Pure Water Drop Research Articles

The Onset and Early Stages of Dynamic Wetting of Superspreading and Non-Superspreading Trisiloxane Surfactant Solutions on Hydrophobic Surfaces

The onset and early stages of dynamic wetting on different hydrophobic surfaces is investigated experimentally for aqueous solutions of two commercial trisiloxane surfacants of similar chemical structure, one of which exhibits superspreading behaviour, in order to investigate the spreading dynamics independently of the surface activity. Superspreading, or the ability of a surfactant solution to spread on a surface beyond the state determined by thermodynamic equilibrium, has been investigated for more than 30 years however its physical mechanism remains poorly understood to date despite its important applications in the formulation of agrochemicals. Surfactant solutions were prepared by dissolving S233 and S240 surfactants (Evonik Industries AG, Essen, Germany) into de-ionised water at a weight concentration of 0.1%. Drops of surfactant solutions and pure water were deposited on three horizontal substrates with different wettability (equilibrium contact angle of water ranging between 55∘ and 100∘), and observed from below with a high-frame rate camera to visualise the advancing contact line. The spreading ratio of drops as a function of time was extracted from high-speed videos by digital image processing. Results reveal that the superspreading solution exhibits an intermittent spreading rate, as well as peculiar features of the contact line, which are not observed for the non-superspreading solution, and confirm the superspreading effect becomes less significant when the surface energy of the substrate is decreased.

Full-field vapor concentration and temperature field measurement of an evaporating ethanol–water binary sessile drop by tomographic laser absorption spectroscopy

Quantitative measurement of full-field, spatially resolved temperature and concentration field of pure water and ethanol–water binary evaporating drop is realized, in which the gas-phase and interfacial temperature and concentration are accurately captured. By studying the vapor field of evaporating drop under different heating temperatures, it is demonstrated that the method can achieve a spatial resolution below 100 μm and a time resolution of <10 s. Simultaneous gas-phase temperature and concentration field measurements reveal the occurrence of buoyancy convection in the gas phase. Through the analysis of interfacial temperature and concentration distribution, it is observed that in the process of pure water drop evaporation, both buoyancy-driven convection and thermal Marangoni convection exist, while in the ethanol–water binary drop evaporation, the solutal Marangoni flow convection and thermal Marangoni convection are coupled, and the buoyancy convection is suppressed. The interfacial temperature and gas-phase water vapor concentration can be obtained from the water vapor measurement, and combined with the activity coefficient models. The liquid-phase mole fraction of water at the interface and its distribution are also obtained, such that the liquid-phase mole fraction distribution of ethanol at the interface can be obtained, and finally, the concentration of ethanol vapor near the interface is obtained. The full-field, high-resolution measurement of evaporated drops is of substantial significance for in-depth understanding of the evaporation process. The measurement of ethanol–water binary drop evaporation provides a new research perception and method exploiting the spectral dimension, providing both quantitative and qualitative observations for the study of multi-component drop evaporation.

Controlling the impact dynamic behavior of a water-in-oil compound drop using the dielectrowetting effect

Controlling the impact dynamic behavior of a water-in-oil compound drop is vital for applications such as the enhancement of printing resolution and the preparation of innovative liquid-separation materials and devices. A phase diagram is constructed between the impact behavior, Weber number, and equilibrium contact angle of the water drop. The dynamic impact behavior of water-in-oil compound drops can be effectively controlled by changing the surface wettability using the dielectrowetting effect. The maximum spreading factor βm at different wettabilities shares a similar scale law βm∝We0.17 for 45.75 < We < 299.62 and βm∝We0.23 for 385.38 < We < 1183.19. Finally, the relationship between the velocity and radius of the rebounded drops was consistent with the results for pure water drops. Our findings indicate that the impact behavior of water-in-oil compound drops can be controlled at the right time scales using a dielectrowetting substrate; this is applicable in industries such as printing and liquid separation.

