Questions about internal ballistics: cartridge pressure and caliber vs velocity

[atl5029]

I have some questions about cartridge pressure vs caliber and how it affects bullet velocity...

I am familiar with the concept of bolt thrust, where for 2 cartridges operating at the same pressure, the one with the larger diameter case head will put a greater rearward force on the bolt due to Force=Pressure x area. Therefore it seems logical that the same thing should happen on the other end of the cartridge: a bigger caliber bullet would have more area for the pressure to act upon, resulting in a larger force to push it out of the barrel, and hence greater velocity. This is not often what we see however. Usually bigger rounds based on the same case as a smaller round are slower. Now I realize several things account for this. First, bigger caliber bullets are usually heavier, and therefore require more force to have an equal acceleration down the barrel. And second, a larger caliber bullet will have more bearing surface, which translates into a larger frictional force against forward motion.

But what about cartridges of comparable caliber and bullet weight? I have been reading a lot lately about cartridges based on the 300 RUM case, and it seems like people are getting some really high velocities with the larger calibers above .338. I have read accounts on this forum of people reaching just over 3000 FPS with a 416-300 RUM with a 340 grain Cutting Edge bullet. And I know With a 338 Edge and 300 grain bullets, people often get around 2900 FPS, so it seems like despite the heavier bullet of 40 grains and larger bearing surface, the larger diameter bullet is able to be pushed faster. Does anyone have any solid insight into the mechanics of how these phenomena work? I would love a good physics/mechanics discussion.

 

[BryanLitz]

There's a relationship between velocity and bullet weight that I think you'll find useful.

Consider a cartridge that pushes a 155 grain bullet at 3000 fps. For this example, caliber doesn't matter, but this is characteristic 308 Win. Now, from the same rifle, at the same chamber pressure, what velocity can we expect from a different weight bullet, say 185 grains?

The solution is reached via energy balance. The solution can be reformulated by setting the kinetic energy equal at the muzzle [1/2*M1*V1^2 = 1/2*M2*V2^2] where M1 and V1 are the mass and velocity of the first bullet, and M2 V2 is the mass and velocity of the second bullet.

Knowing M1, V1, and M2, we solve for V2 and it looks like this:

V2 = V1*sqrt(M1/M2)

So in the example of our 155 bullet at 3000 fps, how fast could the same rifle shoot a 185 at the same pressure?

V2 = 3000*sqrt(155/185) = 2746

Based on experience shooting both these bullets in 308 Win, I can say that 2746 fps for the 185 is reasonable. Of course, as things like this go, there are many complicating factors which make this formulation imperfect, such as the greater volume occupied by the 185 grain bullet, it's longer bearing surface, the burn rate of the powder, etc. But as a rule of thumb, the above relationship will get you in the ballpark.

Regarding your question of how the solid bullet can achieve surprisingly high MV, I can offer two thoughts. First, the bearing surface and drive band (seal tight band in the case of CE) are deliberately sized to balance friction with sealing the bore. In other words, you have reduced friction which acts to increase MV.

Another reason for the high MV of the 416 cal 340 grain bullet is this: 300 grains is a 'heavy weight' bullet for 338 caliber, while 340 grains is relatively light weight for 416 cal. In fact, if you were to scale up a 300 grain 338 caliber bullet to 416 caliber, it would weigh: 300*(416/338)^3 = 559 grains! If you were to shoot this 416 caliber bullet which is proportional to the 300 grain 338, of course the MV would be way down.

One final way to consider this is related to the difference in surface area between these two bullets. A .416 caliber bullet has 0.136 sq inches of area for the pressure to act on. The .338 caliber bullet has only 0.090 sq inches. In other words, the larger caliber bullet has 1.5 times more area for the pressure to act on. Balance this with the fact that the bullet is only 40 grains heavier, which is 1.13 times the weight of the 338 bullet, and it's clear that the 340 grain .416 bullet should be pushed quite a bit faster.

Again there are many details which act to complicate things, but there are some guiding principals and matters of scale that can make a little sense of things.

Interesting discussion.

Take care,
-Bryan