what is the equation that governs evaporation rate in a closed system?

Question

supposed I have a container that I can close entirely. Also suppose that we put some water into the container and leave the rest as air. Then we close the container and start to heat it.

What is the formula that governs the evaporation rate of the water and the total amount of water in vapor form? I would be very interested in the physics behind the derivation of the formula, so if someone can suggest a textbook or webpage, I would be much obliged.

thanks

Answer 1

In a closed system the amount of water in vapor form (at equilibrium) will simply be the volume available for vapor (total volume of container minus volume of liquid) times the density of the saturated vapor at your given temperature. These values can be looked up in a steam table (density is the inverse of specific volume). The evaporation rate of the water is a much tougher nut to crack, as I'll explain below.

The evaporation rate is influenced by:
1) The temperature of the water at the air-water surface
2) The humidity of the air
3) The area of the air-water surface
4) The temperature of the air (more on this below)

In a real-world situation of evaporating water, none of these four quantities above remains constant because the process of evaporation itself changes them. Water evaporating takes quite a lot of heat away -- 540 calories per gram -- when it evaporates. That's enough to cool down 540 grams of water by a degree, or 50 grams of water a little more than ten degrees. If you are not very careful to replace the lost heat energy during the evaporation, the temperature will go down. And even then the temperature right at the surface will be lower than elsewhere in the water and it will depend on water currents convecting heat and the ability to keep the temperature constant at a given temperature.

For a similar reason, the air near the surface of the water will become more saturated with water as the water evaporates. The evaporation rate will depend on airflow past the water/air surface, or in the closed system model the diffusion rates of saturated water vapor in air.

Irving Langmuir developed a way to measure vapor pressure
by measuring the evaporation rate, which we can flip around backwards to approximate what you'd like to derive. His reasoning is that the rate at which molecues are lost due to evaporation to a gas with no partial pressure of the evaporating substance is the same as the rate at which molecules of the substance would hit the surface if it were in equilibrium with the vapor (because in equilibrium the evaporation rate and re-condensation rate cancel each other out). His expression is:

(mass loss rate)/(unit area) = (vapor pressure - ambient partial pressure)*sqrt( (molecular weight)/(2*pi*R*T) )

THIS IS AN APPROXIMATION! But it's usually close enough for government work. Or engineers, we don't need to measure to 4 decimal places. The rate itself, if graphed over time, would show a decrease that could only be modeled by a partial differential equation. If you REALLY want to figure it out exactly take a look at the books I've listed.

Answer 2

Calculating Evaporation Rate

Answer 3

This Site Might Help You.

RE:
what is the equation that governs evaporation rate in a closed system?
supposed I have a container that I can close entirely. Also suppose that we put some water into the container and leave the rest as air. Then we close the container and start to heat it.

What is the formula that governs the evaporation rate of the water and the total amount of water in vapor...

Answer 4

Evaporation Rate Formula

Answer 5

Evaporation is kinetic process, and exact formula does not exist.

However, in many situations it is reasonable to assume that 100% of molecules of water vapor, which hit the surface stick to the surface.
Under such assumption the rate of condesation is equal to total microscopic flow of molecules of vapor.

Now:
becuse of principle of microscopic equilbrium, the number of molecules which are evaporated into phase volume
dΩ(x,px,py,pz) = dx³dp³
must be exacly equal to the number of molecules which condensate from corresponding volume
dΩ(x,px,py,-pz) = dx³dp³.
Under 100% condensation assumption, the latter is simply equal to the number of molecules of saturated vapor which occupy this volume:
dN = n exp(-E/T)dΩ = n exp(-p²/(2mT))dΩ,
where n is concentration of molecules in saturated vapor n = Na/V = Na P/(RT).

The pressure of saturated vapor is given by
Clausius-Clapeyron equation:
http://en.wikipedia.org/wiki/Clausius-Clapeyron_relation

Answer 6

hi, can i calculate evaporation rate if i have vapor pressure? tnx