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Article 1:
Okay, ready to get into some more maths?
Remember Boyle's Law
says, "The volume of a given mass of gas is inversely
proportional to the absolute pressure if the temperature
remains constant?" So, what happens if we change the
temperature? Well, there's a new law that comes into play
called the General Gas Law which states the relation between
Pressure and Temperature
P1 / P2 = T1 / T2.
P1 = Initial Pressure
P2 = Final Pressure
T1 = Initial Temperature (Absolute)
T2 = Final Temperature (Absolute)
Note: Temperature means "Absolute Temperature". Since the
temperature at which molecules stop moving is -460 degrees
Fahrenheit, also known as, Absolute Zero, we have to add 460
to whatever temperature above 0 degrees Fahrenheit that we
want to work with. This is known as the Rankine Scale. So,
50 degrees Fahrenheit (50 + 460) equals 510 Rankine. With me
so far? Okay, let's get back to the math.
Let's put an electric heat wrap on the tank but not turn it
on yet. Now, let's say that the ambient temperature in the
BattleBox is 85 degrees (trust me it feels like 100 in there
:-p) and we have 2500 psi in our 88 ci tank at this
temperature. Before the match starts we flip the switch that
turns on the heat wrap and (for argument's sake) it gets the
tank up to 170 degrees Farenheit. Sounds like we just
doubled the temperature so the pressure should be double,
right? Well, not quite, remember we are working with
absolute temperatures here. So, the absolute temperature at
the beginning of the match is really 545. At 170 degrees
Fahrenheit the absolute temperature is only 630. Not even
close to double the temperature. So, if we apply the General
Gas Law 2500 (P1) / x (P2) = 545 (T1) / 630 (T2) we get x =
2890 psi.
Well, now that we have more pressure in the same amount of
space I would bet that it would have an effect upon how many
shots we can get out of our system. Replacing 2890 for 2500
in the equation above we get 1017.28 ci at 250 psi available
to us instead of only 880. If we finish the equation we get
a total of 7 (well 6.97 but who's counting?) shots. That
gives us 7 foward swings and 7 reloads. That gives us one
whole extra chance to smack the snot out of the opponent.
Had the answer been 6.5 we could have gotten 7 swings but
only 6 reloads so we'd be dragging a limp hammer around the
box until the match was over.
Now because we are dealing with BattleBots rules here the
Technical Regulations say that a bot can carry no more than
2500 psi of N2 or HPA on board at any time (8.2.2.a of Tech
Reg 2.2). This is why it is stated in the Technical
Regulations section 8.9.5 Pneumatic Heaters NOT Allowed.
Okay, now that we know that the heaters are not allowed, and
we know the relation of Pressure to Temperature, what would
the temperature of the gas be after one shot if the ambient
temperature of the gas starts out at 85 degrees Fahrenheit?
Well, this one is gonna take a little more math because the
pressures are different on both sides of the regulator and
we need to know how much gas gets used after one shot.
First, lets determine how many units of Atmosphere we have
available in the tank by multiplying the pressure by the
volume:
2500 x 88 = 220000
Now let's figure out how much of that gets used up when we
fire our weapon. We know that the volume on the push stroke
is 75.36 ci and we are running it at 250 psi. Now we
multiply those together:
250 x 75.36 = 18840
So now we now have (220000 - 18840) 201160 units left that
are stuffed into an 88 ci tank
201160 / 88 = 2285.91
We now have 2285.91 psi of HPA left in the tank after one
shot. That means that we just dropped in pressure so, by the
General Gas Law, there must be a corresponding drop in
temperature of the gas. (Remember to add 460 to the
temperatures!)
2500 (P1) / 2285.91 (P2) = 545 (T1) / (460 + x (T2))
1.094 = 545 / 460 + x
x + 460 = (545 / 1.094)
x + 460 = 498.17
x = 498.17 - 460
x = 38.17 degrees Fahrenheit
So, the temperature of the gas dropped almost 47 degrees
after just one shot. Now the tank itself won't be that cold
because of thermodynamics but that is some serioulsy nasty
math that we don't want to get into at this point. Now
remember, this is just after one actuation of our cylinder.
We still need to reload. So, if we apply the same math to
the reload function (I'll let you do it on your own to see
if you get the same thing) we get a gas temperature of -5.57
degrees Fahrenheit.
Theoretically, you could work out all twelve actuations and
get down pretty close to absolute zero but in reality it
never comes close.
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