presence of surfactant also leads to a decrease in
Jρ for the liquid layer [
44]. The decrease in
Jρ is due to the competing effect of two factors—a decrease in the instantaneous area of the free surface of a drop occupied by water molecules, and a weakening of the molecular bonds between water molecules and surfactant molecules. The integral value of the two parameters for the entire free surface of the liquid decreases the equilibrium partial pressure of water vapor
ρvs and, in accordance with Equation (11), the evaporation rate.
To assess the influence of nonisothermality, the temperature of the free surface of the drop
Ts was measured using a thermal imager and an optical lens providing a tenfold increase in the image (
Figure 10a). Evaporation leads to a decrease in
Ts. The minimum drop temperature corresponds to the maximum height, i.e., the center of the drop (
Figure 10b). Since the diameter of the water with surfactant drop is 1.8–2 times larger than the pure water drop diameter, the height is significantly less. As a result, the temperature difference Δ
Tw =
Tw −
Ts is lower by 0.4–0.7 °C for a water with surfactant droplet than for water alone. In other words, a drop with a lower height has higher values of
Ts and
JT. When adding a surfactant, the increase in
JT (the effect of nonisothermality) is 2–3%.
With high temperature heating (
Tw = 90 °C), the droplet diameter increases by 50–60% due to the addition of surfactant. Due to the fast evaporation rate on the heated wall, the temperature
Ts depends much more on the drop height. For a water with surfactant droplet, the average temperature
Ts (
Figure 11 and
Figure 12 for
t = 7 s) for the droplet surface is 4–5 °C higher than that for water during almost the entire evaporation time. This effect of nonisothermality (
JT) leads to a 14–17% increase in the evaporation rate of a droplet of water with surfactant, compared to a pure water drop (according to the water vapor equilibrium curve).
Figure 11 and
Figure 12 show thermal images of the surface of a drop located on a heated wall, as well as temperature distributions of
Ts along the 0
L axis. As the wall temperature and
Ts increase, the difference Δ
Ts increases (comparison with
Figure 10a). The small height of the drop and the high heat flux from the wall contribute to the temperature field uniformity over the drop surface.
The temperature profile of the droplet surface at a high wall temperature (
Tw = 90 °C) is qualitatively similar to that at an unheated wall. (
Figure 10b). The minimum temperature is at the center of the drop and the maximum temperature is at the edges. This distribution is associated with a negligible effect of free convection in the liquid, i.e., the temperature field is determined by the conductive heat transfer due to the small values of the Rayleigh numbers. In this case, the temperature
Ts is determined by the height of the droplet (
Ts ~ 1/
h). The surfactant results in a higher temperature
Ts and a more uniform temperature distribution over the droplet surface.
4.4. Assessing the Impact of Key Factors on Evaporation of the Water Droplet with Surfactant
To evaluate the effect of several factors on droplet evaporation, it is convenient to exclude the influence of the droplet base diameter, which strongly depends on the surfactant. For this purpose, it is convenient to attribute the value of the evaporation rate to the diameter of the droplet.
Figure 13 provides experimental data on the dependence of
J/2
Rd on the wall temperature
Tw. Points are given for pure water (points 1), water with surfactant 1000 ppm (points 2) and water with surfactant 10,000 ppm (points 3).
For droplet evaporation at room temperature, as well as at small wall heating (Tw = 313 °K), a water drop evaporates faster than a water with surfactant one (by 14–17% at Tw = 293 °K). At a temperature of Tw = 333 °K, the evaporation rates for points 1–3 are close to each other. At a wall temperature of Tw = 363 °K, a droplet of water with surfactant with a concentration of 10,000 ppm evaporates 25–27% faster. Thus, the simultaneous influence of several factors considered above leads to an inversion of J/2Rd c with increasing temperature. The calculated curve 4 is constructed for the maximum value of the contact angle. This angle corresponds to the experiment with the maximum angle, which is taken as an average value over time (θ = (97° + 20°)/2 = 60°). The curve of angle change over time has a quasi-linear character. Curve 5 is plotted for the lowest contact angle over time (θ = (35° + 20°)/2 = 27°). The drop radius is also taken as the time average in the calculation and processing of the experimental data. The diffusion coefficient of vapor in air is determined at a temperature T = (Ts + Ta)/2, where Ts is the temperature of the free surface of the drop, Ts is the temperature of the external air, and the temperature Ts is measured with a thermal imager. The equilibrium partial pressure of water vapor and the value of the vapor density are determined from the temperature of the free surface of the drop Ts.
Based on the above analysis, diagrams illustrating the influence of various factors on the evaporation of a drop are presented in
Figure 14.
In
Figure 14, the influence of various factors is denoted as follows:
Jd is the influence of the droplet diameter,
Jρ is the influence of the equilibrium partial pressure of water vapor,
Jc is the influence of thermogravitational convection, and
JT is the influence of non-isothermality. Both at
Tw = 20 °C and at
Tw = 90 °C, the predominant effect on evaporation is the growth of the droplet diameter due to the surfactant (
Jd). At high temperature,
Jc and
JT are comparable. The decrease in the evaporation rate (
Jρ) as a result of a decrease in the partial pressure of vapor at room temperature is double that at high-temperature heating. As the temperature
Ts increases, the effect of
Jρ on evaporation decreases. The influence of the natural convection of gas (
Jc ∼ (
Rd)
0.5) in Equation (5) noticeably manifests itself at high temperatures only (in
Figure 10b, 18–21%) and at room temperature, thermogravitational convection can be neglected. As the wall temperature increases from 20 °C to 90 °C, the effect of
JT on the enhancement of evaporation increases approximately six times (from 2–3% to 14–17%). Thus, the
J/2
Rd inversion with increasing wall temperature (when comparing the evaporation rate of a water drop and a drop of water with surfactant) is associated with a strong dependence of all these key factors on temperature.
At
Tw = 90 °C, the experimental points
J/2
Rd are located 67–71% above the calculated curves. According to
Figure 14b, the influence of gas convection leads to a 67–70% excess of the experimental values of
J/2
Rd, compared to the calculated diffusion model (Equation 12), if the measured free surface temperature
Ts is taken for the diffusion model. As mentioned above, 40% corresponds to an increase in the evaporation rate due to the buoyancy of water vapor. In
Figure 14, this factor is not specified, since it has the same effect on a water drop with or without a surfactant. The increase in the evaporation rate due to thermogravitational convection is approximately 30% (40% + 30% = 70%). The effect of nonisothermality is expressed in a decrease in temperature
Ts compared to the wall temperature
Tw. Thus, for
Tw = 90, 60, 40, and 21 °C, the average temperature for the droplet surface
Ts is approximately 78–80 °C, 54–55 °C, 38–39 °C, and 19–19.5 °C, respectively. At the maximum wall temperature (
Tw), the decrease in
ρvs due to the decrease in
Ts is 40–50% (the maximum effect of nonisothermality on droplet evaporation).
At Tw = 60 °C, the influence of convection and nonisothermality decreases. At Tw = 40 °C, the calculation is slightly lower than the experimental points. During evaporation without heating (Tw = 20 °C), the Rayleigh number for the gas and liquid phases is low (there is no natural convection, since there is a small droplet height and small temperature gradients due to droplet evaporation), and the experimental data are quite accurately modeled by the diffusion model.