Experimental study of the process of gravitational settling of a monodisperse cluster of drops under the influence of an external air flow

. A technique and an experimental setup for studying the dynamics of gravitational settling of a monodisperse cluster of distilled water droplets under external influence of an air flow are presented. A qualitative picture of the dynamics of deposition of a monodisperse cluster of 60 drops is presented. It is shown that with an increase in the speed of the ascending airflow from 0 to 1 m/s, the average sedimentation velocity of a monodisperse droplet cluster decreases from 3.8 m/s to 2.8 m/s. The results of comparing the deposition rate of a cluster and individual droplets included in it are presented.


Introduction
In nature and in various technical applications, many processes include the generation and evolution of a liquid-droplet aerosol cloud during its gravitational settling.Examples include the formation of atmospheric precipitation from thunderclouds, the spread of toxic components during the separation of spent stages of liquid-propellant launch vehicles [1], the emergency release of aviation fuel [2], the processes in the implementation of aviation technologies for extinguishing fires [3].
When a liquid-droplet aerosol cloud settles in a gravity field, it evolves, during which the shape of the cloud, the velocity of the center of mass, concentration, shape, size of drops, and other parameters change.The change in the characteristics of the aerosol cloud occurs due to the processes of dynamic interaction of drops with the carrier medium, coagulation and crushing of drops during mutual collisions.
One of the effective methods of extinguishing large fires in remote areas is the release of refrigerant from an aircraft or helicopter.The cooling agent is usually finely atomized water, which, when in contact with fire, implements a volume-surface model of interaction.Fine spraying of water provides cooling of the combustion zone and, due to evaporation, blocks the access of oxygen to combustion.The optimization of the use of this method is based on the laws of fine cloud deposition and its interaction with the environment near the fire site.These patterns are the basis for the development of effective modes of using aviation in firefighting, taking into account real conditions and factors [4].
Basic information on the use of helicopters in firefighting is contained in the instructions of the relevant ministries and technical reports on test results, for example [5,6].A comprehensive account of various factors is carried out with a very significant simplification of the problem [7].The main volume of publications on the problems of fire extinguishing with the use of aviation refers to the technical aspects of the implementation of the process of refrigerant discharge into the fire [8,9].One of the important factors to consider when refrigerant is discharge is the effect of updrafts caused by a fire, as well as crosswind, which can affect the process of deposition and dispersion of an aerosol cloud.Highly dispersed droplets can be carried away by the ascending air flow from the source of ignition without reaching the irrigated surface.As a result of the impact of a crosswind, the angle of deposition of the aerosol cloud will change significantly and, as a result, the area of irrigation.In addition, the wind can influence the shape and size of droplets, changing their aerodynamic properties.For example, strong winds can lead to the formation of large droplets as a result of their coagulation, as well as small droplets as a result of their crushing.
This paper presents the results of an experimental study of the dynamics of gravitational settling of a monodisperse cluster of droplets, taking into account the external influence of an ascending air flow.

Experimental technique and scheme of the experimental setup
For an experimental study of the process of gravitational settling of a monodisperse cluster of drops under the external influence of an air flow, an original setup was developed [10].On figure 1   Container 1 is filled with the investigated liquid, on the lower part of the container there is a branch pipe 13, which is closed by a shut-off valve 14.On the upper part of the tank there is a branch pipe 8, which is closed by a tap 16.On the lower part of the vessel 1, 60 equidistant capillaries 2 are located vertically at a distance of 3.4 mm from each other.Injection needles, caliber 30G, were used as capillaries.A system for slowly increasing the pressure of the liquid in the vessel to the value p1 (the pressure at which drops form on the capillaries occurs) and for creating a pressure pulse with amplitude p2 (the pressure at which drops detach from the capillaries) is fixed on the stand 10 with the help of brackets 11.The system for creating a given pressure includes a syringe 5 and a piston which is connected to a micrometric screw 6.The micrometric screw 6 is driven by a micromotor-reducer 7, the speed of which is controlled by a device for the duration of electrical impulses.The internal cavity of the syringe 5 is connected through a flexible hose 3 and a shut-off valve 4 with a container 1. Pressure in vessel 1 is controlled by pressure gauge 15.
The value of pressure p1, at which hanging drops were formed at the ends of capillaries 2, was obtained from the results of experiments with varying the pressure in the falling glass 1.The value of pressure p2, corresponding to the stable detachment of the entire cluster of drops from the capillaries, was obtained experimentally and is determined by the formula: To calculate the diameter of the formed monodisperse droplets, we use the Tate law [11], according to which the critical condition for the detachment of a droplet from a capillary is the equality of the forces of gravity and surface tension acting on the droplet: where: m is the mass of the drop; g is the free fall acceleration; σ, surface tension coefficient; f =0.6, empirical coefficient; d, external (for wetting liquids) or internal (for non-wetting liquids) capillary diameter.Substituting in (2) the mass of a spherical drop m = ρ(πD 3 /6), we obtain formula (3) for calculating the drop diameter D: ( During the experiments, the formation of a compact cluster of monodisperse droplets of a given size was carried out as follows.Through the branch pipe 13 and the open shut-off valve 14, vessel 1 is filled under pressure with the test liquid.In the process of liquid supply, air is displaced through the pipe 8 and the open shut-off valve 16, as well as through the capillaries 2. After the vessel 1 is completely filled, the liquid flows out through the branch pipe 8 and capillaries 2. Slowly increasing the pressure of the solution in the vessel to the value p1 using a syringe 5 and a micrometric screw 6 with a motor-reducer 7 at a given (slow) mode of operation of the device of the duration of electrical impulses.As a result, stable drops 9 are formed, which hang on the ends of capillaries 2 (Figure 2).By creating a pressure pulse with amplitude p2 using a syringe 5 and a micrometer screw 6 with a motorreducer 7, at a given (fast) mode of operation of the device of the duration of electrical pulses, drops are simultaneously detached from the capillaries with the formation of a compact monodisperse cluster.Fixation of the initial monodisperse cluster 18 (a photograph of the initial cluster is shown in figure 3) was carried out on camera 19 (Nikon D600).The change in the size of the cluster during the deposition process was recorded on two machine vision cameras 20 (MER2-502-79U3C).Two methods were used to experimentally determine the size of a single drop.
1. Determining the size of a single drops from video frames.On Figure 4 shows a photograph of a single drop of distilled water with a diameter of D=2.23 mm.A G26 needle with an inner diameter of 0.26 mm and an outer diameter of 0.46 mm was used as a capillary.2. Determination of the size of a single drop by the mass method [12].For this, a receiving container with high-precision scales was additionally installed (not shown in figure 1).Calculations of the size of a single drop using the mass method were carried out with the assumption that during free fall the shape of the drop retains a spherical shape, and then, taking into account the specific gravity of water, we have: where M is the total mass of N drops, measured using a balance.The ascending air flow was created using a bladeless fan 20, the flow rate was varied in the range uas = (0÷1.3)m/s.The lateral air flow was created using a slot fan 21 and its speed was varied in the range usi = (0÷5) m/s.To measure the air flow velocity, a DT-8880 digital hot-wire anemometer was used, the measurement error of which is ±0.2 m/s.When measuring the air velocity in the selected area (0÷200) cm, the maximum deviation was 8%.
Consider the process of settling a drop of water with a diameter D in air.In accordance with Newton's law, the equation of motion of a drop has the form: where: u is the settling rate; i F , forces acting on the drop; ( ) , mass of a spherical drop with diameter D and density ρ .In the projection onto the OX axis directed vertically upwards, equation ( 6) will be written in the following form: where: 1 F mg = , gravity; , where: ρ Re μ uD = ; , coefficient of dynamic viscosity of air.

