How a Fire Piston Works (Boyle's Law) 
by Ken Costello

The Fire Piston is an amazing device that dramatically demonstrates how increases in pressure affect the heat of a gas in an enclosed space. This analysis is based on the Fire Piston Demonstration Unit available from Midwest Native Skills, however the principals can be applied to any other experiment where the pressure of a gas can be suddenly increased in an enclosed space.
The Fire Piston Demonstration Unit had a cylinder volume of 9ml. This volume is reduced to approximately .5 ml when the plunger is depressed yielding a compression ratio of 18:1, thus when the plunger is depressed the unit compresses the air inside the cylinder 18 times its normal pressure.
Using standard atmospheric pressure of 14.7 psi this would yield a pressure of 264.6 psi (265 psi with rounding). However, this is only if it is done slowly and the temperature after compressing is the same. In the case of the fire piston, we do it fast so the effort (work) we put into it is converted to heat. So this extra temperature is going to increase the pressure even more. The work (energy) put into it is the pressure on the piston handle times the distance that it travels (Work = Force x Distance). Remember however, the pressure in the cylinder is also increasing as we push the plunger down, so it's not simple multiplication result. To get the real answer we need to turn to Boyle's Law.

Boyle's Law
In PV=nRT, I normally use these values:
P = pressure in atmospheres (Since 1 atm=14.7 psi, you can divide psi by
14.7 to get atmospheres)
V = Liters
T = Kelvin (Celsius +273 degrees)
R = This is a conversion factor to change temperature and moles into
pressure x volume. It's value here is 0.0821.
  n = moles of gas present.
At zero degrees Celsius (32oF), a mole of gas at 1 atmosphere of pressure occupies 22.4 Liters (22,400ml). So the number of moles in 9 ml would be the fraction 9ml / 22,400ml.
 
9ml / 22,400ml.= 0.0004017 moles of gas
This 0.0004 moles of the gas (air in this case) is about 0.012 grams of air.
Other data needed:
Flash Point of the Cotton tinder = 210oC (410oF)
Self Ignition Point of the Cotton Tinder = 407oC (764oF)


Rearranging the Equation
Using algebra, the PV = nRT formula can be solved for R
 R = PV/nT.
Situation #1 is when the plunger is at the top of the cylinder|
Situation #2 is when we push the plunger all the way down into the cylinder
The values for situation 1 might be written as R=P1V1/n1T1. The values for situation 2 could be written as R=P2V2/n2T2.  Since R is the same for both of the equations, they can be set equal to each other.

P1V1/n1T1 = P2V2/n2T2

Solving the Equation
We now have to look at our specific case starting (the "sub 1" formulas) and our end conditions (the "sub 2" formulas) for the gas in the fire piston. Let’s say the volume starts at 9 ml and ends as 0.5 ml. Let’s also assume the temperature of the gas (the air inside the Fire Piston) goes from room temperature 20°C (68°F) up to 210°C (flash point of cotton). Remember, we have to add 273 to both of these to get them to degrees Kelvin. The moles of air in the piston start and end the same, so “n” does not change and we can drop it from the equation. Let’s say the temperature starts at 14.7 psi (1 atm). So here’s our final equation:

P1V1/T1 = P2V2/T2

(14.7 psi x 9mL) / (20oC + 273oC) = (P2 x 0.5mL) / (210oC + 273oC)

(14.7 psi x 9mL) / (293oK) = (P2 x 0.5mL) / (483oK)

We can solve for P2 by multiplying by 483K and dividing by 0.5mL on both sides.
P2 is then 436 psi.(using the "flash point temperature of the cotton tinder)

If the temperature is the self ignition point of 407°C, then 483K would be replaced by 680K (407+273) and the final pressure would be 614 psi.
That’s a lot of pressure!
I would guess we are talking about flash temperature rather than self ignition temperature, since it's the hot air that ignites it.

How Deep would you have to Go?
We all know that when we dive under water the pressure around us rises resulting from the weight of more and more water over our heads. How deep would you have to dive to feel the same pressures that the inside of the Fire Piston experience when the plunger is slammed down?

Calculating the Answer
One cubic inch is 2.54 centimeters x 2.54 centimeters x 2.54 centimeters or a total of 16.4 cubic centimeters (cc) for the cubic inch. Water weighs one gram for everyone cubic centimeter. So at one inch deep, the weight is 16.4 grams on the square inch area. 10 inches deep would be 164 grams.
A pound is 454 grams. If we divide 454 grams by 16.4 grams per inch deep, we find that it takes 27.7 inches to get a pound of water over that square inch. So for every pound increase in pressure we have to go 27.7 inches deeper down.
So the pressures of 263 psi (straight compression ratio of 18:1), 436 psi (using Boyle's Law and the Flash point of cotton), and 614 psi (using Boyle's Law and the Self Ignition point of cotton) pressures times 27.7 inches each would mean you would have to be 202 yards (606 feet), 335 yards (1005 feet), or 472 yards (1416 feet) respectively below the surface to experience those pressures. That's about 0.1 miles, 0.2 miles, and 1/4 miles.
One quarter mile under water is a lot of pressure! It's amazing we can generate that much pressure by just slamming down on the piston. It's surprising that the pressure doesn't break the acrylic tube!
 

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