Physics of the Squirt

I’m not sure whether we had to work this problem in physic class in high school, but I eventually had to do it at work. I will just post the solution here for future reference in case anybody else has to scratch around and come up with the right computations. It went like this:

I worked for a company that made ink jet printers. These were high-performance jobs that fit into machines that processed bank checks. The machine would feed 20 checks per second into a transport mechanism that would whisk them along at 200 inches per second past various stations. One of the stations was an MICR station that would read the magnetic characters along the bottom edge. Another station would read information off the check using OCR (optical character recognition) technology. A lot of useful information was captured from the speeding checks, and we wanted to immediately put it back onto the checks in various forms. One form was a bar code that could subsequently be read using cheaper technology. We also wrote in numbers and letters on the checks. All this writing was done by ink jet printers.

See the drawing. The printer had a reservoir of ink, and the ink was fed under pressure to a tiny glass nozzle. For simplicity I’m not showing how the stream was broken up into drops and how the drops were deflected to form the patterns printed on the checks. My problem was: “What pressure is needed to get an ink jet at 100 inches per second?”

WaterJet

So you know the velocity. Do you need to know anything else? Apparently only the density of the ink. It works like this. You have a number of parameters and variables:

  • v = velocity of the ink jet
  • A = cross sectional area of the jet
  • P = pressure in the ink reservoir

Where do you go from here? Start with a simple relationship between momentum, force and time:

mv = ft

Mass times velocity = force times time.

If you want a mass m to have a velocity v you can achieve this with a force f applied for a time t.

m is the mass of an arbitrary chunk of ink.

f is the force on this chunk of ink.

t is the time the force acts on this chunk of ink.

Then f = AP.

m = Axρ

x is the distance the ink moves while the force is acting on it.

ρ is the density of the ink.

v = x / t.

And there you have it. A little algebraic manipulation gives you

InkJet

Any time you apply a pressure (pressure difference) P to a liquid of density ρ you can achieve a velocity v.

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