How much water remains as ice when wet clothing is dried in dry wind at 0°C?

[irhum31]

Homework Statement

Wet clothing at 0°C is hung out to dry when the air temperature is 0°C and there is a dry wind blowing. After some time, it is found that some of the water has evaporated and the remainder has frozen. Estimate the fraction of the water originally present in the clothing which remains as ice.

Homework Equations

(At 0°C, specific latent heat of fusion of ice = 336 kJ/kg, specific latent heat of vaporization of water = 2500 kJ/kg, specific heat of water = 4200 J/Kg*K)

The Attempt at a Solution

Well, how do I begin this. I don't know how much heat dry wind delivers.[/B]

 

[stockzahn]

Wet clothing at 0°C is hung out to dry when the air temperature is 0°C ... []

What the heat flux could be from the clothes to the air based on the assumptions made?

 

[irhum31]

What the heat flux could be from the clothes to the air based on the assumptions made?

Shouldn't there be zero heat flux as the temperature is the same? Or did I get it wrong?

 

[stockzahn]

Shouldn't there be zero heat flux as the temperature is the same? Or did I get it wrong?

Correct. So what is the only possible heat source to receive the heat to evaporate water?

 

[irhum31]

Correct. So what is the only possible heat source to receive the heat to evaporate water?

The sun? Not a possibility due to zero heat flux.

I'm guessing this has something to do with the wind, but can't quite nail it.

If there's no temperature change, only phase changes can occur as they do not rely (directly) temperature. (But how on Earth am I supposed to send water to 100C for evaporation?)

Okay, so only phase changes will occur, but I still do not see any heat source.

I'll make a wild guess that it's 88%, if I take the ratio of the heat of fusion and heat of vaporization, see what multiple of water quantity can be transformed into ice for the same heat that is required by vaporization, and convert it to a scale of 100.

This seems wrong in every possible way.

 

[SteamKing]

If there's no temperature change, only phase changes can occur as they do not rely (directly) temperature. (But how on Earth am I supposed to send water to 100C for evaporation?)

Water evaporates even when the ambient temperature is below 100° C.

If the wind is composed of dry air, this suggests its relative humidity (the amount of water vapor in the air) is zero or close to it. Air which is fully saturated with water vapor has a relative humidity of 100%.

https://en.wikipedia.org/wiki/Relative_humidity

 

[irhum31]

Water evaporates even when the ambient temperature is below 100° C.

If the wind is composed of dry air, this suggests its relative humidity (the amount of water vapor in the air) is zero or close to it. Air which is fully saturated with water vapor has a relative humidity of 100%.

https://en.wikipedia.org/wiki/Relative_humidity

Okay, so once the air is saturated, it won't take in any water. So, the amount of water that causes full saturation is our evaporated water. Everything else freezes.

How do I find the amount of water that causes saturation?

 

[stockzahn]

The sun? Not a possibility due to zero heat flux.

I'll make a wild guess that it's 88%, if I take the ratio of the heat of fusion and heat of vaporization, see what multiple of water quantity can be transformed into ice for the same heat that is required by vaporization, and convert it to a scale of 100.

Correct (I calculated 88.15%)

The sun? Not a possibility due to zero heat flux.

If there's no temperature change, only phase changes can occur as they do not rely (directly) temperature. (But how on Earth am I supposed to send water to 100C for evaporation?)

You don't need 100°C for evaporation. A fluid evaporates as long as the partial pressure of the evaporated fluid (i.e. gas) is smaller than its vapour pressure dependent on the temperature. At 100°C the vapour pressure of water is 1 bar, that's why it evaporates very fast at this temperature, but also if the temperature is smaller it will evaporate (only slower) as long as the partial pressure of steam in the air doesn't reach the vapour pressure (otherwise nothing could get dried). As the wind is dry, which means it transports all the evaporated steam away and therefore stays dry, water can change its state from fluid to gas. The required heat is taken from the water itself, therefore part of the water evaporates, whereas the rest freezes - this works, because the vapour pressure over the solid phase of certain matter is smaller than over the fluid phase. But you will find much information about that topic in the internet (vapour pressure, Dalton's law, growth of ice particles in clouds, ...).

 

[irhum31]

Correct (I calculated 88.15%)
You don't need 100°C for evaporation. A fluid evaporates as long as the partial pressure of the evaporated fluid (i.e. gas) is smaller than its vapour pressure dependent on the temperature. At 100°C the vapour pressure of water is 1 bar, that's why it evaporates very fast at this temperature, but also if the temperature is smaller it will evaporate (only slower) as long as the partial pressure of steam in the air doesn't reach the vapour pressure (otherwise nothing could get dried). As the wind is dry, which means it transports all the evaporated steam away and therefore stays dry, water can change its state from fluid to gas. The required heat is taken from the water itself, therefore part of the water evaporates, whereas the rest freezes - this works, because the vapour pressure over the solid phase of certain matter is smaller than over the fluid phase. But you will find much information about that topic in the internet (vapour pressure, Dalton's law, growth of ice particles in clouds, ...).

Well, to be honest I understood very little of what you said, but that's because I haven't touched these topics yet. (I'm self-studying thermodynamics).

Time to explore these new topics! Thank you!

(on a side note, would you happen to know of a good thermodynamics book? The one I'm using isn't so great and is very confusing to work through)

 

[Chestermiller]

Okay, so once the air is saturated, it won't take in any water. So, the amount of water that causes full saturation is our evaporated water. Everything else freezes.

How do I find the amount of water that causes saturation?

The air is not going to get saturated by the water that evaporates from the clothes because there is virtually an infinite body of surrounding air to receive the evaporated water. Besides, if the air is moving at all, the air that contains the evaporated water gets swept away, and is replenished with fresh dry air.

For a good thermo book, try Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics. But, before that, for the questions you are asking, you should first consult a book on Physical Chemistry.

Chet