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Evaporation of Low-Boiling Components from Water Solution Droplets

This paper presents direct measurements of the evaporation dynamics and concentration changes in suspended aqueous droplets of acetone, methanol, and ethanol using a capacitance method, ultimately establishing a generalized correlation for mass loss that incorporates environmental parameters and the Schmidt number to account for internal molecular momentum and mass transfer.

Original authors: Nikolay Shishkin, Alexandr Nazarov

Published 2026-07-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikolay Shishkin, Alexandr Nazarov

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Secret Life of a Floating Drop

Imagine a world where tiny, invisible rivers flow inside a single drop of water, carrying secrets about how fast it will vanish. This is the realm of droplet evaporation, a corner of physics that studies how liquid turns into gas. You might think a drop of water just shrinks evenly until it's gone, but the real story is much more chaotic and fascinating. When a drop contains a mixture of liquids—like water mixed with something that boils easily, such as alcohol or nail polish remover—the "easy-to-boil" part tries to escape first. This creates a tug-of-war: the surface gets depleted of the fast-flying molecules, while the center still has plenty. To understand this, scientists use a few key ideas. Diffusion is like a crowded room where people slowly shuffle from the packed center to the empty edges. Convection is like a fan stirring that room to mix everyone up. And heat transfer is the energy needed to give those molecules enough speed to jump out of the liquid and into the air.

Why does anyone care about a shrinking drop? Because this tiny drama plays out in massive, real-world machines. From the fuel sprays in car engines that need to burn perfectly, to the cooling systems that keep our computers from melting, and even the way medicines are sprayed into our lungs, the behavior of these droplets determines efficiency and safety. If we can predict exactly how a drop changes its recipe as it shrinks, we can build better engines, cooler electronics, and more effective sprays. But to do that, we first have to figure out the rules of the game.

The Paper's Story: Chasing the Vanishing Act

In this study, researchers Nikolay Shishkin and Alexandr Nazarov from the Kutateladze Institute of Thermophysics in Russia decided to play detective with floating drops. They wanted to know: How does the concentration of a "low-boiling" ingredient (like acetone, methanol, or ethanol) change inside a water droplet as it evaporates in a hot wind?

To solve this mystery, they didn't just watch the drops; they "listened" to them. They suspended tiny droplets of water mixed with different alcohols or acetone on a special probe. This probe acted like a high-tech ear, using a capacitance method (basically measuring how the drop stores electrical energy) to track the drop's changing recipe in real-time. As the drop shrank and its mix changed, the electrical signal shifted, allowing the scientists to see the concentration drop moment by moment. They tested these drops in a wind tunnel, blowing air at speeds between 1 and 5 m/s and heating it from 20 to 100°C.

The First Big Discovery: History Doesn't Matter
The team found something surprisingly simple first. If you take a drop with a high concentration of alcohol and another with a low concentration, and you let them evaporate under the same conditions, their paths eventually merge. It's as if the drops have "amnesia" about their past. Once the process starts, the way the concentration drops over time follows the exact same curve, regardless of where they started. The researchers showed that by simply shifting the time on the clock (a "time-shift"), all the different starting points lined up perfectly on a single, smooth curve. This means the evaporation process is independent of the droplet's history; it only cares about what is happening right now.

The Puzzle: Why Do Different Liquids Act Differently?
However, when they tried to use a standard set of math rules (called dimensionless numbers like Reynolds, Prandtl, and Fourier) to predict how fast the drops would lose their volatile ingredients, they hit a snag. They used a complex called the Kossovich number (Ko), which combines the speed of the wind, the heat, and the size of the drop.

When they plotted their data, the drops of acetone behaved differently than the drops of methanol and ethanol. Even though the wind and temperature were the same, the alcohols evaporated at a different rate than the acetone. The standard math, which looked only at the environment outside the drop, couldn't explain why these liquids were acting so differently. It was like having a weather forecast that was perfect for rain but failed to predict snow, even though the temperature was right.

The Solution: Looking Inside the Drop
The scientists realized the missing piece of the puzzle was happening inside the drop, not just outside. They knew that as the fast-flying molecules (the low-boiling components) escaped from the surface, they left a "hole" in the concentration near the edge. To fill that hole, molecules from the center of the drop had to travel outward. This travel happens via diffusion.

Different liquids have different "stickiness" and speed of movement. Acetone molecules move and mix differently than alcohol molecules. To fix their math, the researchers added a new ingredient to their formula: the Schmidt number (Sc). Think of the Schmidt number as a measure of how easily the "ingredients" can shuffle around inside the liquid soup.

When they combined the old environmental math (the Ko complex) with this new "internal shuffle" factor (the Schmidt number), the magic happened. Suddenly, the data for acetone, methanol, and ethanol all collapsed onto a single, perfect curve. The equation they found to describe this was:
Cs = 0.67 – 0.92 · (Ko · Sc¹/³) + 0.33 · (Ko · Sc¹/³)²

This new formula worked so well that the error was less than 4%. It proved that to truly understand how a drop evaporates, you can't just look at the wind blowing on it; you have to understand how the liquid inside is mixing and moving.

What They Ruled Out
The paper explicitly argues against the idea that a single, simple formula based only on external conditions (like wind speed and temperature) is enough to describe all evaporation. They showed that ignoring the internal diffusion processes leads to discrepancies, especially when comparing different types of liquids. They also demonstrated that the "history" of the drop (its initial concentration) doesn't change the fundamental shape of the evaporation curve, only the starting point in time.

