I believe you are on the right track....

First,what is the highest pressure that the tire will stand?

Logically, I would think you want the largest diameter hose and fittings you can get from the tire to the mask.

before one spends too much on parts, why not cobble together a prototype test bed of

- tire

- whatever hose you have on hand,

- ending in the nozzle

- inflate and time how long it takes to deflate.

If I must hazard a guess, it would be minutes.

This is based on my experience with SCUBA, Scott AirPacks, and Oxygen bottles.

Scuba and Scott Airpacks ( virtually the same thing) ... an 80 cu ft tank at ~ 3000 psi will last about 60 minutes or less

SNIP

So I was looking at my old textbook and formulary from my college days, over 24 years ago (!), and I realized that there is a very simple way to tackle the problem of flow in a pipe. There is no need to go into a differential equation or even go solving for implicit equations, because under some conditions it's really easy to treat flow in a pipe as quasi-1 dimensional, meaning that you don't even need to use radial geometry in property distributions, such as pressure, temperature and such, because flow properties for high speed flows can be very simple.

My original idea was to use compressed air to generate a cooling jet of air and give a person enough oxygen, but not necessarily to carry a large tank of air in the back. Not knowing how much a human breathes prevented me from knowing if my idea of using a tyre as a tank was even feasible. So I'm going to go back to the start and define the main problem: create a cooling jet that a human can breathe.

Since I don't have much choice on the materials that I can get (I don't have a metal working shop and a 3D printer or such), I had the idea of using commercial equipment for compressed air tools and tyre inflation equipment such as used in you car. My observation was that if a can of compressed "air" can produce a supersonic jet of gas from a pin-hole, it shouldn't be to hard to use equipment from you hardware store at a pressure about 100 psi or less to generate a similar flow of air. Along the way, the expansion creates the cooling effect which thermodynamically you already paid for by compressing air and radiating that heat to the environment, even before you place the mask on your face - the so called Bell-Coleman Cycle.

The good Prof Marvel pointed out the fact that respiration rates of human beings require more air than a small tyre could handle. Moreover, practical air tanks for scuba diving and medical oxygen tanks operate in a range of pressures between 2000 and 3000 psi and deliver volume rates of air in the order of 5 litres per minute (expanded gas out of the regulator - I assume rated at ambient pressure, 101 kPa). Much higher pressures and volumes of air than what I envisioned. The reason is that if you carry all your air with you, then storage becomes a problem. Humans consume a lot of air to stay alive.

There may no be a way out of carrying a large tank of air, but one thing I know is that I don't want the system to be too heavy or expensive. People will not wear the mask that way. Also, instinctively I knew that any system of O[1 kpsi] (read, "of order of 1000 psi") is between 1 or two orders of magnitude larger than what you needed to create a little jet. But I didn't know how to prove it unless I made calculations. And also I needed to convince myself of the actual requirement for a human being. Would the 2-5 LPM (Litres per minute) of an oxygen tank be enough? And what about Air, not Oxygen? Would my supersonic nozzle needle be able to provide enough? Those are answers I have on paper now.

~ ~ ~ ~

So let's start with human respiration. An average human breathes at a rate of 12 breaths a minute in a state of rest. In a state of distress or exercise, that rate will multiply by a factor of 6. For each breath you have a "Minute Volume" inhaled and then exhaled from the lungs. The average capacity of a human id 0.5 litres. Hence the average consumption rate of air is 6 litres per minute. The reason oxygen tanks give less than that is because that is supplemental oxygen only and you need less volume of air if you are inhaling an oxygen stream. So this answers the question whether a typical oxygen tank is enough. The answer is that it's good for supplemental oxygen only. That leaves scuba equipment as the only equipment rated to provide enough air. But would my needle nozzle be able to do that?

