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I'm currently trying to figure out the following in the simplest possible way:

Say we have a nozzle in a vacuum environment. A gas of a certain pressure is emitted through the nozzle, which has a certain diameter, int the vacuum. The question is now how high is the pressure at a certain distance away from the nozzle exit?

Any help would be greatly appreciated!

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The answer is that it depends on what you mean by pressure.

To understand this imagine you have some gas in a canister at some pressure $P$, and you look at the inside walls of the canister with a microscope so powerful that you can see the gas molecules whizzing about and bouncing off the walls. One of collisions would look like this.

Pressure

The gas molecules have a mass $m$ and some average velocity $v$ so they have a momentum $mv$. When one molecule rebounds from the wall the change of momentum is $2mv$, but the rate of change of momentum is just the force exerted on the wall. So if the number of collisions with the wall per second per unit area is $N$, then the pressure on the wall will be $P = 2Nmv$ (not quite, because not all collisions are at right angles, but let's skip over this).

Anyhow, the point is that pressure is caused by collisions with gas molecules. In a canister of gas (like the room you're in) the molecules are moving in all directions at random so the pressure they produce is the same everywhere. So for example the air pressure on you is the same all over your skin.

But in your example of air escaping from a nozzle the air is not moving at random directions. Instead it's escaping from the nozzle in something like a cone:

Nozzle

Suppose put a box in the path of the escaping gas, and we measure the pressures $P_1$ to $P_4$ on the four sides of the box. If the box were in a gas canister, as we talked about above, all four pressures would be the same because the gas molecule velocities are random so on average equal numbers of gas molecules per second would hit all four sides. But with the gas from the nozzle this clearly isn't true because far more gas molecules will hit side 1 than sides 2, 3 and 4. The pressure $P_1$ would be much greater than $P_2$ and $P_3$ and $P_4$ would be close to zero.

So this is why I started the answer with it depends. You can't simply define a single pressure in a gas stream from a nozzle because the pressure you measure will depend on how you orientate your pressure gauge.

Having said this, the obvious response is to ask what is the pressure $P_1$. The answer is that the number of gas molecules per unit area will decrease as the square of the distance from the nozzle, so the pressure will be:

$$ P_1 \propto \frac{1}{d^2} $$

where the constant of proportionality will be determined by the geometry of the gas flow from the nozzle.

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