Mass flow rate is the amount of a gas that passes through an orifice per unit of time. The three primary parameters that effect the mass flow rate through an orifice are velocity, density, and orifice area. Other factors include gas temperature, orifice entry and orifice exit conditions, both of which are called coefficient of discharge based on the losses around a poppet or turn in the valve... In this thread I'll briefly discuss variables and will happily answer questions if they arise.
Brief mention on how Mass flow rate not only ties into airgun physics but its history.... I used mass flow rate calculations to assist and progress the understanding of balanced valves over at the GTA years ago in the Simplified Balanced Valve thread. My understanding of this calculation helped countless people get reliably tunable balanced valves in their guns for the tuning community, however FX had this figured out, to what degree I cannot say, but they had integrated balanced valves long before the tuning community even knew they had. Since the information went public? Now countless manufacturers are adding these valves to their guns. Coincidence? Maybe..
I also used mass flow rate calculations to help myself and another understand and integrate pilot valves into airguns mechanically while making them incredibly tunable.
The mass flow rate calculations helped paint a picture that shows the nature of whats going on inside both a balanced valve, and a pilot valve in a unit of time the human mind can't quite grasp without math, microseconds.
Velocity:
Lets discuss the Velocity of the air parcel (or rather the mean velocity, as there are many molecules, all moving at different speeds).
We'll use a choked flow formula, because the pressure ratio for airguns is incredibly high, and the flow is very much choked.
- γ (gamma): The adiabatic index (ratio of specific heats). For air, γ ≈ 1.4.
- T1: The absolute temperature of the air in Kelvin.
- p1: Pressure of compressed air inside airgun *but to be exact the average pressure during the shot cycle, which depends on start pressure and plenum volume
- p2: Ambient pressure outside determined by your elevation
- R = Gas constant.
For those curious, the formula to determine if the flow is choked is as follows, P1/P2 > (2/(γ + 1))^((γ + 1)/(γ - 1))
Using the above data for my gun, the velocity of air is approximately 769 m/s...or 2522 fps. Changing the pressure does not increase the velocity, as the parameters that determine velocity in choked flow are properties of the gas, or in our case compressed air, and its temperature. Super heating the air or super cooling the air would have great effects, or switching to helium the gas used to propel the projectile.
The above implies my gun can shoot a projectile upwards of 2,000+ fps on compressed air given the right conditions and a light enough projectile. Some may feel the need to challenge this take, and if so, be my guest, as these velocities have already been achieved in airguns by several members of the GTA community.
As noted, the only way to manipulate velocity of the gas itself, is changing the gas to a lighter or heavier one, or changing its temperature, changing the orifice diameter doesn't change the velocity of air either, however it will greatly effect the mass flow rate. Meaning you cannot increase the velocity of air alone with different calibers or porting, only the mass flow rate (volumetric) of air.
Density: Next up is Density, what determines density? We'll use an approximation based on ideal gas behavior, which some may disagree with, however, if you have a better method please share! Ultimately calculating the exact density isn't the purpose of this discussion, rather understanding its makeup.
Using the following:
- Pressure (P): 2000 psi (converted to pascal) = 13,789,520 pa *(This would be the average pressure present during the shot cycle)
- Temperature (T): 295 K (kelvin converted from 72F)
- Specific Gas Constant for Air (R): 287 J/(kg·K)
We'll convert pressure to pascals.
1 psi = 6894.76 Pa
P=2000 psi×6894.76 Pa/psi=13,789,520 Pa
The above formula gives us a density of 162.1 kg/m3
The few ways to manipulate density is to
A) Change pressure (or change plenum volume to change pressure drop during shot cycle)
B) Change composition of gas
C) Change temperature which also has an effect on pressure
This means that as the density increases (more pressure) the mass air flow rate will also increase, assuming the Oririce area and velocity variables are left unchanged, which ultimately means the same airgun at a higher pressure can potentially eject more air mass over the same unit of time compared to the same airgun at a lower pressure, provided you're creating more dwell to reach the plateau at the higher pressure.
Orifice area This is the porting through your airgun. The porting is limited either by design of the gun, or its caliber. You can port larger than the caliber however gains will quickly diminish to the point where you will experience losses. When porting any airgun, one must account for the CSA (cross-sectional area) of any obstructions within the pathway, be it a poppet stem, or the projectile probe.
CSA is calculated as such.
CSA = π⋅(d/2)²
Where d is the diameter, and CSA is cross sectional area. Lets calculate my valves throat csa, my poppet stem csa, and make sure I have enough flow for my port csa.
Valve throat
: 3.1415*(.238/2)² = .04374 sqaure inches Poppet stem:
3.1415*(.047/2)² = .00174 square inches
Valve exit port:
3.1415*(.225/2)² = .0398 square inches
Now lets subtract the poppet stem csa from the valve throat csa, and we have get .042 in
². Most tuners say you want around 5-10% more flow at this location than the rest of the system, likewise with the valve entrance, however I cannot confirm nor deny, but I generally try for a bit over, and in my case, its about 5.5%. What you don't want, is a negative value which would result in not only a slight reduction due to the restricted orifice area, but also because of additional wasted space ahead of the throat which is intended for more flow than one would see with a choke at the valve seat, resulting in undesirable pressure loss.
The other conditions: Other conditions aren't as impactful yet should be mentioned, such as how many turns, radius of turn, porting surface quality, which often do matter to the most intense of enthusiasts and tuners trying to get every last ounce of performance from their airguns, this is basically Cd, or coefficient of discharge, the efficiency of the flow rate achieved vs calculated in perfect conditions. Temperature can play a large factor however for it to have extreme effects on mass flow rate, you'd have to have extreme shifts in temperature beyond what is reasonable inside the action of an airgun, I'm sure our airguns would eject a more intense mass of air if super heated to a few million celcius opposed to our commonly comfortable range of 0-40c which we shoot at, but even then I am sure seasoned shooters notice the shift in their guns performance from winter to summer, which can partially be chalked up to this key variable that determines the mass flow rate of your airgun. -Matt
