Carbon Catch-and-Release

Around 15 years ago, my father Steve and I were browsing in T. J. Maxx (which I'll forever think of as "Temporal Jail: Maximum Security" thanks to Jasper Fforde) and came across a soda siphon, aka the seltzer bottles that seem to only appear in old movies for comedians to spray each other with. We decided to buy one, and we've continued using them to this day (after spraying each other with it a couple times, naturally).

If you've ever seen one used, the force of the water when it sprays out is impressive, and I was curious if I could derive how fast it's moving. In looking for an answer, I found the Hagen-Poiseuille equation, which relates a change in pressure to the flow of a liquid:

where μ is the viscosity of the water, L the length of the pipe, Vdot the volume flow rate, and r the radius of the pipe. The pressure difference is simply the pressure we add when we charge up the bottle. The CO2 cylinders that go in the bottle (yes, we buy 100 at a time) hold 8 grams of gas each, to charge 1 liter of water. To find the pressure that results in, we can use the Ideal Gas Law:

where n is the number of moles of gas, R is a constant, and T is the temperature. We can get the number of moles from the molar mass of carbon and oxygen. Putting everything together, and solving the differential equation gives

where Vtot is the total volume of the bottle, and V0 is the initial amount of water. I tried plotting this, but I was having some trouble with the units – I think I missed a factor of 1000 somewhere between the millipascals, (kilo)grams, and centimeters. I wound up fudging it to get more reasonable results:

This says that the bottle will be empty in about 8 seconds (and probably not 8 milliseconds, as I initially calculated). For the velocity we get

For reference, 1 cm/s at the top of this graph is about 2 feet every minute, which is way too slow, but leaving in the factor of 1000 is way too fast! I'm not sure where I went wrong with this, but it may be that the Hagen-Poiseuille equation is not the right one to use here – Wikipedia says it "describes the pressure drop due to the viscosity of the fluid", and so it may not apply to the flow due to the difference in pressure. Oh, well – Another of Nate's "that didn't work!" posts.