Ring shells obtained from pure water drops evaporating on a soluble substrate

When a pure water drop evaporates on a soluble substrate, the final ring deposit takes the form of an inclined wall for small and flat drops and of a hollow rim for other geometries. We report here on experiments of evaporation of pure water and salty water drops on salt single crystals and inert substrates, which permit to define precisely the experimental conditions necessary to obtain such structured deposits. Their growth is not a consequence of particular thermal or chemical conditions but of the peculiar geometry of the experiments. Beside the advection of the solute toward the edge by the coffee-stain effect induced by the pinning of the drop, the role of nucleation site played by the salt substrate is a necessary condition for the hollow or inclined rim to appear. These experiments have also shown to which extent the final dryout microstructure is reminiscent of the salt crystallography. Finally, these experiments demonstrate the potentiality of the phenomenon for surface-patterning.

Influence of Lubricant Addition on Heat Exchange Regimes During Spray Cooling

Spray cooling is used in a variety of industries, including metallurgy, electronics, and medicine. Ease of spray generation, as well as the ability to cool relatively large surfaces or precise regions of interest, are among the key assets. Spray cooling can provide a relatively high heat flux, especially if it is accompanied by liquid evaporation. Additionally, it can be applied not only as a coolant but also as a transport medium for lubricating fluid in specific applications, such as hot die forging. Typically, the working fluid is a multi-component mixture with improved cooling and lubricating properties. Once the bulk liquid evaporates, particles or dissolved lubricant can settle on the hot solid substrate. Existence of components with various physico-chemical properties on the surface (binders, surfactants, dispersed particles, etc.) can significantly affect the spray impact, as well as the outcome of cooling regimes. Nevertheless, the accompanying thermal-hydraulic effects are still not well understood. An experimental facility for observing the impact of the multi-component spray, and characterizing heat transfer was designed for this study at the Institute for Fluid Mechanics and Aerodynamics. As a working fluid, varied mixtures of water and industrial white lubricant were used in different ratios. The visualization of spray impact and identification of the main hydrodynamic regimes were achieved using a high-speed video system. The inverse heat conduction problem was used to calculate the heat flux during continuous cooling from 445°C to 100°C taking into account temperature readings from within the substrate. Since the substrate was entirely insulated on all sides (except on the sprayed surface), the boundary conditions were well defined. The experimental data for pure water drops were compared with the results obtained using lubricating additives. It is discovered that even minimal volumes of lubricant augment the heat flux dramatically, particularly at rather high wall temperatures. The spray cooling process is accompanied by an extensive foaming near the wall and as a result, the Leidenfrost point is moved to a higher temperature, thus suppressing film boiling regime.

Validation of a desktop-type magnet providing a quasi-microgravity space in a room-temperature bore of a high-gradient trapped field magnet (HG-TFM)

The concept of a high-gradient trapped field magnet (HG-TFM), which incorporates a hybrid system of two (RE)BaCuO superconducting bulk components with different functions, was proposed in 2021 by the authors based on the results of numerical simulations. The HG-TFM as a desktop-type magnet can be a more effective way to generate a higher magnetic field gradient product of Bz · dBz /dz (>−1400 T2 m−1, as calculated for a pure water), which can realize a quasi-microgravity space applicable for Space Environment Utilization on a laboratory scale. In this study, to validate the quasi-microgravity space in the HG-TFM, a prototype HG-TFM apparatus has been built using a slit-bulk TFM and stacked full-TFM (without slits) with inner diameters of 36 mm. After field-cooled magnetization from 8.60 T at 21 K, a trapped field of B T = 8.57 T was achieved at the center (i.e. at the bottom of a room temperature bore of 25 mm diameter outside the vacuum chamber), and consequently, a maximum Bz · dBz /dz = −1930 T2 m−1 was obtained at the intermediate position between the slit-bulk TFM and the stacked full-TFM. Magnetic levitation was demonstrated successfully for bismuth particles and a pure water drop, which validates the quasi-microgravity environment in the HG-TFM. Based on numerical simulation results of the trapped field profile, it is concluded that the reason for the instability of the levitated targets is because of the repulsive magnetic force applied along the horizontal plane. The levitating state can be controllable, for example, by changing the operating temperature, which would allow objects to levitate statically along the central axis.

