Some Notes on Pressure and Muzzle Velocity

[denton]

If you're developing a software package like QuikLoad, you get to do quite a bit of math. You have to figure out the area under the pressure curve, with time on the X axis, for one thing. It's a challenge, which is probably why we don't have a lot of cheap, effective load modeling software in the market.

But if all you need is proportionality, the process is much simpler. Here is a scatterplot of data from my 8x57:

The relationship is obviously linear, and the math confirms that a straight line model accounts for 99% of the variation over the given range. Random error in the measurement systems probably accounts for the other 1%. I've done several of these, always with the same result. I suppose there are cases that aren't quite so neat.

So, if muzzle velocity is actually driven by the area under the pressure curve, how can muzzle velocity have a linear relationship with just peak pressure? Simple. The peak pressure is the main driver of area under the curve. Also, these measurements are taken over a finite range, small enough that any curvature in the model is likely to be small.

The big take-away is this: The bullet has no source of energy other than compressed gas, and the peak gas pressure is linearly correlated with muzzle velocity over a broad range. It follows that if you're exactly following a published recipe, and you're getting more muzzle velocity than the published load, you are also getting more peak pressure than the published load.

Unexpectedly high muzzle velocity comes from unexpectedly high pressure.

 

[Mosin Bubba]

Just by coincidence, I was actually looking this up out of curiosity last night (does say 10% more velocity necessarily mean 10% more pressure?) and couldn't find any good links on it. Thanks for posting this.

 

[lysanderxiii]

Just by coincidence, I was actually looking this up out of curiosity last night (does say 10% more velocity necessarily mean 10% more pressure?) and couldn't find any good links on it. Thanks for posting this.

No. It will be the slope of that line.

 

[Llama Bob]

The big take-away is this: The bullet has no source of energy other than compressed gas, and the peak gas pressure is linearly correlated with muzzle velocity over a broad range. It follows that if you're exactly following a published recipe, and you're getting more muzzle velocity than the published load, you are also getting more peak pressure than the published load.

Unexpectedly high muzzle velocity comes from unexpectedly high pressure.

This is absolutely true. It's why you should load to velocity (assuming equal barrel lengths), not load to charge weight.

However, one nit: I don't think the relationship is fundamentally linear. It should be fundamentally a squared relationship - the increase in pressure is the square of the increase in velocity. The reasoning is as follows:

force on the bullet is pressure x base area (definition of pressure)
base area is constant
therefore force is proportional to pressure
force integrated over barrel length is energy (definition of work - don't do the integral in time, it's far more confusing)
barrel length is constant
therefore energy is proportional to force which is proportional to pressure
energy = mass x velocity^2 (definition of kinetic energy)
bullet mass is constant
energy is proportional to velocity^2
by substitution pressure is proportional to velocity^2

We can see that's basically true in your data - velocity increases by a factor of 1.2ish and pressure by a factor of 1.5ish which is a lot like 1.2^2.

Now the question in my mind is why we don't see the curvature of the squared function over this relatively large range. You do see it in simulation.

 

[lysanderxiii]

This is absolutely true. It's why you should load to velocity (assuming equal barrel lengths), not load to charge weight.

However, one nit: I don't think the relationship is fundamentally linear. It should be fundamentally a squared relationship - the increase in pressure is the square of the increase in velocity. The reasoning is as follows:

force on the bullet is pressure x base area (definition of pressure)
base area is constant
therefore force is proportional to pressure
force integrated over barrel length is energy (definition of work - don't do the integral in time, it's far more confusing)
barrel length is constant
therefore energy is proportional to force which is proportional to pressure
energy = mass x velocity^2 (definition of kinetic energy)
bullet mass is constant
energy is proportional to velocity^2
by substitution pressure is proportional to velocity^2

We can see that's basically true in your data - velocity increases by a factor of 1.2ish and pressure by a factor of 1.5ish which is a lot like 1.2^2.

Now the question in my mind is why we don't see the curvature of the squared function over this relatively large range. You do see it in simulation.

Unaccounted for energy losses, like friction and heat.

 

[Andrew Leigh]

Mmmmm, you are talking about internal ballistics here and discounting the effect that the inertia has on the loaded round when the bullet engages the lands? QuickLoad does the same, save for their caveat of adding a certain quantum of pressure when one is seated on the lands or if one is shooting a mono.