Well I prefer to think in terms of kg of oxygen per second, because that is compatible with thermodynamics and fluid mechanics formulas. The net volume flow rate of air for a human

~~V~~ (I'm using a strike-through dash instead of a dot on top to indicate rate per second due to format limitations)

~~V~~ = 6 L/min = 0.1 L/s

and if there are 1000 litres per cubic metre then

~~V~~ = 1 x10

^{-4} m

^{3}/s

The density of air at sea level in the Standard Atmosphere is 1.225 kg/m

^{3}, which means that the mass flow rate

~~m~~ is

~~m~~ = 1.225 x 10

^{-4} kg /s

So basically a human breathes only a little over 1 tenth of a thousandth of a kilogram of air per second on average. 7 if very distressed. Air is very light, but very bulky.

~ ~ ~

Now I stumbled on a very useful equation which I'll use right now: The mass flow rate for expanding flows. The equation is a product of the zeroth (continuity) law of thermodynamics, the first law of thermodynamics (energy), the definition of speed of sound, and the equation of state for an ideal gas.

For basic supersonic flows and compressible flows where the temperature is not changing a lot (no combustion) you may treat air to be a calorically perfect gas, meaning that a rise in temperature correspond to a linear change in energy and enthalpy of the air by way of constants known as "specific heat".

Also we are assuming that the system we're studying is adiabatic - meaning there is no heat transfer going into or out of the system - which for fast flows is a reasonable assumption. Also we will be assuming that diffusion of mass (viscosity, friction) diffusion of heat (conduction) are negligible and all work done on the system is reversible. The above conditions imply we are assuming isentropic flow, which takes care of the 2nd law of thermodynamics.

The mass flow rate for expanding compressible flows in a duct starting from a stagnation chamber

A DeLaval converging diverging nozzle (flow is to the right). M = 1 at the throat. The nozzle I propose (needle) ends at the throat.

Note how pressure and temperature decrease as the flow expands. The far left of the nozzle is connected to the stagnation chamber (tank).

You need the pressure at the throat to be less than 53% of the pressure at the tank (P

_{o}) to "choke" the flow and reach M=1

The flow starts at a pressure chamber called the stagnation chamber at a pressure P

_{o} and temperature T

_{o}. Then the flow accelerates continuously through a deLaval nozzle (shown) with the flow becoming supersonic at the narrowest part of the nozzle. If the flow is adiabatic T

_{o} becomes a property of the gas at any pint of the flow - like a measure of thermal energy content, if you will, and if the flow is friction free dissipation free (isentropic) then P

_{o} is also a measure of constant "pressure energy" as well.

Gamma, γ, is the ratio of specific heats, at constant pressure C

_{p} and at constant volume, C

_{v}γ = C

_{p}/C

_{v} ~ 1.4

which is about 1.4 for temperatures close to ambient temperature (say 20 C). I will assume this is constant, but in practice γ will be a bit higher for cooling temperatures. Generally we don't care unless we have temperatures in the thousands of Celsius (combustion) when it goes down to about 1.3. Kinda gives you an idea how stable that is.

R is the specific gas constant for air , 287 J/(KgK)

In the equation above, assuming γ = 1.4, the Mach number-dependent term, α approaches the value 1.728 as M approaches 1 at the throat. So the mass flow rate equation becomes

~~m~~ = √(1.4) P

_{o} A / [1.728 √(R T

_{o} ]

where A* is the area at the throat, which for the basketball needle is ¼πD

^{2}, where D = 1 mm diameter. Plugging everything in, and assuming that the stagnation chamber (tank) has had a long time to cool down after being filled with air, so T

_{o} = 293 K = 20 C = 68 F ( a cool day), then

~~m~~ = 1.8545 x 10

^{-9} P

_{o}So now I can calculate how much mass of air will flow for a given stagnation chamber pressure. You have many choices here. I could start with asking what pressure I need to pass the 6 litres per minute through the needle,

~~m~~ = 1.225 x 10

^{-4} kg /s. And if I plug the numbers in, P

_{o} turns out to be a very low value of 53921 Pa (53.9 kPa) about 7.81 psi, which is lower than ambient pressure (1 atmosphere) at 101 kPa.

Now what?