Molecular dynamics simulations of nanoparticle-laden drop–interface electrocoalescence behaviors under direct and alternating current electric fields

Electrocoalescence is an energy-efficient and environmentally friendly process for separating water-in-oil emulsions. In this study, nanoparticle-laden drop–interface electrocoalescence behaviors under direct current (DC) and alternating current (AC) electric fields were numerically investigated using molecular dynamics methods. Good qualitative agreement was obtained between numerical results and experimental validation work. In this report, the influences of the electric field strength, nanoparticle concentration, and electric field frequency on the electrocoalescence process are systematically examined, analyzed, and discussed from the perspective of intermolecular interactions. The coupling of the hydration effect of ions and the strong interactions between silicon dioxide (SiO2) nanoparticles and water molecules was found to give rise to partial drop–interface electrocoalescence. The critical cone angle at which a transition from complete to partial electrocoalescence occurs was found to be 34.41°. A larger water intermolecular distance and a greater tendency for partial coalescence was observed in the nanoparticle-laden (NP-laden) droplets than in a pure water drop. Compared to a DC electric field, an AC electric field tended to cause complete drop–interface electrocoalescence, and the efficiency of complete coalescence was found to increase with increasing frequencies ranging from 10 to 200 GHz (GHz). The results of this work will be potentially useful for optimizing the design of compact and efficient oil–water separators.

Marangoni convection in an evaporating water droplet

The existence of Marangoni flow in an evaporating pure water drop has remained an open question. In this study, we report the occurrence of Marangoni convection within a water drop resting on a copper rod as it evaporates into the air. Through careful preparation of the test liquid suspension, we have observed convective structures inside the water drop. We also developed a mathematical model representing the evaporation system and solved it numerically to further strengthen support for the experimental data. Our study is different from previous studies on droplet evaporation in the sense that we allowed the droplet to evaporate spontaneously in the normal laboratory condition without heating the substrate. The substrate was neither externally heated nor did radiation from the illumination light stimulate a Marangoni flow. The velocities in the midsection of the droplet were measured using particle tracking velocimetry in both the sessile and pendant configurations. The typical velocities in the center of the droplet were found to be 10 mm/s experimentally and even higher velocities (about double that) were predicted by the mathematical model. This high value stands out among the thermocapillary-driven flows observed in an autonomously evaporated drop of water under ambient conditions.

Insights into the evolution of the thermal field in evaporating sessile pure water drops

We investigate the evolution of the thermal field during evaporation, a fundamental aspect of evaporating sessile drops. With numerous reports in the literature investigating the contact line dynamics, we aspire to identify generalized features in the evolution of the thermal field and ultimately correlate these with the contact line dynamics. Considering a broad range of experimental parameters such as substrate wettability, substrate temperature, initial volume of the drop, and ambient relative humidity results in a wide range of evaporation rates, in turn affecting the strength of internal convective flows. Infrared thermography is utilized to extract the thermal field at the liquid–vapor interface, and optical imaging is used to record the evolution of drop shape during evaporation. We observe that the onset and presence of a convective cell as a cold spot at the interface highlights a non-axisymmetry in the thermal field. In consequence, a hitherto unreported asymmetry in the internal flow field is observed, as evidenced by the particle image velocimetry. Among the multitude of experiments conducted, we report four distinct trends in the evolution of interfacial temperature difference depending on the presence and duration of the presence of the convective cell, which are elucidated by discussing the evolution of maximum and minimum temperatures at the interface. The interplay between heat conducted into the drop and heat released due to evaporation can result in a momentary decrease in temperature of the drop, which is not reported previously. Lastly, a theoretical estimate for the temperature difference within the drop is extracted using vapor diffusion model and energy balance during evaporation. Comparison of this theoretical temperature difference with experimental observations highlights the influence of internal convective flows in homogenizing the thermal field within the drop.

Experimental study of the contact line friction coefficient

Industrial applications use surfactants to control surface tension and wettability of surfaces. Instead of studying prepared solutions, we focus on the movement of the contact line after adding the surfactant. We placed the pedant drop with surfactant to a sessile drop of pure water and filmed the spreading process to obtain friction coefficient of the contact line. Our results confirmed that adding surfactants to sessile droplets drives the contact line movement. Small concentrations of LHS and SDBS as well as high concentrations of SDBS pin droplet and increase the hysteresis. Friction coefficient for cationic CTAB and zwitterionic LHS drops with the concentration increase. Anionic SDBS decreases contact angle, making the surface more hydrophilic and intensifying hysteresis.

