Transient droplet evaporation. A droplet of water is falling down a tower, and w

Question

Transient droplet evaporation. A droplet of water is falling down a tower, and we want to find how long it should take to evaporate. The humidity of the air in the tower is W_ambient= 0.01 (this is 0.01 kg moisture/kg dry air, so the kg cancel, and so we use 0.01). The saturation humidity is W_AS =0.02 at 298 K, and we use this because the air will be saturated at the droplet surface. The Froessling equation estimates that the Sherwood number (Sh) equals to 2 for this problem where the droplet is assumed to be perfectly spherical and the air in the column "static"'. The diameter (d) of the bubble is 2.0 mm at time =0. According to Hogler if temperature variations during evaporation are neglected the shrinking of the droplet may be modelled as Where is density of water, D is the diffusion coefficient of water in air (=2.9 - 10'^-5 m^2/s), and d = d(t) is the bubble diameter, which is of course changing with time. Estimate the time when the diameter will be 1.O mm. 380 sec 5.2 sec 47 sec 0.12 sec 98 sec

Explanation / Answer

we have dm/dt = - Sh *pi*d*rho* D*( Was- Wamb)

we also know that m = rho* V = rho * pi*d3/ 6

so it terms of d we have,

have d( rho * pi*d3/ 6) /dt = - Sh *pi*d*rho* D*( Was- Wamb)

Integrating we get

Now at t= 0, d= 2 mm = 0.002 m

So , C= 2x10-6

Now , we have

We need to find the time when d = 1mm = 0.001 m

So plugging the values we get,

T = 0.12 secs

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