Results of experimental studies
During the experimental studies, the following parameters were measured: a) Single droplet diameter.b) Initial volume concentration of a cluster of monodisperse droplets.
c) The velocity of the center of mass of the cluster along the trajectory of its fall.
d) Velocity of a single drop entering the cluster along the fall trajectory.
In the cluster, the droplet diameter varied from 1.8 to 2.6 mm, depending on the capillary diameters.Medical needles of different calibers were used as capillaries.A qualitative picture of the evolution of the shape of a monodisperse water cluster in the process of its precipitation is shown in Figure 5.This figure shows the results of the evolution of a monodisperse cluster as a result of its gravitational settling in the atmosphere, the droplet diameter in the cluster is 2.3 mm, and the initial distance between drops is 3.4 mm.
As can be seen from figure 5, during the deposition of a monodisperse cluster of droplets, its gradual expansion occurs, both in the horizontal plane and in the vertical one.In this case, the figure shows how the spherical cluster of drops is first deformed into an ellipsoid, elongated in the direction of motion (S = 10÷130 cm), and then rapidly expands in the horizontal plane (S = 130÷180 cm).Starting from the distance Scr, the complete destruction of the monodisperse cluster of drops occurs (the rate of its deposition corresponds to the deposition rate of a single drop).The error in determining the center of mass of a monodisperse cluster of drops from video frames was approximately ±4 cm.On figure 6 shows the dependence graphs for the rate of sedimentation of the center of mass of a monodisperse cluster of 60 drops of distilled water with a diameter of D = 2.3 mm (figure 6 curve 1) and for a single drop of the same diameter (figure 6 curve 2).For comparison, figure 6, curve 3 shows the results for the deposition rate of a single drop, calculated by solving differential equation (6).As can be seen from the figure, the cluster velocity exceeds the velocity of a single drop.The influence of the ascending air flow on the rate of sedimentation of the center of mass of a cluster of 60 monodisperse drops of distilled water is shown in figure 7.
To assess the influence of the lateral air flow on the rate of gravitational settling of a monodisperse cluster of drops, experiments were carried out to measure the effect of the lateral flow velocity on the angle of deviation of drops from the normal of their settling.The maximum deviation of the drop cluster during gravitational settling from the vertical without droplet coagulation was 3.5 deg. at a blowing flow velocity of 4 m/s.At a lateral blowing velocity of more than 4 m/s, the effect of coagulation of some of the droplets in the cluster is noticeable.The study received financial support from the Russian Science Foundation (project no.22-19-00307).
force; CD, drag coefficient.The drag coefficient CD is determined by the relationship: 92 0.147 Re ) , 0.3 Re 1028 Re 0.44, Re 1028

Fig. 5 .
Fig. 5. Video sequence of the process of deposition of a monodisperse cluster from 60 drops of distilled water.

Fig. 6 .
Fig. 6.Dependence of the settling rate on the distance for 1, the center of mass of a monodisperse cluster of drops; 2, single drop (experimental points); 3, single drop (theoretical data).

Fig. 7 .
Fig. 7. Change in the speed of the center of mass of a drop cluster during its gravitational settling depending on the speed of the ascending air flow.