Drop Size Dependence of the Apparent Surface Tension of Aqueous Solutions in Hexane Vapor as Studied by Drop Profile Analysis Tensiometry

Surface tension experiments were performed using the drop profile analysis tensiometry method. The hexane was injected into the measuring cell at certain times before the formation of the solution drop. The influence of the capillary diameter and solution drop size on the measured apparent dynamic surface tension was studied. The amount of hexane transferred from the vapor phase to the drop was estimated. For large pure water drops, it was shown that the ageing of the drop in the hexane vapor during a long time resulted in the formation of a liquid hexane phase covering the drop, but the volume of this phase did not exceed 0.5 mm3. On the contrary, for surfactant solution drops the volume of the hexane phase covering the drop was essentially larger. Experiments with solution drops were performed to measure the surface tension within a wide range of surfactant concentration. It was found that the dependencies of dynamic surface tension on the C13DMPO and C14EO8 solutions concentration exhibit maxima at concentrations of about 1–2 μmol/L for C14EO8 and 2–5 μmol/L for C13DMPO at ageing times of 100 to 1000 s; these maxima were shown to exist also at equilibrium. This phenomenon is presumably ascribed to the competitive character of simultaneous adsorption of hexane and surfactant.

Interfacial Dilational Viscoelasticity of Adsorption Layers at the Hydrocarbon/Water Interface: The Fractional Maxwell Model

In this communication, the single element version of the fractional Maxwell model (single-FMM or Scott–Blair model) is adopted to quantify the observed behavior of the linear interfacial dilational viscoelasticity. This mathematical tool is applied to the results obtained by capillary pressure experiments under low-gravity conditions aboard the International Space Station, for adsorption layers at the hydrocarbon/water interface. Two specific experimental sets of steady-state harmonic oscillations of interfacial area are reported, respectively: a drop of pure water into a Span-80 surfactant/paraffin-oil matrix and a pure n-hexane drop into a C13DMPO/TTAB mixed surfactants/aqueous-solution matrix. The fractional constitutive single-FMM is demonstrated to embrace the standard Maxwell model (MM) and the Lucassen–van-den-Tempel model (L–vdT), as particular cases. The single-FMM adequately fits the Span-80/paraffin-oil observed results, correctly predicting the frequency dependence of the complex viscoelastic modulus and the inherent phase-shift angle. In contrast, the single-FMM appears as a scarcely adequate tool to fit the observed behavior of the mixed-adsorption surfactants for the C13DMPO/TTAB/aqueous solution matrix (despite the single-FMM satisfactorily comparing to the phenomenology of the sole complex viscoelastic modulus). Further speculations are envisaged in order to devise combined FMM as rational guidance to interpret the properties and the interfacial structure of complex mixed surfactant adsorption systems.

Field synergy analysis on the mechanism of heat transfer enhancement by using nanofluids

A numerical simulation investigation on the flow of nanofluids in a circular tube was carried out. The simulations have two parts. Firstly, pressure drop of pure water (single-phase base fluid) flowing in a circular tube was calculated in order to verify the correctness of the simulation model with experimental results. Secondly, numerical simulations for nanofluids were conducted with consideration of different mass fractions of nanoparticles. The influencing mechanisms of nanoparticles for base fluid were further discussed. On one hand, adding nanoparticles causes changes in the fluid velocity distribution in fluid region which further leads to changed fluid flow properties in the radial direction. On the other hand, nanoparticles increase the heat transfer performance of the whole fluid with changed temperature gradient. The combination of the changed velocity field and temperature gradient field leads to the enhancement of the heat transfer process in nanofluids. The results show that the enhancement of heat transfer by nanofluids is achieved by the augmented synergy of the temperature gradient field and the velocity field.

Numerical analyses of sulfur dioxide transport by an atmospheric circulating drop

The absorption of sulfur dioxide by an atmospheric circulating drop was numerically investigated at low Reynolds number. The effects of Reynolds number and solid nucleus (spherical) on sulfur dioxide absorption were particularly focused in the present study. Mass transfer of sulfur dioxide of a pure water drop was simulated by utilizing radial diffusion and internal vortex mechanisms. With an increased Reynolds number, the sulfur dioxide transport via internal vortex is significantly enhanced. Sulfur dioxide, with the lowest concentration, moves from drop center, through the centerline, and towards the vortex center. Meanwhile, vortex strength in the drop gradually decreases as the solid nucleus grows and also due to flow retardation effect at solid surface. For drop film with relative thickness of λ = 0.5, internal circulation significantly affects the sulfur transport process. The lowest concentration point is located at the front stagnation point of the solid particle. In contrast, rear stagnation point hosts extra sulfur dioxide by convection (besides pure radial diffusion) which is originated from solid surface retardation. For drop film with relative thickness of λ = 0.3, the strength of internal circulation is reduced, but still affects sulfur dioxide transport to a lesser extent. With drop film further decreases to λ = 0.1, absorption of sulfur dioxide is almost only controlled by radial diffusion, as the particle is so large that, the one-dimensional diffusion through liquid film is gaining significance when compared with internal convection. The thickness of drop film obviously affects strength of internal circulation and the sulfur transport.

