Need help with the math

[OldNSalty]

I know if you have 2 tanks at different pressures and you try and fill one from the other they will eventually reach equilibrium-higher pressure tank will come down and lower pressure tank will come up.

My question is, how do you calculate it?

Say you have a steel 72 rated at 2250psi with 1500psi in it and an al80 that is full. Is there a way to calculate what the final psi will be if I use a fill whip?

 

[TSandM]

It's two simultaneous equations in two unknowns.

The volume transferred from one tank equals the volume entering the other one, and the ending pressures have to be the same.

I'm assuming the 72 is rated at 2250, which means it holds 72 cf at 2475. This means that every 100 psi is worth 2.9 cf.
In the Al80, every 100 psi is worth roughly 2.5 cf.

So, if x is the increase in pressure in the 72, and y is the decrease in pressure in the 80, you can say:

x (2.9 cf/100psi) = y (2.5 cf/100 psi)

In addition, if the starting pressure in the 72 is A, and the starting pressure in the 80 is B, you can also say:

A + 100x = B - 100y

Solving:

x = [y(2.5cf/100psi)]/2.9cf/100 psi
x = .85y

Therefore: A + 100(.85Y) = B - 100Y, or
A + 85y = B - 100y, or
185y = B -A

Using a specific example, assume your 72 starts at 1500 psi, and your 80 is full at 3000.

185 y = 3000 - 1500
y = 1500/185 = 8.1; so you would take 810 psi out of the 80, and add 690 psi to the 72.

 

[iztok]

What TSandM said.

This is basically the same problem as you are dealing with transferring heat and ending temperature of two objects made of different heat index materials.

 

[knotical]

A slightly different approach (with the algebra hidden behind the scenes), but with similar results.

Let's assume for the moment that the steel 72 actually holds 72 cubic feet at 2250 psi, and the aluminum 80 actually holds 80 cu ft at 3000 psi.

A full 72 would be pressurized to 153 atmospheres (since 2250 divided by 14.7 is 153).
Similarly, a full 80 would be pressurized to 3000 / 14.7 or 204 atm.

The actual internal volume of the 72 is 72 / 153 or about 0.47 cubic feet
And the 80 is 80 / 204 or 0.392 cubic feet.

The total volume of the combined cylinders is about 0.862 cubic ft.

The 72 at 1500 psi has 72 X 1500 / 2250 or 48 cubic feet.

Now after equalizing, a total of 80 + 48 = 128 cubic feet of air is crammed into 0.862 cubic feet yielding a pressure of 128 / 0.862 or 148.5 atm.
148.5 X 14.7 = 2183 psi.

For increased accuracy, you should consider the actual capacities of the specific cylinders (and there are lots of variations).
For example, many common 80's hold 77.4 cubic ft at 3000. And if I recall correctly, many steel 72's only approached 72 when plus rated, and held about 65 cubic feet at 2250.

 

[ianr33]

...............

 

[iztok]

Yes, doing this in metric is really quite simple as we would be dealing with pressure (bar) and internal volume of the tanks.

Many things (to me as I grew up in Europe) are much simpler to do in metric

Anyways good job posting an alternative way.