Friday, September 23, 2011

Condensation Rate

in Condensation Point we considered the temperature at which water vapor will begin to condense into water droplets, thus making a cloud. We did not consider how fast this condensation will take place. Consider air with 20 g/kg of water vapor (that's 20 g of water vapor mixed with each 1 kg of dry air to make 1.020 kg of moist air). This air rises rapidly, expands, and cools to a point where its saturation concentration of water vapor is only 10 g/kg. Does the excess 10 g/kg condense into droplets immediately, or does it take some time, in the same way that the original evaporation took time?

As we saw in Latent Heat, the evaporation of water requires 2.2 kJ of heat for each gram of evaporating water. Because we must put energy into the water to make it evaporate, evaporation takes place slowly. In the case of condensation, however, the exact opposite is the case: condensation liberates 2.2 kJ of heat for each gram of water that condenses. Condensation takes place much more quickly, but it cannot take place instantly. In order for condensation to take place, water vapor molecules must bump into one another and stick together. A dust particle helps accelerate the condensation process by providing a surface upon which water molecules can condense. Until such time as all condensation is complete, the water vapor concentration remains greater than the saturation concentration, and we say the water vapor is supersaturated.

The cloud chambers of early high energy physics experiments used supersaturated water vapor to detect charged sub-atomic particles. A cloud chamber consists of a piston with a glass top. We fill the piston with moist air and pull the piston down rapidly, so that the air cools by adiabatic expansion and becomes supersaturated. When a charged particle, such as a cosmic ray, passes through the chamber, water condenses into a trail along its path. Indeed, cosmic rays may play a part in promoting cloud formation in our atmosphere. The CLOUD experiment is an effort by high energy physicists to apply their experience with cloud chambers to the study of cosmic rays and cloud formation, especially cloud formation at high altitudes where the air is thin and the water vapor is scarce.

Even in a cloud chamber, however, supersaturated water vapor does not endure for long. A useful cloud chamber has a piston going up and down several times a second because the water vapor condenses on its own within a fraction of a second. In our Circulating Cells program, we will check the water vapor concentration of the cells every hundred seconds or so. For the purpose of our simulation, therefore, we will assume that condensation within a cell is complete within a hundred seconds. When we find a cell with 20 g/kg of water vapor and a saturation concentration of 10 g/kg, we will allow 10 g/kg to condense into droplets.

Not only do we expect to encounter moist air rising and cooling, we will also have cloudy air falling and warming. As it warms, the saturation concentration increases, so it is possible for some or all of the water in the droplets to evaporate again. Because evaporation rate is proportional to the surface area of water, the tiny droplets of a cloud will evaporate quickly. The 20-μm diameter droplets of a cloud provide 3000 cm2 of surface area for each gram of water they contain. A 1-cm deep puddle, meanwhile, provides only 1 cm2/g. We expect cloud droplets to evaporate three thousand times more quickly than a 1-cm deep puddle. A 1-cm deep puddle will evaporate in less than ten thousand seconds, so a cloud will evaporate in less than thirty seconds. For the purpose of our simulation, therefore, we will assume that the evaporation of cloud droplets is complete within a hundred seconds.

Combining these two assumptions together, we see that whenever our simulation encounters a gas cell with water vapor concentration greater than the saturation concentration, we will remove the excess water vapor and turn it into cloud droplets. Conversely, whenever we have cloud droplets with water vapor concentration less than the saturation concentration, we will remove however many cloud droplets we can until the water vapor concentration is again equal to the saturation concentration.

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