Re: How would I calculate the pressure in a bottle by blowing air in it?

Greg,

What you are asking about is what fluid dynamicists call "stagnation
pressure" or sometimes "total pressure" (I'll use the terms
interchangeably). The total pressure of a fluid is the pressure that would
result from isentropically slowing the fluid to rest. Isentropic means both
adiabatic (no heat transfer with the surroundings) and reversible (no
losses such as friction, etc.), so the total pressure is really a reference
value that doesn't exist in reality. However it will closely approximate
the pressure you'd really see if you stagnated a fluid, either in a bottle
like you describe or at the leading edge of a wing or other body, as long
as there wasn't too much heat transfer or irreversibility. It's also a very
useful reference value for propulsion engineers since it can be shown that
the performance of any Brayton cycle engine (such as a turbofan, turbojet,
ramjet, scramjet, etc.) will be adversely affected by any total pressure
drop through the cycle. Total pressure drops occur due to losses such as
friction, shock waves, and heat addition at high mach number, so designers
pay close attention to the equations that involve total pressure. I'm not
going to go into further detail since this is truly a huge topic in the
aerospace field. In my opinion it is the single most critical concept for
propulsion engineers to understand, and it is treated heavily in most
college level courses on aerospace propulsion. 

Just to be perfectly clear, the total pressure is *not* the same thing as
static pressure, which is the pressure you would actually sense with a
pressure gauge. If you put a pressure probe in a moving flow such that its
opening was parallel to the direction of the flow, you would measure the
static pressure, not the total pressure. If you also know the velocity (or
really the Mach number) of the flow, you could calculate the total pressure
by the equation:

Pt = P * (1 + 0.5 * (gamma-1) * M ^ 2) ^ (gamma / (gamma-1))

where:
Pt = total or stagnation pressure
P = static pressure
gamma = ratio of specific heats of the fluid
M = Mach number
Note that gamma and M are both unitless, so if you enter Pt will have the
same units that you used for P.

Frequently in the real world we know the ratio of the total pressure (or a
close approximation) to the static pressure by using a device known as a
pitot tube. From this information and knowing the physical properties of
the fluid (gamma), we can calculate the Mach number. If we know a few more
things about the fluid (temperature and gas constant), we can calculate the
velocity by using a couple other equations:

M = V/a

and

a = sqrt(gamma * R * T)

where:
V = velocity of the fluid
a = speed of sound in the fluid
R = gas constant
T = temperature of the fluid

Now, I've kind of answered the opposite of your question by starting with
the stagnation pressure and calculating the velocity, but if you knew the
velocity and pressure generated by your air gun, the temperature of the air
(in the air jet, not the surroundings - they won't necessarily be the
same), and a couple of physical properties of the air, you can approximate
the pressure in the bottle by calculating the total pressure. I'm more
interested in the ramjet though, so I'll do an example of a ramjet where
the flow is totally stopped in the combustor (not realistic, but that's
what you asked).

First, let's suppose the ramjet is traveling at Mach 2.0 at an altitude of
10 kilometers. At that altitude, the ambient pressure is about 26.5 kPa and
the temperature is about 223.26 K. The ratio of specific heats of air is
about 1.4. We can calculate the total pressure of the incoming flow:

Pt = 26500 * (1 + 0.5 * (1.4-1) * 2.0 ^ 2) ^ (1.4 / (1.4-1))

This gives us a value of about 207000 Pa or 207 kPa. Keep in mind though
that ramjets use shock waves to slow the flow down in the inlet, and shock
waves lower the total pressure of the flow. Oates (see reference below)
calculates the total pressure loss through a type of inlet called a
"Kantrowitz-Donaldson Inlet" (don't sweat the details, just accept that
this would be a typical real-world value). For a Mach number of 2.0, the
total pressure ratio through the inlet is about 0.834. In other words, the
total pressure of the flow going through the inlet is reduced by about 17
percent. In our example, the total pressure after the inlet would be:

20700 * 0.834 = 173000 Pa

The pressure at sea level in the standard atmosphere is about 101375 Pa, so
the total pressure after the ramjet inlet is about 1.7 times that, or 1.7
atmospheres. If the ramjet isentropically slowed the flow to zero velocity,
this is the pressure that would result. This is not a particularly large
number, mainly because the ambient pressure at that altitude is only about
a quarter of the pressure at sea level and the Mach number is relatively
low. Ramjets are probably capable of being operated up to Mach 3 or 3.5
before a scramjet engine becomes more efficient, and since the Mach number
is squared in the total pressure equation, the total pressure goes up a lot
as you increase the Mach number. 

I hope this helps. I have to admit I had a hard time with this question
because this topic is quite deep and I've really just glossed over it. If
I've left out something important I'll be glad to answer any follow-up
questions to clarify things.

David Coit

References:

Introduction to Flight, 3rd ed., John D. Anderson, Jr.
Aerothermodynamics of Gas Turbine and Rocket Propulsion, 3rd ed., Gordon C.
Oates