Evaporation: the speed of falling droplets

WATER ON THE MOVE

In today's post we will study how droplets of water move in the air. We will start with the most simple? case which is when the mass of air surrounding the droplets does not move.

Droplets are the stable form of water when surrounded by air. The surface tension of the water forces the droplet to keep a spherical shape. The diameter of this sphere, on the surface of Earth, has a maximum size of 4 mm in radius.? Droplets can also be much? smaller.

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Terminal velocity.

When a small droplet of water is standing still in the middle of still standing air, it is affected by two forces

1- the elastic force of its surface, also called surface tension which? tends to keep the droplet in its spherical shape

2-? the force of gravity? (in blue in the figure) which affects all bodies on Earth and which tends to accelerate the droplet downwards with an acceleration of 9.8 meters per second squared.?As a result of this force and this acceleration the droplets start to move downwards with increasing speed.

?But as soon as there is? some speed, a third force appears; the drag force (in red), also called force of friction.

?3-? the force of friction with the air. This force works opposite to the direction of movement of the droplet so it is opposed to the force of gravity. That means it tends to reduce the downwards acceleration of gravity.

?Contrary to the force of gravity, which is constant, the force of friction grows with the speed of the droplet.? At a certain point the droplet reaches the so-called terminal speed which corresponds to a force of friction which is equal to the force of gravity (right side of the figure).? At this point, the sum of external forces on the droplet is zero since both the gravity and the friction are equal and opposite in direction.? With zero forces acting on the droplet, the droplet is not accelerating anymore and remains moving with constant speed, namely the terminal speed referred above.

This is valid for all objects falling down in the atmosphere:? they accelerate until they reach a critical velocity. After that they continue falling with? constant speed.

The critical velocity is different for different shapes and sizes.? In the case of our droplets we can focus on the spherical shape and we will look at the critical speed for different sizes.

The force of gravity depends on the mass and therefore on the volume of a droplet, so it is proportional to the radius of the droplet elevated to the cubic power. The force of friction depends on the surface of the droplet? so it is proportional to the radius of the droplet elevated to the square. These different rules of proportionalities imply that if we make droplets of? smaller diameter, their corresponding critical velocities will also be smaller.?

Terminal velocities of different sizes of droplets? have been measured. Below you have some of them corresponding to different sizes of droplets in Nature. The terminal velocities are expressed both in meters per second and in kilometers per hour.? I have added a last column expressing the time required to travel a distance of 100 m with the corresponding terminal velocity

The extremely small values for terminal velocities in cloud droplets and in condensation nuclei explain why clouds don't fall down from the sky and why condensation nuclei can stay so high in the atmosphere for so long. In fact the small droplets of the cloud are falling but the air in which they are contained might be moving upwards with? the same or even faster speed.

For the purpose of Enhancing? Water Evaporation Techniques (EWET), I retain the middle part of this table, namely droplets with sizes between 0.1 mm and 1 mm in diameter.

?Indeed in this range we have:

- A relatively low demand on energy and technology to overcome the surface tension required for the formation of the droplet

- A comparatively large surface of exposure per unit of volume of water

- A sufficiently long period of time during the fall in the air allowing the water droplets to evaporate completely?

In the next post I will discuss how things get more complex when there is wind and how a nebulisation of droplets can create its own wind.