So this is not a physical answer, because the back pressure is higher than the stagnation chamber's pressure. The reason this is happening is that the mass flow rate formula is using "absolute pressure" and it's dumb enough not to know when the nozzle is choked. However if the nozzle was in orbit around earth in a near perfect vacuum, the 7.81 psi pressure would be enough to drive a Mach 1 flow through the needle. The way to fix this is to simply raise the stagnation pressure so the flow can take place through the nozzle. It turns out that theory predicts that at the throat with area A*, the pressure P* needs to be less than 53 % less that the stagnation pressure P

_{o} to drive the flow. We could tackle the problem by increasing the stagnation pressure so much, that you insure the pressure ratio will be less than 53% at the exit of the needle.

Why not just increase the mass flow rate by a factor of 10? In choked flow the Mach number at the throat will always be M*=1. And we know that maximum respiration rate of a human is six times higher than the 6 L/ min. So if I just multiply

~~m~~ by 10, then P

_{o} = 539 kPa, which corresponds to 78.2 psi. The ratio of exit to stagnation pressure should be less than 19%, I'll make calculations to see if this holds, but for the moment, seems reasonable.

So at near 80 psi, a 1 mm basketball needle, should (barring any errors) produce a M=1 flow of air equaling 10 times the respiration rate of a human at rest. Enough air to run around the block.

~ ~ ~

It all sounds very nice, but is there a caveat? Yes there is: the flow is very cold.

Following an adiabatic path in a duct the stagnation temperature to local (ie static) pressure relation is given by

The stagnation temperature To to local static temperature relation

for an adiabatic expanding/compressing flow

When I plug the numbers in, the flow cools down from T

_{o}= 293 K to T*= 244 K = -29 C or -20 F. This is a bit of an issue, but one must remember that cooling is a function of expansion and compression. That is the temperature just at the tip of the needle. If the flow slows down to M = 0.33 (which happens very fast) the temperature will increase to 286 K = 12 C or 55C, which is not so bad. What this means is that I have to design a special diffuser - which can't be just a tiny cone, because I don't want to "extend" the DeLaval nozzle such that the flow continues to expand and cool down. If the flow was too warm at the nozzle , then I would have done that, building a tiny rocket nozzle "bell" from the throat A

_{exit} > A*with a bigger area, and the mass flow rate would have increased with the area (decreasing the pressure of the tank faster) but giving me more cooling.

~ ~ ~ ~

So this is what I have so far folks. The needle will give 10 times the slowest respiration rate, or 66% more than the maximum respiration rate of a human with a tiny stream of air at -29 C for a tank with a pressure of near 80 psi. The tyre will not really cut it for this experiment, I'm afraid (I haven't bought the tyre yet anyhow), unless I do the experiment for a non-choked case at M = 0.33 (the mass flow rate may not be enough though). The good news is that I can use readily available hardware. The bad news is that I must adapt hardware to reduce the pressure from O[1000 psi] to O[100 psi] and carry a large heavy tank on my back.

To be honest, there is no real need to carry all the oxygen with you, if you can have a battery pack do the job of compression for you. It might be noisy, but 80 psi is something even the battery operated compressor can handle. A hand bicycle pump can do 120 psi. A self contained, all pneumatic device is an attractive idea, but one that may complicate the system more than it needs to. My original idea was to filter the air. So I think that I will revise that situation and see if a filtered system, such as an oxygen concentrator could fit the bill. If I could compress on the fly, that would preclude dealing with very large pressures and heavy tanks, no to mention pressure regulators. It may come down to carrying several battery packs on a bandolier

I welcome any further suggestions, though. And if someone can figure out a convenient way to carry a self contained tank I'll hear it, just note that having high pressure equipment near your head is never a good idea, so one must "decant" from a high pressure tank, to a low pressure tank, to serve as stagnation chamber!

~~~

Cheers, and thank you for your attention! I'm having a lot of fun stomping my old academic grounds mentally. At least I got to re-read my text books during the pandemic!

J. Wilhelm