The reality is that pressure in a cartridge, in the chamber of a rifle, is not linear but has an exponential component. My personal tests on a 7.62mm show a marked increase in pressure as the bullet passes 1.00mm (0.040") from the lands.

It is a misconception that as one seats further out that the pressure drops ....... well I suppose it is actually true, if no external forces are acting on the cartridge. The one vital fact that we miss is that what we shoot is confined to a chambers with a jump to the lands and therein lays the rub.

 

[denton]

This is absolutely true. It's why you should load to velocity (assuming equal barrel lengths), not load to charge weight.

However, one nit: I don't think the relationship is fundamentally linear. It should be fundamentally a squared relationship - the increase in pressure is the square of the increase in velocity. The reasoning is as follows:

force on the bullet is pressure x base area (definition of pressure)
base area is constant
therefore force is proportional to pressure
force integrated over barrel length is energy (definition of work - don't do the integral in time, it's far more confusing)
barrel length is constant
therefore energy is proportional to force which is proportional to pressure
energy = mass x velocity^2 (definition of kinetic energy)
bullet mass is constant
energy is proportional to velocity^2
by substitution pressure is proportional to velocity^2

We can see that's basically true in your data - velocity increases by a factor of 1.2ish and pressure by a factor of 1.5ish which is a lot like 1.2^2.

Now the question in my mind is why we don't see the curvature of the squared function over this relatively large range. You do see it in simulation.

I think your reasoning is impeccable. And in light of it, I still don't fully understand why we get the result we do. Just as you say, the relationship should be curvilinear.

One that is even more puzzling is that I have a 30-06 data set from Ken Oehler going from about 30 KPSI to 90 KPSI, powder charge and MV. The powder:MV relationship is straight as string linear over the whole range.

One more mystery to solve.

 

[olafhardtB]

Pressure is not constant. It varies with lots of stuff. A few are gas temperature, volume of space behind bullet, reaction products behind bullet and barrel friction.
Gas temperature: I think there is a tendancy for the combustion products to re-react with each other. These reactions consume or release heat or not. This may account for some of the rises seen in pressure traces seen as the bullet passes down the barrel.
Volume: As the bullet passes down the bore the volume increases significantly behind it. This is the gas containment volume.
Reaction products: The products formed during the reactions vary with extreme variations in pressure and these reactions can reverse or react with other products. This varies the number of molecules of gas available.
I believe we deceive ourselves trying to predict all this stuff. Almost all of the chemistry studied and taught occurrs at low temperatures and pressures. We play with " ideal" gasses ignoring friction, assuming equillibrium etc. Strain gauges and piezo-electrical are better than crusher devices and probably the best we have but they just aren'that good. In the information and digital age but there are often cases where we just guess.
Get good manuals, avoid extremes and shoot or shoot factory ammo. Wear safety equipment. The gun may still blow up. To quote some Russian I read about: "Is gun. Is dangerous!"

 

[Jim Watson]

In a bygone day, before Quickload and when Guns & Ammo was a serious publication, there was a rule of thumb that for SMALL changes, velocity increased with increases in powder charge but pressure increased by twice as much. 5% more powder would get you 5% more velocity but at the cost of 10% more pressure. The first Vihtavuori pamphlet I had said +5% powder => +4% velocity, +10% pressure.

 

[denton]

It is a misconception that as one seats further out that the pressure drops ....... well I suppose it is actually true, if no external forces are acting on the cartridge. The one vital fact that we miss is that what we shoot is confined to a chambers with a jump to the lands and therein lays the rub.

This is a great point: All pressure and velocity bets are off for a load that does not specify COL.

 

[Llama Bob]

In a bygone day, before Quickload and when Guns & Ammo was a serious publication, there was a rule of thumb that for SMALL changes, velocity increased with increases in powder charge but pressure increased by twice as much. 5% more powder would get you 5% more velocity but at the cost of 10% more pressure. The first Vihtavuori pamphlet I had said +5% powder => +4% velocity, +10% pressure.

This is a very good rule of thumb that embodies two more fundamental observations:

• pressure increases as the square of the velocity (or proportional to the energy)
• cartridge efficiency increases linearly (or slightly less than linearly) with charge weight