Evaporation kinetics of pure water drops: Thermal patterns, Marangoni flow, and interfacial temperature difference

We report a systematic study of the role of Marangoni convection in the evaporation kinetics of pure water drops, considering the influence of the heating regime and surface wettability. Marangoni flows were induced via heating under constant wall temperature (uniform heating) and constant heat flux (local heating) regimes below the drops. To visualize the thermal patterns emerging during the evaporation, we employed infrared thermography and we captured the evolution of the drop profile with a CCD camera to follow the evaporation kinetics of each drop. We observed a strong correlation between the temperature difference within the drop and the evolution of the drop shape during different modes of evaporation (i.e., constant radius, angle, or stick-slip) resulting in different Marangoni flow patterns. Under uniform heating, stable recirculatory vortices due to Marangoni convection emerged at high temperature, but they faded at later stages of the evaporation process. On the other hand, in the localized heating case, the constant heat flux resulted in a rapid increase in the temperature difference within the drop capable of sustaining Marangoni flows throughout the evaporation. Surface wettability was found also to play a role in both the emergence of the Marangoni flows and the evaporation kinetics. In particular, recirculatory flows in drops on hydrophobic surfaces were stronger when compared to flows on hydrophilic surfaces for both uniform and local heating. To quantify the effect of the heating mode and the importance of Marangoni flows, we calculated the evaporative flux for each case and found it to be much higher in the localized heating case. Evaporative flux depends on both diffusion and natural convection of the vapor phase to the ambient. Hence, we estimated the Grashof number for each case and found a strong relation between natural convection in the vapor phase and heating regime or Marangoni convection in the liquid phase. Subsequently, we demonstrate the limitation of the previously reported diffusion-only model in describing the evaporation of heated drops.

Evaporation of water-ethanol binary sessile drop on fluoropolymer surfaces: Influence of relative humidity

Experimental results on the diffusive drop evaporation behavior of 2 μL pure water, pure ethanol and ethanol-water binary droplets under constant relative humidity and temperature conditions were reported on two flat fluoropolymer surfaces of TEFLON-FEP and PTFE. Contact angles, droplet heights, contact radius were monitored with a camera as a function of time. A roughly linear relationship between the initial static contact angles on fluoropolymer surfaces and ethanol concentration was found similar to the previously published results. It was determined that constant contact angle mode dominated for most of the duration of drop evaporation process after the initial stages. Volumes of the binary droplets were found to decrease nonlinearly over time for all of the compositions. The most important effect of the RH increase of the medium was found to be the increase of the adsorption quantity of water vapor from the environment onto pure ethanol and also binary ethanol-water droplets. For pure ethanol droplet under high RH conditions, the rate of evaporation declined after the evaporation of most of the ethanol in 250 s, then the remaining droplet evaporated with a much slower rate which was close to, but not equal to the rate of pure water drop evaporation, indicating that ethanol was present in the droplet until to end. The same behavior was seen for ethanol-water binary droplets. Evaporation rate of the binary droplet containing 25% ethanol by wt. was observed to be close to that of pure water and 75% ethanol by wt. was found to be close to that of pure ethanol with minor differences. However, evaporation of a binary droplet containing 50% ethanol by wt. was more complicated and it is hard to distinguish ethanol and water evaporation zones for this composition and there was a long transition stage where ethanol and water evaporated together for long durations. It was found that droplet life-times were generally longer on flat TEFLON-FEP substrate than that of slightly rough PTFE due to the higher initial contact angles on the former. The change of V(2/3) values with time for pure water drop evaporation was determined to be linear under all RH conditions obeying the constant contact angle mode. The values of (dV(2/3)/dt) decreased linearly with the increase in RH for pure water. Similar, but a non-linear decrease of the (dV(2/3)/dt) slopes were determined for ethanol-water and pure ethonol droplet evaporations indicating the adsorption of water from the environment.