How Sure Are They?
The authors are quite confident in their findings. They didn't just simulate this on a computer; they measured it directly in a lab with real droplets. They tested multiple liquids (acetone, methanol, ethanol) across a wide range of temperatures and wind speeds. The fact that their new formula, which includes the Schmidt number, fits all their experimental data with such a small error margin (under 4%) suggests that their explanation is robust. They have provided a "criterial generalization," meaning a universal rule that works for these specific mixtures under the conditions they tested.

In short, this paper tells us that while the wind outside a drop is important, the secret to predicting how fast a drop vanishes lies in the invisible dance of molecules mixing inside it. By listening to the drop's electrical heartbeat and accounting for that internal dance, we can finally predict the future of a floating drop with great accuracy.

Technical Summary: Evaporation of Low-Boiling Components from Water Solution Droplets

Problem Statement
The evaporation of multicomponent liquid droplets, particularly those containing low-boiling components mixed with water, is a critical process in various technological applications, including fuel injection, evaporative cooling, and pharmaceuticals. A primary challenge in modeling these systems is that the evaporation of the more volatile component occurs at a higher rate than the solvent, leading to a continuous change in the solution's concentration over time. This dynamic shift alters the droplet's physical properties and evaporation rate, making the process complex. While existing literature addresses temperature fields and hydrodynamics within binary droplets, there is a need for accurate engineering models that link the temporal evolution of component concentration directly to environmental parameters. Previous attempts to generalize experimental data using the Kossovich number ($Ko$) showed promise but revealed discrepancies between different liquid types (specifically acetone versus alcohols), suggesting that current similarity criteria fail to account for internal droplet processes such as diffusion and convective flows.

Methodology
The study employed direct experimental measurements of average component concentration in suspended droplets using a capacitance method.

Experimental Setup: Experiments were conducted in a wind tunnel with an outlet diameter of 52 mm. Droplets of aqueous solutions containing acetone, methanol, or ethanol were suspended from a probe. The airflow velocity ranged from 1 to 5 m/s, and temperatures varied between 20°C and 100°C, with air humidity kept below 2%.
Measurement Technique: A coaxial capacitor probe connected to a high-frequency generator was used. The dielectric permittivity of the binary liquid changes with concentration, causing a shift in the probe's capacitance and generator frequency. This allowed for real-time registration of concentration dynamics. The method accounted for simultaneous changes in droplet size (diameter) to isolate concentration changes, achieving an average error of approximately 8%.
Data Analysis: The authors utilized a "time-shift" method to generalize kinetic curves, based on the principle that droplet evaporation history is independent of its past state. They analyzed mass loss from the droplet surface and calculated the Nusselt ($Nu$) and Sherwood ($Sh$) numbers. To generalize data across varying temperatures and velocities, the study initially used a criterion complex involving Reynolds ($Re$), Prandtl ($Pr$), Fourier ($Fo$), and Kutateladze ($Ku$) numbers. Recognizing discrepancies between liquid types, the authors subsequently introduced the Schmidt number ($Sc$) to account for internal molecular momentum transfer and mass diffusion.

Key Results

Independence of History: Under similar environmental conditions, the evaporation processes for aqueous solutions of acetone, methanol, and ethanol were found to be identical and independent of their initial concentration history. Kinetic curves for different initial concentrations could be superimposed using a time-shift method.
Concentration Dynamics: The decrease in low-boiling component concentration ( $C_s$ ) over time was approximated by a unified quadratic dependence. For pure liquids, the concentration decrease followed a specific linear trend, while solutions showed a distinct rate of mass loss.
Limitations of Existing Criteria: The use of the $Ko$ complex (incorporating $Re$, $Pr$, $Fo$, and $Ku$) successfully generalized data for specific liquid types but failed to unify acetone and alcohol data simultaneously. A significant discrepancy was observed between the experimental mass transfer rates and those predicted by the universal Ranz–Marshall formula.
Role of Internal Diffusion: The study identified that the $Ko$ complex primarily reflects environmental parameters and neglects internal droplet processes. The introduction of the Schmidt number ($Sc$), which characterizes the ratio of momentum diffusivity to mass diffusivity, was necessary to account for the diffusion resistance of the volatile component moving from the droplet core to the surface.
Generalized Correlation: By incorporating $Sc^{1/3}$ into the similarity group, the authors achieved a high-quality generalization of all experimental data across different liquids. The resulting correlation (Equation 12) describes the concentration decrease with a standard deviation of points not exceeding 4%.

Significance and Claims
The paper claims that a complete generalization of evaporation data for binary mixtures requires a similarity complex that extends beyond external environmental parameters. While the $Ko$ complex (based on $Re$, $Pr$, $Fo$, and $Ku$) is effective for describing diameter changes and concentration reduction under constant conditions, it is insufficient for unifying data across different liquid types with varying diffusion coefficients.

The authors assert that the inclusion of the Schmidt number ($Sc$) is essential because it accounts for the internal processes of the droplet, specifically the relative role of molecular momentum transfer and the mass transfer of impurities within the droplet volume. By integrating $Sc$ into the criterion group, the study provides a sufficiently accurate, unified correlation for the evaporation of low-boiling components (acetone, methanol, ethanol) from water droplets across a wide range of airflow temperatures and velocities. This approach bridges the gap between external heat/mass transfer conditions and the internal diffusion limitations that govern the evaporation rate of multicomponent droplets.

The Secret Life of a Floating Drop

The Paper's Story: Chasing the Vanishing Act

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