DMC Newsletter Geek Challenge - May 2010:
Devise and perform an experiment to determine the volumetric density of liquid water in the near-ground atmosphere during a hard rain. Simply put, if you could freeze time and collect all of the liquid water in a 1 cubic meter box of atmosphere at eye level during a rain storm, how much liquid water would you have?
We know that there is a lot of water vapor in the atmosphere during a rain storm, but we are only interested in the liquid amount. While driving when wipers can barely keep up it seems there is a lot of liquid, but there is never so much that a person has a hard time breathing while standing in the rain. How would one quantify it?
Two readers responded with correct answers to the question: Gene Szafranski of Wonderware North, and Brent Nelson of Texas A&M University.
Both respondents recognized that the key to solving the question of the volumetric density of the rainy air is to analyze the flow rate equation at the interface between the falling low-density rain and a rising completely dense puddle. With no water going anywhere else, the rate that that surface water rises must equal the rate that the rain drops fall. For any given collection area, the rain falling in must equal the water rising in the area. Rain falls into the collection area travelling at high velocity but with low density.
Volumetric Rate = Collection Area X Rain Drop Velocity X Density
If the collection area were a straight-sided container, the volumetric rate of water entering the container would cause the water level to rise by the following equation:
Water Rise Rate = Volumetric Rate / Collection Area
Combining these equations, the Collection Area and Volumetric Rate drop out, and what's left is:
Density = Water Rise Rate / Rain Drop Velocity
So, to experimentally determine the water rise rate is easy. All that is required is a rain gauge. Since rain gauges are generally built to measure several inches of rain, a measurement over a shorter duration can be obtained with a larger collection area more with precise volume. I like an 8" funnel and 50ml graduated for this purpose.
A good hard rain measures out to about 1 inch per hour.
Next, we have to figure the velocity of a raindrop. Gene Googled it and said it's between 9 and 13 m/s. Brent suggests you can measure the velocity experimentally using Particle Image Velcimerty, if you have about $50,000 of equipment. I found a less expensive way. I parked my car a short distance from a parking lot street light and observed the rain fall past the light. While I couldn't track an individual drop visually, I could track the general falling mass of drops visually, and I used a stop-watch to time how long it took to reach the ground from the light. My measurement was 0.8 seconds, and on a dry day I estimated that the light is 24 feet (7.3m) high. So my observed rain drop velocity is 9 m/s.
So now it is possible to calculate the density of rain drops in the air. Density = 1 inch/hour divided by 9m/s.
Converting units, that's 0.000007 m/s divided by 9 m/s=0.00000078 or 7.8e-7.
Since there are 1 billion cubic millimeters in a cubic meter, if you froze time and collected the rain inside one cubic meter of rainy air, you would have about 780 cubic mm of water, or almost a cubic cm.
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