Effect of Anionic Surfactant on Wettability of Shale and Its Implication on Gas Adsorption/Desorption Behavior

During the fracking process in shale, an interaction occurs between shale and fracking fluid that contains a cocktail of chemicals. One of the chemicals used in fracking fluid is often surfactant, which is generally used as a viscofier. However, surfactants also have the potential of significantly influencing the wettability and thus gas desorption–key factors affecting ultimate gas recovery from shale reservoirs. Even though a few studies discussed the ability of surfactants to alter wettability in shale, the implication of that change in adsorption/desorption behavior has never been experimentally investigated beyond hypothetical inferences. In this study, the influence of the wettability change by anionic surfactant on gas adsorption/desorption behavior in shale was investigated through a series of experiments. Baseline wettability readings of two shale samples were established by measuring the contact angles (BG-1 = 22.7°, KH-1 = 35°) between a drop of pure water placed on their polished surfaces, ind...

Three-dimensional microplate formation with evaporating nanoparticle suspensions on superhydrophobic surfaces

The concept of evaporation driven self-assembly of nanoparticles is exploited to fabricate three dimensional microstructures on textured superhydrophobic surfaces. The evaporative droplets always remain in Cassie–Baxter state during the process of drying. However, the wetting transition from Cassie to Wenzel state has been observed for pure water drop on similar surfaces. After complete evaporation, residual particles form a (micro-) plate like structure. The area of the final structure is smaller than the initial contact area of the drop. Evaporating droplets are visualized from two directions to keep track of the drop radius and apparent contact angle.

Self-Propulsion and Shape Restoration of Aqueous Drops on Sulfobetaine Silane Surfaces.

The motion of droplets on typical surfaces is generally halted by contact line pinning associated with contact angle hysteresis. In this study, it was shown that, on a zwitterionic sulfobetaine silane (SBSi)-coated surface, aqueous drops with appropriate solutes can demonstrate hysteresis-free behavior, whereas a pure water drop shows spontaneous spreading. By adding solutes such as polyethylene glycol, 2(2-butoxy ethoxy) ethanol, or sodium n-dodecyl sulfate, an aqueous drop with a small contact angle (disappearance of spontaneous spreading) was formed on SBSi surfaces. The initial drop shape was readily relaxed back to a circular shape (hysteresis-free behavior), even upon severe disturbances. Moreover, it was interesting to observe the self-propulsion of such a drop on horizontal SBSi surfaces in the absence of externally provided stimuli. The self-propelled drop tends to follow a random trajectory, and the continuous movement can last for at least 10 min. This self-propelled random motion can be attributed to the combined effects of the hysteresis-free surface and the Marangoni stress. The former comes from the total wetting property of the surface, while the latter originates from surface tension gradient due to fluctuating evaporation rates along the drop border.

Marangoni Flow Induced Evaporation Enhancement on Binary Sessile Drops.

The evaporation processes of pure water, pure 1-butanol, and 5% 1-butanol aqueous solution drops on heated hydrophobic substrates are investigated to determine the effect of temperature on the drop evaporation behavior. The evolution of the parameters (contact angle, diameter, and volume) during evaporation measured using a drop shape analyzer and the infrared thermal mapping of the drop surface recorded by an infrared camera were used in investigating the evaporation process. The pure 1-butanol drop does not show any thermal instability at different substrate temperatures, while the convection cells created by the thermal Marangoni effect appear on the surface of the pure water drop from 50 °C. Because 1-butanol and water have different surface tensions, the infrared video of the 5% 1-butanol aqueous solution drop shows that the convection cells are generated by the solutal Marangoni effect at any substrate temperature. Furthermore, when the substrate temperature exceeds 50 °C, coexistence of the thermal and solutal Marangoni flows is observed. By analyzing the relation between the ratio of the evaporation rate of pure water and 1-butanol aqueous solution drops and the Marangoni number, a series of empirical equations for predicting the evaporation rates of pure water and 1-butanol aqueous solution drops at the initial time as well as the equations for the evaporation rate of 1-butanol aqueous solution drop before the depletion of alcohol are derived. The results of these equations correspond fairly well to the experimental data.