Question

A 1-mm-diameter methanol droplet takes 1 min for complete evaporation at atmospheric condition. What will be the time taken for a 1µm-diameter methanol droplet for complete evaporation at same conditions based on the scaling analysis?

Answer 1

Time taken by the
`1\mu m` diameter droplet is 60 ns

Solution:

As per the question:

Diameter of the droplet, d = 1 mm = 0.001 m

Radius of the droplet, R = 0.0005 m

Time taken for complete evaporation, t = 1 min = 60 s

Diameter of the smaller droplet, d' =
`1* 10^(- 6) m`

Diameter of the smaller droplet, R' =
`0.5* 10^(- 6) m`

Now,

Volume of the droplet, V =
`(4)/(3)\pi R^(3)`

Volume of the smaller droplet, V' =
`(4)/(3)\pi R'^(3)`

Volume of the droplet ∝ Time taken for complete evaporation

Thus

`(V)/(V') = (t)/(t')`

where

t' = taken taken by smaller droplet

`((4)/(3)\pi R^(3))/((4)/(3)\pi R'^(3)) = (60)/(t')`

`((4)/(3)\pi 0.0005^(3))/((4)/(3)\pi (0.5* 10^(- 6))^(3)) = (60)/(t')`

t' =
`60* 10^(- 9) s = 60 ns`